diff --git a/.github/workflows/c6-cluster-coefficients-numerics.yml b/.github/workflows/c6-cluster-coefficients-numerics.yml new file mode 100644 index 000000000..0331e4e39 --- /dev/null +++ b/.github/workflows/c6-cluster-coefficients-numerics.yml @@ -0,0 +1,62 @@ +name: C6 cluster coefficient numerics (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_cluster_coefficients_numerics_20260930/**' + - '.github/workflows/c6-cluster-coefficients-numerics.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-controls-and-replay: + runs-on: ubuntu-latest + timeout-minutes: 20 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify manifest and main-resident pins, replay exact controls and GH40, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c6_cluster_coefficients_numerics_20260930').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']): + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + pins = json.loads((root / 'SOURCE_MAP.json').read_text()) + for entry in pins['sources']: + if not entry['current_required']: + continue + data = pathlib.Path(entry['path']).read_bytes() + blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']: + raise ValueError('consumed source identity mismatch on this tree: ' + entry['path']) + script = str(root / 'coefficients.py') + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=900) + if out.returncode != 0: + raise RuntimeError('check failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:]) + for m in ['cubic-sign', 'antiderivative', 'typed-boundary']: + rc = subprocess.run([sys.executable, *flags, script, '--check', '--mutant', m], cwd=root, capture_output=True, timeout=900).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'certified': False})) + PY + git diff --exit-code diff --git a/.github/workflows/c6-remote-coefficient-numerics.yml b/.github/workflows/c6-remote-coefficient-numerics.yml new file mode 100644 index 000000000..fdb6f61ce --- /dev/null +++ b/.github/workflows/c6-remote-coefficient-numerics.yml @@ -0,0 +1,67 @@ +name: C6 remote coefficient numerics (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_remote_coefficient_numerics_20260930/**' + - '.github/workflows/c6-remote-coefficient-numerics.yml' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/c6_cluster_law_20260929/PROOF.md' + - 'frontiers/remote_collision_20260928/PROOF.md' + - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-controls-and-replay: + runs-on: ubuntu-latest + timeout-minutes: 20 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify manifest and main-resident pins, replay exact controls and sample kernels, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c6_remote_coefficient_numerics_20260930').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']): + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + pins = json.loads((root / 'SOURCE_MAP.json').read_text()) + for entry in pins['sources']: + if not entry['current_required']: + continue + data = pathlib.Path(entry['path']).read_bytes() + blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']: + raise ValueError('consumed source identity mismatch on this tree: ' + entry['path']) + script = str(root / 'remote.py') + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=900) + if out.returncode != 0: + raise RuntimeError('check failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:]) + for m in ['hermite-sign', 'det-abs', 'weight-indicator']: + rc = subprocess.run([sys.executable, *flags, script, '--check', '--mutant', m], cwd=root, capture_output=True, timeout=900).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'certified': False})) + PY + git diff --exit-code diff --git a/.github/workflows/c6-residual-closure.yml b/.github/workflows/c6-residual-closure.yml new file mode 100644 index 000000000..be39b64ba --- /dev/null +++ b/.github/workflows/c6-residual-closure.yml @@ -0,0 +1,72 @@ +name: C6 residual closure checks + +'on': + pull_request: + paths: + - 'reviews/c6_residual_closure_20260930/**' + - '.github/workflows/c6-residual-closure.yml' + - 'frontiers/spectral_cluster_closure_20260929/PROOF.md' + - 'frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md' + - 'frontiers/spectral_cluster_closure_20260929/README.md' + - 'frontiers/c6_cluster_law_20260929/PROOF.md' + - 'frontiers/c6_cluster_law_20260929/README.md' + - 'frontiers/c6_cluster_law_20260929/SOURCE_MAP.json' + - 'frontiers/c6_palm_route_20260929/PROOF.md' + - 'frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md' + - 'frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md' + - 'frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json' + - 'frontiers/downstream_gate_20260925/GRAPH.json' + - 'frontiers/downstream_gate_20260925/SELECTOR_REGION.json' + - 'frontiers/downstream_gate_20260925/hard_gate.py' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-probe: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('reviews/c6_residual_closure_20260930').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + script = str(root / 'residual_check.py') + expected = (root / 'RESULTS.json').read_bytes() + mutants = ['allow-symlink', 'no-hash', 'stale-fingerprint', 'drop-required-edge', 'executed-flag', 'controlling-true', 'tail-reversed', 'pair-identity-broken', 'route-c-broken', 'drop-review-needle', 'installed-drift', 'snapshot-omitted', 'record-unreviewed', 'reviewed-text-drift'] + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], + check=True, capture_output=True, timeout=300).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in mutants: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], + capture_output=True, timeout=300).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/c6-witness-collision-reconciliation.yml b/.github/workflows/c6-witness-collision-reconciliation.yml new file mode 100644 index 000000000..adc6c7d16 --- /dev/null +++ b/.github/workflows/c6-witness-collision-reconciliation.yml @@ -0,0 +1,74 @@ +name: C6 witness-collision reconciliation checks + +'on': + pull_request: + paths: + - 'reviews/c6_witness_collision_reconciliation_20260929/**' + - '.github/workflows/c6-witness-collision-reconciliation.yml' + - 'frontiers/c6_*/**' + - 'frontiers/d5_dimension_lift_20260929/**' + - 'frontiers/elder_dimension_lift_20260928/**' + - 'frontiers/remote_collision_20260928/**' + - 'frontiers/remote_window_20260924/**' + - 'frontiers/downstream_gate_20260925/GRAPH.json' + - 'frontiers/downstream_gate_20260925/SELECTOR_REGION.json' + - 'imports/lifetime_parent_20260925/**' + - 'reviews/d5_reconciliation_20260929/**' + - 'reviews/c6_fourier_completion_20260929/**' + - 'reviews/elder_dimension_lift_claude_20260928/**' + - 'reviews/d5_remote_collision_grok_20260928/**' + - 'reviews/d5_offpin_second_moment_20260928/**' + - 'PROOF_INDEX.md' + - 'reviews/candidates_pending_20260928/CANDIDATES.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-probe: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('reviews/c6_witness_collision_reconciliation_20260929').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + script = str(root / 'reconciliation_check.py') + expected = (root / 'RESULTS.json').read_bytes() + mutants = ['allow-symlink', 'no-hash', 'drop-edge', 'stale-fingerprint', 'executed-flag', 'close-residual', 'drop-lower-bound', 'regional-strict', 'clone-live-node', 'open-cell-left'] + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], + check=True, capture_output=True, timeout=300).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in mutants: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], + capture_output=True, timeout=300).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/d2-remainder-fulldepth-review.yml b/.github/workflows/d2-remainder-fulldepth-review.yml new file mode 100644 index 000000000..ac73540c3 --- /dev/null +++ b/.github/workflows/d2-remainder-fulldepth-review.yml @@ -0,0 +1,64 @@ +name: D2 remainder full-depth nonauthor review checks + +'on': + pull_request: + paths: + - 'reviews/d2_remainder_full_depth_claude_20260930/**' + - '.github/workflows/d2-remainder-fulldepth-review.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-probe: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify record tree, reviewed source identities, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('reviews/d2_remainder_full_depth_claude_20260930').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text()) + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCE_FILES.json']) + if present != expected_names: + raise ValueError('record tree differs from manifest: ' + repr(present)) + for entry in manifest['files']: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular file required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('record file identity mismatch: ' + entry['path']) + for entry in manifest['reviewed_sources']: + data = pathlib.Path(entry['path']).read_bytes() + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['blob']: + raise ValueError('reviewed source drifted: ' + entry['path']) + script = str(root / 'remainder_review_check.py') + expected = (root / 'RESULTS.json').read_bytes() + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], check=True, capture_output=True, timeout=300).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in manifest['mutants']: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=300).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=300).returncode + if rc != 2: + raise RuntimeError('unknown mutant label not rejected with exit 2') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/extreme-cluster-sampling.yml b/.github/workflows/extreme-cluster-sampling.yml new file mode 100644 index 000000000..d47c6fcba --- /dev/null +++ b/.github/workflows/extreme-cluster-sampling.yml @@ -0,0 +1,38 @@ +name: Extreme whole-cluster sampling checks + +'on': + pull_request: + paths: + - 'frontiers/extreme_cluster_sampling_20260930/**' + - '.github/workflows/extreme-cluster-sampling.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public source objects + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf + git fetch --no-tags origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8 + - name: Verify historical sources, packet and exact controls + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/extreme_cluster_sampling_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/fixed-r-inverse-lifetime.yml b/.github/workflows/fixed-r-inverse-lifetime.yml new file mode 100644 index 000000000..9216a6e52 --- /dev/null +++ b/.github/workflows/fixed-r-inverse-lifetime.yml @@ -0,0 +1,36 @@ +name: Fixed-r inverse-lifetime checks + +'on': + pull_request: + paths: + - 'frontiers/fixed_r_inverse_lifetime_20260930/**' + - '.github/workflows/fixed-r-inverse-lifetime.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public parent model + run: | + set -euo pipefail + git fetch --no-tags origin 7a1cb09a9d58d179393e6d29146252f714b9c9bf + - name: Verify source, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/fixed_r_inverse_lifetime_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/lifetime-moment-boundary.yml b/.github/workflows/lifetime-moment-boundary.yml new file mode 100644 index 000000000..db56ae55c --- /dev/null +++ b/.github/workflows/lifetime-moment-boundary.yml @@ -0,0 +1,33 @@ +name: Lifetime and radial moment exact controls +'on': + pull_request: + paths: + - 'frontiers/lifetime_moment_boundary_20260930/**' + - 'reviews/radial_moment_corollaries_20260930/**' + - '.github/workflows/lifetime-moment-boundary.yml' + push: + branches: [main] + paths: + - 'frontiers/lifetime_moment_boundary_20260930/**' + - 'reviews/radial_moment_corollaries_20260930/**' + - '.github/workflows/lifetime-moment-boundary.yml' + workflow_dispatch: +permissions: + contents: read +jobs: + exact-moments: + runs-on: ubuntu-latest + timeout-minutes: 5 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - run: python -B -S frontiers/lifetime_moment_boundary_20260930/verify.py + - name: Receipt and frozen-source regression controls + run: | + python -B -S -m unittest discover -s frontiers/lifetime_moment_boundary_20260930 -p 'test_*.py' -v + python -B -O -S -m unittest discover -s frontiers/lifetime_moment_boundary_20260930 -p 'test_*.py' -v + - run: git diff --exit-code diff --git a/.github/workflows/radial-rate.yml b/.github/workflows/radial-rate.yml new file mode 100644 index 000000000..fa1a9abe5 --- /dev/null +++ b/.github/workflows/radial-rate.yml @@ -0,0 +1,38 @@ +name: Microscopic radial-rate checks + +'on': + pull_request: + paths: + - 'frontiers/radial_rate_20260930/**' + - '.github/workflows/radial-rate.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch exact public source objects + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf + git fetch --no-tags origin 7a1cb09a9d58d179393e6d29146252f714b9c9bf + - name: Verify sources, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/radial_rate_20260930/verify.py + git diff --exit-code diff --git a/.github/workflows/register-alignment.yml b/.github/workflows/register-alignment.yml new file mode 100644 index 000000000..e8d30be7f --- /dev/null +++ b/.github/workflows/register-alignment.yml @@ -0,0 +1,75 @@ +name: Register alignment checks + +'on': + pull_request: + paths: + - 'reviews/register_alignment_20260930/**' + - '.github/workflows/register-alignment.yml' + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md' + - 'coefficients/side24_v1/PROOF.md' + - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md' + - 'frontiers/full_price_20260924/PROOF.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'reviews/pr22_fixed_annulus_nonauthor_20260925/**' + - 'reviews/p15_full_price_nonauthor_20260926/**' + - 'reviews/side24_v1_coefficient_claude_20260929/**' + - 'frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md' + - 'reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md' + - 'frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md' + - 'frontiers/downstream_gate_20260925/GRAPH.json' + - 'frontiers/downstream_gate_20260925/hard_gate.py' + - 'PROOF_INDEX.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-probe: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify sources, replay both modes, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('reviews/register_alignment_20260930').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + script = str(root / 'alignment_check.py') + expected = (root / 'RESULTS.json').read_bytes() + mutants = ['allow-symlink', 'no-hash', 'stale-fingerprint', 'drop-review-source', 'controlling-true', 'executed-flag', 'skip-interface-premise', 'holder-reversed', 'baseline-drift', 'duplicate-drift', 'region-unlinked', 'review-metadata-only'] + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script], + check=True, capture_output=True, timeout=300).stdout + if out != expected: + raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags)) + for m in mutants: + rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], + capture_output=True, timeout=300).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/two-scale-cluster-geometry.yml b/.github/workflows/two-scale-cluster-geometry.yml new file mode 100644 index 000000000..5dae60249 --- /dev/null +++ b/.github/workflows/two-scale-cluster-geometry.yml @@ -0,0 +1,41 @@ +name: Two-scale cluster geometry checks + +'on': + push: + branches: ['main'] + paths: + - 'frontiers/two_scale_cluster_geometry_20260929/**' + - '.github/workflows/two-scale-cluster-geometry.yml' + pull_request: + paths: + - 'frontiers/two_scale_cluster_geometry_20260929/**' + - '.github/workflows/two-scale-cluster-geometry.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-source: + runs-on: ubuntu-latest + timeout-minutes: 10 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Fetch immutable non-base source objects + run: | + set -euo pipefail + git fetch --no-tags --depth=1 origin 7a44130b41b827577424a73c4feac4cde20b5568 + git fetch --no-tags --depth=1 origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8 + - name: Verify all sources, exact controls and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S frontiers/two_scale_cluster_geometry_20260929/verify.py + git diff --exit-code diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md b/frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md new file mode 100644 index 000000000..845881d40 --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md @@ -0,0 +1,149 @@ +# Numerical values of the planar near cluster coefficients `alpha_1`, `alpha_2` + +**Object:** CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1. +**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 30 September 2026. +**Disposition:** numerical evaluation. **Not a proof, not a certified enclosure.** Floating-point quadrature and Monte +Carlo values with reported standard errors, for the explicit finite-dimensional Gaussian integrals that the cluster-law +candidates define. Every packet on the C6 cluster line lists "no numerical value of `nu(1)`, `nu(2)`" as a non-claim; +this note supplies values for the *near* coefficients only, and says exactly what they are conditional on. +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX`, `GRAPH`, claim, catalog, prize or source body changes. +Same GitHub account as every lane; zero organizational-independence credit. + +## 1. What is evaluated + +Planar model of [LP] (`d = 2`), torus side `L`, variance-one periodized Gaussian field with spectral weights +`a_n = exp(-2 pi^2 |n|^2 / L^2) / sum_j exp(-2 pi^2 |j|^2 / L^2)` ([LP] §2), pins `f(M) = b`, `f(S) = b - k r^3`, +`grad f(M) = grad f(S) = 0`, original weight `W_r` and full normalizer `Z_r`. The near coefficients of the cluster law +are, in the notation of [CL] (1.3) and (3.11) at `d = 2` (equivalently Theorem F of [CUB]), + + alpha_j = (1/z_0) integral_(R^4) h_0(0, a, beta, c) w_0(s, a, beta) 1{ n(s, a, beta, c) = j } ds da dbeta dc, j = 1, 2, + +where `h_0` is the contact density of `(f_zz, f_xxz, f_xzz, f_zzz)(0)` given the contact pins +`U_0 = (f, f_x, f_xx, f_xxx, f_z, f_xz)(0) = (b, 0, 0, 12k, 0, 0)`, `z_0 = lim Z_r / r^2`, `w_0 = [-6k(s - beta/2) - +a^2/4]_+ [a^2/4 - 6k(s + beta/2)]_+` is the pin weight, and `n` is the number of extra critical points of the pinned cubic +`F_0` with height in `(-k, 0)`. In the cluster law, `nu(2) = alpha_2` and `nu(1) = alpha_1 + k integral_X Lambda`; the +remote part `k integral_X Lambda` ([RM]) is **not** evaluated here. + +Everything below is conditional on these formulas, i.e. on [CL] Theorem N (Math-#159, nonauthor-accepted and merged 30 September 2026) or on +[CUB] Theorem F (merged, Claude-accepted) for the near coefficients as *lower bounds* of the global intensities. The +numbers do not accept either source. + +## 2. Exact structure of the contact regression (checked exactly, control `exact_structure`) + +Because `a_n` depends only on `|n|` through a Gaussian, `K_L` is the periodization of `K(z) = exp(-|z|^2/2)`; the jet +covariances at one site are the continuum values up to `O(exp(-L^2/2))`, and the lattice sums at `L = 24` and `L = 12` +reproduce them to `1.4e-13` (control `lattice_vs_continuum`). With `Cov(d^alpha f, d^beta f) = (-1)^|beta| d^(alpha+beta) K(0)` +and `d^(2m) exp(-x^2/2)|_0 = (-1)^m (2m-1)!!`: + +| jets | covariance | +|---|---| +| `Var f, Var f_x, Var f_z, Var f_xz` | `1` | +| `Var f_xx, Var f_zz, Var f_xxz, Var f_xzz` | `3` | +| `Var f_xxx, Var f_zzz` | `15` | +| `Cov(f, f_xx) = Cov(f, f_zz) = Cov(f_x, f_xzz) = Cov(f_z, f_xxz)` | `-1` | +| `Cov(f_xx, f_zz) = Cov(f_xxx, f_xzz) = Cov(f_xxz, f_zzz)` | `1`, `3`, `3` | +| `Cov(f_x, f_xxx) = Cov(f_z, f_zzz)` | `-3` | + +Parity splits the ten jets into the even block `(f, f_xx, f_xz, f_zz)` and the odd block `(f_x, f_z, f_xxx, f_xxz, f_xzz, +f_zzz)` with zero cross-covariance ([CUB] Theorem B). Exact regression (rational arithmetic): + +- Even block: `A = f_zz | (f, f_xx, f_xz) = (b, 0, 0)` is `N(-b, 2)`. Hence `p_b(0) = phi(b/sqrt2)/sqrt2` and + `z_0 = 36 k^2 m_(2,b)`, `m_(2,b) = E[A^2 1{A < 0}] = (b^2 + 2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2)`; in particular + `z_0 = 36 k^2` exactly at `b = 0` and `m_(2,1) = 3 Phi(1/sqrt2) + sqrt2 phi(1/sqrt2) = 2.72014...`. +- Odd block: `(a, beta, c) = (f_xxz, f_xzz, f_zzz) | (f_x, f_z, f_xxx) = (0, 0, 12k)` is `N(0, diag(2, 2, 6))` for every + `k` and `b`: the regression coefficient of every output on the `f_xxx = 12k` pin vanishes (for `f_xzz` it is + `[-1, 3] . [[1, -3], [-3, 15]]^(-1) = [-1, 0]`), and the parity zeros kill the rest. + +Therefore + + alpha_j(k, b) = [ phi(b/sqrt2) / ( sqrt2 . 36 k^2 m_(2,b) ) ] . J_j(k), J_j(k) = E_( (a,beta,c) ~ N(0, diag(2,2,6)) ) [ I_j(a, beta, c; k) ], + +and the ratio `alpha_2/(alpha_1 + alpha_2) = J_2/(J_1 + J_2)` is independent of `b` ([CUB] Theorem B), while its `k` +dependence is genuine (the jet law does not scale with `k`). + +## 3. The `s`-integral, exactly + +With `B = beta - a^2/(12k)`, `D = (c - a beta/(4k) + a^3/(72k^2))/2` and the sheared cubic +`G(u, Z) = C(u) + (s + Bu) Z^2/2 + (D/3) Z^3` ([CUB] (C2)–(C3), [CL] (3.12)), the pin weight on the typed domain +`s < -|B|/2` is `w_0 = 9k^2 (4s^2 - B^2)`, and the classifier ([CUB] Theorem C) reads, with + + g(s) = -2 s^3 + 3 B s^2 - B^3 - 12 k D^2 = 12k ( T_*(s) - D^2 ), T_*(s) = -(s - B)^2 (B + 2s) / (12k), + +`n = 2` iff `s > B` and `g(s) > 0`; `n = 1` iff `g(s) < 0`; `n = 0` iff `s <= B` and `g(s) > 0`. So + + I_j(a, beta, c; k) = integral_(s < -|B|/2) 9k^2 (4s^2 - B^2) 1{n = j} ds + +is a sum over at most two intervals with endpoints among the real roots of the cubic `g` (found by bracketed +bisection to machine precision), of the exact antiderivative `12 k^2 s^3 - 9 k^2 B^2 s`. Closed forms used as +controls: the [LM] cubic `(a, beta, c) = (0, -2k, 0)` gives `(I_1, I_2) = (0, 48 k^5)` (`n = 2` exactly on `(B, B/2)`); +the family `(0, 0, c)` gives `(72 k^3 D^2, 0)` with `D = c/2`. The classifier is checked against a direct count of the +critical points of `G` (quadratic in `u`, heights in `(-k, 0)`) on 4000 random typed samples (control `classifier`), and +`I_j` against brute-force `s`-quadrature during development (agreement `1e-4`, the quadrature error). Lemma 3.6 of [CL] +(`-s <= 2|B| + (32 k D^2)^(1/3)` on `{n >= 1}`) is the reason the outer integrand is a polynomial times a Gaussian and +the outer integral converges. + +## 4. Numerical values + +Outer integration over `(a, beta, c) ~ N(0, diag(2, 2, 6))`: tensor Gauss–Hermite of orders 40 and 60 (64,000 and +216,000 nodes) and Monte Carlo with `10^6` samples (fixed seed), standard errors from the sample variance. The +integrand `I_j` is **not** bounded: it grows polynomially in the jets (on the `B = 0` family `I_1 = 18 k^3 c^2`), as +Lemma 3.6 of [CL] allows, and it has kinks where a root of `g` crosses `-|B|/2` or `B`. Against a Gaussian weight a +polynomially growing, piecewise-smooth integrand is integrable, but no convergence rate for Gauss–Hermite is proved +here; the two orders are reported as two deterministic estimates, and the Monte Carlo standard error is the only +statistical error indication. **All precision statements below are empirical.** + +| `k` | `J_1` GH40 / GH60 / MC (`±` s.e.) | `J_2` GH40 / GH60 / MC (`±` s.e.) | `J_2/(J_1+J_2)` | near Palm excess `2J_2/(J_1+2J_2)` | +|---|---|---|---|---| +| `1/2` | 35.652 / 35.676 / 35.559 ± 0.086 | 1.121 / 1.109 / 1.115 ± 0.0062 | 0.0301 | 0.0585 | +| `1` | 169.63 / 169.61 / 169.37 ± 0.28 | 3.249 / 3.255 / 3.281 ± 0.022 | 0.0188 | 0.0370 | +| `2` | 1104.2 / 1103.7 / 1102.6 ± 1.7 | 8.421 / 8.641 / 8.875 ± 0.073 | 0.0078 | 0.0154 | + +`J_j(k) = E_(N(0, diag(2,2,6)))[I_j]`; the ratio columns are `b`-free. Prefactor `p_b(0)/z_0` and the coefficients `alpha_j = (p_b(0)/z_0) J_j` (GH60 values; MC in `RESULTS.json`): + +| `k` | `b` | `z_0 = 36 k^2 m_(2,b)` | `p_b(0)/z_0` | `alpha_1` | `alpha_2` | +|---|---|---|---|---|---| +| `1/2` | `0` | 9 | 0.031344 | 1.118 | 0.0347 | +| `1/2` | `1` | 24.4813 | 0.008974 | 0.3202 | 0.00995 | +| `1` | `0` | 36 | 0.007836 | 1.329 | 0.0255 | +| `1` | `1` | 97.9251 | 0.0022435 | 0.3805 | 0.0073 | +| `2` | `0` | 144 | 0.001959 | 2.162 | 0.0169 | +| `2` | `1` | 391.7 | 0.00056088 | 0.619 | 0.00485 | + +Reading: at `k = 1`, `b = 0`, `alpha_1 ≈ 1.33` and `alpha_2 ≈ 0.026`; about 1.9% of nonempty near clusters are two-point clusters, and the near part of the Palm excess `2 nu(2)/(nu(1) + 2 nu(2))` of [C6] Corollary P is at most about 3.7% (the remote singleton part of `nu(1)` only lowers it). The two-point share decreases in `k` (3.0% at `k = 1/2`, 0.78% at `k = 2`). Empirical precision (GH60 against MC, `10^6` samples): + +| `k` | `J_1`: GH60 − MC (in MC s.e., relative) | `J_2`: GH60 − MC (in MC s.e., relative) | +|---|---|---| +| `1/2` | +0.1162 (+1.4 s.e., 0.33%) | -0.00615 (-1.0 s.e., 0.6%) | +| `1` | +0.237 (+0.8 s.e., 0.14%) | -0.02519 (-1.1 s.e., 0.8%) | +| `2` | +1.05 (+0.6 s.e., 0.10%) | -0.2336 (-3.2 s.e., 2.6%) | + +So `J_1` (hence `alpha_1`) is reproduced by the two independent methods to about `0.1`–`0.3%`, within `1.4` MC standard +errors; `J_2` (hence `alpha_2`) to about `0.5`–`0.8%` at `k = 1/2, 1` (within `1.1` s.e.) and to about `2.7%` at `k = 2`, +where GH60 lies `3.2` s.e. below MC and GH40 further below: the Gauss–Hermite sequence is still moving there, so the MC +value with its standard error is the better estimate of `J_2(2)` and the third digit of `alpha_2(2)` is not established. +These are observed discrepancies, not error bounds. + +Finite `L`: the integrations above use the continuum jet covariance, which the lattice sums reproduce to `1e-13` at +`L = 24` (the parent's SIDE24 application) and `L = 12`. At `L = 6` the lattice jet covariances deviate from the continuum +table by up to `8.8e-4` (absolute), so `L = 6` is *not* covered by the continuum computation: the script propagates the +actual `L = 6` lattice covariance through the even/odd regressions (floating point) and the order-40 quadrature, and +`RESULTS.json` (`lattice_L6`) records the resulting `alpha_j` and their relative deviations from the continuum values, +of order `1e-4`–`1e-3`; `--check` replays the `k = 1` case. The full run is `coefficients.py` (writes `RESULTS.json`); the +replay mode `--check` re-verifies the exact structure, the closed forms, the classifier and the order-40 quadrature +against `RESULTS.json` (tolerance `1e-6`, floating point), and three mutants (`cubic-sign`, `antiderivative`, +`typed-boundary`) exit 1. + +## 5. What these numbers are not + +Not certified: no interval arithmetic; the Monte Carlo standard errors and the Gauss–Hermite order comparison are the +only error indications. Not the full `nu(1)`: the remote singleton coefficient `k integral_X Lambda` of [RM] is a +separate two-site Gaussian integral, not evaluated. Not a statement about `d >= 3`: the odd/even block structure and +the `N(0, diag(2, 2, 6))` law are planar. Not an acceptance of [CL] or [CUB]: if either near-coefficient formula fails +review, these are numbers for a formula, not for the model. Not a rate, not a finite-`r` statement. + +## 6. Provenance + +Sources and their exact bytes are in `SOURCE_MAP.json`: [LP] and [CUB] on `main`; [CL] at the head of Math-#159 (not +required on `main`). `coefficients.py` is standard-library Python; `RESULTS.json` is its output rounded to six +significant digits. The [LM] cubic `(−3k/2, 0, −2k, 0)` and the classifier are the same objects as in [CUB] and [CL]; +no new mathematics is claimed here beyond the exact regression identities of §2, which are elementary. diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/README.md b/frontiers/c6_cluster_coefficients_numerics_20260930/README.md new file mode 100644 index 000000000..5e7d05e10 --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/README.md @@ -0,0 +1,16 @@ +# C6 cluster coefficient numerics (CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1) + +Numerical values (floating point, with standard errors; **not certified**) of the planar near cluster coefficients +`alpha_1`, `alpha_2` defined by the C6 cluster-law candidates ([CL] Math-#159 (1.3)/(3.11) at `d = 2`, [CUB] Theorem F) +for the exact periodized Gaussian kernel of the parent [LP]. Scientific effect NONE. + +- `NOTE.md`: what is evaluated, the exact contact-regression structure (`A ~ N(-b, 2)`, odd block `N(0, diag(2,2,6))`, + `z_0 = 36 k^2 m_(2,b)`), the exact interval structure of the `s`-integral, the values, and what they are not. +- `coefficients.py`: standard library only. `python -B -S coefficients.py` recomputes `RESULTS.json` (several minutes); + `python -B -S coefficients.py --check` replays the exact controls and the order-40 quadrature against `RESULTS.json`; + `--check --mutant NAME` exits 1 for `cubic-sign`, `antiderivative`, `typed-boundary`. +- `RESULTS.json`: `J_j(k)` (Gauss–Hermite 40/60, Monte Carlo `10^6` with standard errors), the exact prefactor + `p_b(0)/z_0`, and `alpha_j(k, b)` for `k in {1/2, 1, 2}`, `b in {0, 1}`. +- `SOURCE_MAP.json`, `SOURCE_FILES.json`: pins and manifest. + +Not evaluated: the remote part `k integral Lambda` of `nu(1)`; anything for `d >= 3`; any rate or enclosure. diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json b/frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json new file mode 100644 index 000000000..2648314f8 --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json @@ -0,0 +1,172 @@ +{ + "certified": false, + "exact_structure": { + "cov_fx_fxxx": "-3", + "mu_A": "-b", + "odd_cov_diag": [ + 2, + 2, + 6 + ], + "odd_mean": [ + 0, + 0, + 0 + ], + "var_A": 2, + "var_fxxx": "15" + }, + "lattice_L6": { + "k=1": { + "J1_gh40": 169.6136140680646, + "J2_gh40": 3.249752944819857, + "alpha1_gh40": 1.32905, + "alpha2_gh40": 0.0254643, + "muA": 0.0, + "muJ": [ + 0.0, + 3.99757e-15, + 0.0 + ], + "prefactor_b0": 0.00783576, + "rel_dev_alpha1_vs_continuum_gh40": -0.000152308, + "rel_dev_alpha2_vs_continuum_gh40": 7.00985e-05, + "varA": 2.00004 + }, + "k=1/2": { + "J1_gh40": 35.649991916363994, + "J2_gh40": 1.1211979306327606, + "alpha1_gh40": 1.11738, + "alpha2_gh40": 0.0351418, + "muA": 0.0, + "muJ": [ + 0.0, + 1.99879e-15, + 0.0 + ], + "prefactor_b0": 0.031343, + "rel_dev_alpha1_vs_continuum_gh40": -9.38401e-05, + "rel_dev_alpha2_vs_continuum_gh40": 2.09463e-05, + "varA": 2.00004 + }, + "k=2": { + "J1_gh40": 1104.0768381368005, + "J2_gh40": 8.42232807714671, + "alpha1_gh40": 2.16282, + "alpha2_gh40": 0.0164988, + "muA": 0.0, + "muJ": [ + 0.0, + 7.99515e-15, + 0.0 + ], + "prefactor_b0": 0.00195894, + "rel_dev_alpha1_vs_continuum_gh40": -0.000176894, + "rel_dev_alpha2_vs_continuum_gh40": 0.00011526, + "varA": 2.00004 + } + }, + "lattice_vs_continuum_max_dev": { + "12": 1.95399e-14, + "24": 1.36779e-13, + "6": 0.000878343 + }, + "method": "exact contact regression + interval-exact s-integral + Gauss-Hermite / Monte Carlo", + "object": "CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1", + "per_k": { + "1": { + "J1_gh40": 169.63498712445082, + "J1_gh60": 169.611, + "J1_mc": 169.374, + "J1_mc_se": 0.282859, + "J2_gh40": 3.2494396407422346, + "J2_gh60": 3.25538, + "J2_mc": 3.28057, + "J2_mc_se": 0.02219, + "palm_near_gh60": 0.0369675, + "ratio2_gh60": 0.0188318 + }, + "1/2": { + "J1_gh40": 35.652399345763925, + "J1_gh60": 35.6757, + "J1_mc": 35.5595, + "J1_mc_se": 0.0856231, + "J2_gh40": 1.1211449404513953, + "J2_gh60": 1.10852, + "J2_mc": 1.11467, + "J2_mc_se": 0.00618859, + "palm_near_gh60": 0.0585084, + "ratio2_gh60": 0.0301358 + }, + "2": { + "J1_gh40": 1104.2431167706125, + "J1_gh60": 1103.68, + "J1_mc": 1102.63, + "J1_mc_se": 1.68663, + "J2_gh40": 8.421135809095846, + "J2_gh60": 8.64125, + "J2_mc": 8.87486, + "J2_mc_se": 0.0727029, + "palm_near_gh60": 0.0154176, + "ratio2_gh60": 0.00776868 + } + }, + "prefactor": { + "k=1,b=0": { + "alpha1_gh60": 1.32906, + "alpha1_mc": 1.32721, + "alpha2_gh60": 0.0255091, + "alpha2_mc": 0.0257064, + "m2_b": 1.0, + "p_b0_over_z0": 0.00783597, + "z0": 36.0 + }, + "k=1,b=1": { + "alpha1_gh60": 0.380523, + "alpha1_mc": 0.379993, + "alpha2_gh60": 0.00730347, + "alpha2_mc": 0.00735999, + "m2_b": 2.72014, + "p_b0_over_z0": 0.00224351, + "z0": 97.9251 + }, + "k=1/2,b=0": { + "alpha1_gh60": 1.11821, + "alpha1_mc": 1.11457, + "alpha2_gh60": 0.0347454, + "alpha2_mc": 0.034938, + "m2_b": 1.0, + "p_b0_over_z0": 0.0313439, + "z0": 9.0 + }, + "k=1/2,b=1": { + "alpha1_gh60": 0.320155, + "alpha1_mc": 0.319112, + "alpha2_gh60": 0.00994791, + "alpha2_mc": 0.0100031, + "m2_b": 2.72014, + "p_b0_over_z0": 0.00897403, + "z0": 24.4813 + }, + "k=2,b=0": { + "alpha1_gh60": 2.1621, + "alpha1_mc": 2.16004, + "alpha2_gh60": 0.0169281, + "alpha2_mc": 0.0173858, + "m2_b": 1.0, + "p_b0_over_z0": 0.00195899, + "z0": 144.0 + }, + "k=2,b=1": { + "alpha1_gh60": 0.619027, + "alpha1_mc": 0.61844, + "alpha2_gh60": 0.00484668, + "alpha2_mc": 0.0049777, + "m2_b": 2.72014, + "p_b0_over_z0": 0.000560877, + "z0": 391.7 + } + }, + "schema": 1, + "scientific_effect": "NONE" +} diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..a96c69cf8 --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_FILES.json @@ -0,0 +1,45 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "certified": false, + "review_status": "AUTHOR_SIDE_NUMERICAL_NOTE", + "files": [ + { + "path": "NOTE.md", + "bytes": 11018, + "sha256": "f20c1ac96b00edc071e428bd2e5dc4c91b8546a3ee0693367b783109b26aa518", + "git_blob": "845881d40520d4acbe15ee196a1ba0d5412eafa1" + }, + { + "path": "README.md", + "bytes": 1317, + "sha256": "1c47f7009455e1a4e72eb3f40e45def4fd4f530b61a87eb610d080bbed9bfbc0", + "git_blob": "5e7d05e10c3f61d2629bd5cf1ab47957860bf40f" + }, + { + "path": "RESULTS.json", + "bytes": 4095, + "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb", + "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f" + }, + { + "path": "SOURCE_MAP.json", + "bytes": 3651, + "sha256": "29ab4b897055724afcebb52d570f8271df60fe770b5a38bcb933bebffcedb7e6", + "git_blob": "443970cadfd2c06222f46fdf4582047f0ce13459" + }, + { + "path": "coefficients.py", + "bytes": 18825, + "sha256": "bcd63f0806a58a76a85189372bc6eee99fc3847c74dcc34906ef63464a64b97b", + "git_blob": "bd45a9fdecf300111b646644384dcead20c2e529" + } + ], + "nonauthor_reads": [ + { + "lane": "xAI/Grok", + "review_id": 5360559751, + "scope": "Slices A+B accepted as identities; Slice C empirical, not certified" + } + ] +} diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_MAP.json new file mode 100644 index 000000000..443970cad --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/SOURCE_MAP.json @@ -0,0 +1,55 @@ +{ + "schema": 1, + "object": "CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "2479a7c2bcadf441517cadd1adff2cc07019b49d", + "note": "Numerical evaluation only. The formulas evaluated are (1.3)/(3.11) of [CL] at d = 2 (equivalently Theorem F of [CUB]) with the exact kernel of [LP] section 2. No source theorem is consumed as accepted; [CL] (Math-#159) was nonauthor-accepted (OpenAI review 5360192822, source-exposed) and merged to main on 30 September 2026 with the evaluated PROOF.md bytes unchanged; it is now pinned on main like the other sources. The numbers are floating-point quadrature and Monte Carlo values with reported standard errors, not enclosures. xAI/Grok review 5360559751 (30 September 2026, head 0b35e76) accepts Slices A and B as identities and reads Slice C as correctly empirical; recorded under nonauthor_reads without changing review_status (numerical note, not a theorem).", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "2479a7c2bcadf441517cadd1adff2cc07019b49d", + "role": "section 2: the exact periodized Gaussian kernel K_L and its spectral weights a_n; section 3: contact frame U_0 and target v_0 = (b, 0, 0, 12k, 0, 0); (5.4)-(5.5): the full normalizer z_0 = lim Z_r / r^2", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "id": "CUB", + "commit": "2479a7c2bcadf441517cadd1adff2cc07019b49d", + "role": "Theorem C (exact classifier of the extra window critical points of the sheared cubic; xAI-accepted at algebraic scope) and Theorem F (finite positive alpha_1, alpha_2; Claude-accepted); Theorem B (birth-height parity factorization)", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_law_20260929/PROOF.md", + "bytes": 70131, + "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5", + "id": "CL", + "commit": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "role": "(1.3) the near coefficient integral, (3.11) the reduced weight, section 1.4 (planar reduction m = p_0), Lemma 3.6 (saddles only; s-range) and section 3.8 (sheared form); Math-#159, nonauthor-accepted (OpenAI review 5360192822) and merged to main on 30 September 2026 with PROOF.md bytes unchanged", + "current_required": true, + "on_main": true + } + ], + "nonauthor_acceptance": false, + "governing_status_changed": false, + "main_at_update": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "nonauthor_reads": [ + { + "lane": "xAI/Grok", + "review_id": 5360559751, + "head": "0b35e76ff05676e6ad5c89171d65807668c8de15", + "date": "2026-09-30", + "disposition": "ACCEPT Slices A and B as identities (one-site covariances, even/odd regressions, b-free share, exact antiderivative and the two closed controls); Slice C read as correctly stated empirical precision, not certified; classifier-versus-direct table not re-audited; not a second acceptance of [CL]/[CUB]; same GitHub account, organizational-independence credit 0" + } + ] +} diff --git a/frontiers/c6_cluster_coefficients_numerics_20260930/coefficients.py b/frontiers/c6_cluster_coefficients_numerics_20260930/coefficients.py new file mode 100644 index 000000000..bd45a9fde --- /dev/null +++ b/frontiers/c6_cluster_coefficients_numerics_20260930/coefficients.py @@ -0,0 +1,357 @@ +"""Numerical evaluation of the planar near cluster coefficients alpha_1, alpha_2 (CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1). + +Standard library only. NOT a proof and NOT a certified enclosure: floating-point quadrature and Monte Carlo with +reported standard errors. Scientific effect NONE. + + python -B -S coefficients.py # full run: writes RESULTS.json (GH40, GH60, MC 1e6 per k) + python -B -S coefficients.py --check # quick replay: exact identities, closed forms, GH40 against RESULTS.json + python -B -S coefficients.py --check --mutant NAME # rc 1 for each name in MUTANTS +""" +import argparse, json, math, os, random, sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('cubic-sign', 'antiderivative', 'typed-boundary') +MUTANT = None +K_VALUES = (F(1, 2), F(1), F(2)) +B_VALUES = (F(0), F(1)) + +# ------------------------------------------------------------------ exact part: continuum Gaussian kernel jets +JETS = {'f': (0, 0), 'fx': (1, 0), 'fz': (0, 1), 'fxx': (2, 0), 'fxz': (1, 1), 'fzz': (0, 2), + 'fxxx': (3, 0), 'fxxz': (2, 1), 'fxzz': (1, 2), 'fzzz': (0, 3)} + +def hermite_moment(m): + """d^m/dx^m exp(-x^2/2) at 0 = (-1)^(m/2) (m-1)!! for even m, 0 for odd m.""" + if m % 2: return F(0) + v = F(1) + for j in range(1, m, 2): v *= j + return v if (m // 2) % 2 == 0 else -v + +def exact_cov(): + """Cov(d^alpha f(0), d^beta f(0)) = (-1)^|beta| d^(alpha+beta) K(0), K(z) = exp(-|z|^2/2).""" + C = {} + for u, au in JETS.items(): + for v, av in JETS.items(): + sign = -1 if (av[0] + av[1]) % 2 else 1 + C[(u, v)] = sign * hermite_moment(au[0] + av[0]) * hermite_moment(au[1] + av[1]) + return C + +def solve(A, B): + n = len(A); m = len(B[0]); Ab = [A[i][:] + B[i][:] for i in range(n)] + for i in range(n): + piv = next(r for r in range(i, n) if Ab[r][i] != 0) + Ab[i], Ab[piv] = Ab[piv], Ab[i]; p = Ab[i][i] + Ab[i] = [x / p for x in Ab[i]] + for r in range(n): + if r != i and Ab[r][i] != 0: + f = Ab[r][i]; Ab[r] = [x - f * y for x, y in zip(Ab[r], Ab[i])] + return [row[n:] for row in Ab] + +def regression(C, targets, outputs): + """(coefficient matrix K, conditional covariance) with mean = K . target values.""" + S_tt = [[C[(r, c)] for c in targets] for r in targets] + S_ot = [[C[(r, c)] for c in targets] for r in outputs] + S_oo = [[C[(r, c)] for c in outputs] for r in outputs] + Kt = solve(S_tt, [list(col) for col in zip(*S_ot)]) # S_tt^-1 S_ot^T + K = [list(col) for col in zip(*Kt)] # S_ot S_tt^-1 + cov = [[S_oo[i][j] - sum(K[i][t] * S_ot[j][t] for t in range(len(targets))) for j in range(len(outputs))] + for i in range(len(outputs))] + return K, cov + +def require(ok, msg): + if not ok: raise ValueError(msg) + +def exact_structure(): + C = exact_cov() + K_even, cov_even = regression(C, ['f', 'fxx', 'fxz'], ['fzz']) + require(K_even == [[F(-1), F(0), F(0)]], 'even block: mean of f_zz given (f, f_xx, f_xz) = (b,0,0) is -b') + require(cov_even == [[F(2)]], 'even block: conditional variance of f_zz is 2') + K_odd, cov_odd = regression(C, ['fx', 'fz', 'fxxx'], ['fxxz', 'fxzz', 'fzzz']) + require(all(K_odd[i][2] == 0 for i in range(3)), 'odd block: no dependence on the f_xxx = 12k pin') + require(K_odd[0][0] == 0 and K_odd[2][0] == 0 and K_odd[1][1] == 0, 'odd block: parity zeros') + require(cov_odd == [[F(2), F(0), F(0)], [F(0), F(2), F(0)], [F(0), F(0), F(6)]], + 'odd block: (f_xxz, f_xzz, f_zzz) | pins ~ N(0, diag(2, 2, 6))') + return {'mu_A': '-b', 'var_A': 2, 'odd_mean': [0, 0, 0], 'odd_cov_diag': [2, 2, 6], + 'var_fxxx': str(C[('fxxx', 'fxxx')]), 'cov_fx_fxxx': str(C[('fx', 'fxxx')])} + +# ------------------------------------------------------------------ lattice sums for the exact periodized kernel +def lattice_cov(L): + N = int(math.ceil(L * 2.6)) + 3; c = 2 * math.pi ** 2 / L ** 2 + Z = 0.0; M = {} + for n1 in range(-N, N + 1): + for n2 in range(-N, N + 1): + w = math.exp(-c * (n1 * n1 + n2 * n2)); Z += w + for p in range(0, 7, 2): + for q in range(0, 7, 2): + M[(p, q)] = M.get((p, q), 0.0) + w * n1 ** p * n2 ** q + C = {} + for u, au in JETS.items(): + for v, av in JETS.items(): + p, q = au[0] + av[0], au[1] + av[1] + if p % 2 or q % 2: C[(u, v)] = 0.0; continue + coef = (2j * math.pi / L) ** (au[0] + au[1]) * (-2j * math.pi / L) ** (av[0] + av[1]) + C[(u, v)] = (coef * M[(p, q)] / Z).real + return C + +# ------------------------------------------------------------------ the s-integral of the pin weight over {n = j} +def cubic_g(s, B, k, D2): + sgn = 1 if MUTANT == 'cubic-sign' else -1 + return -2 * s ** 3 + 3 * B * s ** 2 + sgn * B ** 3 - 12 * k * D2 + +def cubic_roots(B, k, D2): + g = lambda s: cubic_g(s, B, k, D2) + pts = sorted({-1e3 * (1 + abs(B) + k + D2), B, 0.0, 1e3 * (1 + abs(B) + k + D2)}) + while g(pts[0]) <= 0: pts[0] *= 2 + while g(pts[-1]) >= 0: pts[-1] *= 2 + roots = [] + for lo, hi in zip(pts[:-1], pts[1:]): + glo, ghi = g(lo), g(hi) + if glo == 0: roots.append(lo); continue + if glo * ghi < 0: + for _ in range(200): + mid = 0.5 * (lo + hi); gm = g(mid) + if gm == 0: lo = hi = mid; break + if glo * gm < 0: hi, ghi = mid, gm + else: lo, glo = mid, gm + roots.append(0.5 * (lo + hi)) + return sorted(set(roots)) + +def shear(a, b, c, k): + B = b - a * a / (12 * k); D = (c - a * b / (4 * k) + a ** 3 / (72 * k * k)) / 2 + return B, D + +def typed_top(B): + return -abs(B) if MUTANT == 'typed-boundary' else -abs(B) / 2 + +def n_classifier(s, a, b, c, k): + B, D = shear(a, b, c, k) + if not s < typed_top(B): return 0 + g = cubic_g(s, B, k, D * D) + if s > B: return 2 if g > 0 else 1 + return 1 if g < 0 else 0 + +def n_direct(s, a, b, c, k): + """critical points of G(u, Z) = C(u) + (s + Bu) Z^2/2 + (D/3) Z^3 off Z = 0 with height in (-k, 0).""" + B, D = shear(a, b, c, k); Cu = lambda u: 2 * k * u ** 3 - 1.5 * k * u - k / 2; hs = [] + if abs(D) > 1e-14: + A2 = 6 * k + B ** 3 / (2 * D * D); A1 = B * B * s / (D * D); A0 = B * s * s / (2 * D * D) - 1.5 * k + if abs(A2) < 1e-14: us = [-A0 / A1] if abs(A1) > 1e-14 else [] + else: + disc = A1 * A1 - 4 * A2 * A0 + us = [] if disc < 0 else [(-A1 + sg * math.sqrt(disc)) / (2 * A2) for sg in (1, -1)] + for u in us: + w = s + B * u; Z = -w / D + if abs(Z) > 1e-12: hs.append(Cu(u) + w * Z * Z / 2 + D * Z ** 3 / 3) + elif abs(B) > 1e-14: + u = -s / B; Z2 = 3 * k * (1 - 4 * u * u) / B + if Z2 > 1e-14: hs += [Cu(u), Cu(u)] + return sum(1 for h in hs if -k < h < 0) + +def antiderivative(s, B, k): + if MUTANT == 'antiderivative': return 12 * k * k * s ** 3 - 9 * k * k * B * s + return 12 * k * k * s ** 3 - 9 * k * k * B * B * s # of 9 k^2 (4 s^2 - B^2) + +def I_j(a, b, c, k): + B, D = shear(a, b, c, k); D2 = D * D; top = typed_top(B) + roots = cubic_roots(B, k, D2); g = lambda s: cubic_g(s, B, k, D2) + bps = [-math.inf] + roots + [math.inf]; pos, neg = [], [] + for lo, hi in zip(bps[:-1], bps[1:]): + if lo == -math.inf and hi == math.inf: mid = 0.0 + elif lo == -math.inf: mid = hi - 1.0 - abs(hi) + elif hi == math.inf: mid = lo + 1.0 + abs(lo) + else: mid = 0.5 * (lo + hi) + (pos if g(mid) > 0 else neg).append((lo, hi)) + def clip(lo, hi, a_, b_): + lo2, hi2 = max(lo, a_), min(hi, b_) + return antiderivative(hi2, B, k) - antiderivative(lo2, B, k) if hi2 > lo2 else 0.0 + I1 = sum(clip(lo, hi, -1e300, top) for lo, hi in neg) + I2 = sum(clip(lo, hi, B, top) for lo, hi in pos) if B < top else 0.0 + return I1, I2 + +# ------------------------------------------------------------------ prefactor p_b(0) / z_0, closed form +def phi(x): return math.exp(-x * x / 2) / math.sqrt(2 * math.pi) +def Phi(x): return 0.5 * math.erfc(-x / math.sqrt(2)) + +def prefactor(k, b): + """p_b(0) / z_0 with A ~ N(-b, 2): p_b(0) = phi(b/sqrt2)/sqrt2, z_0 = 36 k^2 E[A^2 1{A<0}].""" + t = b / math.sqrt(2) + m2 = (b * b + 2) * Phi(t) + math.sqrt(2) * b * phi(t) + return phi(t) / math.sqrt(2) / (36 * k * k * m2), m2 + +# ------------------------------------------------------------------ quadrature +def gauss_hermite(n): + """probabilists' Gauss-Hermite nodes/weights (sum of weights 1), by Newton on normalized Hermite functions.""" + nodes = [] + for i in range(1, n // 2 + 1): + if i == 1: x = math.sqrt(2 * n + 1) - 1.85575 * (2 * n + 1) ** (-1 / 6) + elif i == 2: x = nodes[-1] - 1.14 * n ** 0.426 / nodes[-1] + elif i == 3: x = 1.86 * nodes[-1] - 0.86 * nodes[0] + elif i == 4: x = 1.91 * nodes[-1] - 0.91 * nodes[-2] + else: x = 2 * nodes[-1] - nodes[-2] + for _ in range(100): + p0 = 1.0 / math.pi ** 0.25; pm, pc = p0, math.sqrt(2) * x * p0 + for j in range(2, n + 1): + pm, pc = pc, math.sqrt(2 / j) * x * pc - math.sqrt((j - 1) / j) * pm + dx = pc / (math.sqrt(2 * n) * pm); x -= dx + if abs(dx) < 1e-15: break + nodes.append(x) + out = [] + for x in sorted(set(nodes + [-v for v in nodes] + ([0.0] if n % 2 else []))): + p0 = 1.0 / math.pi ** 0.25; pm, pc = p0, math.sqrt(2) * x * p0 + for j in range(2, n + 1): + pm, pc = pc, math.sqrt(2 / j) * x * pc - math.sqrt((j - 1) / j) * pm + w = 2.0 / (2 * n * pm * pm) / math.sqrt(math.pi) + out.append((x * math.sqrt(2), w)) + return out + +SIG = (math.sqrt(2.0), math.sqrt(2.0), math.sqrt(6.0)) # odd block: N(0, diag(2, 2, 6)), exact structure + +def float_regression(C, targets, target_vals, outputs): + """Floating-point Gaussian regression (used only for the finite-L lattice covariance).""" + S_tt = [[C[(r, c)] for c in targets] for r in targets] + S_ot = [[C[(r, c)] for c in targets] for r in outputs] + S_oo = [[C[(r, c)] for c in outputs] for r in outputs] + n = len(targets); Ab = [S_tt[i][:] + [S_ot[j][i] for j in range(len(outputs))] for i in range(n)] + for i in range(n): + piv = max(range(i, n), key=lambda r: abs(Ab[r][i])); Ab[i], Ab[piv] = Ab[piv], Ab[i]; pv = Ab[i][i] + Ab[i] = [x / pv for x in Ab[i]] + for r in range(n): + if r != i and Ab[r][i] != 0.0: + f = Ab[r][i]; Ab[r] = [x - f * y for x, y in zip(Ab[r], Ab[i])] + K = [[Ab[i][n + j] for i in range(n)] for j in range(len(outputs))] # S_ot S_tt^-1 + mean = [sum(K[j][i] * target_vals[i] for i in range(n)) for j in range(len(outputs))] + cov = [[S_oo[a][b] - sum(K[a][i] * S_ot[b][i] for i in range(n)) for b in range(len(outputs))] for a in range(len(outputs))] + return mean, cov + +def cholesky(A): + n = len(A); Lm = [[0.0] * n for _ in range(n)] + for i in range(n): + for j in range(i + 1): + v = A[i][j] - sum(Lm[i][k] * Lm[j][k] for k in range(j)) + Lm[i][j] = math.sqrt(v) if i == j else v / Lm[j][j] + return Lm + +def lattice_setup(L, k, b): + """Finite-L contact regression from the lattice covariance: odd-block mean/Cholesky and the prefactor p_b(0)/z_0.""" + C = lattice_cov(L) + muA, covA = float_regression(C, ['f', 'fxx', 'fxz'], [b, 0.0, 0.0], ['fzz']) + muA = muA[0]; sA = math.sqrt(covA[0][0]); t = -muA / sA + m2 = muA * muA * Phi(t) - 2 * muA * sA * phi(t) + sA * sA * (Phi(t) - t * phi(t)) + pf = phi(t) / sA / (36 * k * k * m2) + muJ, covJ = float_regression(C, ['fx', 'fz', 'fxxx'], [0.0, 0.0, 12 * k], ['fxxz', 'fxzz', 'fzzz']) + return {'mu': muJ, 'chol': cholesky(covJ), 'prefactor': pf, 'muA': muA, 'varA': sA * sA, 'covJ': covJ} + +def J_gh_lattice(k, order, setup): + gh = gauss_hermite(order); mu, Lc = setup['mu'], setup['chol']; J1 = J2 = 0.0 + for x1, w1 in gh: + for x2, w2 in gh: + for x3, w3 in gh: + a = mu[0] + Lc[0][0] * x1 + bb = mu[1] + Lc[1][0] * x1 + Lc[1][1] * x2 + c = mu[2] + Lc[2][0] * x1 + Lc[2][1] * x2 + Lc[2][2] * x3 + i1, i2 = I_j(a, bb, c, k); w = w1 * w2 * w3 + J1 += w * i1; J2 += w * i2 + return J1, J2 + +def J_gh(k, order): + gh = gauss_hermite(order); J1 = J2 = 0.0 + for x1, w1 in gh: + for x2, w2 in gh: + for x3, w3 in gh: + i1, i2 = I_j(SIG[0] * x1, SIG[1] * x2, SIG[2] * x3, k); w = w1 * w2 * w3 + J1 += w * i1; J2 += w * i2 + return J1, J2 + +def J_mc(k, n, seed): + rng = random.Random(seed); s1 = s2 = q1 = q2 = 0.0 + for _ in range(n): + i1, i2 = I_j(SIG[0] * rng.gauss(0, 1), SIG[1] * rng.gauss(0, 1), SIG[2] * rng.gauss(0, 1), k) + s1 += i1; s2 += i2; q1 += i1 * i1; q2 += i2 * i2 + m1, m2 = s1 / n, s2 / n + return m1, m2, math.sqrt(max(q1 / n - m1 * m1, 0) / n), math.sqrt(max(q2 / n - m2 * m2, 0) / n) + +def sig6(x): return float('%.6g' % x) + +# ------------------------------------------------------------------ controls +def controls(): + ex = exact_structure() + # lattice sums at L = 24 and 12 agree with the continuum table + devs = {} + for L in (24.0, 12.0, 6.0): + C = lattice_cov(L); dev = 0.0 + for key, val in exact_cov().items(): + dev = max(dev, abs(C[key] - float(val))) + devs[str(int(L))] = dev + require(devs['24'] < 1e-9 and devs['12'] < 1e-9, 'lattice-sum jet covariances match the continuum Gaussian table at L = 24, 12') + require(1e-4 < devs['6'] < 2e-3, 'the finite-L correction at L = 6 is of order 1e-3 (recorded, not negligible)') + dev = devs + # classifier against direct root count on the typed domain + rng = random.Random(5); mism = 0 + for _ in range(4000): + k = rng.choice([0.5, 1.0, 2.0]); a, b, c = (rng.uniform(-4, 4) for _ in range(3)) + B, _ = shear(a, b, c, k); s = -abs(B) / 2 - rng.uniform(0, 6) + mism += n_classifier(s, a, b, c, k) != n_direct(s, a, b, c, k) + require(mism == 0, 'classifier agrees with the direct critical-point count on the typed domain') + # closed forms: [LM] cubic gives (0, 48 k^5); B = 0 family gives (72 k^3 D^2, 0); exact antiderivative identity + for k in (0.5, 1.0, 2.0): + i1, i2 = I_j(0.0, -2 * k, 0.0, k) + require(abs(i1) < 1e-9 and abs(i2 - 48 * k ** 5) < 1e-9 * (1 + 48 * k ** 5), '[LM] cubic mass 48 k^5 on n = 2') + i1, i2 = I_j(0.0, 0.0, 1.0, k) + require(abs(i2) < 1e-9 and abs(i1 - 18 * k ** 3) < 1e-9 * (1 + 18 * k ** 3), 'B = 0 family mass 72 k^3 D^2 on n = 1') + kq, Bq = F(1), F(-2); Fq = lambda s: 12 * kq * kq * s ** 3 - 9 * kq * kq * Bq * Bq * s + require(Fq(Bq / 2) - Fq(Bq) == 48, 'exact antiderivative of 9k^2(4s^2 - B^2) over (B, B/2) at k = 1, B = -2') + require(sum(w for _, w in gauss_hermite(40)) - 1 < 1e-12 and abs(sum(x * x * w for x, w in gauss_hermite(40)) - 1) < 1e-10, + 'Gauss-Hermite weights normalized') + return ex, devs + +def main(): + global MUTANT + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--check', action='store_true'); ap.add_argument('--mutant', choices=MUTANTS) + args = ap.parse_args(); MUTANT = args.mutant + try: + ex, dev = controls() + if args.check: + ref = json.load(open(os.path.join(HERE, 'RESULTS.json'))) + for k in K_VALUES: + J1, J2 = J_gh(float(k), 40) + rec = ref['per_k'][str(k)] + for got, name in ((J1, 'J1_gh40'), (J2, 'J2_gh40')): + require(abs(got - rec[name]) <= 1e-6 * (1 + abs(rec[name])), 'GH40 replay matches RESULTS.json for ' + name) + st6 = lattice_setup(6.0, 1.0, 0.0); J6 = J_gh_lattice(1.0, 40, st6); rec6 = ref['lattice_L6']['k=1'] + for got, name in ((J6[0], 'J1_gh40'), (J6[1], 'J2_gh40')): + require(abs(got - rec6[name]) <= 1e-6 * (1 + abs(rec6[name])), 'L = 6 GH40 replay matches RESULTS.json for ' + name) + print(json.dumps({'passed': True, 'mode': 'check', 'scientific_effect': 'NONE'}, sort_keys=True)) + return 0 + out = {'schema': 1, 'object': 'CL-C6-CLUSTER-COEFF-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', + 'certified': False, 'method': 'exact contact regression + interval-exact s-integral + Gauss-Hermite / Monte Carlo', + 'exact_structure': ex, 'lattice_vs_continuum_max_dev': {L: sig6(v) for L, v in dev.items()}, 'per_k': {}, 'prefactor': {}, + 'lattice_L6': {}} + for k in K_VALUES: + kf = float(k) + g40 = J_gh(kf, 40); g60 = J_gh(kf, 60); mc = J_mc(kf, 1000000, seed=2026) + rec = {'J1_gh40': g40[0], 'J2_gh40': g40[1], 'J1_gh60': sig6(g60[0]), 'J2_gh60': sig6(g60[1]), + 'J1_mc': sig6(mc[0]), 'J2_mc': sig6(mc[1]), 'J1_mc_se': sig6(mc[2]), 'J2_mc_se': sig6(mc[3]), + 'ratio2_gh60': sig6(g60[1] / (g60[0] + g60[1])), 'palm_near_gh60': sig6(2 * g60[1] / (g60[0] + 2 * g60[1]))} + out['per_k'][str(k)] = rec + for b in B_VALUES: + pf, m2 = prefactor(kf, float(b)) + out['prefactor'][f'k={k},b={b}'] = {'p_b0_over_z0': sig6(pf), 'm2_b': sig6(m2), 'z0': sig6(36 * kf * kf * m2), + 'alpha1_gh60': sig6(pf * g60[0]), 'alpha2_gh60': sig6(pf * g60[1]), + 'alpha1_mc': sig6(pf * mc[0]), 'alpha2_mc': sig6(pf * mc[1])} + st6 = lattice_setup(6.0, kf, 0.0); J6 = J_gh_lattice(kf, 40, st6) + out['lattice_L6'][f'k={k}'] = {'J1_gh40': J6[0], 'J2_gh40': J6[1], 'prefactor_b0': sig6(st6['prefactor']), + 'alpha1_gh40': sig6(st6['prefactor'] * J6[0]), 'alpha2_gh40': sig6(st6['prefactor'] * J6[1]), + 'rel_dev_alpha1_vs_continuum_gh40': sig6(st6['prefactor'] * J6[0] / (prefactor(kf, 0.0)[0] * g40[0]) - 1), + 'rel_dev_alpha2_vs_continuum_gh40': sig6(st6['prefactor'] * J6[1] / (prefactor(kf, 0.0)[0] * g40[1]) - 1), + 'muJ': [sig6(x) for x in st6['mu']], 'varA': sig6(st6['varA']), 'muA': sig6(st6['muA'])} + print(f'k={k}: done', file=sys.stderr, flush=True) + path = os.path.join(HERE, 'RESULTS.json') + json.dump(out, open(path, 'w'), indent=2, sort_keys=True); open(path, 'a').write('\n') + print(json.dumps({'passed': True, 'mode': 'full', 'scientific_effect': 'NONE'}, sort_keys=True)) + return 0 + except ValueError as exc: + print(json.dumps({'passed': False, 'error': str(exc)}, sort_keys=True)); return 1 + +if __name__ == '__main__': + sys.exit(main()) diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md new file mode 100644 index 000000000..47fc0fedf --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/NOTE.md @@ -0,0 +1,234 @@ +# Numerical values of the remote singleton coefficient `k integral_X Lambda` (planar, parent kernel) + +**Object:** `CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical +note. **Scientific effect:** NONE. **Certified:** no (floating-point quadrature with convergence checks; not an +enclosure). Companion of the near-coefficient note `frontiers/c6_cluster_coefficients_numerics_20260930` +(Math-#168), which evaluated `nu_near(1) = alpha_1` and `nu(2) = alpha_2` and listed the remote part of `nu(1)` as +not evaluated. This note evaluates that part. + +## 1. What is evaluated + +[CL] (1.4) writes the one-point cluster coefficient as + + nu(1) = nu_near(1) + k integral_X Lambda(x; b, k, u) dx, Lambda = sum_j Lambda_j, + +with `Lambda_j` the contact kernel of [RM] (13): + + Lambda_j(x) = p_(Y_x | U_0 = v_0)(0, b) / z_0 . E[ w_0 F_j(H_x) | U_0 = v_0, Y_x = (0, b) ], + +`U_0 = (f, f_x, f_z, f_xx, f_xz, f_xxx)(0) = (b, 0, 0, 0, 0, 12k)` the contact observation, `A_0 = f_zz(0)` the soft +curvature, `w_0 = (6k)^2 A_0^2 1{A_0 < 0}` ([RM] (10) at `d = 2`), `z_0 = E[w_0 | U_0 = v_0] = 36 k^2 m_(2,b)` +([RM] (11), [LP] (5.4)) with `m_(2,b) = E[A_0^2 1{A_0 < 0} | U_0]` taken under the same kernel as the numerator +(continuum: `A_0 | U_0 ~ N(-b, 2)` exactly; `L = 6`: the periodized conditional law, `m_(2,0) = 1.0000175`), `Y_x = (f_x, f_z, f)(x)`, `H_x` the Hessian at `x`, `F_j(H) = |det H| 1{index H = j}`. The +field is the centered unit-variance Gaussian field on `X = R^2/(L Z^2)` with the exact kernel of [LP] section 2, +`K_L(z) = sum_n exp(-|z + Ln|^2/2) / sum_n exp(-|Ln|^2/2)`. Orientation: the continuum kernel is isotropic, so +continuum quantities (`Lambda_inf`, `c_hole`, sample kernels, profiles) hold for every pin direction `u`; the +square torus is not rotation-invariant, and every `L`-periodized value in this note is computed with the pin +direction `u = e_1` aligned with a lattice axis (the image lattice is written in pin-frame coordinates). No other +orientation is computed. Everything is a finite-dimensional Gaussian computation: the 13-vector of jets (seven at `0`, six at `x`) has covariance +`Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| (d^(alpha+beta) K_L)(t - s)`, with +`d^a_x d^c_z exp(-|z|^2/2) = (-1)^(a+c) He_a(x_1) He_c(z) exp(-|z|^2/2)` (probabilists' Hermite polynomials). + +[CL] section 5.1 shows `k integral_X Lambda = lim_(rho -> 0) k integral_(D_rho) Lambda` exists, with +`k integral_(|x| < rho) Lambda <= C rho^2`; so `Lambda` is integrable at the pin and the torus integral of the +kernel is the right object. A direct scaling check is consistent with this: at a critical point `x` of the contact +field at level `b` with `|x| = eps` small, the Euler identity `x . grad f = 3 C + A_0 z^2 + 4 q + ...` for the cubic +part `C`, the soft quadratic `A_0 z^2 / 2` and the quartic part `q` forces `A_0 z^2 = O(eps^4)`, i.e. `A_0 = O(eps^2)`; +the density of `Y_x` at `(0, b)` is `O(eps^(-6))` while `w_0 |det H_x| = O(eps^4 . eps^2)`, so `Lambda = O(1)` along +such a cone. (A first guess `A_0 = O(eps)` would give `Lambda ~ eps^(-2)` and a logarithmically divergent integral; +it is wrong exactly because of the Euler identity.) Two qualifications: the division by `z^2` is legitimate only on +cones `|z| >= c |x|`, so this is a scaling check on such cones and gives no pointwise bound towards the axis +(where the computed kernel is in any case exponentially small, section 4); and the integrability of `Lambda` at the +pin used by this note rests on [CL] section 5.1's integral bound `k integral_(|x| < rho) Lambda <= C rho^2`, not on +this heuristic. No uniform pointwise `Lambda = O(1)` is asserted. The computed values are bounded and, in fact, +`Lambda(x) -> 0` as `x -> 0` in every sampled direction (section 4). + +## 2. Exact structure + +**Far field.** As `|x| -> infinity` the two sites decouple: `p_(Y_x | U_0)(0, b) -> phi(b)/(2 pi)` (`f`, `f_x`, +`f_z` independent with unit variance), `E[w_0 | U_0] / z_0 -> 1`, and `H_x | (f = b, grad f = 0)` has `h_1 = f_xx`, +`h_3 = f_zz ~ N(-b, 2)` independent (`Var f_xx = 3`, `Cov(f_xx, f_zz) = 1`, `Cov(f, f_xx) = -1`, regression on +`f = b` removes the common part), `h_2 = f_xz ~ N(0, 1)`. Hence + + Lambda_inf,j(b) = phi(b)/(2 pi) . E[ F_j(H) | f = b, grad f = 0 ], + +the ordinary height-resolved density of index-`j` critical points at level `b`, per unit height and area. It is +`k`-free; the `k`-dependence of `Lambda(x)` enters only through the conditioning on `f_xxx(0) = 12k` and dies with +the correlations. It is also closed-form: with `h_1 = a + c`, `h_3 = a - c` (`a ~ N(-b, 1)`, `c ~ N(0, 1)`, `h_2 ~ +N(0, 1)` independent) `det H = a^2 - rho^2`, `rho^2 ~ Exp(mean 2)`, and + + E F_1 = sqrt2 e^(-b^2/4), + E F_2 = (b^2+1) Phi(b) + b phi(b) - 2 Phi(b) + sqrt2 e^(-b^2/4) Phi(b/sqrt2), E F_0 = E F_2 |_(b -> -b), + E|det H_b| = b^2 - 1 + 2 sqrt2 e^(-b^2/4), Lambda_inf(b) = phi(b) E|det H_b| / (2 pi), + +so `Lambda_inf(0) = (2 sqrt2 - 1)/(2 pi)^(3/2) = 0.1160934862...`, and `integral Lambda_inf(b) db = 2/(pi sqrt3)`, the +total critical-point density. The quadrature reproduces these closed forms to `9e-13` (absolute), a +second exact end-to-end control of the Hessian integrator with a non-centered mean. + +**Exact control of the Hessian integrator.** For the unconditional planar Hessian (`Var h_1 = Var h_3 = 3`, +`Cov(h_1, h_3) = 1`, `Var h_2 = 1`) write `h_1 = a + c`, `h_3 = a - c` with `a ~ N(0, 2)`, `c ~ N(0, 1)` independent; +then `det H = a^2 - (c^2 + h_2^2) = a^2 - rho^2` with `rho^2 ~ Exp(mean 2)`, so +`E|det H| = 2 E[(rho^2 - a^2)_+] = 4 E exp(-a^2/2) = 4/sqrt3`, `E F_1 = 2/sqrt3`, `E F_0 = E F_2 = 1/sqrt3`, +`E det H = 0`. (Total critical-point density `2/(pi sqrt3) = 0.3676`, maxima `1/(2 pi sqrt3) = 0.0919`: the +Longuet-Higgins values for this spectrum.) The integrator reproduces these to `9.7e-11` (absolute). + +**Coordinates adapted to the cone `det H = 0`.** The integrals `E[A_0^2 1{A_0 < 0} |det H| 1{index j}]` over the +conditional Gaussian law of `(A_0, H_x)` are computed with `A_0` integrated out in closed form +(`E[A^2 1{A < 0}] = (m^2 + s^2) Phi(-m/s) - m s phi(m/s)` for `A | H ~ N(m, s^2)`, `m` affine in `H`) and the +three Hessian coordinates written as `a = tr H / 2` and polar `(rho, theta)` in the traceless plane `(c, h_2)`. +Then `det H = a^2 - rho^2`, the kink of `|det H|` is the coordinate surface `rho = |a|`, and the index is read off +from `rho < |a|` (definite; sign of `a` separates minimum and maximum) or `rho > |a|` (saddle). The `a`-range is +split at `0`, the `rho`-range at `|a|`, the `theta` nodes are those of the periodic trapezoid rule under an +analytic map concentrating them at the angular position of the (generally offset and anisotropic) conditional +mean. Apart from two truncations, each piece of the integrand is then smooth: the Gaussian ranges are cut at +`cut = 9` (the `a`-range at `cut` standard deviations; the `rho`-range endpoint `rho_max = |mean| + cut . max +marginal sd` of the transverse pair, which the `|a|` split can cross; and a Mahalanobis cut `q > 2 cut^2` inside +the `rho`-loop). These are numerical truncations whose omitted weighted mass depends on the actual conditional +covariance, mean and normalization; no universal relative size is claimed. What can be said in general is that +the transverse cap sits at standardized distance at least `cut/sqrt2 = 6.4` (the largest eigenvalue of a `2 x 2` +covariance is at most twice its largest marginal variance), so the omitted Gaussian mass is at most of order +`e^(-20)` times polynomial factors, and the `a`-cut and the Mahalanobis cut are at `9` and `sqrt2 . 9` +standard deviations; an absolute bound for the omitted Gaussian domain of the exact integral is the Maxwell-tail +majorant of the OpenAI lane (Math-#174 comment 5907931352, `3e-16` for the `b = 0` control). No +geometric-convergence theorem is claimed for the implemented rule; what is claimed is the observed behaviour on +the exact controls (`E|det H| = 4/sqrt3` to `1e-10` at the default order and `3e-15` at the higher order; the +closed-form far field to `1e-12`) and the order comparisons of section 5. + +**Ill-conditioned points.** Near the pin the conditional law of `Y_x` given `U_0` has variances down to `|x|^6`, +and the double-precision regression can lose positivity, either in the Hessian block (near the axis, where +`Lambda` is below `10^(-20)`) or in the Schur complement `Var(A_0 | H_x)` (near the pin in the transverse sector, +where `Lambda` is small but not negligible: at `r = 0.05` transverse a clamped `Var(A_0 | H_x)` would misreport +`Lambda` by a factor ten). The full `4 x 4` conditional covariance of `(A_0, H_x)` is therefore Cholesky-factored, +and any nonpositive pivot sends the point (`54` in the whole run) to a 50-digit decimal regression. + +## 3. Method + +`Lambda_j(x)` is evaluated on midpoint grids (spacing `h`, reflection symmetry `z -> -z` used) and integrated by +the midpoint rule: + +- continuum kernel, hole constant `c_hole,j = integral_(R^2) (Lambda_j - Lambda_inf,j) dx` on `[-6, 6]^2` + (`h = 0.15`). The sixth-order jet correlations decay only like `He_6(|x|) e^(-|x|^2/2)` (`0.32` at distance 4, + `4.4e-4` at 6), so the cutoff is checked directly: `Lambda/Lambda_inf - 1` is `3.6e-5` (axis), `3.1e-6` (45 deg), + `4.4e-7` (transverse) at `|x| = 5`, and `4.8e-9`, `4.4e-10`, `3.8e-11` at `|x| = 6`; from these sampled + profiles the neglected exterior is estimated at less than `10^(-7)` on `c_hole` (an estimate from the sampled + rays, not an integrated tail bound); +- periodized kernel at `L = 6`, direct integral over the fundamental domain `[-3, 3]^2` (`h = 0.15`); +- periodized kernel at `L = 12`, direct integral (`h = 0.2`), compared with `L^2 Lambda_inf + c_hole`. + +Then `integral_X Lambda ~ L^2 Lambda_inf + c_hole` for `L = 12, 24` (the periodization correction being +`O(e^(-18))` at `L >= 12`), and the remote singleton mass is `k` times that. + +## 4. Values + +**Far field** (`k`-free), per unit height and area: + +| `b` | `Lambda_inf` | minima `j = 0` | saddles `j = 1` | maxima `j = 2` | `p_inf = phi(b)/(2 pi)` | +|---|---|---|---|---|---| +| `0.0` | 0.116093 | 0.01315 | 0.0897936 | 0.01315 | 0.0634936 | +| `1.0` | 0.0848309 | 0.000850618 | 0.0424155 | 0.0415649 | 0.0385108 | + +(At `b = 0`: `E|det H_0| = 1.82843` with `h_1, h_3 ~ N(0, 2)`, `h_2 ~ N(0, 1)`.) + +**Correlation hole** `c_hole = integral_(R^2) (Lambda - Lambda_inf) dx` (dimension of an area times a density +per height; negative means depletion): + +| `k` | `b` | `c_hole` | `j = 0` | `j = 1` | `j = 2` | `c_hole / Lambda_inf` (effective depleted area) | +|---|---|---|---|---|---|---| +| `0.5` | `0.0` | -1.08414 | -0.302824 | -0.88442 | 0.103108 | -9.338 | +| `1.0` | `0.0` | -1.63115 | -0.313786 | -1.3325 | 0.0151367 | -14.05 | +| `2.0` | `0.0` | -2.1382 | -0.330282 | -1.75684 | -0.0510768 | -18.42 | +| `0.5` | `1.0` | -0.750629 | -0.00830817 | -0.260797 | -0.481524 | -8.849 | +| `1.0` | `1.0` | -1.18448 | -0.0055866 | -0.464948 | -0.713947 | -13.96 | +| `2.0` | `1.0` | -1.56814 | -0.00268732 | -0.636506 | -0.928945 | -18.49 | + +The pinned pair depletes its neighbourhood: along the pin axis the contact field is `b + 2k x_1^3 + ...` to leading +order, a monotone ramp, so window-level critical points near the pins live in the transverse sector only +(`Lambda(x)/Lambda_inf` at `k = 1, b = 0`: on the axis `2e-22` at `|x_1| = 1.5`, `1.6e-6` at `2`, `0.037` at +`2.5`, `0.68` at `3`, `1.02` at `4`; transversally `0.21` at `|z| = 1`, `0.68` at `2`, `1.00` at `3`; the ray +profiles at angles `0, 30, 45, 60, 90` degrees are in `RESULTS.json`, `ray_profile`). Near the pin `Lambda -> 0` +(`near_zero_profile`: `Lambda(0.05, 60 deg) = 0.001`, `Lambda(0.05, 90 deg) = 0.000104`, `Lambda(0.4, 90 deg) = 0.0079` at `k = 1, b = 0`; on the +axis and at `30` degrees it is below `10^(-20)`). + +**Remote singleton mass** `R(k, b, L) = k integral_X Lambda dx` and the assembled `nu(1)`, with `alpha_1` from the +companion note ([NUM] = Math-#168 `RESULTS.json`, merged, pinned; GH60 values; `alpha_2` likewise): + +| `k` | `b` | `L` | `integral_X Lambda` | `R = k integral_X Lambda` | `alpha_1` (#168) | `nu(1) = alpha_1 + R` | `nu(2) = alpha_2` | `nu(2)/nu(1)` | +|---|---|---|---|---|---|---|---|---| +| `0.5` | `0.0` | `6` (direct) | 3.05732 | 1.52866 | 1.11821 | 2.64687 | 0.0347454 | 0.0131 | +| `0.5` | `0.0` | `12` | 15.6333 | 7.81666 | 1.11821 | 8.93487 | 0.0347454 | 0.00389 | +| `0.5` | `0.0` | `24` | 65.7857 | 32.8929 | 1.11821 | 34.0111 | 0.0347454 | 0.00102 | +| `1.0` | `0.0` | `6` (direct) | 2.54206 | 2.54206 | 1.32906 | 3.87112 | 0.0255091 | 0.00659 | +| `1.0` | `0.0` | `12` | 15.0863 | 15.0863 | 1.32906 | 16.4154 | 0.0255091 | 0.00155 | +| `1.0` | `0.0` | `24` | 65.2387 | 65.2387 | 1.32906 | 66.5678 | 0.0255091 | 0.000383 | +| `2.0` | `0.0` | `6` (direct) | 2.17973 | 4.35945 | 2.1621 | 6.52155 | 0.0169281 | 0.0026 | +| `2.0` | `0.0` | `12` | 14.5793 | 29.1585 | 2.1621 | 31.3206 | 0.0169281 | 0.00054 | +| `2.0` | `0.0` | `24` | 64.7316 | 129.463 | 2.1621 | 131.625 | 0.0169281 | 0.000129 | +| `0.5` | `1.0` | `6` (direct) | 2.23698 | 1.11849 | 0.320155 | 1.43865 | 0.00994791 | 0.00691 | +| `0.5` | `1.0` | `12` | 11.465 | 5.73251 | 0.320155 | 6.05267 | 0.00994791 | 0.00164 | +| `0.5` | `1.0` | `24` | 48.112 | 24.056 | 0.320155 | 24.3762 | 0.00994791 | 0.000408 | +| `1.0` | `1.0` | `6` (direct) | 1.8121 | 1.8121 | 0.380523 | 2.19262 | 0.00730347 | 0.00333 | +| `1.0` | `1.0` | `12` | 11.0312 | 11.0312 | 0.380523 | 11.4117 | 0.00730347 | 0.00064 | +| `1.0` | `1.0` | `24` | 47.6781 | 47.6781 | 0.380523 | 48.0587 | 0.00730347 | 0.000152 | +| `2.0` | `1.0` | `6` (direct) | 1.53019 | 3.06037 | 0.619027 | 3.6794 | 0.00484668 | 0.00132 | +| `2.0` | `1.0` | `12` | 10.6475 | 21.295 | 0.619027 | 21.9141 | 0.00484668 | 0.000221 | +| `2.0` | `1.0` | `24` | 47.2945 | 94.589 | 0.619027 | 95.208 | 0.00484668 | 5.09e-05 | + +For `L = 6` the direct periodized integral and the approximation `36 Lambda_inf + c_hole` are both given +(`RESULTS.json`, `remote_total`); they differ by between `2.4e-3` and `6.4e-2` relative, most at `k = 2`. At +`L = 6` the depleted region (radius about `3`) fills the fundamental domain, so the depletions around the image +pins overlap and the additive approximation `L^2 Lambda_inf + c_hole` is not valid; the table uses the direct +periodized integral for `L = 6`. (The periodization also changes the one-site covariances by up to `8.8e-4` at +`L = 6`; that is included in the direct value.) For `L = 12` the direct integral at `k = 1, b = 0` is `15.0866` +against `15.0863` from the far field plus hole, so the additive form is accurate to `2e-5` there. No direct +`L = 24` integral was run: at `L = 24` the additive form is expected, not measured, to be at least as accurate. + +**Reading.** For the parent torus sizes the remote singletons dominate `nu(1)` (`R ~ L^2 Lambda_inf`: by a factor +`50` at `L = 24`, `k = 1, b = 0`), so the two-point cluster probability relative to a nonempty window is +`nu(2)/(nu(1) + nu(2)) ~ 3.83e-04` at `k = 1, b = 0, L = 24`, against the near-only share `0.019` reported in +#168. The near share is the right object for the cluster geometry; the global share is the right object for the +finite-torus count. The pins do not only deplete: along the `45`-`60` degree rays `Lambda/Lambda_inf` overshoots to +`1.24`-`1.48` at `|x| = 2`-`2.5` (`k = 1, b = 0`) before settling, so the displaced critical points pile up in a +ring; the net hole is nevertheless negative for every `(k, b)` computed, and grows with `k` (steeper ramp +`2k x_1^3`, stronger `f_xxx` conditioning) and shrinks with `b`. + +## 5. Precision (empirical, not bounds) + +- Quadrature: at the sample points the default orders `(24, 32, 24)` differ from `(32, 48, 32)` by at most `3e-9` + relative wherever `Lambda > 10^(-2) Lambda_inf`, by `5e-4` at `(0.15, 0.15)` where `Lambda = 2e-5 Lambda_inf`, and + by up to `4.0e-01` at points where `Lambda < 10^(-16) Lambda_inf` (the recorded maximum, carried entirely by such + points: the conditional law of `H_x` is extremely anisotropic there and the value is exponentially small anyway). + On the exact controls the default order is accurate to `9.7e-11` (unconditional Hessian) and + `9e-13` (far field, both `b`). The `h = 0.3` hole constant changes by `4e-08` between the + two orders. +- Grid: at `k = 1, b = 0` the hole constant is `-1.629661` (`h = 0.3`), `-1.631148` (`h = 0.15`), + `-1.631335` (`h = 0.1`). Measured: the tabulated `h = 0.15` value differs from the `h = 0.1` value by + `1.88e-4`. Model (NONRIGOROUS, this one parameter set only): a three-point power law `A(h) = A_0 + C h^p` + fitted to the non-nested spacings `0.3, 0.15, 0.1` gives `p = 2.63` (`Lambda` has a conical, + direction-dependent limit at the pin), `A_0 = -1.63143`, and model residuals `2.9e-4` at `h = 0.15` and + `9.8e-5` at `h = 0.1`; these are extrapolations, not error bounds, and nothing is inferred from them for the + other `(k, b)` rows. The `h = 0.3` run on `[-5, 5]^2` differs from the `[-6, 6]^2` run by `9.5e-5`; because + the cell count is rounded (`n = round(5/0.3) = 17`, effective spacing `5/17 = 0.294` against `0.3`), that + difference mixes grid placement with the exterior annulus and is not a pure exterior-tail measurement (the + exterior estimate rests on the sampled profiles of section 3). `--check` replays this summary arithmetic + from the stored numbers. +- The periodized `L = 12` direct integral (`h = 0.2`) agrees with far field plus hole to `2.0e-05` relative. +- The Monte Carlo cross-check of the unconditional control (`2 . 10^5` samples, seed 2026) gives + `E|det H| = 2.3032 +- 0.0058`, `4/sqrt3 = 2.3094` lying at `1.1` standard errors. + +These are observed discrepancies. No error bound is claimed. + +## 6. What the numbers are not + +Not a proof and not an enclosure. They are values of the formula [RM] (13) integrated as [CL] (1.4) prescribes; +[RM] is an accepted source, [CL] nonauthor-accepted and merged; the numbers do not re-review either. Planar only +(`d = 2`); no rate, no finite-`r` statement; the assembled `nu(1)` inherits the empirical precision of #168's +`alpha_1` (about `0.3%` or better). No register, catalog or STATUS change. Same GitHub account as every lane; zero +organizational-independence credit. The author will not merge. + +## 7. Provenance + +`remote.py` (standard library; `--check` replays the exact controls, the far field, sixteen sample kernels at two +parameter pairs for the continuum and the `L = 6` kernel, one decimal-regression point and the near-zero profile +against `RESULTS.json`; three mutants `hermite-sign`, `det-abs`, `weight-indicator` must exit 1), `RESULTS.json` +(full run, about 36 minutes), `SOURCE_MAP.json` (pins [LP], [RM], [CL], [RC] on `main` +`7d0c89a`), `SOURCE_FILES.json`, workflow `c6-remote-coefficient-numerics.yml`. diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/README.md b/frontiers/c6_remote_coefficient_numerics_20260930/README.md new file mode 100644 index 000000000..79b060628 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/README.md @@ -0,0 +1,23 @@ +# Remote singleton coefficient numerics (planar, parent kernel) + +**Object:** `CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical +note (floating-point quadrature with convergence checks; no proof, no enclosure). **Scientific effect:** NONE. + +Evaluates the contact kernel `Lambda_j(x; b, k, u)` of [RM] (13) in the plane for the parent kernel +`K(z) = exp(-|z|^2/2)` and its `L`-periodization, its far-field value, the correlation-hole constant +`c_hole = integral_(R^2) (Lambda - Lambda_inf) dx`, and the remote singleton mass `k integral_X Lambda dx` +that enters `nu(1)` by [CL] (1.4). Companion of the near-coefficient note (Math-#168), whose values of +`alpha_1 = nu_near(1)` are quoted to assemble `nu(1)`. + +- `NOTE.md` — statement of what is evaluated, exact structure, method, values, precision, non-claims. +- `remote.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against + `RESULTS.json`, point-wise tolerance `1e-6` relative to cover interpreter differences), full run regenerates + `RESULTS.json` (about 36 minutes); `--mutant {hermite-sign, det-abs, weight-indicator}` must exit 1. +- Orientation: continuum values hold for every pin direction `u`; all `L`-periodized values are for `u = e_1` + aligned with a lattice axis. +- `RESULTS.json` — all numbers, grids and convergence checks. +- `SOURCE_MAP.json` — pinned sources on `main` (blob identities); `SOURCE_FILES.json` — packet manifest. +- Workflow `.github/workflows/c6-remote-coefficient-numerics.yml` replays manifest, pins, both check modes and the + mutants. + +Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge. diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json b/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json new file mode 100644 index 000000000..3e4a7ddb8 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json @@ -0,0 +1,5438 @@ +{ + "certified": false, + "controls": { + "far_field": { + "0.0": { + "E_F_j": [ + 0.2071067811866195, + 1.4142135623732415, + 0.20710678118661952 + ], + "lambda": 0.11609348619094405, + "lambda_j": [ + 0.013149962564175739, + 0.08979356106259258, + 0.01314996256417574 + ], + "rho_j": [ + 0.013149962564175728, + 0.08979356106259259, + 0.01314996256417573 + ] + }, + "1.0": { + "E_F_j": [ + 0.022087746264714315, + 1.1013906298184966, + 1.079302883553834 + ], + "lambda": 0.08483094979588068, + "lambda_j": [ + 0.0008506175936846622, + 0.042415474897939356, + 0.04156485730425666 + ], + "rho_j": [ + 0.0008506175936846623, + 0.04241547489793938, + 0.04156485730425671 + ] + } + }, + "mc_absdet_unconditional": { + "mean": 2.3031855264703522, + "n": 200000, + "se": 0.005762874124208739, + "seed": 2026 + }, + "one_site_exact": { + "Cov_f_fxx": "-1", + "Cov_fx_fxxx": "-3", + "Cov_fxx_fzz": "1", + "Var_fxx": "3", + "Var_fxxx": "15", + "Var_fxz": "1" + }, + "one_site_lattice_max_dev": { + "L12": 7.003755912451762e-58, + "L24": 5.551522854989634e-244, + "L6": 0.0008783433650769723, + "continuum": 0.0 + }, + "reflection_symmetry_max_rel_dev": 8.38164154450512e-16, + "unconditional_hessian": { + "E_F_j": [ + 0.5773502692137803, + 1.1547005384275606, + 0.5773502692137802 + ], + "E_absdet": 2.309401076855121, + "E_det": 0.0 + } + }, + "exact_regression_points": 54, + "far_field": { + "0.0": { + "E_F_j_given_level": [ + 0.2071067811866195, + 1.4142135623732415, + 0.20710678118661952 + ], + "lambda_inf": 0.11609348619094405, + "lambda_inf_j": [ + 0.013149962564175739, + 0.08979356106259258, + 0.01314996256417574 + ], + "p_inf": 0.06349363593424098 + }, + "1.0": { + 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amendments, applied" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364176857, + "scope": "reassessment at a04fc81; formulation ACCEPT stands; truncation wording repaired" + } + ] +} diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json new file mode 100644 index 000000000..0d3f8b889 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_MAP.json @@ -0,0 +1,99 @@ +{ + "schema": 1, + "object": "CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "note": "Numerical evaluation only. The formula evaluated is the contact kernel Lambda_j(x; b, k, u) of [RM] (13), summed and integrated over the torus as in [CL] (1.4)-(1.5) (the remote singleton part k integral_X Lambda of nu(1)), at d = 2 with the exact kernel of [LP] section 2. No source theorem is consumed as accepted beyond its recorded disposition; the numbers are deterministic floating-point quadrature values with reported convergence checks, not enclosures. The companion near-coefficient note (Math-#168) merged to main on 30 September 2026 (02772ec) and is now pinned as [NUM]. OpenAI Sol review 5363739676 (30 September 2026, head f4d8bb9) accepts the closed-form structure of section 2 as identities; recorded under nonauthor_reads without changing review_status. OpenAI Astra review 5364077220 (head f4d8bb9) accepts the Slice B formulation with two wording amendments, applied.", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "section 2: the exact periodized Gaussian kernel K_L (periodization of exp(-|z|^2/2)); section 3: contact frame U_0 and target v_0 = (b, 0, 0, 12k, 0, 0); (5.4)-(5.5): the full normalizer z_0 = lim Z_r / r^2", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/remote_window_20260924/PROOF.md", + "bytes": 18355, + "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", + "id": "RM", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "Theorem A (fixed-remote mean measure) and the contact kernel Lambda_j(x; b, k, u) of (13): p_(Y_x | U_0 = v_0)(0, b) / z_0 times E[w_0 F_j(H_x) | U_0 = v_0, Y_x = (0, b)] with w_0 = (6k)^2 (det A_0)^2 1{A_0 < 0} ((10)) and z_0 = E[w_0 | U_0 = v_0] ((11)); this is the object evaluated", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_law_20260929/PROOF.md", + "bytes": 70131, + "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5", + "id": "CL", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "(1.4): nu(1) = nu_near(1) + k integral_X Lambda dx, nu(2) = nu_near(2); section 5.1 (5.2): existence of k integral_X Lambda = lim_(rho -> 0) k integral_(D_rho) Lambda with error <= C rho^2, which is what makes the torus integral of the bounded kernel the right object; Math-#159, nonauthor-accepted (OpenAI review 5360192822), merged", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/remote_collision_20260928/PROOF.md", + "bytes": 26002, + "sha256": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2", + "git_blob": "7b48a88e2af54e759e89a8c8219573bb450a65ea", + "id": "RC", + "commit": "7d0c89a5bb77a52a7a661d7292ba5d955bd0a66d", + "role": "Corollary E: Q_r^W{N_j(E) >= 1} = k r^3 integral_E Lambda_j + O(r^4 |E|) on fixed remote regions, the probability reading of the same kernel (context only)", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json", + "bytes": 4095, + "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb", + "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f", + "id": "NUM", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "near coefficients alpha_1(k, b) (GH60) and alpha_2(k, b) quoted in NOTE.md section 4 to assemble nu(1) = alpha_1 + k integral_X Lambda; Math-#168, merged to main on 30 September 2026 (02772ec) with RESULTS.json bytes unchanged from the quoted head 0b35e76; xAI read 5360559751 accepted its Slices A and B as identities", + "current_required": true, + "on_main": true + } + ], + "nonauthor_acceptance": false, + "governing_status_changed": false, + "main_at_update": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "nonauthor_reads": [ + { + "lane": "OpenAI/GPT-5.6 Sol", + "review_id": 5363739676, + "head": "f4d8bb9c30b5848996fceb18d213433cfe9956b9", + "date": "2026-09-30", + "disposition": "ACCEPT Slice A (closed-form far field, covariance rule, index split, unconditional controls, cone-adapted coordinates); the near-pin Euler paragraph read as a scaling heuristic only, with [CL] section 5.1 supplying the integrability bound; explicitly not an acceptance of the grids, the then-stale RESULTS.json, the precision claims or merge readiness; same account, credit 0" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364077220, + "head": "f4d8bb9c30b5848996fceb18d213433cfe9956b9", + "date": "2026-09-30", + "disposition": "ACCEPT Slice B (cone-coordinate conditional-Gaussian formulation, closed-form A_0 integration, corrected full-normalizer interface, 4x4 positivity check) with two wording amendments (truncation breakpoints in the cubature; angular qualification of the Euler scaling check), applied in the successor commit; not an acceptance of numerical precision or merge readiness; same account, credit 0" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364176857, + "head": "a04fc81a8753769b8228419d33998aed0679fd30", + "date": "2026-09-30", + "disposition": "successor reassessment: the two Slice B amendments resolved, the formulation ACCEPT stands; one further wording repair for the truncation sentence (no universal relative bound; exterior figure an estimate), applied in the successor commit; Slice C still open; same account, credit 0" + }, + { + "lane": "OpenAI GPT-6 Astra Pro", + "review_id": 5364643055, + "head": "a8c6d7918061c62a2e523b6cda34e9b78c9ac320", + "date": "2026-09-30", + "disposition": "AMEND (reporting only): two stale NOTE section 5 sentences; applied in the successor commit (Rc = 5 minus Rc = 6 at h = 0.3 is 9.46e-5, not 2.1e-3; fitted-error figures labelled nonrigorous model extrapolations for k = 1, b = 0 only with the measured 1.88e-4 finest-grid difference; unequal effective Rc = 5 spacing 5/17 noted; L = 24 improvement stated as expectation, not measurement). Does not certify quadrature accuracy or the table.", + "slice": "C (empirical precision reporting)" + } + ] +} diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/remote.py b/frontiers/c6_remote_coefficient_numerics_20260930/remote.py new file mode 100644 index 000000000..9553e3a95 --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/remote.py @@ -0,0 +1,805 @@ +#!/usr/bin/env python3 +"""Numerical evaluation of the remote contact kernel Lambda_j(x; b, k, u) of [RM] (13) for the +parent kernel K(z) = exp(-|z|^2/2) (and its L-periodization) in the plane d = 2, together with +its far-field value, the correlation-hole constant and the torus integral k * integral_X Lambda dx +that enters nu(1) = nu_near(1) + k integral_X Lambda ([CL] (1.4)). + +Standard library only. Deterministic nested Gauss quadrature; no random numbers except in the +optional Monte Carlo cross-check (fixed seed). Nothing here is a proof or an enclosure. + + python -B -S remote.py # full run, writes RESULTS.json (long: tens of minutes) + python -B -S remote.py --check # exact controls + replay of a small subset against RESULTS.json + # (Python >= 3.11; point-wise replay tolerance 1e-6 relative covers + # interpreter/libm differences between 3.11 and 3.14) + python -B -S remote.py --check --mutant NAME # must exit 1 + +Conventions. Coordinates x = (x1, z): x1 along the pin axis u, z transverse. Jets are indexed by +derivative orders (a, c) = d^a/dx1^a d^c/dz^c. Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| +(d^(alpha+beta) K)(t - s). Contact observation U_0 = (f, f_x, f_z, f_xx, f_xz, f_xxx)(0) = +(b, 0, 0, 0, 0, 12k); A_0 = f_zz(0); Y_x = (f_x, f_z, f)(x) = (0, 0, b); H_x = (f_xx, f_xz, f_zz)(x). +w_0 = 36 k^2 A_0^2 1{A_0 < 0}; z_0 = E[w_0 | U_0] = 36 k^2 m_(2,b). +Lambda_j(x) = p_(Y_x | U_0)(0, 0, b) * E[w_0 F_j(H_x) | U_0, Y_x] / z_0, F_j(H) = |det H| 1{index j}. +""" +import argparse +import json +import math +import os +import random +import sys +from decimal import Decimal, getcontext +from fractions import Fraction + +getcontext().prec = 50 + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('hermite-sign', 'det-abs', 'weight-indicator') +MUT = None + +SITE0 = [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0)] # f, f_x, f_z, f_xx, f_xz, f_zz, f_xxx at 0 +SITEX = [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2)] # f, f_x, f_z, f_xx, f_xz, f_zz at x +OBS0 = [0, 1, 2, 3, 4, 6] # U_0 within the 13-vector +OBSX = [8, 9, 7] # Y_x = (f_x, f_z, f)(x) +TARGETS = [5, 10, 11, 12] # A_0 = f_zz(0), h1 = f_xx(x), h2 = f_xz(x), h3 = f_zz(x) + + +def require(cond, msg): + if not cond: + print('FAIL: ' + msg) + sys.exit(1) + + +# ----------------------------------------------------------------------------- Hermite / kernel +def hermite(n, t): + """Probabilists' Hermite polynomial He_n(t): (d/dt)^n exp(-t^2/2) = (-1)^n He_n(t) exp(-t^2/2).""" + h0, h1 = 1, t + if n == 0: + return h0 + for m in range(1, n): + h0, h1 = h1, t * h1 - m * h0 + return h1 + + +def _exp(t): + return t.exp() if isinstance(t, Decimal) else math.exp(t) + + +def _sqrt(t): + return t.sqrt() if isinstance(t, Decimal) else math.sqrt(t) + + +def _log(t): + return t.ln() if isinstance(t, Decimal) else math.log(t) + + +def kernel_derivative(a, c, x1, z, L=None, images=2): + """(d^a/dx1^a d^c/dz^c) K_L(x1, z) for K(z) = exp(-|z|^2/2), K_L its normalized L-periodization. + L = None: continuum kernel (single image). Works in float or Decimal arithmetic (type of x1). + Orientation: (x1, z) are pin-frame coordinates and the image lattice is L Z^2 in these SAME coordinates, + i.e. every periodized value in this note is for the pin direction u = e_1 aligned with a lattice axis. + The continuum kernel is isotropic, so its values hold for every u; the periodized ones do not ([RM]: + Lambda_j depends on u through the frame), and no other orientation is computed here.""" + if L is None: + return (-1) ** (a + c) * hermite(a, x1) * hermite(c, z) * _exp(-(x1 * x1 + z * z) / 2) + tot = 0 * x1 + norm = 0 * x1 + for m1 in range(-images, images + 1): + for m2 in range(-images, images + 1): + s1, s2 = x1 + L * m1, z + L * m2 + tot += (-1) ** (a + c) * hermite(a, s1) * hermite(c, s2) * _exp(-(s1 * s1 + s2 * s2) / 2) + norm += _exp(-(L * L) * (m1 * m1 + m2 * m2) / 2 + 0 * x1) + return tot / norm + + +def cov_entry(alpha, beta, dx1, dz, L=None): + """Cov(d^alpha f(s), d^beta f(t)) with t - s = (dx1, dz).""" + sign = (-1) ** (alpha[0] + alpha[1]) + if MUT == 'hermite-sign': + sign = (-1) ** (beta[0] + beta[1]) + return sign * kernel_derivative(alpha[0] + beta[0], alpha[1] + beta[1], dx1, dz, L) + + +def joint_covariance(x1, z, L=None): + """13 x 13 covariance of the two-site jet vector (SITE0 at 0, SITEX at x); float or Decimal.""" + jets = [(0, a) for a in SITE0] + [(1, a) for a in SITEX] + n = len(jets) + zero = 0 * x1 + if L is not None and isinstance(x1, Decimal): + L = Decimal(L) + C = [[zero] * n for _ in range(n)] + for i in range(n): + si, ai = jets[i] + for j in range(n): + sj, aj = jets[j] + d1 = (x1 if sj == 1 else zero) - (x1 if si == 1 else zero) + d2 = (z if sj == 1 else zero) - (z if si == 1 else zero) + C[i][j] = cov_entry(ai, aj, d1, d2, L) + return C + + +# ----------------------------------------------------------------------------- exact one-site table +def hermite_zero_exact(n): + """He_n(0) exactly: 0 for odd n, (-1)^(n/2) (n-1)!! for even n.""" + if n % 2: + return Fraction(0) + v = Fraction(1) + for m in range(1, n, 2): + v *= m + return v * (-1) ** (n // 2) + + +def exact_one_site_cov(alpha, beta): + return (-1) ** (alpha[0] + alpha[1]) * hermite_zero_exact(alpha[0] + beta[0]) * hermite_zero_exact(alpha[1] + beta[1]) + + +# ----------------------------------------------------------------------------- linear algebra +def cholesky(A): + n = len(A) + zero = 0 * A[0][0] + Lm = [[zero] * n for _ in range(n)] + for i in range(n): + for j in range(i + 1): + s = A[i][j] - sum((Lm[i][k] * Lm[j][k] for k in range(j)), zero) + if i == j: + if s <= 0: + raise ValueError('matrix not positive definite (pivot %r at %d)' % (s, i)) + Lm[i][j] = _sqrt(s) + else: + Lm[i][j] = s / Lm[j][j] + return Lm + + +def chol_solve(Lm, b): + n = len(Lm) + zero = 0 * Lm[0][0] + y = [zero] * n + for i in range(n): + y[i] = (b[i] - sum((Lm[i][k] * y[k] for k in range(i)), zero)) / Lm[i][i] + x = [zero] * n + for i in reversed(range(n)): + x[i] = (y[i] - sum((Lm[k][i] * x[k] for k in range(i + 1, n)), zero)) / Lm[i][i] + return x + + +def submatrix(C, rows, cols): + return [[C[i][j] for j in cols] for i in rows] + + +def condition(C, obs, values, targets): + """Gaussian regression: law of C[targets] given C[obs] = values (zero prior means). + Returns (mean, cov, obs_logdensity_at_values).""" + Coo = submatrix(C, obs, obs) + zero = 0 * C[0][0] + values = [v + zero for v in values] + Lm = cholesky(Coo) + alpha = chol_solve(Lm, values) + logdet = 2 * sum((_log(Lm[i][i]) for i in range(len(obs))), zero) + quad = sum((values[i] * alpha[i] for i in range(len(obs))), zero) + logdens = float(-quad / 2 - logdet / 2) - 0.5 * len(obs) * math.log(2 * math.pi) + mean = [] + cov = [] + Cto = submatrix(C, targets, obs) + for r, t in enumerate(targets): + mean.append(sum((Cto[r][i] * alpha[i] for i in range(len(obs))), zero)) + solved = [chol_solve(Lm, Cto[r]) for r in range(len(targets))] + for r, t in enumerate(targets): + row = [] + for s, u in enumerate(targets): + row.append(C[t][u] - sum((Cto[r][i] * solved[s][i] for i in range(len(obs))), zero)) + cov.append(row) + return mean, cov, logdens + + +# ----------------------------------------------------------------------------- quadrature rules +def gauss_legendre(n): + nodes, weights = [], [] + for i in range(1, n + 1): + x = math.cos(math.pi * (i - 0.25) / (n + 0.5)) + for _ in range(100): + p0, p1 = 1.0, x + for m in range(2, n + 1): + p0, p1 = p1, ((2 * m - 1) * x * p1 - (m - 1) * p0) / m + dp = n * (x * p1 - p0) / (x * x - 1.0) + dx = p1 / dp + x -= dx + if abs(dx) < 1e-15: + break + p0, p1 = 1.0, x + for m in range(2, n + 1): + p0, p1 = p1, ((2 * m - 1) * x * p1 - (m - 1) * p0) / m + dp = n * (x * p1 - p0) / (x * x - 1.0) + nodes.append(x) + weights.append(2.0 / ((1.0 - x * x) * dp * dp)) + return nodes, weights + + +def gauss_hermite(n): # retained for the Monte Carlo-free control of the normalization; unused in the main integral + """Probabilists' Gauss-Hermite: sum w_i g(t_i) ~ E g(T), T ~ N(0, 1).""" + nodes, weights = [], [] + for i in range(1, n + 1): + if i == 1: + x = math.sqrt(2 * n + 1) - 1.85575 * (2 * n + 1) ** (-1.0 / 6.0) + elif i == 2: + x = nodes[-1] - 1.14 * n ** 0.426 / nodes[-1] + elif i == 3: + x = 1.86 * nodes[-1] - 0.86 * nodes[0] + elif i == 4: + x = 1.91 * nodes[-1] - 0.91 * nodes[-2] + else: + x = 2.0 * nodes[-1] - nodes[-2] + for _ in range(200): + h0, h1 = 1.0, x + for m in range(1, n): + h0, h1 = h1, x * h1 - m * h0 + dh = n * h0 + dx = h1 / dh + x -= dx + if abs(dx) < 1e-14: + break + h0, h1 = 1.0, x + for m in range(1, n): + h0, h1 = h1, x * h1 - m * h0 + nodes.append(x) + weights.append(math.factorial(n) / (n * n * h0 * h0)) + return nodes, weights + + +SQRT2PI = math.sqrt(2 * math.pi) + + +def phi(t): + return math.exp(-t * t / 2) / SQRT2PI + + +def Phi(t): + return 0.5 * math.erfc(-t / math.sqrt(2)) + + +def m2_negative(m, s): + """E[A^2 1{A < 0}] for A ~ N(m, s^2).""" + if MUT == 'weight-indicator': + return m * m + s * s + if s <= 0.0: + return m * m if m < 0 else 0.0 + t = m / s + return (m * m + s * s) * Phi(-t) - m * s * phi(t) + + +class Rules: + """Quadrature orders: n_a Gauss-Legendre nodes on each side of a = 0, n_theta trapezoid nodes on the circle, + n_rho Gauss-Legendre nodes on each of [0, |a|] and [|a|, rho_max]; Gaussian ranges cut at cut standard deviations.""" + + def __init__(self, n_a=24, n_theta=32, n_rho=24, cut=9.0): + self.gl_a = gauss_legendre(n_a) + self.n_theta = n_theta + self.gl_rho = gauss_legendre(n_rho) + self.cut = cut + self.orders = [n_a, n_theta, n_rho] + + +def weighted_determinant_integrals(mu, S, rules): + """(A, h1, h2, h3) ~ N(mu, S). Returns dict with + G[j] = E[ A^2 1{A<0} |det H| 1{index H = j} ], F[j] = E[ |det H| 1{index H = j} ], D = E det H (control). + Coordinates: a = (h1 + h3)/2, c = (h1 - h3)/2, so det H = a^2 - c^2 - h2^2 = a^2 - rho^2 with (c, h2) = + rho (cos theta, sin theta). The cone det H = 0 is the coordinate surface rho = |a|; the a-range is split at 0 + (index by the sign of the trace) and the rho-range at |a|. Apart from the truncations (a-range at cut standard + deviations; rho_max = |mean| + cut * max marginal sd of the transverse pair, i.e. at standardized distance at + least cut/sqrt2 since the largest eigenvalue is at most twice the largest marginal variance; the Mahalanobis cut + q > 2 cut^2 below) each piece of the integrand is smooth. The omitted weighted mass of these truncations depends + on the conditional covariance, mean and normalization (no universal relative bound is claimed); the accuracy of + the nested Gauss-Legendre / trapezoid rule is established empirically on the exact controls, not by a + convergence theorem. The general (non-isotropic) Gaussian density + of (a, c, h2) is evaluated directly at the nodes; A is integrated out in closed form through m2_negative.""" + # linear map (h1, h2, h3) -> w = (a, c, h2) + T = [[0.5, 0.0, 0.5], [0.5, 0.0, -0.5], [0.0, 1.0, 0.0]] + SHH = [[S[i][j] for j in (1, 2, 3)] for i in (1, 2, 3)] + muH = [mu[1], mu[2], mu[3]] + muw = [sum(T[i][k] * muH[k] for k in range(3)) for i in range(3)] + TS = [[sum(T[i][k] * SHH[k][j] for k in range(3)) for j in range(3)] for i in range(3)] + Sw = [[sum(TS[i][k] * T[j][k] for k in range(3)) for j in range(3)] for i in range(3)] + Lw = cholesky(Sw) + logdet = 2.0 * sum(math.log(Lw[i][i]) for i in range(3)) + norm = math.exp(-0.5 * logdet) / (2 * math.pi) ** 1.5 + # A | w regression: A = muA + wA . (w - muw), residual sd sA + SAH = [S[0][j] for j in (1, 2, 3)] + SAw = [sum(T[i][k] * SAH[k] for k in range(3)) for i in range(3)] + wA = chol_solve(Lw, SAw) + sA2 = S[0][0] - sum(SAw[i] * wA[i] for i in range(3)) + if sA2 <= 0.0: + raise ValueError('Var(A_0 | H) not positive: %r' % sA2) + sA = math.sqrt(sA2) + muA = mu[0] + # inverse via Cholesky for the quadratic form + Linv = [[0.0] * 3 for _ in range(3)] + for j in range(3): + e = [1.0 if i == j else 0.0 for i in range(3)] + col = chol_solve(Lw, e) + for i in range(3): + Linv[i][j] = col[i] + P = Linv # S_w^{-1} + cut = rules.cut + sa = math.sqrt(Sw[0][0]) + sc, s2 = math.sqrt(Sw[1][1]), math.sqrt(Sw[2][2]) + rho_max = math.hypot(muw[1], muw[2]) + cut * max(sc, s2) + G = [0.0, 0.0, 0.0] + F = [0.0, 0.0, 0.0] + # theta nodes: periodic trapezoid rule in u, theta = theta0 + u - beta sin u, concentrated at the angular + # position theta0 of the (c, h2) mean when the Gaussian is offset and narrow (analytic periodic map: the + # trapezoid rule keeps its geometric convergence; the Jacobian 1 - beta cos u is included in the weights) + rmu = math.hypot(muw[1], muw[2]) + smax = max(sc, s2) + width = min(1.0, smax / max(rmu, 1e-300)) + beta = max(0.0, 1.0 - rules.n_theta * width / (8 * math.pi)) + theta0 = math.atan2(muw[2], muw[1]) if rmu > 0 else 0.0 + cs = [] + for i in range(rules.n_theta): + u = -math.pi + 2 * math.pi * (i + 0.5) / rules.n_theta + th = theta0 + u - beta * math.sin(u) + cs.append((math.cos(th), math.sin(th), 1.0 - beta * math.cos(u))) + wtheta = 2 * math.pi / rules.n_theta + a_lo, a_hi = muw[0] - cut * sa, muw[0] + cut * sa + for lo, hi in ((a_lo, min(0.0, a_hi)), (max(0.0, a_lo), a_hi)): + if hi <= lo: + continue + mid, half = 0.5 * (lo + hi), 0.5 * (hi - lo) + for ta, wa in zip(*rules.gl_a): + a = mid + half * ta + wt_a = wa * half + aa = abs(a) + jpos = 0 if a > 0 else 2 # det > 0: sign of the trace decides minimum (0) / maximum (2) + segs = [] + if aa < rho_max: + segs.append((0.0, aa, jpos)) + segs.append((aa, rho_max, 1)) + else: + segs.append((0.0, rho_max, jpos)) + da = a - muw[0] + for lo_r, hi_r, j in segs: + if hi_r <= lo_r: + continue + mid_r, half_r = 0.5 * (lo_r + hi_r), 0.5 * (hi_r - lo_r) + accG = accF = 0.0 + for tr, wr in zip(*rules.gl_rho): + rho = mid_r + half_r * tr + adet = a * a - rho * rho + if MUT != 'det-abs': + adet = abs(adet) + base = wr * rho * adet + for cth, sth, jac in cs: + c = rho * cth + h2 = rho * sth + dc = c - muw[1] + d2 = h2 - muw[2] + q = (P[0][0] * da * da + P[1][1] * dc * dc + P[2][2] * d2 * d2 + + 2 * (P[0][1] * da * dc + P[0][2] * da * d2 + P[1][2] * dc * d2)) + if q > 2 * cut * cut: + continue + dens = math.exp(-0.5 * q) * jac + mA = muA + wA[0] * da + wA[1] * dc + wA[2] * d2 + accF += base * dens + accG += base * dens * m2_negative(mA, sA) + G[j] += wt_a * half_r * accG + F[j] += wt_a * half_r * accF + scale = norm * wtheta + G = [g * scale for g in G] + F = [f * scale for f in F] + D = (muw[0] ** 2 + Sw[0][0]) - (muw[1] ** 2 + Sw[1][1]) - (muw[2] ** 2 + Sw[2][2]) + return {'G': G, 'F': F, 'D': D} + + +# ----------------------------------------------------------------------------- Lambda +def m2b(b): + return m2_negative(-b, math.sqrt(2.0)) + + +def _conditioning(x1, z, b, k, L, exact): + """Density of Y_x at (0, 0, b) given U_0 (log) and the conditional law (mu, S) of (A_0, H_x) given + U_0 and Y_x. exact = True recomputes the Gaussian regression in 50-digit decimal arithmetic (used + automatically where the double-precision regression loses positivity; the near-axis points, where + Lambda is exponentially small).""" + if exact: + x1, z, bb, kk = Decimal(repr(x1)), Decimal(repr(z)), Decimal(repr(b)), Decimal(repr(k)) + else: + bb, kk = b, k + C = joint_covariance(x1, z, L) + u0 = [bb, 0 * bb, 0 * bb, 0 * bb, 0 * bb, 12 * kk] + # one-site contact law of A_0 = f_zz(0) given U_0 under the SAME kernel (continuum: exactly N(-b, 2); + # periodized: the L-dependent conditional law), so that z_0 = 36 k^2 E[A_0^2 1{A_0 < 0} | U_0] is the + # normalizer of the field actually used ([RM] (11)) + meanA, covA, _ = condition(C, OBS0, u0, [5]) + m2 = m2_negative(float(meanA[0]), math.sqrt(float(covA[0][0]))) + meanY, covY, _ = condition(C, OBS0, u0, OBSX) + LY = cholesky(covY) + dev = [0 - meanY[0], 0 - meanY[1], bb - meanY[2]] + a = chol_solve(LY, dev) + logp = float(-sum((dev[i] * a[i] for i in range(3)), 0 * bb) / 2 - sum((_log(LY[i][i]) for i in range(3)), 0 * bb)) - 1.5 * math.log(2 * math.pi) + obs = OBS0 + OBSX + vals = u0 + [0 * bb, 0 * bb, bb] + mu, S, _ = condition(C, obs, vals, TARGETS) + mu = [float(v) for v in mu] + S = [[float(v) for v in row] for row in S] + # the quadrature needs the (h1, h2, h3) block AND the Schur complement Var(A_0 | H_x) positive; a + # nonpositive pivot anywhere in the full 4 x 4 conditional covariance means the double-precision + # regression has lost the tiny residual variances and the caller must redo it in decimal arithmetic + cholesky(S) + return logp, mu, S, m2 + + +def lambda_at(x1, z, b, k, rules, L=None): + """Returns dict: Lambda_j (j = 0, 1, 2), Lambda (sum), the unweighted remote density rho_j = + p * E[F_j | obs] (what the same point would carry without the pin weight), logp, coupling.""" + try: + logp, mu, S, mm = _conditioning(x1, z, b, k, L, False) + exact = False + except ValueError: + logp, mu, S, mm = _conditioning(x1, z, b, k, L, True) + exact = True + EXACT_COUNT[0] += 1 + p = math.exp(logp) + W = weighted_determinant_integrals(mu, S, rules) + lam = [p * W['G'][j] / mm for j in range(3)] + rho = [p * W['F'][j] for j in range(3)] + fsum = W['F'][0] + W['F'][1] + W['F'][2] + coupling = (W['G'][0] + W['G'][1] + W['G'][2]) / (mm * fsum) if fsum > 0 else None + return {'lambda_j': lam, 'lambda': sum(lam), 'rho_j': rho, 'rho': sum(rho), 'logp': logp, 'exact_regression': exact, + 'coupling': coupling} + + +def lambda_far(b, rules): + """Far-field value: p = phi(b)/(2 pi), H | (f = b, grad f = 0): h1, h3 ~ N(-b, 2) independent, + h2 ~ N(0, 1); A_0 independent of H. Lambda_inf,j = phi(b)/(2 pi) E[F_j(H)].""" + mu = [-b, -b, 0.0, -b] + S = [[2.0, 0.0, 0.0, 0.0], [0.0, 2.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0], [0.0, 0.0, 0.0, 2.0]] + W = weighted_determinant_integrals(mu, S, rules) + p = phi(b) / (2 * math.pi) + lam = [p * W['G'][j] / m2b(b) for j in range(3)] + return {'lambda_j': lam, 'lambda': sum(lam), 'rho_j': [p * W['F'][j] for j in range(3)], 'E_F_j': W['F']} + + +def far_field_closed_form(b): + """Exact far field. With h1 = a + c, h3 = a - c, a ~ N(-b, 1), c ~ N(0, 1), h2 ~ N(0, 1) independent: + det H = a^2 - rho^2, rho^2 ~ Exp(mean 2). E[(rho^2 - a^2)_+ | a] = 2 exp(-a^2/2), E[(a^2 - rho^2)_+ | a] = + a^2 - 2 + 2 exp(-a^2/2), and E[exp(-a^2/2) 1{a < 0}] = exp(-b^2/4) Phi(b/sqrt2)/sqrt2. Hence + E F_1 = sqrt2 e^(-b^2/4) (saddles) + E F_2 = (b^2+1) Phi(b) + b phi(b) - 2 Phi(b) + sqrt2 e^(-b^2/4) Phi(b/sqrt2) (maxima, a < 0) + E F_0 = the same with b -> -b (minima) + E|det H_b| = b^2 - 1 + 2 sqrt2 e^(-b^2/4), + and Lambda_inf,j(b) = phi(b) E F_j / (2 pi). At b = 0: Lambda_inf = (2 sqrt2 - 1)/(2 pi)^(3/2). Integrating + Lambda_inf over b gives 2/(pi sqrt3), the total critical density (Longuet-Higgins).""" + e = math.exp(-b * b / 4) + F1 = math.sqrt(2) * e + F2 = (b * b + 1) * Phi(b) + b * phi(b) - 2 * Phi(b) + math.sqrt(2) * e * Phi(b / math.sqrt(2)) + F0 = (b * b + 1) * Phi(-b) - b * phi(b) - 2 * Phi(-b) + math.sqrt(2) * e * Phi(-b / math.sqrt(2)) + p = phi(b) / (2 * math.pi) + return {'lambda': p * (b * b - 1 + 2 * math.sqrt(2) * e), 'lambda_j': [p * F0, p * F1, p * F2], 'E_absdet': b * b - 1 + 2 * math.sqrt(2) * e} + + +def unconditional_hessian_controls(rules): + """H unconditional: Var f_xx = Var f_zz = 3, Cov = 1, Var f_xz = 1. With h1 = a + c, h3 = a - c + (a ~ N(0, 2), c ~ N(0, 1) independent) det H = a^2 - (c^2 + h2^2) = a^2 - rho^2 with rho^2 ~ Exp(mean 2), + so E|det H| = 2 E[(rho^2 - a^2)_+] = 4 E exp(-a^2/2) = 4/sqrt3, E F_1 = 2/sqrt3, E F_0 = E F_2 = 1/sqrt3, + E det H = 0. Total critical density 2/(pi sqrt3), maxima 1/(2 pi sqrt3) (Longuet-Higgins).""" + mu = [0.0, 0.0, 0.0, 0.0] + S = [[2.0, 0.0, 0.0, 0.0], [0.0, 3.0, 0.0, 1.0], [0.0, 0.0, 1.0, 0.0], [0.0, 1.0, 0.0, 3.0]] + W = weighted_determinant_integrals(mu, S, rules) + return {'E_absdet': sum(W['F']), 'E_F_j': W['F'], 'E_det': W['D']} + + +def mc_absdet(n, seed=2026): + """Monte Carlo cross-check of E|det H| for the unconditional Hessian (fixed seed).""" + rng = random.Random(seed) + acc = acc2 = 0.0 + for _ in range(n): + g1, g2, g3 = rng.gauss(0, 1), rng.gauss(0, 1), rng.gauss(0, 1) + h1 = math.sqrt(2.0) * g1 + g3 # Var 3 + h3 = math.sqrt(2.0) * g2 + g3 # Var 3, Cov(h1, h3) = 1 + h2 = rng.gauss(0, 1) + v = abs(h1 * h3 - h2 * h2) + acc += v + acc2 += v * v + mean = acc / n + var = acc2 / n - mean * mean + return mean, math.sqrt(var / n) + + +# ----------------------------------------------------------------------------- grids +def hole_integral(b, k, rules, Rc=5.0, h=0.2, lam_inf=None): + """2 * sum over the midpoint grid on [-Rc, Rc] x (0, Rc] of (Lambda(x) - Lambda_inf) h^2, by index. + Continuum kernel.""" + n2 = int(round(Rc / h)) + h = Rc / n2 # effective spacing: the cells tile [-Rc, Rc] x (0, Rc] exactly + n1 = 2 * n2 + tot = [0.0, 0.0, 0.0] + tot_rho = [0.0, 0.0, 0.0] + minlam, maxlam = float('inf'), float('-inf') + for i in range(n1): + x1 = -Rc + (i + 0.5) * h + for j in range(n2): + z = (j + 0.5) * h + r = lambda_at(x1, z, b, k, rules) + for q in range(3): + tot[q] += r['lambda_j'][q] - lam_inf['lambda_j'][q] + tot_rho[q] += r['rho_j'][q] - lam_inf['rho_j'][q] + minlam = min(minlam, r['lambda']) + maxlam = max(maxlam, r['lambda']) + scale = 2.0 * h * h + return {'hole_j': [scale * t for t in tot], 'hole': scale * sum(tot), + 'hole_unweighted_j': [scale * t for t in tot_rho], 'hole_unweighted': scale * sum(tot_rho), + 'grid': {'Rc': Rc, 'h': h, 'points': n1 * n2, 'symmetry': 'z -> -z'}, + 'lambda_min_on_grid': minlam, 'lambda_max_on_grid': maxlam} + + +def torus_integral(b, k, rules, L, h=0.2): + """integral over the fundamental domain [-L/2, L/2]^2 of Lambda_L(x) dx (periodized kernel), by index. + The pin direction u is the lattice axis e_1 (see kernel_derivative).""" + n2 = int(round(L / (2 * h))) + h = L / (2 * n2) # effective spacing: the cells tile [-L/2, L/2] x (0, L/2] exactly + n1 = 2 * n2 + tot = [0.0, 0.0, 0.0] + for i in range(n1): + x1 = -L / 2 + (i + 0.5) * h + for j in range(n2): + z = (j + 0.5) * h + r = lambda_at(x1, z, b, k, rules, L) + for q in range(3): + tot[q] += r['lambda_j'][q] + scale = 2.0 * h * h + return {'integral_j': [scale * t for t in tot], 'integral': scale * sum(tot), + 'grid': {'L': L, 'h': h, 'points': n1 * n2, 'symmetry': 'z -> -z', 'u': 'e_1'}} + + +# ----------------------------------------------------------------------------- controls +EXACT_COUNT = [0] +# replay tolerance for point-wise kernel values: libm/interpreter differences (Python 3.11 versus 3.12-3.14 were +# observed at 8e-9 relative on a near-pin sample) stay far below this; the mutants move values by O(1) +REPLAY_TOL = 1e-6 +SAMPLE_POINTS = [(0.15, 0.15), (0.5, 0.3), (-0.7, 0.9), (1.0, 0.0), (0.0, 1.0), (2.0, 1.5), (-1.5, 0.5), (3.0, 3.0)] + + +def controls(rules): + out = {} + # (a) exact one-site Hermite table versus the floating kernel derivatives at 0 (continuum and L = 24, 12) + table = {} + maxdev = {'continuum': 0.0, 'L24': 0.0, 'L12': 0.0, 'L6': 0.0} + for a in SITE0: + for c in SITE0: + ex = exact_one_site_cov(a, c) + table['%d%d,%d%d' % (a + c)] = str(ex) + fl = cov_entry(a, c, 0.0, 0.0) + maxdev['continuum'] = max(maxdev['continuum'], abs(fl - float(ex))) + for L, key in ((24.0, 'L24'), (12.0, 'L12'), (6.0, 'L6')): + maxdev[key] = max(maxdev[key], abs(cov_entry(a, c, 0.0, 0.0, L) - float(ex))) + require(table['20,20'] == '3' and table['30,30'] == '15' and table['10,30'] == '-3' and table['00,20'] == '-1' + and table['20,02'] == '1' and table['11,11'] == '1' and table['00,00'] == '1', 'exact Hermite one-site table') + require(maxdev['continuum'] < 1e-12 and maxdev['L24'] < 1e-9 and maxdev['L12'] < 1e-9, 'lattice one-site covariances at L = 24, 12') + require(1e-4 < maxdev['L6'] < 2e-3, 'L = 6 one-site deviation is of the expected size') + out['one_site_exact'] = {'Var_fxx': table['20,20'], 'Var_fxxx': table['30,30'], 'Cov_fx_fxxx': table['10,30'], + 'Cov_f_fxx': table['00,20'], 'Cov_fxx_fzz': table['20,02'], 'Var_fxz': table['11,11']} + out['one_site_lattice_max_dev'] = maxdev + # (b) unconditional Hessian: E|det H| = sqrt3, E F_2 = sqrt3/4, E F_1 = sqrt3/2, E det = 0 + U = unconditional_hessian_controls(rules) + require(abs(U['E_absdet'] - 4 / math.sqrt(3)) < 1e-9, 'E|det H| = 4/sqrt(3) (exact: a^2 - rho^2 decomposition)') + require(abs(U['E_F_j'][2] - 1 / math.sqrt(3)) < 1e-9 and abs(U['E_F_j'][0] - 1 / math.sqrt(3)) < 1e-9 + and abs(U['E_F_j'][1] - 2 / math.sqrt(3)) < 1e-9, 'index split 1/sqrt3, 2/sqrt3, 1/sqrt3') + require(abs(U['E_det']) < 1e-12, 'E det H = 0 (Euler characteristic of the torus)') + out['unconditional_hessian'] = U + # (c) far field: E[w_0 | U_0] / z_0 = 1 and Lambda_inf = phi(b) E|det H_b| / (2 pi) + far = {} + for b in (0.0, 1.0): + r = lambda_far(b, rules) + far[str(b)] = r + rr = lambda_at(8.0, 0.0, b, 1.0, rules) + require(abs(rr['lambda'] - r['lambda']) < 1e-9 * r['lambda'], 'Lambda(8, 0) equals the far-field value, b = %r' % b) + require(abs(rr['coupling'] - 1.0) < 1e-9, 'pin weight decouples at |x| = 8') + out['far_field'] = far + # (d) z -> -z symmetry and a few sample values + sym = 0.0 + for (x1, z) in SAMPLE_POINTS[:4]: + r1 = lambda_at(x1, z, 0.0, 1.0, rules) + r2 = lambda_at(x1, -z, 0.0, 1.0, rules) + sym = max(sym, abs(r1['lambda'] - r2['lambda']) / r1['lambda']) + require(sym < 1e-10, 'reflection symmetry z -> -z') + out['reflection_symmetry_max_rel_dev'] = sym + # (e) k-dependence enters only through the f_xxx = 12k conditioning: at |x| = 8 Lambda is k-free + a1 = lambda_at(8.0, 0.0, 0.0, 0.5, rules)['lambda'] + a2 = lambda_at(8.0, 0.0, 0.0, 2.0, rules)['lambda'] + require(abs(a1 - a2) < 1e-9 * a1, 'far field is k-free') + return out + + +def sample_table(b, k, rules, L=None): + rows = [] + for (x1, z) in SAMPLE_POINTS: + r = lambda_at(x1, z, b, k, rules, L) + rows.append({'x': [x1, z], 'lambda': r['lambda'], 'lambda_j': r['lambda_j'], 'rho': r['rho'], 'coupling': r['coupling'], 'exact_regression': r['exact_regression']}) + return rows + + +def ray_profile(b, k, rules): + """Lambda(x)/Lambda_inf along rays from the pin at angles 90 (transverse), 60, 45, 30, 0 degrees.""" + far = lambda_far(b, rules)['lambda'] + rows = [] + for ang_deg in (90, 60, 45, 30, 0): + th = math.radians(ang_deg) + for rad in (0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0): + x1, z = rad * math.cos(th), rad * math.sin(th) + if ang_deg == 0: + z = 0.0 + r = lambda_at(x1, z, b, k, rules) + rows.append({'angle_deg': ang_deg, 'r': rad, 'ratio': r['lambda'] / far, 'ratio_unweighted': r['rho'] / far}) + return rows + + +def near_zero_profile(b, k, rules): + rows = [] + for rad in (0.05, 0.1, 0.2, 0.4): + for ang_deg in (0, 30, 60, 90, 120, 150, 180): + th = math.radians(ang_deg) + x1, z = rad * math.cos(th), rad * math.sin(th) + if ang_deg in (0, 180): + z = 0.0 + r = lambda_at(x1, z, b, k, rules) + rows.append({'r': rad, 'angle_deg': ang_deg, 'lambda': r['lambda'], 'lambda_j': r['lambda_j'], 'exact_regression': r['exact_regression']}) + return rows + + +# ----------------------------------------------------------------------------- main +def full_run(fast=False): + rules = Rules() + hi_rules = Rules(32, 48, 32) + res = {'object': 'CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, + 'orientation': 'continuum values (Lambda_inf, c_hole, samples, profiles) hold for every pin direction u (isotropic kernel); every L-periodized value is for u = e_1 aligned with a lattice axis', + 'python_requirement': '>= 3.11; point-wise replay tolerance 1e-6 relative', + 'quadrature': {'coordinates': 'a = tr H / 2, (c, h2) polar; splits at a = 0 and rho = |a|', + 'a_gauss_legendre_per_side': 24, 'theta_trapezoid': 32, 'rho_gauss_legendre_per_side': 24, 'cut_sd': 9.0, + 'high_order_check': [32, 48, 32]}} + res['controls'] = controls(rules) + mc = mc_absdet(200000) + res['controls']['mc_absdet_unconditional'] = {'mean': mc[0], 'se': mc[1], 'n': 200000, 'seed': 2026} + # quadrature-order check at the sample points + qdev = 0.0 + for (x1, z) in SAMPLE_POINTS: + a = lambda_at(x1, z, 0.0, 1.0, rules)['lambda'] + c = lambda_at(x1, z, 0.0, 1.0, hi_rules)['lambda'] + qdev = max(qdev, abs(a - c) / c) + res['quadrature']['max_rel_dev_vs_high_order_at_samples'] = qdev + res['samples'] = {} + res['samples_L6'] = {} + res['near_zero_profile'] = {} + res['ray_profile'] = {} + res['far_field'] = {} + res['hole'] = {} + res['torus'] = {} + res['remote_total'] = {} + ks = (0.5, 1.0, 2.0) + bs = (0.0, 1.0) + grid_h = 0.4 if fast else 0.15 + for b in bs: + far = lambda_far(b, rules) + res['far_field'][str(b)] = {'lambda_inf': far['lambda'], 'lambda_inf_j': far['lambda_j'], 'E_F_j_given_level': far['E_F_j'], + 'p_inf': phi(b) / (2 * math.pi)} + for k in ks: + key = 'k=%s,b=%s' % (k, b) + res['samples'][key] = sample_table(b, k, rules) + res['samples_L6'][key] = sample_table(b, k, rules, 6.0) + res['near_zero_profile'][key] = near_zero_profile(b, k, rules) + res['ray_profile'][key] = ray_profile(b, k, rules) + H = hole_integral(b, k, rules, Rc=6.0, h=grid_h, lam_inf=far) + res['hole'][key] = H + T6 = torus_integral(b, k, rules, 6.0, h=grid_h) + res['torus'][key] = {'L=6': T6} + rt = {} + for L in (6.0, 12.0, 24.0): + approx = L * L * far['lambda'] + H['hole'] + rt['L=%g' % L] = {'integral_X_Lambda_approx': approx, 'k_integral': k * approx} + rt['L=6']['integral_X_Lambda_periodized_direct'] = T6['integral'] + rt['L=6']['k_integral_direct'] = k * T6['integral'] + res['remote_total'][key] = rt + print('done', key, 'hole', H['hole'], 'L6 direct', T6['integral'], 'approx', 36 * far['lambda'] + H['hole'], flush=True) + # grid-convergence, quadrature-order and tail checks for one parameter set; direct periodized L = 12 integral + far0 = lambda_far(0.0, rules) + far0_hi = lambda_far(0.0, hi_rules) + Hc = hole_integral(0.0, 1.0, rules, Rc=6.0, h=0.3, lam_inf=far0) + Hc_hi = hole_integral(0.0, 1.0, hi_rules, Rc=6.0, h=0.3, lam_inf=far0_hi) + Hf = hole_integral(0.0, 1.0, rules, Rc=6.0, h=0.1, lam_inf=far0) if not fast else Hc + Ht = hole_integral(0.0, 1.0, rules, Rc=5.0, h=0.3, lam_inf=far0) + T12 = torus_integral(0.0, 1.0, rules, 12.0, h=0.2) + res['grid_convergence'] = {'k=1.0,b=0.0': {'hole_h=0.3,Rc=6': Hc['hole'], 'hole_h=0.3,Rc=6,high_order': Hc_hi['hole'], + 'hole_h=0.3,Rc=5': Ht['hole'], 'hole_h=0.1,Rc=6': Hf['hole'], + 'hole_h=%s,Rc=6' % grid_h: res['hole']['k=1.0,b=0.0']['hole'], + 'hole_j_h=0.1,Rc=6': Hf['hole_j'], + 'torus_L=12_direct_h=0.2': T12['integral'], + 'torus_L=12_approx': 144 * far0['lambda'] + res['hole']['k=1.0,b=0.0']['hole'], + 'lambda_inf_high_order': far0_hi['lambda']}} + res['exact_regression_points'] = EXACT_COUNT[0] + return res + + +def check_run(): + rules = Rules() + ctl = controls(rules) + path = os.path.join(HERE, 'RESULTS.json') + require(os.path.exists(path), 'RESULTS.json present') + with open(path) as fh: + ref = json.load(fh) + for b in (0.0, 1.0): + far = lambda_far(b, rules) + require(abs(far['lambda'] - ref['far_field'][str(b)]['lambda_inf']) < 1e-10, 'far-field replay b = %r' % b) + closed = far_field_closed_form(b) + require(abs(far['lambda'] - closed['lambda']) < 1e-10 and max(abs(far['lambda_j'][j] - closed['lambda_j'][j]) for j in range(3)) < 1e-10, + 'far field equals its closed form, b = %r' % b) + require(abs(ref['far_field']['0.0']['lambda_inf'] - (2 * math.sqrt(2) - 1) / (2 * math.pi) ** 1.5) < 1e-10, 'Lambda_inf(0) = (2 sqrt2 - 1)/(2 pi)^(3/2)') + for key, k in (('k=1.0,b=0.0', 1.0), ('k=0.5,b=1.0', 0.5)): + b = 0.0 if key.endswith('0.0') else 1.0 + rows = sample_table(b, k, rules) + for r, rr in zip(rows, ref['samples'][key]): + require(abs(r['lambda'] - rr['lambda']) <= REPLAY_TOL * max(abs(rr['lambda']), 1e-6), 'sample replay %s at %r' % (key, r['x'])) + rows6 = sample_table(b, k, rules, 6.0) + for r, rr in zip(rows6, ref['samples_L6'][key]): + require(abs(r['lambda'] - rr['lambda']) <= REPLAY_TOL * max(abs(rr['lambda']), 1e-6), 'L = 6 sample replay %s at %r' % (key, r['x'])) + # a near-axis point that needs the decimal regression + r = lambda_at(0.1, 0.0, 0.0, 1.0, rules) + require(r['exact_regression'] and r['lambda'] < 1e-20, 'near-axis point handled by the decimal regression and exponentially small') + for row in ref['near_zero_profile']['k=1.0,b=0.0']: + if row['r'] == 0.1 and row['angle_deg'] == 0: + require(abs(row['lambda'] - r['lambda']) <= REPLAY_TOL * max(abs(r['lambda']), 1e-30), 'near-zero replay') + gs = grid_summary(ref) + print(json.dumps({'check': 'ok', 'controls': list(ctl.keys()), 'grid_summary': gs, 'mutant': MUT})) + + +def grid_summary(ref): + """Replay of the NOTE section 5 summary arithmetic from the stored grid_convergence block (k = 1, b = 0): + the Rc = 5 / Rc = 6 difference at h = 0.3, the finest-grid difference, the three-point power-law model + A(h) = A_0 + C h^p on the non-nested spacings 0.3, 0.15, 0.1 (a floating extrapolation, not a bound), the + default/high-order difference and the L = 12 direct/additive discrepancy.""" + g = ref['grid_convergence']['k=1.0,b=0.0'] + a3, a15, a1 = g['hole_h=0.3,Rc=6'], g['hole_h=0.15,Rc=6'], g['hole_h=0.1,Rc=6'] + d_rc = g['hole_h=0.3,Rc=5'] - a3 + d_fine = a15 - a1 + ratio = (a3 - a15) / (a15 - a1) + f = lambda p: (0.3 ** p - 0.15 ** p) / (0.15 ** p - 0.1 ** p) - ratio + lo, hi = 1.0, 5.0 + require(f(lo) * f(hi) < 0, 'power-law exponent bracketed in (1, 5)') + for _ in range(200): + mid = 0.5 * (lo + hi) + if f(lo) * f(mid) <= 0: + hi = mid + else: + lo = mid + p = 0.5 * (lo + hi) + C = (a3 - a15) / (0.3 ** p - 0.15 ** p) + A0 = a15 - C * 0.15 ** p + out = {'rc5_minus_rc6_h0.3': d_rc, 'h0.15_minus_h0.1': d_fine, 'model_p': p, 'model_A0': A0, + 'model_residual_h0.15': abs(a15 - A0), 'model_residual_h0.1': abs(a1 - A0), + 'default_minus_high_order_h0.3': a3 - g['hole_h=0.3,Rc=6,high_order'], + 'L12_direct_minus_additive_rel': (g['torus_L=12_direct_h=0.2'] - g['torus_L=12_approx']) / g['torus_L=12_direct_h=0.2'], + 'certified': False} + require(abs(d_rc + 9.46e-5) < 5e-8, 'Rc = 5 minus Rc = 6 at h = 0.3 is -9.46e-5 (NOTE section 5)') + require(abs(d_fine - 1.875e-4) < 5e-7, 'h = 0.15 minus h = 0.1 is 1.875e-4 (NOTE section 5)') + require(abs(p - 2.63) < 5e-3 and abs(A0 + 1.63143) < 5e-5, 'power-law model p = 2.63, A_0 = -1.63143 (NOTE section 5)') + require(abs(out['model_residual_h0.15'] - 2.86e-4) < 5e-6 and abs(out['model_residual_h0.1'] - 9.83e-5) < 5e-7, + 'model residuals 2.9e-4 (h = 0.15) and 9.8e-5 (h = 0.1) (NOTE section 5)') + require(abs(out['default_minus_high_order_h0.3']) < 5e-8, 'default/high-order difference below 5e-8 (NOTE section 5)') + require(abs(out['L12_direct_minus_additive_rel'] - 2.04e-5) < 1e-7, 'L = 12 direct/additive discrepancy 2.0e-5 relative (NOTE section 4)') + return out + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--fast', action='store_true', help='coarser grid (development only)') + ap.add_argument('--mutant', choices=MUTANTS) + args = ap.parse_args() + MUT = args.mutant + if args.check: + check_run() + return + res = full_run(fast=args.fast) + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + json.dump(res, fh, indent=1, sort_keys=True) + fh.write('\n') + print('wrote RESULTS.json') + + +if __name__ == '__main__': + main() diff --git a/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md b/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md new file mode 100644 index 000000000..56348f676 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md @@ -0,0 +1,236 @@ +# Addendum: an explicit unit-square law for the whole extreme doublet + +Object: OA-EXTREME-CLUSTER-SAMPLING-20260930-D1. +Author: OpenAI / GPT-6 Astra Pro. New conditional derivation; nonauthor review OPEN. +Scientific effect NONE. PROOF.md remains byte-identical. This addendum consumes +that manuscript's E12-E19/E29 as candidate inputs; their review is not inferred +from finite tests. It introduces no additional Gaussian source assumption. + +## 1. The law and its exact domain + +Consider the limiting ordinary extreme cluster of PROOF Theorem2, conditioned +on being a doublet. Its outer-root shape has density Q/J on S2_outer, with +J=1083417/280. Write p=-u, A=12p^2-3 and ell=v/A. Define + + rho=|Z_companion/Z_outer| in (0,1), + w=2ell, + z=(1+rho)(w-1)/(2rho). (D1) + +The map is a smooth bijection, up to the boundary null sets already stated in +PROOF, between S2_outer and the OPEN UNIT SQUARE 0p_c, equivalent to w>1 for 01,00 directly, +and eta_in(u,v)=(-p,A ell) has absolute Jacobian A. At fixed w, + + |partial p/partial rho|/2=(w^2-1)/(8rho^2 w). + +Together with Q=A^3(4ell^2-1)(4p ell-1), the density in (rho,w) is + + 81(1+rho)(w-1)^7(w+1)^7 + *(rho*w-rho+w+1)^4(rho*w+rho+w-1)^4 + /(4096rho^11 w^9). (D6) + +Insert w=1+2rho*z/(1+rho) and its Jacobian2rho/(1+rho) to obtain D2. These +identities can be checked by direct substitution; test_ratio_law.py also uses +independent exact two-variable differentiation of the inverse map to check +L=Q|det d(u,v)/d(rho,z)| on72 rational interior cases. Integration of D2 is J +because this is a bijective change of variables from the source outer sector, +not because of numerical normalization. + +## 3. A new small-companion asymptotic + +Let f_rho be the marginal density of rho under the doublet law. Set + + H0=integral_0^1 z^7(z+1)^4 dz=6401/3960, + Csmall=82944 H0/J=137647104/3972529. (D7) + +The rational factor multiplying165888 rho z^7(z+1)^4 in D2 tends uniformly +to1 as rho->0. Its denominator is bounded away from zero on the closed square. +Therefore + + f_rho(rho) ~ (275294208/3972529) rho, + P(rho0, then the microscopic extreme threshold->infinity, then epsilon->0. +It is not a uniform triple-scale assertion for the original field. + +More precisely, conditional on rho Zsmall^2, + eta_in/rho^3 -> 2 Zsmall^2(Zsmall+1). (D10) + +Equivalently eta_in/epsilon^3 has limit2 Qsmall^3 Zsmall^2(Zsmall+1). The small +companion thus approaches the upper height boundary with a CUBIC radius-ratio +scale while the outer height remains nonzero. This describes normalized height +marks, not an elder-pairing or actual finite-r persistence-lifetime theorem. + +The limiting outer-height density in D10 is particularly simple: + + e^3(1+sqrt(e))^4/(2H0), 01-, continuity of D2 and integration on the bounded +z interval give + + f_rho(1-)=857223/2889112-(945/361139)log2. (D12) + +For verification L(1,z)=81 z^7(z+2)^7/[8(z+1)]. Put x=z+1 and expand +(x^2-1)^7/x on[1,2]: its integral is2571669/2240-(81/8)log2. Dividing by J +gives D12. The equal-radius boundary itself still has probability zero. + +## 4. A rejection sampler with a certified ideal-real acceptance rate + +Define + + a(rho,z)=(rho*z+1)^4(rho*z+rho+1)^7 + /[(rho+1)^5(2rho*z+rho+1)^9]. (D13) + +It satisfies00 limit is taken before the new radius->infinity limit. + +Isolation: dedicated new runtime directory; upstream repository checkout is +not available (public raw-host DNS failed). GitHub writes will be additive on +a new branch. No frozen packet or another writer branch is edited. + +Execution: foreground inline; no subagent or independent-provider activity is +invented. Dylan's explicit autonomous continuation supplies the task authority. + +1. [x] Write fail-first rational tests of the root involution, anchor and constants. +2. [x] Implement stdlib Fraction controls with a separate Laurent integration + certificate and rational lower/upper bounds for log(2). +3. [x] Prove and document the root-sector partition, once-per-cluster change of + variables, maximum/minimum tail constants, joint anchored law, height + ordering, and deliberately independent-replica maximum consequence. +4. [x] Test source custody against real Git fixtures, including replacement refs, + noncommit objects, symlinks, nested cwd and duplicate/empty inventories. +5. [x] Verify both Python modes, deterministic outputs, semantic negative controls + and complete flat packet inventory. Publish all exact source pins. +6. [x] Create PR169, read back exact bytes/identities, request nonauthor + algebra/integration and probability reviews, and observe initial-head checks. + +Review focus: strict height endpoints; denominator-zero and equal-radius sets; +point versus configuration multiplicity; physical radius versus |Z|; the +order of limits; latent parameter/orientation bias; finite controls versus +analytic proofs; all source and provider-exposure limitations. + +Scientific effect NONE. No self-acceptance, self-merge, global-register or prize +change. No finite-r spatial rate, growing-volume result, numerical Gaussian +coefficient or independence inside the original field is promised. + +Ruling: no duplicate review of PR166 A/B/C. Its explicit maximum-coefficient +and ordinary-cluster sampling nonclaims define this separate consumer. +Ruling: preserve source commits even when a metadata-only peer successor appears; +new target source authentication remains pinned to the consumed mathematical bytes. + +7. [x] Derive the separate companion-ratio unit-square law, its small-ratio/height + asymptotic and ideal-real rejection sampler; add fail-first exact controls. +8. [ ] Publish the additive successor with original main proof/code unchanged, + request separate Slice D review, and read its actual hosted results. + +Ruling: the companion sampler is explicitly ordinary floating point, while all +normalization/envelope/acceptance constants are checked by exact rational algebra. +Ruling: adopt the peer's physical-order counterexample as credited support only; +it validates the scope boundary and is not an independent mathematical verdict. diff --git a/frontiers/extreme_cluster_sampling_20260930/PROOF.md b/frontiers/extreme_cluster_sampling_20260930/PROOF.md new file mode 100644 index 000000000..19a6044f2 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/PROOF.md @@ -0,0 +1,486 @@ +# Extreme whole-cluster sampling: exact maximum/minimum tails and the anchored doublet law + +Object: OA-EXTREME-CLUSTER-SAMPLING-20260930-v1. +Author: OpenAI / GPT-6 Astra Pro. Foreground continuation authorized by Dylan. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No earlier proof, review, global register, Boolean, +prize or integration disposition is changed. Same-provider/source exposure is +not organizational independence. Finite checks below are not analytic acceptance. + +## 1. Exact object and source boundary + +Fix d>=2, L>0, b in R, k>0 and one axial/transverse orthonormal frame. This +paper concerns the ALREADY-FORMED limiting microscopic cubic measure M, not +the full finite-r Gaussian field and not its remote singleton component. +The limit that forms M is taken BEFORE the radius t tends to infinity. + +Sources and their complete identities are in SOURCES.json: + +[R] PR166 RADIAL_TAIL.md, exact head80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, +blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4, 12533 bytes. Consume the exact +root-parametrized integral R4-R12, its Gaussian majorant, the point tail R13, +and the height moment identities H3-H4. Its separate reviews B/C bind these +unchanged bytes. This paper adds a once-per-cluster selector; R does NOT already +claim the coefficient of the maximum or an ordinary cluster-selected H1 law. + +[M] PR166 TWO_SCALE_LAW.md at the same head, bloba32fd5f7d941bbe1fe943df045b1e0fbec8d691c, +17139 bytes: only the definition M1-M2 of the explicit microscopic measure and +its embedding in the fixed original frame. The new two-scale convergence theorem +of that file is NOT a premise of the calculation here. + +[SC] PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the explicit Gaussian/spectral M, +its finite nonempty mass, and the contact-density bound. This conversation +AUTHORED SC; citing it does not independently revalidate its parent premises. + +[CUB] PR158 PROOF.md atbfcc67dc8957b3ed1fbb789e5a0305e98d7359c8, +blobbb446d08db8a944537a743ad550b88c1c2ad5758: the exact deterministic cubic +classifier only. Its Gaussian-sector statement is not consumed. This conversation +authored CUB. R and M are also OpenAI-authored, in a separate active lane. + +All Gaussian conditions remain conditional on those exact sources. No whole +review disposition is inferred from these source-local uses. + +For theta=(s,h,O,tau), the hard curvatures satisfy 00 is finite, with total mass +A_micro=a1+a2>0, and n belongs to {0,1,2} almost everywhere. + +The unsheared cubic and its physical limiting root position are + + P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2+(a/2)(X^2-1/4)Z + +(beta/2)XZ^2+(c/6)Z^3, + z_i=X_i e_axis+Z_i e_soft(O), R_i=|z_i|, + eta_i=-P(X_i,Z_i)/k in (0,1). (E2) + +Only additional strict-window roots are included, not the pins. Every such +root has full limiting index d-1. Define R_max on nonempty configurations; +on doublets also define R_min. Let + + F(t)=integral sum_i 1{R_i>t} dM, + G(t)=M(n>0,R_max>t), + G2(t)=M(n=2,R_max>t), B2(t)=M(n=2,R_min>t). (E3) + +F is a POINT intensity. G is a CLUSTER tail. Exactly, for every t, + + F(t)=G(t)+B2(t). (E4) + +This elementary multiplicity identity is the distinction missing from a +naive transfer of the point-selected law to whole clusters. + +## 2. Main constants and results + +Let J_cusp and gamma(a) be the positive finite integral and physical-shear +factor from R12: + + gamma(a)=sqrt(1+(a/(12k))^2), + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) da dxi dh dO. + +Set + + C0=(216/11) k^7 (c_m/z0) J_cusp, + I=246528/35, + J=1083417/280, + D=27066286003/223205220-(79298560/4782969) log(2), + I_max=I-D. (E5) + +Here D is a universal scalar shape integral, NOT a hard Hessian or the cubic +coefficient called D in CUB/R. In Sections3-4 the latter is written D_cub. + +**Theorem 1 (whole-cluster tail coefficients).** All the constants multiplying +C0 below are strictly positive, and + + F(t) ~ C0 I t^-11, [the prior R result] + G(t) ~ C0 (I-D) t^-11, + G2(t) ~ C0 J t^-11, + B2(t) ~ C0 D t^-11, + M(n=1,R_max>t) ~ C0 (I-D-J) t^-11. (E6) + +In particular the maximum/point coefficient ratio is the exact universal number + + theta_star=(I-D)/I + =1545114756173/1572181042176 + +(10841600/4605999147)log(2). (E7) + +This ratio is NOT asserted to be a temporal extremal index of the original +field. There is no stationary sequence of pin pairs or mixing assertion here. + +Under the ordinary microscopic cluster probability law +pi=A_micro^-1 1{n>0}dM, conditioning on R_max>t gives + + P_pi(n=2 | R_max>t) -> J/(I-D), + P_pi(K_t=2 | R_max>t) -> D/(I-D), + E_pi[K_t | R_max>t] -> I/(I-D), (E8) + +where K_t is the number of points above threshold t. A double cluster need +not have both of its points above the threshold. Exact rational/logarithmic +interval certificates, not numerical Gaussian integration, give + + 0.984415773209943375 < theta_star < 0.984415773209943376, + 0.558034234065747870 < J/(I-D) < 0.558034234065747871, + 0.015830939745347846 < D/(I-D) < 0.015830939745347847. (E9) + +**Theorem 2 (anchored extreme configuration).** Section6 specifies the full +weak limit of a cluster under pi(. | R_max>t), with its positions divided by t, +its original normalized heights, and its maximum-radius root distinguished. +The radial ratio has Pareto exponent11 and is independent of its anchored shape, +sign and cusp parameters. The limiting doublet is antipodal, has radius ratio +rho in (0,1), and its farther saddle has strictly GREATER downward height eta, +i.e. it lies lower in the height window. Shape/count/height/radius-ratio laws +are universal for these fixed-parameter microscopic extremes; the orientation +and tail amplitude generally retain model dependence. + +Theorems1-2 are new consumers, not part of R's already reviewed point theorem. + +## 3. Companion-root involution and the once-per-cluster shape domain + +Use R's shear invariants + + u=X+aZ/(12k), x=-s>0, + b0(u)=3-12u^2, v=xZ^2/k, + B=k b0(u)/Z^2, D_cub=k(v-b0(u)u)/Z^3. (E10) + +A strict typed window root is parametrized exactly by Z!=0 and + + S={-3/20. (E11) + +The letter S denotes this bounded SHAPE region, not the conditioned saddle. + +Put b=b0(u) temporarily and define + + L=4v^2-8buv+b^2, + t_c=(4v^2-b^2)/L, + u_c=[2bv-u(4v^2+b^2)]/L, v_c=v t_c^2. (E12) + +Off L=0, (u_c,v_c,t_c Z) is the other extra stationary root. Its membership +in S determines whether it is also counted. Thus n=2 exactly when (u_c,v_c) +is in S. L=0 is the degree-drop locus: the second root in this parametrization +is at infinity rather than a finite second point. It is an algebraic null set. +No contribution is assigned to a fictitious point there. + +Proof of E12. At Z'=tZ the stationary line and conic reduce to + + b u'+(v-bu)t=v, 12(u')^2+b t^2=3. + +When b!=0, substitute u'=(v-(v-bu)t)/b. The resulting quadratic has one root +1; its other root is the ratio of its constant to its leading coefficient: + + t_c=(12v^2-3b^2)/(12(v-bu)^2+b^3) + =(4v^2-b^2)/(4v^2-8buv+b^2). + +Substitution gives u_c and then v_c. At b=0 the direct equations give the +same formula wherever L!=0; the b=0 slice is null in any case. Since the +construction swaps the two roots, applying it twice returns the original +(u,v), with t_c t_cc=1. Direct useful identities are + + b0(u_c)=b t_c^2, + v_c-b0(u_c)u_c=(v-bu)t_c^3. (E13) + +These preserve the same x,B,D_cub under E10 with Z'=t_cZ. + +### 3.1 Explicit doublet and anchor domains + +For doublets b<0. Write p=-u, A=12p^2-3=-b>0. Define + + V_+(p)=[36p^2-12p-3+3 sqrt((2p-1)^3(18p+7))]/4, + L_p=A/2, U_p=3+6p. + +Up to boundary null sets, the doublet region splits into + + S2_outer: + 1/2=0 cannot meet v<-b. +For b<0, u=-p with 1/20. If p<1, +H_p(A)=18(p-1)(2p-1)^2(2p+1)<0 and A=1, U_p<=A and +H_p(U_p)=36(p-1)(2p+1)^2>=0, so the interval is (L_p,U_p). + +Both counted roots are on the negative branch of the hyperbola. The stationary +line minus that strictly concave branch is strictly convex and is negative +at Z=0 (its line intercept is -v/A<-1/2). If it has two zeros, zero lies strictly +between them: a strictly convex function with two zeros is positive outside +their interval. Hence t_c<0 on doublets. Since 4v^2-b^2>0, L<0 there. Consequently + + |t_c|>1 iff v>b u=A p, + |t_c|<1 iff v0, so +L_p=1, Ap-U_p=3(p-1)(2p+1)^2>=0. These +observations prove E14. Thus S_star contains exactly one asymptotic maximum +anchor per cluster, apart from the harmless tie curve. + +### 3.2 Height ordering and the intensity-change identity + +On a doublet put ell=v/A (>1/2). A direct substitution in E12 gives + + eta_c-eta=2(ell-p)(4p ell-1)^3/(4ell^2-8p ell+1)^2. (E17) + +Here p>1/2 and 4p ell>1. The outer anchor has ell

0 over a nonempty open set. I-D-J is positive because the single-root shape +sector is also nonempty and open. + +For later height moments define D_j=integral_(S2_inner) eta^j Q du dv. An explicit +primitive for the same rationalization is + + P_j(p,v)=sum_(ell=0..j) binom(j,ell)(1/2-p)^(j-ell)/6^ell + *[16p v^(ell+4)/(ell+4)-4A v^(ell+3)/(ell+3) + -4p A^2 v^(ell+2)/(ell+2)+A^3 v^(ell+1)/(ell+1)]. (E24) + +E22-E23 then give + + D_0=27066286003/223205220-(79298560/4782969)log2, + D_1=1259212131941593/15982386572880 + -(3186360320/387420489)log2, + D_2=72706616008642669/1315122095139840 + -(15686696960/3486784401)log2. (E25) + +For certified decimal bounds use log2=2 sum_(j>=0)3^(-(2j+1))/(2j+1). +After N terms the omitted positive remainder is bounded above by + + 2*3^(-(2N+1))/[(2N+1)(1-1/9)]. (E26) + +All interval endpoints are rational. The code uses60 terms, propagates coefficient +signs and positive divisions, and rounds the displayed lower/upper decimals +OUTWARD. No floating-point quadrature is used to establish E9 or E25. + +## 5. Proof of the maximum and minimum tails + +R's exact root-resolved integral is, with the physical radius retained, + + (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12 + h0(A0,Rot_O tau(a,a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4Z^2)+2k(v-b0u)/Z^3,xi)) + du dv dZ da dxi dh dO. (E27) + +It counts a cluster once for each of its qualifying roots. For G(t), multiply +by 1{R_i>t} times the indicator that this root has the greatest ACTUAL physical +radius, dividing by the number of ties if necessary. The sum over roots is +exactly 1{R_max>t}. This is a Borel operation on a finite configuration and +introduces no extra determinant or normalizer. + +Put Z=t z. For almost every shape, off the algebraic exceptional sets L=0, +eta_c in{0,1} and |t_c|=1, the actual maximum indicator tends to 1_(S_star). +Indeed, R_i/t->gamma(a)|z| and R_c/t->gamma(a)|t_c z| if a counted companion +exists. The bounded u and u_c offsets disappear, but a is NOT integrated away: +its gamma factor remains inside the radius condition. + +The dominating argument is exactly the source R11-R12 argument with one more +factor between0 and1. For t>=3, |u|<=3/2 and R_i>t imply +|z|>1/(2gamma(a)). Orthogonal Gaussian norm equivalence bounds the density by +C exp[-c(|h|^2+a^2+|xi|^2)]. The z-integral of |z|^-12 over that set costs +only C gamma(a)^11; S is bounded, H(h) is polynomial, and the resulting +Gaussian integral is finite. Thus dominated convergence applies, including +the unbounded a and hard/tensor coordinates. No inverse hard eigenvalue enters. + +The limiting z-integral on either sign, with the radius condition retained, +is gamma(a)^11/11. Both signs, the source factors108 and the cusp integral +yield C0. The shape integral is I_max=integral_(S_star)Q=I-D, proving +G(t)~C0(I-D)t^-11. Multiplying by 1{n=2} gives shape S2_outer and coefficient +C0 J for G2. E4 and the known point tail then give the B2 coefficient C0 D. +Subtract G2 from G to get the singleton coefficient. This proves E6, and all +ratios E7-E9 follow since the denominators have strictly positive limits. + +There is no assumption that the seed root's |Z| ordering equals physical +radius ordering at finite t. It need only agree almost everywhere in the limit; +the exact physical selector was used before taking that limit. + +Moment consequences: under the appropriate normalized nonempty/doublet measure, +R_max and R_min both have moment threshold11. For R_min the doublet condition is +understood. The capped unnormalized moments have exact boundary coefficients + + integral_(n>0) min(R_max,t)^11 dM ~11 C0(I-D)log t, + integral_(n=2) min(R_min,t)^11 dM ~11 C0 D log t. (E28) + +These follow from Tonelli and E6. They are not uniform finite-r spatial moments. + +## 6. Full anchored extreme law and its universal consequences + +Distinguish the actual maximum-radius root, using any Borel tie rule. Under +pi(. | R_max>t), retain its seed shape(u,v), q=R_max/t, sign epsilon=sign Z, +and remaining parameters(a,h,O,xi). The limiting joint probability is the product +of + + Q(u,v)1_(S_star)/I_max du dv; + 11q^-12 dq on q>1; + the uniform sign law on{-1,1}; + H(h)gamma(a)^11 h0(A0,Rot_O tau_cusp)/J_cusp da dxi dh dO. (E29) + +Proof: insert a bounded continuous function of these variables into E27 with +the exact maximum selector. The same majorant works. The limiting change +q=gamma(a)|z| produces gamma(a)^11 q^-12 dq, so normalization by the maximum +coefficient gives E29. The maps and the selector are continuous off the +already excluded null curves. This proves weak convergence, not total variation +of spatial laws. No uniformity in escaping model parameters is asserted. + +Let + + e_*(a,O)=[-a e_axis/(12k)+e_soft(O)]/gamma(a). + +If the selected shape is a singleton, the limiting marked configuration is +one point at epsilon q e_* with height eta=u+1/2+v/6. If it is a doublet, +put rho=|t_c|<1, eta_c=u_c+1/2+v_c/6. The full configuration, in positions +divided by the threshold t, is + + {(epsilon q e_*, eta, d-1), + (-epsilon q rho e_*, eta_c, d-1)}. (E30) + +E17 proves eta_c1. +Combining E31 with P(n=2)=J/I_max gives D/I_max in E8 once more. This reconciles +the two-point-cluster frequency with the much smaller double-exceedance frequency. + +### 6.1 The maximum anchor's height is not the old intensity-selected height + +Let H_eta(e) be R's nonnegative height numerator (H1-H2), with integral I. +The height density for the maximum anchor is instead + + [H_eta(e)-6 integral_(1/2)^1 Q(-p,6(e+p-1/2)) + 1{A(p)p<6(e+p-1/2)0, E6 gives + + P(max_(i<=n) R_max(C_i) <= a_n x) + =[1-P_pi(R_max>a_n x)]^n -> exp(-x^-11). (E34) + +This is the standard iid extreme-value consequence with the newly identified +cluster coefficient. Normalizing by the POINT tail coefficient instead gives +exp(-theta_star x^-11); it is not the same normalization. No new coefficient +convergence rate, simultaneous n/r limit, within-field Poisson assertion or +stationary-sequence extremal-index theorem follows from E34. + +## 8. Review obligations and nonclaims + +A: E10-E19, companion involution, strict-window partition, tie/degree-drop +boundaries, exact Jacobian/multiplicity and the height ordering. +B: E20-E26 and E33, the rationalized Laurent integral, all logarithmic constants, +certified rational intervals and the independent finite integration controls. +C: E27-E34, actual physical maximum selection, domination, full anchored weak +law, the differing sampling measures, min/max/exceedance coefficients and iid +scope. These are not waived by tests or parent review acceptance. + +No global scientific-status change, numerical evaluation of J_cusp, finite-r +uniform spatial tail/moment, new elder-pairing law, dependence theorem across pin +pairs, growing-volume theorem, or Lean formalization is claimed. Counterexamples +to shortcuts: F counts both above-threshold roots whereas G counts the cluster +once; replacing G by F sets theta_star=1 incorrectly. Omitting the maximum +selector and summing over roots gives the prior point-intensity height law, not +E32. Uniformly choosing one root inside each uniformly selected cluster is yet +another sampling convention and must not be identified with point intensity. Dropping log2 from E21 changes a positive exact shape integral. A single +1/r rescaling still cannot retain the remote singleton population. diff --git a/frontiers/extreme_cluster_sampling_20260930/README.md b/frontiers/extreme_cluster_sampling_20260930/README.md new file mode 100644 index 000000000..e5c71f7ec --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/README.md @@ -0,0 +1,130 @@ +# Extreme whole-cluster sampling + +OpenAI / GPT-6 Astra Pro. New source-bound conditional proof candidate. +**Nonauthor analytic review OPEN. Scientific effect NONE. No self-merge.** + +PR166 identified the point-INTENSITY radius tail and extreme-height law. This +packet keeps the same limiting microscopic measure and selects each extreme +CLUSTER once. Those are different probability laws: a cluster with both roots +above a threshold contributes twice to point intensity but once to a maximum. + +Start with PROOF.md. It supplies an explicit companion-root involution, the +larger-radius anchor domain, exact maximum/minimum tail coefficients, a joint +anchored extreme law, strict height ordering in doublets, and certified universal +constants. Generic Palm/anchor and iid extreme-value principles are credited in +RECONNAISSANCE.md; no claim of inventing those general principles is made. + +## Main candidate results + +Let C0 be the common positive spectral/Gaussian amplitude defined in PROOF E5. +Put + + I=246528/35, + J=1083417/280, + D=27066286003/223205220-(79298560/4782969)log(2). + +The point tail is the prior C0 I t^-11. New proposed tails are + + cluster maximum: C0(I-D)t^-11; + doublet maximum: C0 J t^-11; + doublet minimum: C0 D t^-11. + +Thus maximum/point coefficient ratio is approximately0.9844157732099434; +a cluster conditioned to have an extreme maximum is a doublet with probability +about0.5580342340657479, whereas BOTH points exceed that same threshold with +probability about0.01583093974534785. These are not interchangeable statistics. +All displayed decimals have rational interval certificates, not quadrature fits. +No numerical evaluation of the Gaussian amplitude C0 is claimed. + +The maximum anchor's limiting downward height has mean about0.8195869887849185, +not the old point-intensity mean. On an extreme doublet its companion lies in +the opposite direction, is closer, and has a strictly smaller downward height. +The cluster radial maximum ratio is independently Pareto(11). See the full +parameter/shape law and qualifications in PROOF E29-E33. + +## Exact scope + +Every theorem here concerns the ALREADY-FORMED limiting microscopic measure: +**first r->0, then t->infinity**. Fixed d>=2,L>0,b real,k>0 and frame. The full +original endpoint normalizer is retained through its source z0. The remote +singleton population is not part of this measure. No finite-r uniform spatial +moment, convergence rate, original-field independence, numerical Gaussian +coefficient, new elder pairing, Lean formalization or global-register change. + +The source use is explicitly conditional: RADIAL root integral/majorant and +point theorem, MICRO definition M1-M2 only, SC's spectral measure/finite nonempty +mass, and CUB's deterministic classifier. MICRO's separate two-scale theorem, +PR159's alternative proof and PR157's exponential theorem are not premises. +The author of this thread authored SC/CUB; no independent revalidation is implied. + +## Reproduction + +Python standard library, compatible with Python3.11+. Full project source +verification requires a Git checkout with all four exact commit/path identities: + + python -B -S frontiers/extreme_cluster_sampling_20260930/verify.py + +The hosted workflow fetches the named public source commits, verifies every +historical path/blob/byte/SHA256, checks the complete packet, runs both Python +modes and the semantic negative controls. Git replacement refs are disabled, +annotated tags/trees cannot masquerade as commits, and source ancestors/leaves +are checked for actual regular-file/tree modes. + +Standalone replay without actual upstream Git objects: + + python -B -S verify.py --local-only + +This explicitly reports source_pins_checked=false. It still runs all real-Git +fixture regressions, but those authenticate test fixtures, not project sources. + +`extremes.py` is the independent exact control entry point. It uses Fraction +arithmetic for companion roots, strict domains, the anchor partition, Laurent +integration and outward log(2) enclosures. `test_extremes.py` includes an independent +line/conic root oracle, exact automatic differentiation of the involution Jacobian, +and dense interpolation checks of the Laurent integrand. Tests do not prove the +Gaussian domination, multiplicity change of variables, or weak convergence. + +## Review requested + +A: PROOF E10-E19, companion roots, type/window/degree-drop/tie boundaries, + exact anchor partition, weighted involution and height ordering. +B: E20-E26/E33, both strip integrals, Laurent rationalization, log(2) constants, + interval certificates and the distinct height sampling laws. +C: E27-E34, exact PHYSICAL maximum selector before the limit, Gaussian domination, + whole-cluster weak law, min/max/double-exceedance probabilities and iid scope. + +Pickups and verdicts must bind exact proof bytes and declare source exposure. +Neither a prior source verdict nor green code is acceptance of this new consumer. + +## Separate companion-ratio addendum + +COMPANION_RATIO.md adds a unit-square parametrization of the limiting extreme +DOUBLE cluster. Its explicit positive rational density determines both heights +and the companion/maximum radius ratio rho, without an unknown Gaussian integral. +It gives the further small-ratio law + + P(rho0, then extreme radius->infinity, THEN epsilon->0. No uniform triple-scale +statement about the original field is made. + +An ideal-real rejection algorithm has exact acceptance probability +3972529/137647104. `ratio_law.sample_doublet` implements its scalar shape/height +part with ordinary floating-point pseudorandom numbers and a bounded trial budget: + + import random + from ratio_law import sample_doublet + marks = sample_doublet(random.Random(20260930)) + +It does not simulate the Gaussian orientation/amplitude or a finite-r field, +and its rounding error is not interval-enclosed. Sixteen added test methods +bring the suite to82 per Python mode. An exact counterexample credited to the +supporting recovery lane pins the distinction between finite-scale physical +radius ordering and the |Z| ordering valid in the second limit. + +This addendum is a SEPARATE Slice D review obligation. It leaves PROOF.md, +extremes.py, the original RESULTS and all source identities unchanged. An +acceptance of A/B/C is not automatically an acceptance of the new addendum. diff --git a/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md b/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..12bfcc069 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/RECONNAISSANCE.md @@ -0,0 +1,57 @@ +# Reconnaissance memo — 30 September 2026 UTC + +Scientific effect NONE. Limited literature reconnaissance, not a novelty or +priority audit. No retrieved paper supplies the Gaussian/pinned-field premises +of the new manuscript; those come from the exact GitHub sources. + +## Connector attempts + +Consensus.search was actually invoked on the two listed Primary and Gadget +accounts with queries about Gaussian critical-point clusters, Palm distributions, +regular variation and extreme cluster marks (years2015-2026/2018-2026). Both +returned the 30-search monthly quota error and an October1 reset. No search +result or paper record was obtained from that connector and none is fabricated. + +## Primary web searches and access depth + +Queries included: +- site.arxiv.org tail processes anchored cluster distributions extremal index regular variation +- site.arxiv.org Gaussian critical points repulsion Palm conditioning short range saddles + +Opened primary arXiv abstract/metadata records only: + +1. Hrvoje Planinic, "Palm theory for extremes of stationary regularly varying + time series and random fields", arXiv:2104.03810v2 (14 March2023). + https://arxiv.org/abs/2104.03810 + Relevant concept: exceedance-selected cluster laws carry a size bias relative + to an anchored/typical cluster law. This generic distinction predates this + packet. The new work is the exact companion involution, shape partition and + coefficients for the specified limiting pinned-Gaussian cubic measure. + +2. Gunter Last, "Tail processes and tail measures: An approach via Palm calculus", + arXiv:2112.15380v3 (9 January2023). + https://arxiv.org/abs/2112.15380 + Relevant concept: tail/exceedance measures, anchor maps and Palm calculus. + No temporal extremal index is inferred here: our original field is not turned + into a stationary regularly varying sequence of pin pairs by analogy. + +3. Dmitry Beliaev, Valentina Cammarota, Igor Wigman, "No repulsion between critical + points for planar Gaussian random fields", arXiv:1911.03455v3 (27 November2020). + https://arxiv.org/abs/1911.03455 + Related but different: short-range critical-point behavior in an unconditioned + planar isotropic model. It is not a proof of the present exact pinned, + determinant-tilted, already-formed microscopic extreme-cluster law. + +No full text of these papers was read for this memo. No exhaustive absence claim +is justified by these searches. The iid Frechet consequence in PROOF Section7 is +standard once its regularly varying tail has been derived; only its source-specific +coefficient is newly calculated here. + +## Source exposure and coordination + +This conversation authored PR158/CUB and PR162/SC. PR166/RADIAL and MICRO are +another OpenAI author lane, with their separate nonauthor records. Sharing a +provider or GitHub login confers no organizational independence. New consumers +require their own review. Main#95 pickup5901916663 and PR166 comment5902055460 +identify the nonoverlapping frontier and shared candidate formulas. No claimed +reviewer/writer lane was duplicated and no parent proof was changed. diff --git a/frontiers/extreme_cluster_sampling_20260930/RESULTS.json b/frontiers/extreme_cluster_sampling_20260930/RESULTS.json new file mode 100644 index 000000000..6cfef6d63 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/RESULTS.json @@ -0,0 +1,69 @@ +{ + "constants": { + "companion_ratio_11th_moment_given_double": { + "lower": "0.028369119274289249", + "upper": "0.028369119274289250" + }, + "double_cluster_given_extreme": { + "lower": "0.558034234065747870", + "upper": "0.558034234065747871" + }, + "inner_double_shape_mass": { + "lower": "109.769950345687984746", + "upper": "109.769950345687984747" + }, + "maximum_shape_mass": { + "lower": "6933.887192511454872396", + "upper": "6933.887192511454872397" + }, + "maximum_to_point_tail_ratio": { + "lower": "0.984415773209943375", + "upper": "0.984415773209943376" + }, + "mean_exceedance_count": { + "lower": "1.015830939745347846", + "upper": "1.015830939745347847" + }, + "outer_double_shape_mass": { + "lower": "3869.346428571428571428", + "upper": "3869.346428571428571429" + }, + "outer_height_mean": { + "lower": "0.819586988784918536", + "upper": "0.819586988784918537" + }, + "outer_height_second_moment": { + "lower": "0.694438469047655310", + "upper": "0.694438469047655311" + }, + "point_shape_mass": { + "lower": "7043.657142857142857142", + "upper": "7043.657142857142857143" + }, + "two_exceedances_given_extreme": { + "lower": "0.015830939745347846", + "upper": "0.015830939745347847" + } + }, + "inner_integrals": { + "0": [ + "27066286003/223205220", + "-79298560/4782969" + ], + "1": [ + "1259212131941593/15982386572880", + "-3186360320/387420489" + ], + "2": [ + "72706616008642669/1315122095139840", + "-15686696960/3486784401" + ] + }, + "log2_terms": 60, + "mathematical_acceptance": false, + "passed": true, + "rational_shape_cases": 893, + "scientific_effect": "NONE", + "scope": "exact finite identities and rational/logarithmic integrals only", + "tail_power": 11 +} diff --git a/frontiers/extreme_cluster_sampling_20260930/SOURCES.json b/frontiers/extreme_cluster_sampling_20260930/SOURCES.json new file mode 100644 index 000000000..c990aa1c0 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/SOURCES.json @@ -0,0 +1,44 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "new_consumer_review": "OPEN", + "sources": [ + { + "id": "RADIAL", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "use": "Exact R4-R13 root integral, Gaussian domination, point tail; H3-H4 point height moments." + }, + { + "id": "MICRO", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "bytes": 17139, + "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "git_blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c", + "use": "M1-M2 definition and original-frame embedding ONLY, not the two-scale convergence theorem." + }, + { + "id": "SC", + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "use": "Explicit spectral/Gaussian micro measure, finite positive nonempty mass and raw Gaussian density bound. Author-side exposure." + }, + { + "id": "CUB", + "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "use": "Exact typed deterministic cubic classifier ONLY; no Gaussian-sector consumption. Author-side exposure." + } + ] +} diff --git a/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md b/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md new file mode 100644 index 000000000..f9220a2e0 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/VALIDATION.md @@ -0,0 +1,126 @@ +# Validation record — author-side, not analytic acceptance + +Scientific effect NONE. Runtime: Python3.13.5 and Git2.47.3. Shipped code is +standard-library only and uses Python3.11-compatible syntax. The hosted workflow +requests Python3.11.16 and real historical project source objects. + +## Test-first mathematical work + +The initial ten-test scaffold produced ten expected assertion failures before +implementation. One proposed companion fixture was first recalculated from an +independent line/conic intersection and corrected BEFORE the implementation: +for u=-1,v=6 the correct companion is(-13/23,294/529,-7/23). + +After implementation, all ten tests passed. Subsequent mathematical tests bring +the suite to35 tests:893 rational shape/companion/classifier/anchor checks; +independent exact line/conic elimination;32 exact automatic-differentiation +checks of Q(T)|det DT|=Q|t_c|^11; direct polynomial-strip integrals; dense Lagrange +interpolation checks of the rationalized Laurent integrand at four rational +y-values and four height-moment orders; strict endpoint, degree-drop and tie +cases; outward rational log(2) intervals and normalized ratios. + +An exploratory generic symbolic integration timed out. The delivered integral +certificate instead uses the explicit rationalizing substitution and a finite +Laurent polynomial, integrated termwise by Fraction arithmetic. Exploratory CAS +scripts are not runtime dependencies or shipped proof evidence. + +## Test-first custody work + +A no-op verifier scaffold ran31 real-Git/inventory test methods and produced45 +expected assertion failures (including separate subtests). The completed verifier +passed all31. Hostile fixtures cover changed historical identities, source files +whose worktree contents differ, missing/duplicate/unexpected identities, path +traversal, duplicate JSON, ancestor and leaf symlinks, tree/blob/annotated-tag +objects offered as commits, and Git replacement refs that can forge or invalidate +ordinary git-show results. All Git reads disable replacements, and the nested +packet path test exercises --full-tree. No project source tampering is asserted. + +The hosted environment also sets GIT_NO_REPLACE_OBJECTS=1. Replaying the hostile +fixture under that environment first produced two expected failures because its +ordinary git-show precondition inherited the protection. The fixture now removes +that variable ONLY for its deliberately ordinary Git calls; the real verifier +always disables replacements. Replaying the same environment passes both controls. +This fixes the test harness, not the source-authentication policy. + +## Fresh complete local replay + +66 tests PASS under python -B -S and66 under python -B -O -S. Normal/optimized +baseline stdout is byte-identical to RESULTS.json with no stderr. The eight +semantic mutants reject with return code1 in EACH mode on the intended assertions: + +| Mutation | Intended rejected statement | +|---|---| +| drop-companion-sign | signed companion | +| point-is-cluster | one maximum anchor, not point counting | +| close-window | height endpoint must be open | +| drop-log2 | inner integral | +| double-count-inner | inner integral | +| lose-height-jacobian | point integral and height convention | +| wrong-tail-power | cusp exponent ledger | +| uniform-cluster-from-point | cluster law must be unbiassed | + +Unknown mutant labels return2. The complete packet inventory is authenticated +before and after replay. No temporary files or __pycache__ are added to it. + +The decimal enclosures use60 rational terms of the convergent log(2) series, +a rigorous rational remainder upper bound, signed interval arithmetic, positive +divisions and outward decimal rounding. No Gaussian coefficient is numerically +computed and no quadrature is used to certify a decimal. + +## Limitations and corrections + +The local environment could not resolve raw.githubusercontent.com; no full +upstream checkout was obtained. Therefore local-only replay explicitly reports +source_pins_checked=false. Real-Git fixtures test the verifier but do not replace +actual source authentication. The hosted target must fetch and verify all four +SOURCES.json identities, separately from the existing full downstream/formal gates. +No hosted pass is asserted by this author-time file; actual head/run/job readbacks +will be posted on the PR, with revalidation if head or base changes. + +During manuscript self-review, a loose sentence equating a uniformly chosen root +of a uniformly chosen cluster with point intensity was corrected before the +first publication. Point intensity sums over roots and biases cluster sampling; +uniformly choosing a root within a uniformly chosen cluster is a third convention. +The corrected PROOF Section8 explicitly distinguishes them. + +Finite exact tests support algebra and implementation. They do not independently +prove the source Gaussian assumptions, changes of variables with multiplicity, +physical-anchor domination or weak convergence. Those new arguments require +source-bound nonauthor analytic review. Existing Lean package success is not a +Lean formalization of this manuscript. No self-merge or scientific-register fold. + +## Separate unit-square companion addendum — successor evidence + +Original PROOF.md, extremes.py, test_extremes.py, RESULTS.json, SOURCES.json, +verify.py and test_verify.py remain byte-identical to published head7ab0d31. +The new files are COMPANION_RATIO.md, ratio_law.py and test_ratio_law.py. + +Test-first record: the initial ratio scaffold ran12 methods and produced15 +expected assertion failures, including subtests. The sampler scaffold extension +ran15 methods and produced20 assertion failures plus one KeyError because its +stub returned an empty mapping. The implemented15 methods then passed. One more +exact physical-order counterexample from peer support5902401772 was independently +recomputed and added as a regression; it does not confer independent acceptance. + +Fresh normal and optimized replay, both with the hosted replacement-disabling +environment, now each run82 tests and PASS. The16 new methods include72 independent +exact Jacobian comparisons,25 rational companion/height substitutions, explicit +positive height-difference/top-boundary factors, exact endpoint and small-ratio +constants, ideal proposal-density factorization/envelope, reproducible float +sampler/range tests and rejection-budget checks. The original semantic mutants +and deterministic RESULTS identity remain required by verify.py. + +A seeded FLOATING-POINT diagnostic (seed20260930,20000 accepted doublets) observed +mean proposal count35.0511 and mean rho^11=0.030448088971641796. Every sampled height +pair and radius ratio satisfied the strict ranges. The exact theoretical values +are137647104/3972529 for mean proposals and D/J for the eleventh moment. This is +an illustrative Monte Carlo sanity check, not a proof, interval guarantee, or +assertion of exact finite-sample agreement. The exact ideal-real acceptance and +moment identities are established analytically, independently of this diagnostic. + +Initial published-head hosted evidence was directly read: target36655624974 / +job109699228659 SUCCESS; full36655625213 with downstream109699229856, +formal109699230080 and required aggregate109699995661 all SUCCESS. These are +7ab0d31 results ONLY. An addendum successor must obtain its own actual hosted +source authentication and current-base full checks; these predecessor runs do +not substitute for them. No new analytic review is claimed by this record. diff --git a/frontiers/extreme_cluster_sampling_20260930/extremes.py b/frontiers/extreme_cluster_sampling_20260930/extremes.py new file mode 100644 index 000000000..38f3d4537 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/extremes.py @@ -0,0 +1,352 @@ +"""Exact controls for the extreme whole-cluster candidate; standard library only. + +The rational identities and integral certificates are executable. Gaussian +conditioning, changes of variables and weak limits are analytic obligations, +not conclusions that this program certifies. +""" +from __future__ import annotations +import argparse +from fractions import Fraction as F +import json +from math import comb +from typing import Optional + +Poly = dict[int, F] +AffineLog = tuple[F, F] # a + b log(2) +Interval = tuple[F, F] +MUTANTS = ('drop-companion-sign', 'point-is-cluster', 'close-window', + 'drop-log2', 'double-count-inner', 'lose-height-jacobian', + 'wrong-tail-power', 'uniform-cluster-from-point') +_MUTANT: Optional[str] = None + + +def _r(x: F | int) -> F: + if isinstance(x, bool) or not isinstance(x, (F, int)): + raise TypeError('an integer or Fraction is required') + return F(x) + + +def b0(u: F | int) -> F: + u = _r(u) + return 3 - 12*u*u + + +def height(u: F | int, v: F | int) -> F: + return _r(u) + F(1, 2) + _r(v)/6 + + +def in_shape(u: F | int, v: F | int) -> bool: + u, v = _r(u), _r(v) + if _MUTANT == 'close-window': + return -F(3,2) <= u <= F(1,2) and abs(b0(u))/2 <= v <= 3-6*u + return -F(3,2) < u < F(1,2) and abs(b0(u))/2 < v < 3-6*u + + +def shape_weight(u: F | int, v: F | int) -> F: + u, v = _r(u), _r(v) + b = b0(u) + return (4*v*v-b*b)*(b-4*u*v) + + +def companion(u: F | int, v: F | int) -> Optional[tuple[F, F, F]]: + """Other root's (u',v',Z'/Z); None is the linear degree-drop case. + + This algebraic map is defined even outside the strict shape region; + membership must be checked separately. It is not a field root solver. + """ + u, v = _r(u), _r(v) + b = b0(u) + den = 4*v*v - 8*b*u*v + b*b + if den == 0: + return None + t = (4*v*v-b*b)/den + up = (2*b*v-u*(4*v*v+b*b))/den + vp = v*t*t + if _MUTANT == 'drop-companion-sign': + t = abs(t) + return up, vp, t + + +def is_double(u: F | int, v: F | int) -> bool: + if not in_shape(u,v): + return False + c = companion(u,v) + return c is not None and in_shape(c[0], c[1]) + + +def classifier_double(u: F | int, v: F | int) -> bool: + """Independent CUB threshold read on an already counted root.""" + u, v = _r(u), _r(v) + b = b0(u) + return in_shape(u,v) and v < -b and 12*(v-b*u)**2 < (v+b)**2*(2*v-b) + + +def anchor_weight(u: F | int, v: F | int) -> F: + """Asymptotic |Z|-maximum selector; ties have weight one half.""" + if not in_shape(u,v): + raise ValueError('strictly typed in-window root required') + if _MUTANT == 'point-is-cluster': + return F(1) + c = companion(u,v) + if c is None or not in_shape(c[0], c[1]): + return F(1) + ratio = abs(c[2]) + return F(1) if ratio < 1 else F(1,2) if ratio == 1 else F(0) + + +def inner_double(u: F | int, v: F | int) -> bool: + """Polynomial description; strict boundary, no square-root comparison.""" + u, v = _r(u), _r(v) + p = -u + a = 12*p*p-3 + f = -2*v*v+(36*p*p-12*p-3)*v-72*p**3+36*p*p+18*p-9 + return in_shape(u,v) and F(1,2)0 + + +def outer_double(u: F | int, v: F | int) -> bool: + u, v = _r(u), _r(v) + p = -u + a = 12*p*p-3 + return ((F(1,2) Poly: + out: Poly = {} + for poly in polys: + for k, value in poly.items(): + out[k] = out.get(k,F(0)) + value + return {k:v for k,v in out.items() if v} + + +def scale(poly: Poly, c: F | int) -> Poly: + c = _r(c) + return {k:v*c for k,v in poly.items() if v*c} + + +def multiply(left: Poly, right: Poly) -> Poly: + out: Poly = {} + for i,x in left.items(): + for j,y in right.items(): + out[i+j] = out.get(i+j,F(0)) + x*y + return {k:v for k,v in out.items() if v} + + +def power(poly: Poly, n: int) -> Poly: + if isinstance(n,bool) or not isinstance(n,int) or n<0: + raise ValueError('nonnegative integer power required') + result: Poly = {0:F(1)} + base = dict(poly) + while n: + if n & 1: + result = multiply(result,base) + base = multiply(base,base) + n //= 2 + return result + + +def evaluate(poly: Poly, x: F | int) -> F: + x = _r(x) + return sum((v*x**k for k,v in poly.items()),F(0)) + + +def integrate_polynomial(poly: Poly, left: F, right: F) -> F: + if any(k<0 for k in poly): + raise ValueError('ordinary polynomial required') + return sum((v*(right**(k+1)-left**(k+1))/F(k+1) + for k,v in poly.items()),F(0)) + + +def _moment_gap(u: Poly, b: Poly, hi: Poly, lo: Poly, j: int) -> Poly: + """Integrate Q(u,v)*(u+1/2+v/6)^j in v between polynomial bounds.""" + if isinstance(j,bool) or not isinstance(j,int) or not 0<=j<=12: + raise ValueError('moment order must be an integer between 0 and 12') + q = {3:scale(u,-16), 2:scale(b,4), + 1:scale(multiply(u,power(b,2)),4), 0:scale(power(b,3),-1)} + bias = add(u,{0:F(1,2)}) + total: Poly = {} + for ell in range(j+1): + eta = scale(power(bias,j-ell),F(comb(j,ell),6**ell)) + for order, coef in q.items(): + n = order+ell+1 + gap = add(power(hi,n),scale(power(lo,n),-1)) + total = add(total,scale(multiply(multiply(eta,coef),gap),F(1,n))) + return total + + +def point_moment(j: int) -> F: + u: Poly = {1:F(1)} + b = add({0:F(3)},scale(power(u,2),-12)) + hi = add({0:F(3)},scale(u,-6)) + value = F(0) + for left,right,sgn in ((F(-3,2),F(-1,2),-1),(F(-1,2),F(1,2),1)): + poly = _moment_gap(u,b,hi,scale(b,F(sgn,2)),j) + value += integrate_polynomial(poly,left,right) + return value/6 if _MUTANT=='lose-height-jacobian' else value + + +def inner_certificate(j: int) -> Poly: + """Laurent integrand on y in [1/4,1], after rationalizing V_+(p).""" + p: Poly = {-1:F(2,9), 0:F(1,18), 1:F(2,9)} + a = add(scale(power(p,2),12),{0:F(-3)}) + hi: Poly = {-2:F(8,9), -1:F(-4,3), 1:F(4,9)} + lo = multiply(a,p) + gap = _moment_gap(scale(p,-1),scale(a,-1),hi,lo,j) + return multiply(gap,{-2:F(2,9),0:F(-2,9)}) + + +def inner_moment(j: int) -> AffineLog: + poly = inner_certificate(j) + rat = sum((c*(1-F(1,4)**(k+1))/F(k+1) + for k,c in poly.items() if k!=-1),F(0)) + log = 2*poly.get(-1,F(0)) + if _MUTANT=='drop-log2': + log = F(0) + if _MUTANT=='double-count-inner': + rat,log = 2*rat,2*log + return rat,log + + +def outer_double_mass() -> F: + p: Poly = {1:F(1)} + a = add(scale(power(p,2),12),{0:F(-3)}) + u,b,lo = scale(p,-1),scale(a,-1),scale(a,F(1,2)) + first = _moment_gap(u,b,multiply(a,p),lo,0) + second = _moment_gap(u,b,add({0:F(3)},scale(p,6)),lo,0) + return (integrate_polynomial(first,F(1,2),F(1)) + + integrate_polynomial(second,F(1),F(3,2))) + + +def log_two_bounds(n: int=60) -> Interval: + """Rational atanh(1/3) lower sum and a proved geometric upper remainder.""" + if isinstance(n,bool) or not isinstance(n,int) or n<1: + raise ValueError('positive integer number of terms required') + z,power_z = F(1,3), F(1,3) + lower = F(0) + for j in range(n): + lower += 2*power_z/F(2*j+1) + power_z *= z*z + remainder = 2*power_z/(F(2*n+1)*(1-z*z)) + return lower,lower+remainder + + +def affine_bounds(value: AffineLog, log: Interval) -> Interval: + a,b = value + return (a+b*log[0],a+b*log[1]) if b>=0 else (a+b*log[1],a+b*log[0]) + + +def divide_positive(numerator: Interval, denominator: Interval) -> Interval: + if numerator[0]<0 or denominator[0]<=0: + raise ValueError('positive interval division required') + return numerator[0]/denominator[1],numerator[1]/denominator[0] + + +def subtract_fixed(x: F, interval: Interval) -> Interval: + return x-interval[1],x-interval[0] + + +def certified_constants(n: int=60) -> dict[str,Interval]: + log = log_two_bounds(n) + I,J = point_moment(0),outer_double_mass() + inner = affine_bounds(inner_moment(0),log) + max_mass = subtract_fixed(I,inner) + inner1 = affine_bounds(inner_moment(1),log) + inner2 = affine_bounds(inner_moment(2),log) + double_probability = divide_positive((J,J),max_mass) + if _MUTANT=='uniform-cluster-from-point': + double_probability = ((J+inner[0])/I,(J+inner[1])/I) + return { + 'point_shape_mass':(I,I), 'inner_double_shape_mass':inner, + 'maximum_shape_mass':max_mass, 'outer_double_shape_mass':(J,J), + 'maximum_to_point_tail_ratio':(max_mass[0]/I,max_mass[1]/I), + 'double_cluster_given_extreme':double_probability, + 'two_exceedances_given_extreme':divide_positive(inner,max_mass), + 'mean_exceedance_count':divide_positive((I,I),max_mass), + 'companion_ratio_11th_moment_given_double':(inner[0]/J,inner[1]/J), + 'outer_height_mean':divide_positive(subtract_fixed(point_moment(1),inner1),max_mass), + 'outer_height_second_moment':divide_positive(subtract_fixed(point_moment(2),inner2),max_mass), + } + + +def decimal_bounds(interval: Interval, digits: int=18) -> dict[str,str]: + if not 1<=digits<=60: + raise ValueError('digits must lie between 1 and 60') + unit = 10**digits + lo = interval[0]*unit + hi = interval[1]*unit + low = lo.numerator//lo.denominator + high = -((-hi.numerator)//hi.denominator) + def fmt(n: int) -> str: + sign='-' if n<0 else '' + n=abs(n) + return f'{sign}{n//unit}.{n%unit:0{digits}d}' + return {'lower':fmt(low),'upper':fmt(high)} + + +def _need(ok: bool, message: str) -> None: + if not ok: + raise AssertionError(message) + + +def finite_checks() -> int: + _need(not in_shape(F(-1),F(9)),'height endpoint must be open') + _need(companion(F(-1),F(6))==(F(-13,23),F(294,529),F(-7,23)), 'signed companion') + _need(anchor_weight(F(-3,4),F(3))==0,'one maximum anchor, not point counting') + _need(point_moment(0)==F(246528,35),'point integral and height convention') + _need(point_moment(1)/point_moment(0)==F(5771,7062),'point first moment') + _need(inner_moment(0)==(F(27066286003,223205220),-F(79298560,4782969)), 'inner integral') + _need(outer_double_mass()==F(1083417,280),'outer double integral') + const = certified_constants(50) + p2=const['double_cluster_given_extreme'] + _need(F(558,1000)=high:continue + for j in range(1,20): + v=low+(high-low)*F(j,20) + _need(in_shape(u,v) and shape_weight(u,v)>0,'open positive shape') + c=companion(u,v) + _need(is_double(u,v)==classifier_double(u,v),'companion/CUB disagreement') + if c is not None: + up,vp,t=c + _need(b0(up)==b*t*t,'conic invariance') + _need(b*up+(v-b*u)*t==v,'line invariance') + back=companion(up,vp) + _need(back is not None and back[0]==u and back[1]==v and back[2]*t==1,'involution') + if in_shape(up,vp): + _need(t<0,'doublet transverse signs') + _need(anchor_weight(u,v)+anchor_weight(up,vp)==1,'once per cluster') + if abs(t)!=1: + _need(inner_double(u,v)==(abs(t)>1),'inner region') + _need(outer_double(u,v)==(abs(t)<1),'outer region') + _need((height(up,vp)-height(u,v))*(abs(t)-1)>0,'outer saddle is deeper') + cases+=1 + return cases + + +def results() -> dict: + cases=finite_checks() + return {'passed':True,'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'exact finite identities and rational/logarithmic integrals only', + 'rational_shape_cases':cases,'log2_terms':60, + 'tail_power':11, + 'inner_integrals':{str(j):[str(x) for x in inner_moment(j)] for j in range(3)}, + 'constants':{k:decimal_bounds(v) for k,v in sorted(certified_constants().items())}} + + +def main() -> None: + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args() + global _MUTANT + _MUTANT=args.mutant + print(json.dumps(results(),indent=2,sort_keys=True)) + + +if __name__=='__main__':main() diff --git a/frontiers/extreme_cluster_sampling_20260930/ratio_law.py b/frontiers/extreme_cluster_sampling_20260930/ratio_law.py new file mode 100644 index 000000000..49404bf07 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/ratio_law.py @@ -0,0 +1,106 @@ +"""Universal extreme-doublet shape: exact rational controls and a float simulator. + +The density is for the already-formed limiting microscopic extreme doublet, +not finite-r field samples. sample_doublet uses ordinary floating-point random +numbers for experimentation; it is not an exact-real or interval sampler. +""" +from fractions import Fraction as F +from math import comb, sqrt +import random +import extremes as e + + +def _unit(x): + if isinstance(x,bool) or not isinstance(x,(int,F)): + raise TypeError('integer or Fraction required') + x=F(x) + if not 0<=x<=1: raise ValueError('unit interval required') + return x + + +def coordinates(rho,z): + """Continuous square extension of the outer root's (u,v); interior is typed.""" + rho,z=_unit(rho),_unit(z) + den=2*rho*z+rho+1 + p=(2*rho*z*z+2*rho*z+rho+2*z+1)/(2*den) + ell=(1+2*rho*z/(1+rho))/2 + return -p,(12*p*p-3)*ell + + +def acceptance(rho,z): + rho,z=_unit(rho),_unit(z) + return ((rho*z+1)**4*(rho*z+rho+1)**7 / + ((rho+1)**5*(2*rho*z+rho+1)**9)) + + +def density(rho,z): + """Unnormalized positive density Q(u,v)|d(u,v)/d(rho,z)|; integral is J.""" + rho,z=_unit(rho),_unit(z) + return 165888*rho*z**7*(z+1)**4*acceptance(rho,z) + + +def heights(rho,z): + """Downward heights of outer and companion roots, respectively.""" + rho,z=_unit(rho),_unit(z) + den=(rho+1)*(2*rho*z+rho+1) + outer=z*z*(rho*z+1)*(rho*z+2*rho+1)/den + inner=rho**3*z*z*(z+1)*(rho*z+rho+2)/den + return outer,inner + + +def z_normalizer(): + return sum((F(comb(4,j),8+j) for j in range(5)),F(0)) + + +def small_ratio_constant(): + """CDF coefficient P(rho=a:continue + den=(rho+1)*(2*rho*z+rho+1) + return {'radius_ratio':rho,'shape_coordinate':z, + 'height_maximum':z*z*(rho*z+1)*(rho*z+2*rho+1)/den, + 'height_companion':rho**3*z*z*(z+1)*(rho*z+rho+2)/den, + 'proposals':trial} + raise RuntimeError('extreme-doublet rejection sampler exhausted its trial budget') diff --git a/frontiers/extreme_cluster_sampling_20260930/test_extremes.py b/frontiers/extreme_cluster_sampling_20260930/test_extremes.py new file mode 100644 index 000000000..503ca3571 --- /dev/null +++ b/frontiers/extreme_cluster_sampling_20260930/test_extremes.py @@ -0,0 +1,213 @@ +"""Exact algebra controls; these tests do not certify the Gaussian arguments.""" +from fractions import Fraction as F +import unittest +import extremes as e + +class InitialTests(unittest.TestCase): + def test_strict_shape(self): + self.assertTrue(e.in_shape(F(-3,4),F(3))) + def test_companion_exact(self): + self.assertEqual(e.companion(F(-1),F(6)),(F(-13,23),F(294,529),F(-7,23))) + def test_outer_double_mass(self): + self.assertEqual(e.outer_double_mass(),F(1083417,280)) + def test_point_mass(self): + self.assertEqual(e.point_moment(0),F(246528,35)) + def test_point_mean(self): + self.assertEqual(e.point_moment(1)/F(246528,35),F(5771,7062)) + def test_inner_mass(self): + self.assertEqual(e.inner_moment(0),(F(27066286003,223205220),-F(79298560,4782969))) + def test_inner_first_moment(self): + self.assertEqual(e.inner_moment(1),(F(1259212131941593,15982386572880),-F(3186360320,387420489))) + def test_inner_second_moment(self): + self.assertEqual(e.inner_moment(2),(F(72706616008642669,1315122095139840),-F(15686696960,3486784401))) + def test_log_two_lower(self): + lo,hi=e.log_two_bounds(30) + self.assertGreater(lo,F(6931471805599453,10**16)) + def test_log_two_upper(self): + lo,hi=e.log_two_bounds(30) + self.assertLess(hi,F(6931471805599454,10**16)) + self.assertGreater(hi,lo) + + +class GeometryTests(unittest.TestCase): + def test_full_rational_grid(self): + self.assertEqual(e.finite_checks(),893) + def test_degree_drop(self): + self.assertTrue(e.in_shape(F(-5,8),F(27,16))) + self.assertIsNone(e.companion(F(-5,8),F(27,16))) + self.assertEqual(e.anchor_weight(F(-5,8),F(27,16)),1) + def test_equal_radius_doublet(self): + u,v=F(-3,4),F(45,16) + self.assertEqual(e.companion(u,v),(u,v,F(-1))) + self.assertEqual(e.anchor_weight(u,v),F(1,2)) + def test_zero_B_finite_companion_outside_window(self): + c=e.companion(F(-1,2),F(3)) + self.assertEqual(c,(F(1,2),F(3),F(1))) + self.assertFalse(e.is_double(F(-1,2),F(3))) + self.assertEqual(e.anchor_weight(F(-1,2),F(3)),1) + def test_height_and_type_boundaries_are_open(self): + for u,v in ((F(-1),F(9)),(F(-1),F(9,2)),(F(-1,2),F(0))): + self.assertFalse(e.in_shape(u,v)) + def test_invalid_numeric_types(self): + for x in (True,0.5,'1/2'): + with self.subTest(x=x),self.assertRaises(TypeError):e.b0(x) + def test_anchor_rejects_outside_shape(self): + with self.assertRaises(ValueError):e.anchor_weight(2,1) + def test_other_root_via_independent_line_conic(self): + for iu in range(-21,7): + u=F(iu,16);b=3-12*u*u + if not -F(3,2) Path: + """Check spelling before resolve; a symlinked ancestor must not disappear.""" + path=Path(os.path.abspath(path)) + for p in (path,*path.parents): + if p.is_symlink(): raise ValueError('symlink prohibited: '+str(p)) + return path + + +def read_json(path: Path) -> dict: + path=no_symlinks(path) + def unique(pairs): + result={} + for key,value in pairs: + if key in result: raise ValueError('duplicate JSON key: '+key) + result[key]=value + return result + obj=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(obj,dict): raise ValueError('JSON object required: '+str(path)) + return obj + + +def source_path(text: str) -> PurePosixPath: + if not isinstance(text,str) or not text or '\\' in text or '\0' in text: + raise ValueError('invalid path') + p=PurePosixPath(text) + if p.is_absolute() or '..' in p.parts or str(p)!=text or str(p)=='.': + raise ValueError('noncanonical source path: '+text) + return p + + +def check_identity(data: bytes, entry: dict) -> None: + if not isinstance(entry,dict): raise ValueError('identity entry must be an object') + size=entry.get('bytes'); sha=entry.get('sha256'); blob=entry.get('git_blob') + if type(size) is not int or size<0 or not isinstance(sha,str) or not HEX64.fullmatch(sha): + raise ValueError('invalid size/hash metadata') + if not isinstance(blob,str) or not HEX40.fullmatch(blob): raise ValueError('invalid blob identity') + actual_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if len(data)!=size or hashlib.sha256(data).hexdigest()!=sha or actual_blob!=blob: + raise ValueError('identity mismatch: '+str(entry.get('path'))) + + +def git(root: Path, *args: str) -> bytes: + env=os.environ.copy() + for key in ('GIT_DIR','GIT_WORK_TREE','GIT_INDEX_FILE','GIT_OBJECT_DIRECTORY', + 'GIT_ALTERNATE_OBJECT_DIRECTORIES','GIT_NAMESPACE'): + env.pop(key,None) + env['GIT_NO_REPLACE_OBJECTS']='1' + command=['git','--no-replace-objects','--literal-pathspecs',*args] + try: + return subprocess.run(command,cwd=root,env=env,capture_output=True, + check=True,timeout=60).stdout + except (subprocess.CalledProcessError,subprocess.TimeoutExpired) as exc: + raise ValueError('Git authentication failed: '+' '.join(args)) from exc + + +def verify_sources(root: Path, required_ids=None) -> int: + root=no_symlinks(root) + required=SOURCE_IDS if required_ids is None else frozenset(required_ids) + entries=read_json(root/'SOURCES.json').get('sources') + if not isinstance(entries,list) or not entries or not required: + raise ValueError('nonempty exact source inventory required') + ids=[]; paths=[] + for entry in entries: + if not isinstance(entry,dict) or not isinstance(entry.get('id'),str): + raise ValueError('invalid source entry') + ids.append(entry['id']) + commit=entry.get('commit'); path=entry.get('path') + if not isinstance(commit,str) or not HEX40.fullmatch(commit): + raise ValueError('literal full commit identity required') + p=source_path(path); paths.append((commit,path)) + # Reject tree/blob/annotated-tag objects, rather than peeling them. + if git(root,'cat-file','-t',commit)!=b'commit\n': + raise ValueError('declared source object is not a commit') + for count in range(1,len(p.parts)+1): + prefix='/'.join(p.parts[:count]) + raw=git(root,'ls-tree','--full-tree','-z',commit,'--',prefix) + records=[record for record in raw.split(b'\0') if record] + if len(records)!=1: raise ValueError('missing/ambiguous source path: '+prefix) + try: + header,name=records[0].split(b'\t',1) + mode,kind,object_sha=header.split() + except ValueError as exc: raise ValueError('malformed Git tree record') from exc + if name.decode('utf-8')!=prefix: raise ValueError('Git path mismatch') + if count int: + root=no_symlinks(root) + entries=read_json(root/'MANIFEST.json').get('files') + if not isinstance(entries,list) or not entries: raise ValueError('empty packet inventory') + names=[] + for entry in entries: + if not isinstance(entry,dict): raise ValueError('invalid packet identity') + text=entry.get('path'); p=source_path(text) + if len(p.parts)!=1 or text=='MANIFEST.json': raise ValueError('flat packet leaf required') + names.append(text) + path=no_symlinks(root/text) + if not path.is_file(): raise ValueError('missing regular packet file: '+text) + check_identity(path.read_bytes(),entry) + if len(names)!=len(set(names)): raise ValueError('duplicate packet path') + if sorted(p.name for p in root.iterdir())!=sorted(names+['MANIFEST.json']): + raise ValueError('complete packet inventory mismatch') + return len(entries) + + +def main() -> None: + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--local-only',action='store_true') + args=parser.parse_args() + root=no_symlinks(Path(__file__).absolute().parent) + verify_inventory(root) + n_sources=0 if args.local_only else verify_sources(root) + expected=(root/'RESULTS.json').read_bytes() + for flags in (['-B','-S'],['-B','-O','-S']): + subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'], + cwd=root,check=True,timeout=180) + command=[sys.executable,*flags,str(root/'extremes.py')] + result=subprocess.run(command,cwd=root,capture_output=True,check=True,timeout=60) + if result.stdout!=expected or result.stderr: + raise ValueError('deterministic mathematical output mismatch') + for name in MUTANTS+('unknown',): + result=subprocess.run(command+['--mutant',name],cwd=root, + capture_output=True,timeout=60) + if result.returncode!=(2 if name=='unknown' else 1): + raise ValueError('negative control did not reject: '+name) + verify_inventory(root) + print(json.dumps({'passed':True,'source_pins_checked':not args.local_only, + 'source_count':n_sources,'scientific_effect':'NONE', + 'mathematical_acceptance':False},sort_keys=True)) + + +if __name__=='__main__': main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json b/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json new file mode 100644 index 000000000..dd6871344 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/MANIFEST.json @@ -0,0 +1,67 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "NONAUTHOR_REVIEW_OPEN", + "files": [ + { + "path": "PROOF.md", + "bytes": 17213, + "sha256": "5adef725c1c696adae9bc73a525efe0bfe09d5c07e95ee3f067db9e877283630", + "git_blob": "0d4018777d13672c1de1696525512e80b23b7c3c" + }, + { + "path": "README.md", + "bytes": 3763, + "sha256": "cd167a13b3f7250b09635a61aec9b47f48da044c87bafeb86ce7486cb8d63d8b", + "git_blob": "dc87d46d95a7a5d6dcdd7debf1da7422a7d1d80c" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 2222, + "sha256": "38ca0b266398fcfcec2565f2bdd1b53b15494995fcd433f442090887e0a66158", + "git_blob": "47b0f54a2189efcfa9adbaae3ec4e8bb5fa744df" + }, + { + "path": "RESULTS.json", + "bytes": 516, + "sha256": "46ccc0f5e2d8cba11040ca1bc617e9e99e54ca7b19ac85ff4091e2601104efeb", + "git_blob": "ef9ff136e85198dad73d664d1250775baf3e1d7e" + }, + { + "path": "SOURCES.json", + "bytes": 651, + "sha256": "9ca481cd235a0e16ee8175538be0eea2477fbd8340833bc84d6ba995f20cf6e0", + "git_blob": "a7a4ebdd902a11f5daf87e0ea8fd93576aa403fa" + }, + { + "path": "VALIDATION.md", + "bytes": 2500, + "sha256": "b1528945a1d7610b2d3b27fbc907cf6538d887262f634d02b4fe0fc64fc0693c", + "git_blob": "7dc4ffe63a94db6e21164309f089b8ba0b8f1b85" + }, + { + "path": "inverse.py", + "bytes": 6501, + "sha256": "69c8e91e227bde716e237945358b0244d8a414eda9de9d4ce5e4fccb93ee7862", + "git_blob": "62698d649ff1eba92656fb555bf57d603f2bbc37" + }, + { + "path": "test_inverse.py", + "bytes": 6312, + "sha256": "0bcd93e078ee4800647f8af5aaf77894c63912d63dca9d74c49f732432bc5e6f", + "git_blob": "9695c472e7b0f4acc7946c310765d6ec204dc742" + }, + { + "path": "test_verify.py", + "bytes": 4083, + "sha256": "ee6aa8e6b589acb2eb32b482b4e9965d614d7c8416737624cd6df16eaa0b2960", + "git_blob": "1fccbe702825118906bfea6e266973a51225ca78" + }, + { + "path": "verify.py", + "bytes": 5190, + "sha256": "df2f54f44a95f1dc62022145feed2e22dca242bd37266e1eed82b12e519f6355", + "git_blob": "22db229cfc2be9d3486d7759c82c50bea83b14e0" + } + ] +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md b/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md new file mode 100644 index 000000000..0d4018777 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/PROOF.md @@ -0,0 +1,332 @@ +# Fixed-separation lifetime tails and the sharp inverse-moment boundary + +Object OA-FIXED-R-INVERSE-LIFETIME-20260930-v1. +Author: OpenAI / GPT-6 Astra Pro. AUTHOR-SIDE PROOF CANDIDATE; new nonauthor +analytic review OPEN. Scientific effect NONE. No prior proof, status, graph, +Boolean, prize or review disposition is changed. + +## 1. Statement and the precise probability law + +Fix d>=2, L>0, b real, k>0, a unit vector u and 0b) min_t f(gamma(t)), + ell=b-d_f(M). (2) + +An empty class of paths has d_f=-infinity and ell=+infinity (essential class). +On finite deaths this is the actual lifetime, not a gap to a counted witness. +Let F denote failure of the prescribed saddle S to be the ordinary partner. +On the usual distinct-critical-value locus, (2) is the ordinary elder saddle +value. The estimates below apply directly to (2), without counting saddles +or assuming a particular local saddle is the global partner. + +**Theorem 1 (two-sided fixed-r small-lifetime order).** There are constants +00, depending on the fixed parameters, such that +for 0=0, + + E_QW[ell^(-p);ell finite] < infinity iff p<2/3, + E_QW[ell^(-p);F,ell finite] < infinity iff p<2/3. (4) + +For p=0 these denote the finite-event probabilities. The corresponding +conditional expectations have the same finiteness boundary. Multiplication by +the fixed positive scale Delta leaves the threshold unchanged, so (4) also +holds for the normalized lifetime X_r=ell/Delta. At critical and supercritical +powers the capped expectations obey + + E_QW[min(ell^(-1),T)^(2/3);F,ell finite]=Theta(log T), + E_QW[min(ell^(-1),T)^p;F,ell finite]=Theta(T^(p-2/3)) + for p>2/3. (5) + +The same holds without F or after the stated finite-event conditioning. The +constants are not evaluated. There is no claimed exact leading coefficient +or lifetime-density equivalent in (3). Together with elementary finiteness of +positive moments, the full finite-death power-moment domain is q>-2/3: +E[ell^q;finite] and E[ell^q;F,finite] are finite exactly on that range. + +**Scope.** These are statements at EACH FIXED positive r, with epsilon->0. +The constants and epsilon0 may degenerate arbitrarily as r->0. No uniform +integrability as r->0 or limit of inverse moments is asserted. In particular +this is NOT the distant-partner-conditioned law of PR181, where r->0 and a +second extreme-radius limit occur before a small-lifetime limit. Its exponent4 +and this exponent2/3 concern different laws and different limiting operations. + +[P] is the exact source in SOURCES.json. Only its model, observation, weighting +conventions and fixed-point Fourier facts in Sections1-2 are used; the latter +are reconstructed below. The uniform cap theorem, global count laws, elder +candidates PR170/175, positive-moment PR177 and extreme-mark PR181 are NOT proof +premises. The author is exposed to [P]. This is a new consumer, not independent +revalidation of that source's other theorems. + +## 2. Fixed-parameter Gaussian facts, with no contact uniformity + +Poisson summation gives Fourier weights + + a_n=exp(-2 pi^2 |n|^2/L^2)/sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +For every integer m, sum_n sqrt(a_n)(1+|n|)^m is finite. A real sine/cosine +expansion and Minkowski's inequality show that every C^m norm of the field +has moments of every finite order. For orders below1 use a larger moment. +This also supplies a smooth version of the field. + +Any finite list of distinct derivative evaluations at distinct sites has +strictly positive covariance. Indeed a zero-variance linear combination +annihilates every Fourier mode since a_n>0. As a distribution it is a finite +sum of derivatives of delta masses, all of whose Fourier coefficients vanish. +The distribution is zero. Test functions supported near one site with arbitrary +finite jets show that all its coefficients vanish. Symmetric tensor entries +are listed once only; a change of orthonormal frame is invertible on these jets. + +This proves nondegeneracy of the original observation vector O and of the +following additional lists modulo O: + + U=(H_M,H_S), J=(H_M,H_S,D^3 f(M)). (6) + +All conditional covariances here are at FIXED r>0. None is replaced by its +singular r=0 counterpart. Their conditional Gaussian densities are strictly +positive on every compact set and bounded above by C exp(-c|U|^2), or the +analogous bound for J, with constants depending on the fixed observations. + +Full-field regression conditional on U has the form + + f=m_U+G_U, ||m_U||_C3 <= C(1+|U|), + G_U independent of U, E||G_U||_C3^q0. In particular K is NOT presumed +independent of the Hessians. The same argument conditional on J gives an +independent C4 residual and a uniform C4 mean bound on each compact J set. + +The Gaussian density of U is positive on the open product of the required +Hessian cones; W is strictly positive there. All its polynomial moments are +finite. Hence Z in (1) is positive and finite. No r^-2 normalizer estimate is +needed or used. + +For completeness, (2) is measurable: for hh is existence +of a path from M to a point above b with its minimum strictly above h. A +uniform approximation of such a path by torus piecewise-linear paths with +rational vertices retains both strict margins. Thus a countable set of paths +suffices; their endpoint values and compact-path minima are measurable field +functionals. An essential maximum simply has no eligible path. + +## 3. A deterministic local barrier gives a lower bound on lifetime + +Let f be any C3 field on the torus with f(M)=b, grad f(M)=0 and H_M<0. Write +lambda_min for the smallest eigenvalue of -H_M and use K above, so +0= c_L lambda_min^3/K^2, c_L=a_L^2/3>0. (9) + +This holds also as an inequality for ell=+infinity. It proves strict positivity +of every finite lifetime without any global Morse argument. The barrier does +not claim the partner is on that sphere; it bounds all older-reaching paths. + +## 4. Keep the determinant weight: the upper small-lifetime bound + +For any nonnegative measurable function psi, (7) and the Gaussian U density +imply a one-dimensional weighted eigenvalue estimate + + E_Q[W K^2 psi(lambda_min)] + <= C integral_0^infinity lambda psi(lambda) + (1+lambda)^N exp(-c lambda^2)d lambda. (10) + +Here is the full reduction. On H_M<0 diagonalize -H_M with ordered positive +eigenvalues 0=1. Apply (10) with +psi(lambda)=min(1,epsilon/lambda^3), and divide ONCE by Z. For epsilon=s^3<=1, +the portion lambda<=1 is bounded by a constant times the exact integral + + integral_0^1 lambda min(1,s^3/lambda^3)d lambda + =s^2/2+s^3(1/s-1)=(3/2)s^2-s^3. (12) + +The lambda>=1 portion is O(epsilon) by the Gaussian envelope. Therefore +Q^W(00. (13) + +Let C range over a compact box of strictly negative definite (d-1)-matrices, +v over a fixed small box of positive volume, and H_S over a compact box inside +the index-(d-1) cone. We may ensure eigenvalues of C stay between -2 and -1/2, +|det C|>=c_C>0 and |det H_S|>=c_S>0. The entire boxes can be chosen with +nonempty interiors. In these coordinates + + e^T H_M e=-lambda, H_M<0, + |det H_M|=lambda |det C|. (14) + +The change from the independent H_M entries to (lambda,v,C) has absolute +Jacobian1: only H_11 is replaced, with derivative -1 in lambda. We do not set +v=0 on a measure-zero slice, nor shrink its box with lambda. The direction e +adapts smoothly to v,C and has a uniform finite norm bound E. + +Let the third-derivative tensor T at M vary over a fixed compact positive-volume +box for which T[e,e,e]>=a>0 for every C,v in the chosen boxes. Such a box exists: +start with T_111=2a and all other entries zero, choose v small, and then a +sufficiently small full-dimensional tensor neighborhood. Initially allow +0<=lambda<=1. This defines a compact set of complete J targets. By Section2 its +raw conditional density has a strictly positive lower bound m_r on that set. + +Write the conditional full field as m_J+G_J. Choose a fixed residual C4 norm +bound with probability p0>0, possible because the residual has a finite norm +almost surely. It is independent of J. The conditional mean is bounded on the +compact targets; consequently there is a uniform directional fourth-derivative +bound M4>=1 along e on this event, accounting for |e|<=E. This uses a fixed +positive-probability residual event, not the probability of a shrinking Gaussian +ball. M4 and m_r may be very unfavorable as r decreases, which is allowed. + +For the straight path t -> M+t e, Taylor now gives + + f(M+t e) >= b-(lambda/2)t^2+(a/6)t^3-(M4/24)t^4. (15) + +Put B=6/a. If lambda<=6/(M4 B^2), then at t=B lambda the right side minus b is +at least (B^2/4)lambda^3>0. Over the entire interval 0<=t<=B lambda it is at +least -(3B^2/4)lambda^3, hence certainly at least -B^2 lambda^3. Reduce a +fixed lambda0>0 further so that B lambda0 Eb-Delta, not from a count of nearby critical +points. We need not identify the actual death saddle or forbid an even higher +alternative global path: either could only shorten ell further. + +For 0= c integral_0^((epsilon/B^2)^(1/3)) lambda d lambda + = (c/2)B^(-4/3)epsilon^(2/3). (17) + +This is the lower half of (3). It holds for all sufficiently small epsilon at +the fixed model parameters, not just along a sequence or finite numerical grid. + +## 6. Inverse moments and capped growth + +Both normalizing events {ell finite} and {F,ell finite} have positive probability +by (17). For epsilont) <= C t^(-2/3). (18) + +Endpoint strict versus weak inequalities cause no difficulty: upper bounds use +(3), and a lower bound at epsilon=1/(2t) is already enough. For p>0, Tonelli gives + + E[Y^p]=p integral_0^infinity t^(p-1) Q^W(Y>t)dt. + +The portion below a fixed large threshold is finite. The remaining integral +converges precisely when p<2/3. This proves (4). For positive powers, on finite deaths ell<=2||f||_infty; +all weighted positive moments of this norm are finite by the Gaussian norm +and Hessian moments of Section2 and Cauchy-Schwarz. This yields the stated +full power-moment domain without asserting any small-r moment convergence. +Truncating the same integral at T gives the logarithmic and power orders in (5). These are two-sided growth +orders, not exact coefficients; a density or precise regular-variation limit +was not proved by the two-sided comparison. + +## 7. What this closes, and what it does not + +The fixed-r inverse boundary is now determined at candidate-theorem level, +complementing PR177's separately derived positive finite-lifetime moments. +Neither the small-r marked law nor its moment-convergence hypotheses are used +here. In particular the new result cannot be reinterpreted as a failure of +PR181: that law further conditions on a distant replacement partner after +already taking r->0, whereas (3) includes all finite failures at fixed r. + +The exponent2/3 has two independent sources: local cubic escape costs a lifetime +of order lambda^3, while the original Palm weight and eigenvalue volume give +lambda d lambda. Dropping that weight changes the conclusion. As a comparison, +the same proof for the raw Gaussian law conditioned only on the two open Hessian +types, without either determinant tilt, gives exponent1/3; in its upper argument +integral_0^1 min(1,s^3/lambda^3)d lambda=(3/2)s-(1/2)s^3. +This comparison is not a change to the law (1) used in the main theorems. + +No exact lifetime density, leading coefficient, r-uniform constant, joint +r/epsilon limit, convergence of inverse moments as r->0, distant-partner inverse +threshold at finite r, or globally deduplicated persistence-bar intensity is +asserted. Essential-inclusive POSITIVE moments and their convention remain +separate. These proofs are not formalized in Lean and finite tests are not +analytic acceptance. + +Requested review A: fixed-r Gaussian nondegeneracy, regression and weighted +spectral reduction (Sections2-4), including the original full Z. Review B: +full-dimensional Schur sector, retained residual event, older-reaching path, +lower bound and the exact moment conclusions (Sections5-6). Prior source +verdicts or green workflows do not accept this new consumer. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/README.md b/frontiers/fixed_r_inverse_lifetime_20260930/README.md new file mode 100644 index 000000000..dc87d46d9 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/README.md @@ -0,0 +1,72 @@ +# Fixed-r lifetime moments: the inverse boundary is 2/3 + +OpenAI / GPT-6 Astra Pro. **New author-side theorem candidate; nonauthor review +OPEN. Scientific effect NONE. No self-merge.** + +For the exact determinant-weighted maximum/saddle pair-Palm law, hold the two +prescribed pins at any fixed positive embedded separation r. Let ell be the +ACTUAL ordinary elder lifetime, excluding the essential infinite lifetime. +The new proof gives two-sided small-lifetime bounds + + c_r epsilon^(2/3) <= P(0-2/3. Inverse moments of order p are finite exactly when p<2/3, both under +the finite-death restriction and conditional on failure and finite death. +The capped critical inverse moment grows as Theta(log T); above criticality +it grows as Theta(T^(p-2/3)). No exact leading coefficient is asserted. + +This fills the inverse-moment exclusion in the separately merged PR177. It +uses neither PR170/175's marked-limit candidates nor PR181's extreme-selected +law. The latter conditions on distant partners AFTER r->0 and has a different +small-lifetime exponent; it must not be substituted for this fixed-r law. + +## Why the exponent is 2/3 + +A deterministic sphere around the prescribed maximum forces +ell >= c lambda_min^3/K^2, where K controls the third derivative. Conditional +Gaussian norm moments retain K inside the endpoint determinant weight. The +remaining soft-eigenvalue integral is exactly + + integral_0^1 lambda min(1,s^3/lambda^3)d lambda=(3/2)s^2-s^3. + +An open Schur-coordinate sector gives the converse. The Hessian has a soft +curvature -lambda but a positive cubic derivative along an adapted direction. +A short straight path reaches a point above birth with only O(lambda^3) loss. +The original determinant weight supplies lambda, and full jet support supplies +d lambda. Thus the lower probability is of order epsilon^(2/3) as well. +The off-diagonal Hessian entries range over a fixed OPEN box, not a zero-measure +slice or a box shrinking with lambda. + +## Exact boundaries + +All constants can depend on fixed r,d,L,b,k and frame. No r-uniform rate, +interchange of limits, density equivalent, numerical amplitude, inverse-moment +convergence as r->0, or result conditional on a distant partner is claimed. +No source selection theorem, global count asymptotic or hard-fibre bridge is +required: maximin paths and local barriers control the actual lifetime directly. + +PROOF.md contains the full arguments. SOURCES.json binds the one parent model +carrier; only its model/weight conventions and fixed-jet Fourier facts are used, +and the latter are proved again here. Parent asymptotic theorems are not promoted. + +## Replay + + python -B -S frontiers/fixed_r_inverse_lifetime_20260930/verify.py + +The full mode requires the exact parent Git object. For a standalone extraction: + + python -B -S verify.py --local-only + +Local-only explicitly reports that historical project sources are not checked. +Thirty test methods run in each normal/optimized mode; six intentionally wrong +variants must fail per mode. Exact controls include36 Schur matrices across +four hard dimensions,36 Taylor-path cases/612 grid checks,36 local barriers, +and11 source/inventory fixture tests. These finite checks are not acceptance +of the Gaussian integration or the theorem. No new Lean formalization. + +The foreground work follows Dylan's NEW explicit September30 continuation +request, after the September27 stop. No stopped process or timer is restarted. +Review A covers Gaussian support/regression and the upper weighted bound; +review B covers the Schur sector, lower bound and moment domain. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md b/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..47b0f54a2 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/RECONNAISSANCE.md @@ -0,0 +1,37 @@ +# Narrow primary reconnaissance and scope — 30 September 2026 + +Queries actually executed: +- site.arxiv.org Gaussian random fields persistence lifetime distribution critical points small persistence +- site.arxiv.org Gaussian critical points finiteness moments Gass Stecconi 2023 +- site.arxiv.org Gaussian fields Palm small eigenvalue Hessian local maximum saddle persistence + +Primary abstract/metadata pages actually opened: + +1. Louis Gass and Michele Stecconi, The number of critical points of a Gaussian + field: finiteness of moments, arXiv:2305.17586v2, revised19 July2023. + https://arxiv.org/abs/2305.17586 + Context: count-moment finiteness under regularity and jet nondegeneracy. This + is not an inverse ordinary-elder lifetime theorem and is not a premise here. + +2. Dmitry Beliaev, Valentina Cammarota and Igor Wigman, No repulsion between + critical points for planar Gaussian random fields, arXiv:1911.03455v3, + revised27 November2020. https://arxiv.org/abs/1911.03455 + Context: near-diagonal critical-point correlations with index information. + No unconditioned count-correlation formula is substituted for the actual + fixed-pin lifetime law in the new manuscript. + +Access depth was abstract/metadata, not a full-text novelty audit. The cubic +fold scaling, Schur-complement algebra, Gaussian regression, spectral matrix +Jacobian and layer-cake formula are existing general tools. No claim of +inventing them or of exhaustive priority is made. The specific new in-project +argument combines the actual maximin barrier, fixed-r full-jet support and the +original determinant tilt to identify the two-sided inverse-moment boundary. + +Connected repository context read: PR177 README at + d768a1761aee8d6911c14bc7524089e14c8cd81c, blob508e3963304cf323099bc0e118559a1f86f04146, +explicitly excludes inverse moments. That positive-moment work is credited, not +consumed as a premise or independently revalidated. PR181's distant-selected +limit is a different conditioning and is also not a premise. Coordination +pickup main#95/5908596793 and technical notes177/5908600694,5908718757 separate +these lanes. The user requested GitHub/Superpowers; Consensus was not invoked +and no quota-status change is claimed. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json b/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json new file mode 100644 index 000000000..ef9ff136e --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/RESULTS.json @@ -0,0 +1,15 @@ +{ + "critical_capped_growth": "Theta(log T)", + "extreme_partner_conditioning": false, + "fixed_r_small_lifetime_exponent": "2/3", + "inverse_moment_threshold": "2/3", + "mathematical_acceptance": false, + "passed": true, + "path_grid_checks": 612, + "path_parameter_cases": 36, + "scientific_effect": "NONE", + "scope": "finite rational Taylor, Schur and integral controls only", + "supercritical_capped_growth": "Theta(T^(p-2/3))", + "uniform_in_pin_separation": false, + "unweighted_comparison_exponent": "1/3" +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json b/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json new file mode 100644 index 000000000..a7a4ebdd9 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/SOURCES.json @@ -0,0 +1,15 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "use": "Sections1-2 model, exact original observations, full determinant weight and fixed-jet Fourier support. Gaussian facts reconstructed here. No Theorem A/B/C, contact limit, or marked elder law consumed. Author source exposure." + } + ] +} diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md b/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md new file mode 100644 index 000000000..7dc4ffe63 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/VALIDATION.md @@ -0,0 +1,39 @@ +# Actual author validation, separate from analytic acceptance + +The initial eight-method mathematical scaffold failed all eight assertions. +After implementation all eight passed. The expanded mathematical suite has19 +methods; eleven real-Git/source/inventory tests bring the total to30. + +Fresh replay runs all30 methods in normal and optimized Python, including the +six CLI semantic mutants in each mode and the unknown label. Deterministic +baseline output must match RESULTS.json with no stderr. The exact controls +include36 full Schur matrices in hard dimensions1-4 with nonzero off-diagonal +entries,36 Taylor-path parameter sets and612 rational path points,36 sphere +barriers with360 interior points, exact cubic saddle gaps, weighted split +integrals, and the critical/supercritical scalar-model moment formulas. + +The verifier is reused from the author's PR181 driver, narrowed to the single +parent source and new mathematical CLI. Its byte-count field is required here. +The11 real-Git/inventory tests cover the actual nested working directory, byte +and hash drift, missing/duplicate source inventory, noncommit and unsafe paths, +symlink source/payload, historical replacement forgery, extra files and duplicate +JSON. The replacement fixture first verifies that an ordinary Git read sees +forged bytes, then that the protected reader authenticates the original bytes. +These fixtures are not evidence of tampering in the user's repository. + +The upper/lower Gaussian bounds and infinite-dimensional support are analytic +proof obligations, not conclusions of the scalar examples. No original field +is simulated and no Gaussian coefficient is numerically enclosed. No source +proof, review disposition or global status is changed. + +The local attempt to access the GitHub API failed with DNS resolution error. +Local-only replay therefore reports source_pins_checked=false. The hosted +workflow must fetch the ACTUAL parent commit and verify its full path, regular +mode, byte count, SHA256 and Git-blob identity before reporting source success. +Any hosted receipt belongs to the actual published head, not this author-time +record. Existing Lean checks do not formalize the new analytic proof. + +The new explicit owner instruction is the current September30 ChatGPT request +beginning "Continue to Work on closing as many mathematical items as possible". +It follows the September27 stop and authorizes foreground work, not timers. +No pre-stop autonomy grant or automated-loop restart is used as authorization. diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py b/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py new file mode 100644 index 000000000..62698d649 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/inverse.py @@ -0,0 +1,162 @@ +"""Exact controls for the fixed-separation inverse-lifetime theorem candidate. + +Finite rational checks only: this does not simulate a conditioned Gaussian field +or certify the infinite-dimensional support argument. Standard library only. +""" +import argparse +from fractions import Fraction as F +import json + +MUTANTS=('lose-determinant-weight','wrong-cubic-power','wrong-soft-direction', + 'admit-critical-inverse','drop-endpoint-margin','unconditional-tail-shortcut') +_MUTANT=None + + +def rat(x): + if type(x) not in (int,F):raise TypeError('integer or Fraction required') + return F(x) + + +def determinant(matrix): + n=len(matrix) + if any(len(row)!=n for row in matrix):raise ValueError('square matrix required') + a=[[rat(x) for x in row] for row in matrix];out=F(1) + for j in range(n): + i=next((i for i in range(j,n) if a[i][j]),None) + if i is None:return F(0) + if i!=j:a[j],a[i]=a[i],a[j];out=-out + pivot=a[j][j];out*=pivot + for i in range(j+1,n): + q=a[i][j]/pivot + for k in range(j+1,n):a[i][k]-=q*a[j][k] + return out + + +def inverse(matrix): + n=len(matrix) + if any(len(row)!=n for row in matrix):raise ValueError('square matrix required') + a=[[rat(x) for x in row]+[F(i==j) for j in range(n)] for i,row in enumerate(matrix)] + for j in range(n): + i=next((i for i in range(j,n) if a[i][j]),None) + if i is None:raise ValueError('singular matrix') + a[j],a[i]=a[i],a[j];p=a[j][j];a[j]=[x/p for x in a[j]] + for i in range(n): + if i!=j: + q=a[i][j];a[i]=[x-q*y for x,y in zip(a[i],a[j])] + return [row[n:] for row in a] + + +def soft_chart(C,v,lam): + lam=rat(lam);C=[[rat(x) for x in row] for row in C];v=list(map(rat,v));n=len(C) + if not n or len(v)!=n or any(len(row)!=n for row in C) or lam<=0: + raise ValueError('positive soft curvature and a nonempty square hard block required') + if any(C[i][j]!=C[j][i] for i in range(n) for j in range(n)): + raise ValueError('hard block must be symmetric') + if any((-1)**j*determinant([row[:j] for row in C[:j]])<=0 for j in range(1,n+1)): + raise ValueError('hard block must be strictly negative definite') + inv=inverse(C);cv=[sum((inv[i][j]*v[j] for j in range(n)),F(0)) for i in range(n)] + top=sum((x*y for x,y in zip(v,cv)),F(0))-lam + H=[[top]+v]+[[v[i]]+C[i] for i in range(n)] + e=[F(1)]+[-x for x in cv] + if _MUTANT=='wrong-soft-direction':e=[F(1)]+cv + return H,e + + +def barrier(L,lam,K): + L,lam,K=map(rat,(L,lam,K)) + if L<=0 or not 0=1 required') + a=min(F(1),L/4);R=a*lam/K + return R,lam*R*R/3 + + +def path_constants(a,M): + """B=6/a, admissible lambda ceiling, loss bound C and endpoint-gain coefficient.""" + a,M=map(rat,(a,M)) + if a<=0 or M<0:raise ValueError('positive cubic lower bound and nonnegative fourth bound required') + B=6/a;cut=min(F(1),6/(M*B*B)) if M else F(1) + return B,cut,B*B,F(0) if _MUTANT=='drop-endpoint-margin' else B*B/4 + + +def cubic_gap(lam,g): + lam,g=map(rat,(lam,g)) + if lam<=0 or g<=0:raise ValueError('positive lambda and cubic required') + return 2*lam**(2 if _MUTANT=='wrong-cubic-power' else 3)/(3*g*g) + + +def upper_split(s): + """Integral_0^1 lambda min(1,s^3/lambda^3)d lambda, 0=0, 3p!=2 required') + n=2-three_p + return s**n+2*(1-s**n)/n + + +def check(): + def need(ok,msg): + if not ok:raise ValueError(msg) + need(tail_exponent(1)==F(2,3),'retain original maximum determinant weight') + need(cubic_gap(F(1,10),F(2))==F(1,6000),'cubic lifetime scale') + H,e=soft_chart([[F(-2)]],[F(1,3)],F(1,10)) + q=sum((e[i]*H[i][j]*e[j] for i in range(2) for j in range(2)),F(0)) + need(q==-F(1,10),'correct Schur soft direction') + need(not inverse_finite(F(2,3)),'critical inverse moment diverges') + need(path_constants(1,4)[3]==9,'strict endpoint above birth') + need(upper_split(F(1,10))==F(7,500),'retain weighted near-zero eigenvalue integral') + cases=0 + for a in (F(1,2),F(1),F(2)): + for M in (F(0),F(1),F(4),F(20)): + B,ceiling,C,gain=path_constants(a,M) + for frac in (F(1,10),F(1,2),F(1)): + lam=ceiling*frac + endpoint=-lam*(B*lam)**2/2+a*(B*lam)**3/6-M*(B*lam)**4/24 + need(endpoint>=gain*lam**3>0,'Taylor endpoint') + for j in range(17): + t=B*lam*F(j,16) + lower=-lam*t*t/2+a*t**3/6-M*t**4/24 + need(lower>=-C*lam**3,'Taylor path stays above lower barrier') + cases+=1 + return {'passed':True,'mathematical_acceptance':False,'scientific_effect':'NONE', + 'scope':'finite rational Taylor, Schur and integral controls only', + 'path_parameter_cases':cases,'path_grid_checks':17*cases, + 'fixed_r_small_lifetime_exponent':'2/3','inverse_moment_threshold':'2/3', + 'unweighted_comparison_exponent':'1/3', + 'critical_capped_growth':'Theta(log T)', + 'supercritical_capped_growth':'Theta(T^(p-2/3))', + 'uniform_in_pin_separation':False,'extreme_partner_conditioning':False} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args();global _MUTANT;_MUTANT=args.mutant + try:ans=check() + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True));raise SystemExit(1) + print(json.dumps(ans,indent=2,sort_keys=True)) + +if __name__=='__main__':main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py b/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py new file mode 100644 index 000000000..9695c472e --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/test_inverse.py @@ -0,0 +1,121 @@ +from fractions import Fraction as F +import unittest +import inverse as m + +class InitialTests(unittest.TestCase): + def test_schur_chart(self): + H,e=m.soft_chart([[F(-2)]],[F(1,3)],F(1,10)) + self.assertEqual(H,[[F(-7,45),F(1,3)],[F(1,3),F(-2)]]) + self.assertEqual(e,[F(1),F(1,6)]) + def test_barrier(self): + self.assertEqual(m.barrier(F(4),F(1,10),F(2)),(F(1,20),F(1,12000))) + def test_path_constants(self): + self.assertEqual(m.path_constants(F(1),F(4)),(F(6),F(1,24),F(36),F(9))) + def test_spectral_split(self): + self.assertEqual(m.upper_split(F(1,10)),F(7,500)) + def test_exact_cubic_gap(self): + self.assertEqual(m.cubic_gap(F(1,10),F(2)),F(1,6000)) + def test_tilt_exponent(self): + self.assertEqual(m.tail_exponent(1),F(2,3)) + self.assertEqual(m.tail_exponent(0),F(1,3)) + def test_threshold(self): + self.assertEqual([m.inverse_finite(p) for p in (F(0),F(1,3),F(2,3),F(1))],[True,True,False,False]) + def test_critical_cap(self): + self.assertEqual(m.critical_model(),(F(1),F(-2))) + + +class ExtendedTests(unittest.TestCase): + def test_full_scalar_controls(self): + r=m.check();self.assertEqual(r['path_parameter_cases'],36) + self.assertEqual(r['path_grid_checks'],612) + def test_schur_in_arbitrary_hard_dimensions(self): + count=0 + for n in range(1,5): + B=[[F((i+2*j)%5-2,7) for j in range(n)] for i in range(n)] + C=[[-sum(B[k][i]*B[k][j] for k in range(n))-int(i==j) for j in range(n)] for i in range(n)] + for numerator in (1,2,5): + v=[F(numerator*(i+1),11) for i in range(n)] + for lam in (F(1,100),F(1,10),F(1)): + H,e=m.soft_chart(C,v,lam) + He=[sum(H[i][j]*e[j] for j in range(n+1)) for i in range(n+1)] + self.assertEqual(He,[-lam]+[F(0)]*n) + self.assertEqual(m.determinant(H),-lam*m.determinant(C)) + for x in (F(-1),F(2,3)): + y=[F(i-1,3) for i in range(n)];vec=[x]+y + q=sum(vec[i]*H[i][j]*vec[j] for i in range(n+1) for j in range(n+1)) + cv=m.inverse(C);z=[y[i]+x*sum(cv[i][j]*v[j] for j in range(n)) for i in range(n)] + self.assertEqual(q,-lam*x*x+sum(z[i]*C[i][j]*z[j] for i in range(n) for j in range(n))) + count+=1 + self.assertEqual(count,36) + def test_uniform_scalar_barrier_inequality(self): + for L in (F(1,3),F(1),F(4),F(24)): + for K in (F(1),F(3),F(20)): + for ratio in (F(1,100),F(1,3),F(1)): + lam=K*ratio;R,gap=m.barrier(L,lam,K) + self.assertLess(R,L/2) + self.assertLessEqual(-lam*R**2/2+K*R**3/6,-gap) + for j in range(1,11): + t=R*F(j,10) + self.assertLess(-lam*t*t/2+K*t**3/6,0) + def test_scalar_cubic_stationary_point(self): + for lam in (F(1,100),F(1,7),F(1)): + for g in (F(1,2),F(1),F(7)): + t=2*lam/g + self.assertEqual(-lam*t+g*t*t/2,0) + self.assertGreater(-lam+g*t,0) + self.assertEqual(-lam*t*t/2+g*t**3/6,-m.cubic_gap(lam,g)) + def test_lower_path_at_worst_fourth_derivative(self): + for g0 in (F(1,3),F(1),F(3)): + for M in (F(1),F(10),F(100)): + B,cut,C,gain=m.path_constants(g0,M) + for g in (g0,2*g0): + lam=cut/2 + for h in (-M,F(0),M): + val=lambda t:-lam*t*t/2+g*t**3/6+h*t**4/24 + self.assertGreaterEqual(val(B*lam),gain*lam**3) + self.assertTrue(all(val(B*lam*F(j,20))>=-C*lam**3 for j in range(21))) + def test_weighted_split_by_independent_primitives(self): + for s in (F(1,100),F(1,3),F(1)): + direct=s*s/2+s**3*(1/s-1) + self.assertEqual(direct,m.upper_split(s)) + self.assertLessEqual(direct,F(3,2)*s*s) + self.assertGreaterEqual(direct,s*s/2) + def test_random_norm_factor_can_be_absorbed(self): + for A in (F(1),F(3,2),F(10)): + for x in (F(1,100),F(1,3),F(1),F(5)): + self.assertLessEqual(min(F(1),A*x),A*min(F(1),x)) + def test_capped_model_piecewise_integration(self): + for s in (F(1,20),F(1,3),F(1)): + for n in (0,1,3,4,5): + direct=s**(2-n)+2*(1-s**(2-n))/F(2-n) + self.assertEqual(m.cap_model(s,n),direct) + self.assertGreaterEqual(direct,1) + def test_critical_model_logarithm(self): + # For p=2/3, the lower part is1 and the upper integral is -2log(s). + self.assertEqual(m.critical_model(),(1,-2)) + self.assertEqual(F(2)/3,m.tail_exponent()) + def test_wrong_inputs_rejected(self): + for val in (True,0.5,'1/2'): + with self.assertRaises(TypeError):m.cubic_gap(val,1) + for C,v,l in (([[1]],[0],1),([[-1,1],[0,-1]],[0,0],1),([[-1]],[0],0)): + with self.assertRaises(ValueError):m.soft_chart(C,v,l) + with self.assertRaises(ValueError):m.barrier(0,1,2) + with self.assertRaises(ValueError):m.barrier(1,3,2) + with self.assertRaises(ValueError):m.path_constants(0,1) + with self.assertRaises(ValueError):m.upper_split(0) + def test_cli_both_modes_and_mutants(self): + import subprocess,sys + from pathlib import Path + path=str(Path(m.__file__).resolve());baseline=None + for flags in (['-B','-S'],['-B','-O','-S']): + run=subprocess.run([sys.executable,*flags,path],capture_output=True,timeout=25) + self.assertEqual(run.returncode,0,run.stderr);self.assertEqual(run.stderr,b'') + if baseline is None:baseline=run.stdout + self.assertEqual(run.stdout,baseline) + for mutant in m.MUTANTS: + bad=subprocess.run([sys.executable,*flags,path,'--mutant',mutant],capture_output=True,timeout=25) + self.assertEqual(bad.returncode,1,(mutant,bad.stdout,bad.stderr)) + bad=subprocess.run([sys.executable,*flags,path,'--mutant','unknown'],capture_output=True,timeout=25) + self.assertEqual(bad.returncode,2) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py b/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py new file mode 100644 index 000000000..1fccbe702 --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/test_verify.py @@ -0,0 +1,69 @@ +"""Real-Git regressions for the reused, narrowed source-verification driver.""" +import hashlib,json,os,subprocess,tempfile,unittest +from pathlib import Path +import verify as v + + +def entry(path,data,**kw): + return dict(path=path,bytes=len(data),sha256=hashlib.sha256(data).hexdigest(), + git_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest(),**kw) + +class SourceTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.repo=Path(t.name) + self.root=self.repo/'frontiers/packet';self.root.mkdir(parents=True) + self.git('init','-q');self.data=b'source fixture\n' + (self.repo/'P.md').write_bytes(self.data);self.git('add','P.md');self.git('commit','-qm','fixture') + self.sha=self.git('rev-parse','HEAD').decode().strip();self.e=entry('P.md',self.data,id='P',commit=self.sha);self.write() + def git(self,*args): + env=os.environ.copy();env.pop('GIT_NO_REPLACE_OBJECTS',None) + return subprocess.run(['git','-c','user.name=Fixture','-c','user.email=fixture@example.invalid',*args],cwd=self.repo,env=env,capture_output=True,check=True,timeout=15).stdout + def write(self,es=None): + (self.root/'SOURCES.json').write_text(json.dumps({'sources':[self.e] if es is None else es})) + def test_original_nested_source(self):self.assertEqual(v.sources(self.root),1) + def test_identity_drift(self): + for k,x in [('bytes',0),('bytes',True),('sha256','0'*64),('git_blob','0'*40)]: + self.write([dict(self.e,**{k:x})]) + with self.subTest(k=k),self.assertRaises(ValueError):v.sources(self.root) + def test_empty_duplicate_sources(self): + for es in ([],[self.e,self.e]): + self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_noncommit_and_unsafe_path(self): + self.write([dict(self.e,commit=self.git('rev-parse','HEAD^{tree}').decode().strip())]) + with self.assertRaises(ValueError):v.sources(self.root) + self.write([dict(self.e,path='../P.md')]) + with self.assertRaises(ValueError):v.sources(self.root) + def test_replace_cannot_change_history(self): + new=b'forged\n';(self.repo/'P.md').write_bytes(new);self.git('add','P.md');self.git('commit','-qm','change') + self.git('replace',self.sha,self.git('rev-parse','HEAD').decode().strip()) + self.assertEqual(self.git('show',self.sha+':P.md'),new) + self.assertEqual(v.sources(self.root),1) + self.write([entry('P.md',new,id='P',commit=self.sha)]) + with self.assertRaises(ValueError):v.sources(self.root) + def test_symlink_source(self): + (self.repo/'P.md').unlink();(self.repo/'P.md').symlink_to('elsewhere') + self.git('add','P.md');self.git('commit','-qm','link') + self.write([dict(self.e,commit=self.git('rev-parse','HEAD').decode().strip())]) + with self.assertRaises(ValueError):v.sources(self.root) + +class PacketTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.root=Path(t.name) + self.data=b'packet\n';(self.root/'PROOF.md').write_bytes(self.data) + (self.root/'MANIFEST.json').write_text(json.dumps({'files':[entry('PROOF.md',self.data)]})) + def test_original(self):self.assertEqual(v.inventory(self.root),1) + def test_changed_payload(self): + (self.root/'PROOF.md').write_bytes(b'changed') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_extra_file(self): + (self.root/'extra').write_bytes(b'x') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_duplicate_json(self): + (self.root/'MANIFEST.json').write_text('{"files":[],"files":[]}') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_payload(self): + (self.root/'PROOF.md').unlink();(self.root/'PROOF.md').symlink_to('MANIFEST.json') + with self.assertRaises(ValueError):v.inventory(self.root) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/fixed_r_inverse_lifetime_20260930/verify.py b/frontiers/fixed_r_inverse_lifetime_20260930/verify.py new file mode 100644 index 000000000..22db229cf --- /dev/null +++ b/frontiers/fixed_r_inverse_lifetime_20260930/verify.py @@ -0,0 +1,110 @@ +"""Replay the fixed-r inverse-lifetime packet; --local-only excludes historical sources.""" +import argparse +import hashlib +import json +import os +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + +IDS={'P'} +MUTANTS=('lose-determinant-weight','wrong-cubic-power','wrong-soft-direction', + 'admit-critical-inverse','drop-endpoint-margin','unconditional-tail-shortcut') + + +def read_json(path): + if path.is_symlink() or not path.is_file():raise ValueError('regular JSON required') + def unique(pairs): + out={} + for k,v in pairs: + if k in out:raise ValueError('duplicate JSON key') + out[k]=v + return out + data=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(data,dict):raise ValueError('JSON object required') + return data + + +def parts(path): + if not isinstance(path,str) or not path:raise ValueError('nonempty path required') + p=PurePosixPath(path) + if p.is_absolute() or str(p)!=path or any(x in ('.','..') or not re.fullmatch('[A-Za-z0-9_.-]+',x) for x in p.parts): + raise ValueError('unsafe or noncanonical path') + return p.parts + + +def identity(data,e): + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if type(e['bytes']) is not int or len(data)!=e['bytes']: + raise ValueError('size mismatch') + if hashlib.sha256(data).hexdigest()!=e['sha256'] or blob!=e['git_blob']: + raise ValueError('hash mismatch: '+e['path']) + + +def git(root,*args): + run=subprocess.run(['git','--no-replace-objects',*args],cwd=root, + env=dict(os.environ,GIT_NO_REPLACE_OBJECTS='1'), + capture_output=True,timeout=45) + if run.returncode:raise ValueError('Git read failed: '+run.stderr.decode(errors='replace')[:160]) + return run.stdout + + +def sources(root): + es=read_json(root/'SOURCES.json')['sources'] + if not isinstance(es,list) or len(es)!=len(IDS) or {e['id'] for e in es}!=IDS: + raise ValueError('source inventory mismatch') + seen=set() + for e in es: + path=e['path'];ps=parts(path);commit=e['commit'] + if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit): + raise ValueError('full commit identity required') + if (commit,path) in seen:raise ValueError('duplicate source binding') + seen.add((commit,path)) + if git(root,'cat-file','-t',commit).strip()!=b'commit':raise ValueError('not a commit object') + for j in range(1,len(ps)+1): + prefix='/'.join(ps[:j]) + rows=[s for s in git(root,'ls-tree','--full-tree','-z',commit,'--',prefix).split(b'\0') if s] + if len(rows)!=1:raise ValueError('missing or ambiguous source path') + meta,name=rows[0].split(b'\t',1);mode,kind,blob=meta.decode().split() + if name.decode()!=prefix:raise ValueError('source path mismatch') + if j0 and the fixed model parameters. Using the notation of PROOF.md, define + + m_q=integral lambda(theta)^q dmu_fail(theta), + c_q=a-m_q=integral [1-lambda(theta)^q] dmu_fail(theta). + +Because a>0 and 00. QED. + +This is a pair-Palm expectation under the existing marked-law premises. It is not a new Kac–Rice intensity, an unrestricted parameter statement, or an acceptance of those premises. No existing weak-limit scope was defective. The separate rational control checks finite probability bookkeeping, the finite-conditioning denominator, and the difference between clipped loss and signed deficit; it is not a Gaussian-model simulation or a proof of the analytic premises. diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json b/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json new file mode 100644 index 000000000..6ebb1c1ce --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json @@ -0,0 +1,20 @@ +{ + "LOSS_COROLLARY.md": { + "bytes": 3374, + "sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "git_blob": "f12bd82d2a0d0804f4c262d0d080b3efcf269f59" + }, + "check_loss_exact.py": { + "bytes": 2680, + "sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "git_blob": "630938818e6230e9df39211885eb86ddcb7ea4d6" + }, + "control_observation": { + "frozen_proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "loss_corollary_sha256": "8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c", + "rational_probability_cases": 48, + "runner_sha256": "20b5f8e4c1330e1a4e0b0ee3256cebf002ca9f6114b0686b30e9d63ba98441bd", + "scope": "Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.", + "status": "PASS" + } +} diff --git a/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md b/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..003f36db8 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/LOSS_NONAUTHOR_REVIEW.md @@ -0,0 +1,15 @@ +# Nonauthor review of the additive lifetime-loss consequence + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`, 30 September 2026. No drafting or amendment of this consequence was performed by the reviewer. Same-provider/source exposure and shared-account work grant zero organizational-independence credit. + +**VERDICT: ACCEPT (L1)–(L3) of `LOSS_COROLLARY.md`, SHA256 `8d202bf2adbf25cb10ba7b114024e7772f2e6839746dc744e1494027a25a178c`, as conditional consequences of the separately reviewed frozen C28 proof and its original actual-elder event definition. No blocking amendment found.** + +The source check is substantive: MARK_D §1 defines F_r as failure of the actual ordinary elder partner to equal S and identifies finite death with that partner's height on the full-probability Morse/distinct-value locus. MARK_2 uses the same event and expressly equates success with d_f(M)=f(S). Since f(S)=b-kr³, success implies finite death and X_r=1 exactly. No witness-count event is substituted. + +The finite numerator decomposes exactly into Q^W(F_r complement) plus the finite-death failure moment. Substituting Q^W(F_r)=a r³+o(r³) and the frozen proof's failure moment m_q r³+o(r³) gives 1-(a-m_q)r³+o(r³). The essential mass e_r is o(r³), so division by 1-e_r leaves this coefficient unchanged. Positivity and strict upper bound of c_q=a-m_q follow from positive failure mass and limiting fractions strictly between zero and one. + +For the clipped formula, success contributes one and essential deaths also have clipped value one; subtracting its expectation from one leaves the bounded nonnegative loss. For the finite signed deficit the exact identity is Q^W(finite)-E[X_r^q;finite]=(1-e_r)-E[X_r^q;finite]. The candidate correctly distinguishes it from a pointwise nonnegative loss: it is negative on finite X_r>1. The frozen axial-tail bound makes the expectation of this discrepancy negligible at r³ scale. Neither formula silently treats infinity times an indicator as an ordinary finite moment. + +Multiplication by the deterministic positive factor (kr³)^q yields the physical-lifetime expansion at fixed q,k. Both essential-inclusive positive moments remain infinite. These remain weighted pair-Palm expectations, not counts or intensities of distinct replacement bars, and there is no extra failure normalization. + +The additive file leaves frozen `PROOF.md` SHA256 `3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8` unchanged. The separate exact control was inspected and replayed once with `python3 -B -S`; its 48 rational probability-bookkeeping cases passed. The fixture retains separate success, ordinary failure, rare finite long lifetime, and essential atoms, so it checks the conditioning denominator and the distinction between signed and clipped loss. It does not prove any Gaussian-model premise. File identities and the observed receipt are in `LOSS_IDENTITY_AND_CONTROL_OBSERVATIONS.json`. diff --git a/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md b/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..24c571bfb --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/NONAUTHOR_REVIEW.md @@ -0,0 +1,44 @@ +# Nonauthor review of the actual-lifetime moment boundary + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`. +Date: 30 September 2026. + +**VERDICT: ACCEPT the frozen consumer's axial escape lemma, clipped and finite-death positive-moment consequences, and essential-inclusive infinite-moment obstruction, conditional on its exact stated parent and failure-mark premises. No blocking amendment found.** + +The reviewer did not draft, suggest, or edit the candidate proof. This is a separate mathematical read after the authors froze it. The reviewer has source-custody and queue-triage exposure to this research chain, shares the authors' provider and GitHub account, and earns zero organizational-independence credit. This acceptance does not revalidate the Gaussian parent or the draft marked-law sources and does not change their disposition. + +## Exact reviewed object + +- `payload/proof/PROOF.md`: 13,952 bytes; SHA256 `3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8`; Git blob `d5ee8abee120a8bec2f6c470e517ff9db4645183`. +- `payload/proof/SOURCES.json`: 5,148 bytes; SHA256 `45d36269416ba0033edf184a540c32b3007e91fb2031ab8b77ae89eb1942db2e`; Git blob `46a791e3313df1954a332db2da7d3489d29bc48c`. +- The seven retained source bodies P, ERR, BOREL, REC, MARK_D, MARK_2 and CAP match every declared byte count, SHA256 and Git blob. The identity observations are recorded separately in `IDENTITY_AND_CONTROL_OBSERVATIONS.json`. + +The mathematical read covered the complete candidate; P §§1–5 and the actual-death/Morse interface in §8; REC's source identity set and reading rules; MARK_2 §§8.7–8.8; and MARK_D's actual-death definition and §6 marked-law interface. The two source failure-mark theorems remain explicitly assumed inputs. Their full proofs are not re-reviewed here. + +## Reconstructed checks + +1. **The estimand is the actual elder lifetime.** P §8 and both marked-law sources define death by the supremum of path minima over endpoints strictly older than the pinned birth. Empty endpoint sets give death minus infinity. The candidate retains this definition, the original endpoint weight, and one full normalizer. It does not relabel a saddle count or a local witness as the actual death. Under the tilted law the maximum type is enforced. Essential maxima are failures because they cannot die at the finite pinned saddle. + +2. **The clipped consequence follows from the supplied weak law.** For fixed positive q, the clipped power is a bounded continuous function of the finite normalized death mark, with the isolated essential sentinel assigned value one. The source gives vanishing scaled sentinel mass, so its treatment does not create an extra limiting contribution. A finite death remains available even if its position is outside the chart; the candidate correctly does not replace that height by a position sentinel. The limiting fraction lies strictly between zero and one and the limiting failure mass a is positive, hence 0 < m_q < a. Dividing by the convergent failure mass is valid. + +3. **The repeated-Rolle step is valid with C4 regularity.** The Hermite cubic satisfies both values and both zero derivatives. Thus g minus the cubic has zeros of total multiplicity four on [a,c], and a third-derivative zero exists in its interior. The cubic's third derivative is 12k. The fourth-derivative bound therefore gives g''' at least 12k minus 2rH on the extended interval [a,e], which is at least 6k under rH ≤ 3k. No simultaneous vector Rolle zero is used: this is the scalar axial restriction. + +4. **Strict convexity identifies the path minimum.** Since (g')'' = g''' > 0, g' is strictly convex. Its exact zeros at a and c force it below zero between the zeros and above zero to their right. Hence the axial path has its minimum exactly at c, with value b minus kr³. The Hermite extrapolation remainder at e uses five zeros counting multiplicities after subtracting a multiple of (x-a)²(x-c)². Four differentiations are legitimate. Its factor is 4r⁴/24; the cubic value at e is b plus 4kr³. Therefore the asserted lower endpoint margin b plus (7/2)kr³ is correct. This produces one admissible path to an older endpoint and proves death at least f(S); it does not assert that S is the winning partner. + +5. **The weighted joint moments are actually present in the parent.** P(4.1) gives all finite moments of the global C4 norm uniformly at the stated compact parameter scope. The sentence following P(5.3) explicitly gives all finite moments of W_r/r². Hölder therefore bounds E_Q[(W_r/r²)K^s] for every finite s while retaining correlations. P(5.5), read with ERR and REC, supplies the positive full-normalizer floor. There is no assumption that the field norm and weight are independent, and no substitution of a good-event normalizer. + +6. **The rare large-norm estimate has the correct powers.** The axial result implies that an essential death or a finite fraction above one requires K > const/r. Weighted Markov bounds its tilted mass by every positive power of r. For a finite death on the connected torus, some older endpoint exists, every path minimum is at least the global minimum, and birth equals f(M); therefore the finite lifetime is bounded by 2K. Multiplication by r^(-3q) and the failure scaling r^(-3) produces r^(-3q-3). On K > const/r, using K^q 1_B ≤ const r^p K^(q+p) gives the displayed exponent p-3q-3. Arbitrarily high parent moments allow any prescribed positive remainder power. The q+epsilon version supplies uniform integrability at the scaled failure-measure level. + +7. **The finite-death conditioning is legitimate.** The difference between the finite qth moment and the clipped numerator consists only of the finite tail above one and the clipped essential contribution. Both vanish after r^(-3) scaling. Essential probability is smaller than every power, so Q^W(F_r, finite)/r³ tends to the same positive a. This proves both formulas in (MOM) for each fixed q>0. It neither proves inverse moments nor parameter-uniform convergence of the assumed marked law. + +8. **Full affine conditional support follows for the actual model.** P §2 has strictly positive variance for every real Fourier mode, including the constant mode, and summable coefficients in every Cq norm. Finite coefficient neighborhoods and an independent small C2 tail give positive probability near every trigonometric polynomial. Their C2 density gives full unconditioned C2 support. For fixed distinct pins, the value/gradient covariance is positive definite. The regression map A is a continuous right inverse of the pin map L, so T = Id minus AL is a continuous projection onto ker L. Its Gaussian residual is independent of the observations, and the pushforward of full support has support closure(T(B)) = ker L. Thus the canonical pinned law has full relative support on the precise affine pin space for the prescribed values, without additional conditioning or a changed normalizer. + +9. **The essential maximum is a robust, admissible typed configuration.** Disjoint pin balls can be chosen at every fixed r. The M quadratic has a strict negative Hessian. The S quadratic has one positive and d-1 negative directions while its entire support lies below birth by a strict margin. Convex cutoff interpolation with the lower constant stays strictly below b away from M. Exact pinning plus negative Hessian controls a neighborhood of M; the compact complement has a strict value gap. These conditions persist in a relative C2 neighborhood. Both determinant factors stay bounded away from zero there, so the original tilted law gives the neighborhood positive probability. Intersecting the source's full-measure Morse/distinct-value locus preserves that probability. There is no older endpoint, hence this is the actual essential class. + +10. **Positive essential mass and a superpolynomial upper bound are compatible.** The support neighborhood is constructed separately for each fixed r and no uniform probability lower bound is asserted. Its fourth derivatives may become large as the pins coalesce, exactly as the axial lemma requires. On its positive-probability essential subset the extended lifetime is infinity, so every positive essential-inclusive moment is infinite, including after conditioning on failure. This does not conflict with convergence of all finite-death moments or clipped moments. + +## Verification and boundary + +I read the finite control script and replayed it once with `python3 -B -S`. It passed all seven source identities, 54 admissible quartic controls, 18 outside-hypothesis axial negative controls and 12 normalization-exponent controls. These are arithmetic and custody observations, not the justification for the infinite-dimensional support argument, weak convergence, or uniform integrability; those were checked analytically above. + +The candidate's fixed-radius axial lemma and conditional support argument are new consumer arguments. The marked failure limit, its positive mass, the Gaussian model and the reconciled parent interfaces retain their existing assumptions. No essential-class intensity or uniquely counted replacement-bar intensity is inferred. No source file, register, graph, readiness state or external review was changed by this read. diff --git a/frontiers/lifetime_moment_boundary_20260930/PROOF.md b/frontiers/lifetime_moment_boundary_20260930/PROOF.md new file mode 100644 index 000000000..d5ee8abee --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/PROOF.md @@ -0,0 +1,179 @@ +# Actual lifetime moments: axial escape, finite-death moments, and the essential class + +30 September 2026. Additive local analytic proposal, not a parent source edit or acceptance decision. R17 claim `19650bdb-2c21-42b9-b4a9-d67cabb12d4c` / Work Event 319; exact public scope pickup on Math- #175 comment 5907574791. Root owns integration; no branch write is authorized before review. +Authors: the reader_first_independent_review OpenAI subagent and the root OpenAI agent. Root supplied the clipped-moment formulation and independently reconstructed the essential-class support argument; the subagent supplied the axial escape lemma and its finite-death uniform-integrability consequence. This is author-side technical work with zero organizational-independence credit. It does not repeat the completed exact-head reviews of the existing proofs. + +## Source boundary and current ownership + +At 08:44:53 UTC, fresh GitHub reads returned: + +- Math- #175: open draft, head `9c6a73489c5ed2465a3cf65dcc950eeeae074b05`, base `8d48d02a6a580666a879d719a9e61f460baa5f40`. Its marks/lifetime slice already has same-provider review 5363612238. Its consumed `frontiers/concave_fibre_elder_20260930/PROOF.md` is SHA256 `5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e`. +- Math- #170: open draft, head `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`, base `358f2562efbccc59f9549e5a79650e2e936395c3`. Its remaining rare-measure/marks/lifetime slice already has same-provider review 5363625339. Its consumed `frontiers/local_elder_geometry_20260930/PROOF.md` is SHA256 `f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b`. +- Parent P: `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, commit `ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f`, blob `dfed3b8d318a3ab1950957f393307733a4bef3f2`, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. + +Cached bytes were rehashed before this derivation. Exact consumed source bytes are retained under `sources/` beside this note; immutable URLs, commits, Git blob IDs, SHA256 values, byte counts, consumption roles, and every retained reading rule are in [SOURCES.json](SOURCES.json). Relevant source slices are #175 §§1,5–6; #170 §§8.1,8.7–8.8; P §§2,4–5,7–8. P §12 concerns integrated finite-bar intensity moments, which are a different estimand. The full parent identity set includes P, CAP, the congruence erratum ERR, Borel replacement BOREL, and reconciliation REC. Their reading rules remain in force; no new formal alignment or scientific acceptance is claimed here. + +The bounded question is what the existing actual failure-mark weak law implies about moments of the actual lifetime fraction. Its intentional weak scope is not a defect. The conclusion below is stronger than a generic statement that weak convergence needs an extra uniform-integrability assumption: the original pin constraints and already available all-order global moments supply that assumption for the finite-death submeasure. + +## Exact estimand + +Fix the parameters and the original pinned Gaussian law of the relevant source, in dimension 2 for #170 or a fixed dimension at least 3 for #175. Retain the reconciliation embedding restriction r0. + +Keep the original weight and full normalizer + + W_r=F_d(H_M) F_(d-1)(H_S), Z_r=E_Q W_r, + Q_r^W=(W_r/Z_r)Q_r. + +Let F_r be failure of S to be the ordinary elder partner, and let E_r be the essential event d_f(M)=-infinity. Under Q_r^W, M is a maximum; E_r is contained in F_r. Define the extended lifetime fraction + + X_r=(b-d_f(M))/(k r^3) in [0,infinity]. + +For finite deaths it is nonnegative because every path in the death definition includes M, so d_f(M)<=b. Define its clipped version C_r=min(X_r,1), setting C_r=1 on E_r. A location outside the local chart is not a reason to replace an available finite death height by a sentinel. + +Assume the source's failure-mark conclusion: r^-3 Q_r^W(F_r)->a>0 and the finite marked measures converge weakly, with isolated sentinel mass o(r^3), to mu_fail and lifetime fraction lambda(theta)=-h_*(theta)/k in (0,1). Here a=a_fail in #175 and a=alpha_1+alpha_2 in #170. Put + + m_q=integral lambda(theta)^q dmu_fail(theta), q>0. + +Thus 00, + + r^-3 E_QW[C_r^q;F_r] -> m_q, + E_QW[C_r^q | F_r] -> m_q/a. (B1) + +Proof: on finite death marks use the bounded continuous function + + t -> min(max(-t/k,0),1)^q + +of D_r=(d_f(M)-b)/r^3, and assign value 1 at the isolated essential sentinel. Its limit is lambda^q because the limiting death lies in (-k,0). The exceptional position sentinel is irrelevant to this test. Divide by r^-3 Q_r^W(F_r)->a for the second formula. This is only a corollary of the stated weak law. + +## 2. Deterministic axial escape lemma + +Let g(x)=f(x,0) on [a,e]=[-r/2,3r/2] in the embedded chart; put c=r/2. Let H be the supremum of the absolute fourth directional derivative along that segment. If + + r H <= 3 k, (A1) + +there is a path from M to a point with field value greater than b whose minimum is exactly f(S). In particular + + d_f(M)>=f(S), E_r cannot occur, and 0<=X_r<=1. (A2) + +This assertion does not say S is the actual partner. Another path can give an earlier death at a higher level. + +Proof: the exact cubic Hermite interpolant at a,c is + + p(x)=b+k r^3 [2(x/r)^3-(3/2)(x/r)-1/2]. + +It has p(a)=b, p(c)=b-kr^3, p'(a)=p'(c)=0 and p'''=12k. Since g-p has double zeros at a and c, repeated Rolle's theorem gives xi in (a,c) with g'''(xi)=12k. For every x in [a,e], + + g'''(x)>=12k-2rH>=6k>0. + +Therefore g' is strictly convex, has its exact zeros at a,c, is negative on (a,c), and is positive on (c,e]. Consequently g has its minimum on [a,e] at c. + +For clarity the endpoint exceeds b with a strict quantitative margin. The same repeated-Rolle argument for the fourth-order Hermite remainder gives, for some zeta in (a,e), + + g(e)-p(e)=g''''(zeta)(e-a)^2(e-c)^2/24. + +Here p(e)=b+4kr^3 and (e-a)^2(e-c)^2=4r^4, so + + g(e)>=b+4kr^3-Hr^4/6>=b+(7/2)kr^3>b. + +The axial path x from a to e is therefore one admissible older-endpoint path, with minimum g(c)=f(S), proving (A2). Generalized Rolle's theorem here is elementary: subtract the displayed multiple of (x-a)^2(x-c)^2 so that the difference also vanishes at e, and differentiate four times. No probabilistic estimate or hard-fibre geometry enters this lemma. + +## 3. The current parent bounds supply finite-death uniform integrability + +Take K=1+||f||_(C4) in a fixed global coordinate norm and a dimension constant C_d with H<=C_d K for every axial orientation. Define + + B_r={K>3k/(C_d r)}. + +The lemma shows that E_r and {finite death, X_r>1} are contained in B_r. The parent supplies, uniformly for small r at fixed parameters, + + Z_r/r^2 >= z_*>0, + sup_r E_Q[(W_r/r^2) K^s] < infinity for every finite s>=0. (P1) + +The second assertion follows from P(4.1), the all-order W_r/r^2 bound following P(5.3), and Holder's inequality. It retains the weight-field correlations. Its constants can also be taken common on the parent's compact b,k,R sets with k bounded away from zero. + +For every N>0, direct weighted Markov gives + + Q_r^W(B_r) + = E_Q[(W_r/r^2)1_B]/(Z_r/r^2) + <= C_N r^N. (U1) + +For a finite death the torus is connected and an older endpoint exists. Its path minimum is at least min f, so + + 0<=b-d_f(M)<=b-min f<=2||f||_infinity<=2K. + +For q>0 and any N>0, using a Markov power p>N+3q+3 in (P1), + + r^-3 E_QW[X_r^q; finite death, X_r>1] + <= C r^(-3q-3) E_Q[(W_r/r^2) K^q 1_B] + <= C_p r^(p-3q-3) + = O(r^N). (U2) + +Changing p slightly if necessary gives the stated exponent for any prescribed N. The same estimate with q+epsilon supplies the conventional sufficient criterion + + sup_(small r) r^-3 E_QW[X_r^(q+epsilon);F_r,finite death]0. (U3) + +This is an upper bound only. It does not imply the essential event has zero probability at any fixed r. + +Combining (B1), (U2) and (U3) gives the finite-death moment corollary: + + r^-3 E_QW[X_r^q;F_r,d_f(M)>-infinity] -> m_q, + E_QW[X_r^q | F_r,d_f(M)>-infinity] -> m_q/a. (MOM) + +Indeed the difference between the first numerator and the clipped numerator consists of the finite part above 1, bounded by (U2), and the clipped contribution Q_r^W(E_r), bounded by (U3). Also Q_r^W(F_r,finite)/r^3->a. These statements hold for every fixed positive q. They do not assert inverse moments (q<0), convergence uniform in all unbounded parameters, or an intensity for uniquely counted replacement bars. + +## 4. Essential-inclusive moments are nevertheless infinite in the actual model + +For every fixed sufficiently small r>0 and fixed parameter choice, the original tilted law has + + Q_r^W(E_r)>0. (S1) + +Therefore, with the literal extended-value convention, + + E_QW[X_r^q;F_r]=infinity, + E_QW[X_r^q | F_r]=infinity, q>0. (S2) + +This is a model-realized obstruction, not an abstract rare-tail example. It is compatible with the faster-than-every-power upper bound (U3). + +Proof of conditional support: P §2 gives strictly positive variance for every real Fourier mode and convergence of the Fourier expansion in C2 with its expected tail norm tending to zero. Every smooth trigonometric polynomial is in the C2 support: finitely many coefficients can fall in any prescribed neighborhood with positive probability, independently of a tail whose C2 norm is small with positive probability. Smooth trigonometric polynomials are dense in C2 on the torus (smooth approximation followed by Fourier approximation). Thus the unconditioned Gaussian field has full support in the Banach space B=C2(T^d). + +Let L:B->R^(2(d+1)) be the original value/gradient pin map and Sigma its positive-definite covariance. Define the continuous regression right inverse + + A y = Cov(f,Lf) Sigma^-1 y, so L A=Id, + T=Id-A L. + +Then T is a continuous projection of B onto ker L, and Tf is a centered Gaussian field independent of Lf. For the prescribed pin vector y, Q_r is the law of Ay+Tf. Since f has full support in B, Tf has support the closure of T(B)=ker L. Therefore Q_r has full relative C2 support on {h:Lh=y}. This uses the original pin law, not a second conditioning or a restricted normalizer. + +Proof of a stable essential typed configuration: write Delta=kr^3. Choose disjoint small embedded balls about M,S, a constant c00 so q_S=w0>0 on U. Conditional full support gives Q_r(U)>0. Thus + + Q_r^W(E_r)>=E_Q[W_r;U]/Z_r>=w0 Q_r(U)/Z_r>0. + +The prototype itself need not be globally Morse outside the pin balls: the source's almost-sure Morse distinct-value event has Q_r-measure one and intersects U with the same positive measure. There is no older endpoint on U, so d_f(M)=-infinity and E_r is indeed a failure event. No lower bound uniform in r or quantitative probability asymptotic is asserted. + +## Assessment and retained boundaries + +There is no defect in the existing weak mark claims. There is a useful author-side positive extension: the finite-death conditional law has convergence of every fixed positive moment, because an axial path and the parent's existing all-order moments already prove the needed uniform integrability. Clipped positive moments are an immediate bounded-test corollary. With the actual essential value retained as infinity, every positive moment is infinite at each r despite the vanishing essential mass. The finite-death convention must therefore be stated explicitly. + +The source failure measure and cubic mark identification remain required hypotheses. This proposal supplies neither their acceptance nor a provider-distinct review. It changes no scientific status, evidence JSON, formal-required-check contract, or historical source. No repository source was edited, no external message or review was posted, and no Lean execution or full test suite was run. Root retains integration and external coordination ownership. diff --git a/frontiers/lifetime_moment_boundary_20260930/README.md b/frontiers/lifetime_moment_boundary_20260930/README.md new file mode 100644 index 000000000..508e39633 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/README.md @@ -0,0 +1,31 @@ +# Which lifetime moments exist? + +The answer depends on whether the maximum eventually dies. For finite deaths, the existing pinned Gaussian estimates give convergence of **every fixed positive lifetime moment**. Including the essential maximum makes every positive moment infinite, even though its probability becomes smaller than every power of the pin separation. + +Let `X_r=(b-d_f(M))/(k r³)` be the actual elder lifetime divided by the prescribed pin gap, under the original weighted law `Q_r^W`. If `F_r` is pairing failure, the existing marked law has mass `a r³+o(r³)` and limiting fraction `λ∈(0,1)`. Write `m_q=∫λ^q dμ_fail`. + +| Quantity, for fixed `q>0` | Result | +| --- | --- | +| Failure moment, restricted to finite deaths | `r^-3 E[X_r^q; F_r, finite] → m_q` | +| Moment conditional on failure and finite death | `E[X_r^q | F_r, finite] → m_q/a` | +| Finite lifetime tail above the nominal gap | `r^-3 E[X_r^q; finite, X_r>1] = O(r^N)` for each fixed `N>0` | +| Essential maximum | Positive probability at each fixed small `r`; `O(r^N)` for every fixed `N>0` | +| Moment including essential lifetime `∞` | Infinite at every fixed small `r` | + +The key geometric step is short. A cubic matching the two values and two zero axial derivatives reaches a value above the maximum after passing the saddle. A fourth-derivative bound preserves that path. Thus a lifetime above the nominal gap requires an unusually large global derivative norm; the parent's all-order weighted bounds control its contribution. + +[Read the proof](PROOF.md) · [Nonauthor analytic review](NONAUTHOR_REVIEW.md) · [Exact source identities and reading rules](SOURCES.json) · [Companion radial moment results](../../reviews/radial_moment_corollaries_20260930/README.md) + +The [weighted moment and loss corollary](LOSS_COROLLARY.md), with its [separate review](LOSS_NONAUTHOR_REVIEW.md), gives a direct consequence for the full pinned law. Put `c_q=a−m_q>0`. Then the finite-death moment and the moment conditional on finite death both have expansion `1−c_q r³+o(r³)`. The bounded nonnegative loss `E[1−min(X_r,1)^q]` has expansion `c_q r³+o(r³)`. These use actual pairing success, on which `X_r=1` exactly; their finite-death convention is essential. + +These are conditional mathematical consequences of the frozen parent estimates and the marked-law premises from PRs [#170](https://github.com/d6g8k5htny-coder/Math-/pull/170) and [#175](https://github.com/d6g8k5htny-coder/Math-/pull/175). Their review and acceptance obligations remain. The analysis uses actual elder pairing, the full weight and normalizer, and fixed parameters. It proves neither inverse moments nor an intensity for uniquely counted replacement bars. + +## Reproduce the finite checks + +From the repository root, using Python 3.11: + +```sh +python3 -B -S frontiers/lifetime_moment_boundary_20260930/verify.py +``` + +This checks exact rational examples, negative controls, both Python optimization modes, and frozen source bytes. It does not replace the analytic proof, the separate nonauthor review, or the repository's required hosted formal gate. The proof's author-stage operational notes are preserved with its reviewed bytes; publication state belongs to the PR and delivery handoff. Scientific effect: NONE. diff --git a/frontiers/lifetime_moment_boundary_20260930/SOURCES.json b/frontiers/lifetime_moment_boundary_20260930/SOURCES.json new file mode 100644 index 000000000..46a791e33 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/SOURCES.json @@ -0,0 +1,85 @@ +{ + "repository": "d6g8k5htny-coder/Math-", + "review_independence_credit": 0, + "scientific_effect": "NONE", + "scope": "Additive author-side lifetime moment boundary: axial escape, clipped and finite-death moment corollaries, positive-probability essential obstruction. 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the new axial escape lemma does not use it.", + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md" + } + ] +} diff --git a/frontiers/lifetime_moment_boundary_20260930/check_exact.py b/frontiers/lifetime_moment_boundary_20260930/check_exact.py new file mode 100644 index 000000000..a15e2b4df --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/check_exact.py @@ -0,0 +1,97 @@ +#!/usr/bin/env python3 +"""Finite rational controls and source-identity checks, not analytic proof acceptance.""" +from fractions import Fraction as F +from pathlib import Path +from itertools import product +import hashlib +import json + +HERE = Path(__file__).resolve().parent + +def require(ok, message): + if not ok: + raise RuntimeError(message) + +def add(a, b): + return [(a[i] if i < len(a) else F(0)) + (b[i] if i < len(b) else F(0)) for i in range(max(len(a), len(b)))] + +def mul(a, b): + result = [F(0)] * (len(a) + len(b) - 1) + for i, x in enumerate(a): + for j, y in enumerate(b): + result[i + j] += x * y + return result + +def der(p): + return [i * p[i] for i in range(1, len(p))] + +def at(p, x): + return sum(c * x ** i for i, c in enumerate(p)) + +def cubic(b, k, r): + return [b-k*r**3/2, -3*k*r**2/2, F(0), 2*k] + +source_inventory = json.loads((HERE / 'SOURCES.json').read_text()) +source_checks = [] +proof_text = (HERE / 'PROOF.md').read_text() +for source in source_inventory['sources']: + raw = (HERE / source['local_path']).read_bytes() + sha256 = hashlib.sha256(raw).hexdigest() + blob = hashlib.sha1(f'blob {len(raw)}\0'.encode() + raw).hexdigest() + require(len(raw) == source['bytes'], f"size {source['id']}") + require(sha256 == source['sha256'], f"SHA256 {source['id']}") + require(blob == source['git_blob'], f"Git blob {source['id']}") + if source['id'] in {'P', 'MARK_2', 'MARK_D'}: + require(sha256 in proof_text, f"stated source hash {source['id']}") + source_checks.append({'id': source['id'], 'bytes': len(raw), 'sha256': sha256, 'git_blob': blob}) + +quartic_cases = 0 +negative_controls = 0 +for r, k, b in product([F(1,8), F(1,100)], [F(1,3), F(1), F(7,2)], [F(-4), F(0), F(5)]): + a, c, e = -r/2, r/2, 3*r/2 + p = cubic(b, k, r) + require(at(p,a) == b and at(p,c) == b-k*r**3, 'cubic values') + require(at(der(p),a) == 0 and at(der(p),c) == 0, 'cubic gradient pins') + require(der(der(der(p))) == [12*k], 'third derivative normalization') + require(at(p,e) == b+4*k*r**3, 'cubic escape value') + require((e-a)**2*(e-c)**2 == 4*r**4, 'Hermite remainder factor') + bubble = mul([-a, F(1)], [-c, F(1)]) + bubble = mul(bubble,bubble) + for t in [F(-1), F(0), F(1)]: + coeff = t*k/(8*r) + g = add(p, [coeff*v for v in bubble]) + H = abs(der(der(der(der(g))))[0]) + require(r*H <= 3*k, 'good fourth-derivative condition') + require(at(g,a) == b and at(g,c) == b-k*r**3, 'quartic exact values') + require(at(der(g),a) == 0 and at(der(g),c) == 0, 'quartic exact gradients') + require(der(g) == mul([-r*r/4,F(0),F(1)],[6*k,4*coeff]), 'exact derivative factorization') + require(6*k - 6*abs(coeff)*r >= 21*k/4, 'positive derivative factor throughout axial segment') + require(at(g,e) >= b+F(7,2)*k*r**3, 'escape endpoint bound') + require(12*k-2*r*H >= 6*k, 'strict convexity margin') + quartic_cases += 1 + # Dropping the derivative condition is not harmless, even with the exact pins + # and the correct axial max/min types. This is only an axial negative fixture. + coeff = -2*k/r + g = add(p, [coeff*v for v in bubble]) + require(at(g,e) == b-4*k*r**3, 'negative fixture endpoint') + require(r*abs(der(der(der(der(g))))[0]) == 48*k, 'negative fixture outside hypothesis') + require(at(der(der(g)),a) < 0 < at(der(der(g)),c), 'negative fixture axial types retained') + negative_controls += 1 + +power_cases = 0 +for q, desired in product([F(1,2),F(1),F(2),F(29,4)], [F(1),F(4),F(10)]): + markov_power = 3*q + 3 + desired + 1 + require(markov_power - 3*q - 3 == desired + 1, 'weighted tail exponent') + power_cases += 1 + +print(json.dumps({ + 'status': 'PASS', + 'scope': 'Exact finite algebra controls, hypothesis-sensitive negative fixtures, normalization exponents, and source identities only; no Gaussian simulation, infinite-dimensional proof certification, or scientific acceptance.', + 'source_identity_checks': source_checks, + 'good_quartic_cases': quartic_cases, + 'outside_hypothesis_axial_negative_controls': negative_controls, + 'normalization_exponent_cases': power_cases, + 'proof_sha256': hashlib.sha256((HERE/'PROOF.md').read_bytes()).hexdigest(), + 'sources_sha256': hashlib.sha256((HERE/'SOURCES.json').read_bytes()).hexdigest(), + 'runner_sha256': hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), +}, indent=2, sort_keys=True)) diff --git a/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py b/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py new file mode 100644 index 000000000..630938818 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/check_loss_exact.py @@ -0,0 +1,52 @@ +#!/usr/bin/env python3 +"""Exact finite probability bookkeeping; not a Gaussian-model simulation.""" +from fractions import Fraction as F +from itertools import product +from pathlib import Path +import hashlib +import json + +HERE = Path(__file__).resolve().parent +FROZEN = '3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8' + +def require(ok, message): + if not ok: + raise RuntimeError(message) + +require(hashlib.sha256((HERE/'PROOF.md').read_bytes()).hexdigest() == FROZEN, 'frozen proof changed') +cases = 0 +for t,a,lam,q,k in product([F(1,100),F(1,1000)],[F(1,3),F(3,4)],[F(1,4),F(2,3)],[1,2,3],[F(1,3),F(2)]): + # Four abstract atoms: success X=1, ordinary failure X=lam, + # rare finite long lifetime X=2, essential X=infinity. + # t represents r^3 for the algebra. These probabilities are finite + # normalization fixtures, not realizations or tail rates of the model. + failure = a*t + essential, long = t**3, t**4 + ordinary = failure-essential-long + success = 1-failure + require(min(ordinary,essential,long,success) > 0, 'positive atom probabilities') + require(ordinary+essential+long+success == 1, 'full probability normalization') + finite_moment = success + ordinary*lam**q + long*2**q + clipped_moment = success + ordinary*lam**q + long + essential + genuine_loss = ordinary*(1-lam**q) + signed_deficit = ordinary*(1-lam**q) + long*(1-2**q) + coefficient = a*(1-lam**q) + remainder = -essential*lam**q + long*(2**q-lam**q) + require(finite_moment == 1-coefficient*t+remainder, 'success/failure moment decomposition') + require(genuine_loss == 1-clipped_moment, 'bounded actual loss') + require(signed_deficit == 1-essential-finite_moment, 'finite signed deficit') + require(genuine_loss-signed_deficit == long*(2**q-1), 'rare long-tail sign retained') + conditional = finite_moment/(1-essential) + require(conditional-(1-coefficient*t) == (remainder+essential*(1-coefficient*t))/(1-essential), 'finite-conditioning denominator') + physical_moment = success*(k*t)**q + ordinary*(k*t*lam)**q + long*(2*k*t)**q + require(physical_moment == (k*t)**q*finite_moment, 'physical lifetime power scaling') + cases += 1 + +print(json.dumps({ + 'status':'PASS', + 'rational_probability_cases':cases, + 'scope':'Finite normalization, clipped versus signed loss, and deterministic physical scaling only; analytic premises and model-realized probabilities are not tested.', + 'frozen_proof_sha256':FROZEN, + 'loss_corollary_sha256':hashlib.sha256((HERE/'LOSS_COROLLARY.md').read_bytes()).hexdigest(), + 'runner_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), +},indent=2,sort_keys=True)) diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md b/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md new file mode 100644 index 000000000..fe9b9ce49 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/BOREL.md @@ -0,0 +1,147 @@ +# D1 Section 9 repair — Borel elder mark in weighted pair Kac–Rice + +**Object:** D1-SECTION9-BOREL-REPAIR-20260925-v1.1 +**Author lane:** OpenAI / ChatGPT +**Parent:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, Drive file 1foDgiDi4XIKOfbrZEE8dWKV_BkZU8LIb, 40261 B, SHA256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7. +**Disposition:** AUTHOR-SIDE AMENDMENT to Section 9 only; exact-source nonauthor re-review required. +**Scientific effect:** NONE. The parent file is unchanged. Theorem A/C, the cap theorem, residual independence, normalizer, and coefficient (15.2) are not accepted by this amendment. + +## 1. Review finding consumed — and one citation correction + +The nonauthor review on main PR112 correctly identified the load-bearing write-up gap: Section 9 names a monotone-class passage from continuous cylinder weights to the global Borel elder mark but does not explicitly construct the regular conditional law on the whole C^2 field space or state the closure hypotheses of the determining class. + +The review's theorem-number statement is not correct for arXiv:2304.07424v3. Primary-source inspection of the exact v3 PDF/HTML gives: + +- Theorem 2.2: Gaussian unweighted Kac–Rice under C^1 paths and positive-definite point covariance; compact domains have finite expected level-set measure. +- Theorem 6.1: Crofton's formula for rectifiable sets. +- Theorem 7.1: Expected integral on the level set, i.e. the weighted formula for a nonnegative weight g(t,Z(.)), under lower-semicontinuity in the location and mark and continuity of the conditional mark law. Its displayed formula is (7.2). + +Therefore the parent was right to point to Theorem 7.1 for the continuous-cylinder base case. It was also right not to apply Theorem 7.1 directly to the elder indicator: the elder equality/path mark is Borel but need not be lower semicontinuous. The repair below makes the author's separate finite-measure/monotone-class extension explicit. + +Primary source inspected: Armentano–Azaïs–León, *On a general Kac–Rice formula for the measure of a level set*, arXiv:2304.07424v3 (5 Dec 2023), especially Theorems 2.2 and 7.1. No priority claim. + +## 2. Exact replacement for parent Section 9 + +### 9. A source-bound weighted Kac–Rice interface for the global elder mark + +Fix a compact ordered-pair domain D contained in the off-diagonal set of X x X, with positive distance from the diagonal. Write + + G(x,y) = (grad f(x), grad f(y)) in R^(2d). + +By the positive Fourier spectrum and distinct-site distribution argument of parent Section 2, Cov G(x,y) is positive definite at every pair, uniformly on D. The sample field is smooth. The derivative of G with respect to (x,y) is block diagonal, so its normal Jacobian is + + Delta(x,y) = |det H_x det H_y|. + +Armentano–Azaïs–León v3, Theorem 2.2, therefore gives the unweighted zero-count formula for G=0 on D, and finiteness on compact D. We use their Theorem 7.1 only for bounded nonnegative continuous cylinder weights, for which its lower-semicontinuity hypotheses hold automatically. The Borel elder mark is reached by the independent finite-measure argument below, not by claiming it is lower semicontinuous. + +#### 9.1 A regular conditional law on the whole C^2 field + +Let E = C^2(X), equipped with its usual separable Banach topology on the compact torus. For t=(x,y) in D and z in R^(2d), finite-dimensional Gaussian regression gives a smooth conditional mean m_(t,z) and a centered residual Gaussian field R_t such that, as a random element of E, + + law(f | G(t)=z) = law(m_(t,z) + R_t). + +The residual can be constructed from the full Gaussian field by subtracting + + Cov(f,G(t)) Cov(G(t))^(-1) G(t). + +Uniform nonsingularity on D and smooth covariance imply that this subtraction is continuous in (t,z) in every finite C^q mean norm needed here. In particular, + + K(t,z,A) = P{m_(t,z) + R_t in A}, A in Borel(E), + +is a regular conditional probability kernel on E, and for every bounded continuous cylinder functional F, the map + + (t,z) -> integral F(phi) K(t,z,dphi) + +is continuous. This is the whole-field version required below; it is not merely a family of conditional laws for a finite list of cylinder coordinates. + +#### 9.2 Two finite measures on D x E + +On the almost-sure zero set of G, define + + mu(A) = E sum_{t in D : G(t)=0} 1_A(t,f), + +for A in Borel(D x E). Theorem 2.2 makes mu finite. + +Define a second finite Borel measure + + nu(A) = integral_D p_(G(t))(0) + E[ Delta(t) 1_A(t,f) | G(t)=0 ] dt. + +The conditional expectation is defined by the kernel K(t,0,.). Its total mass equals the finite unweighted Kac–Rice integral, so nu is finite. + +For every bounded nonnegative continuous cylinder function h(t,phi), Theorem 7.1 (or equivalently the ordinary Gaussian regression proof of its continuous case) gives + + integral h dmu = integral h dnu. (9.1) + +#### 9.3 Cylinder functions generate the full Borel sigma-algebra + +Choose a countable dense set Q in X. The maps + + phi -> partial^alpha phi(q), q in Q, |alpha| <= 2, + +together with a countable coordinate basis on D, separate points of D x E. Because the C^2 norm on the compact torus is recovered from suprema over the dense set for derivatives through order two, these evaluations generate Borel(D x E). + +Let H be the bounded real Borel functions h for which (9.1) holds. Since mu and nu are finite, H + +1. is a vector space, +2. contains the constants, +3. is closed under bounded pointwise convergence by dominated convergence. + +It contains the algebra of bounded continuous cylinder functions. The functional monotone-class theorem therefore implies that H contains every bounded Borel function on D x E. + +This is the missing closure. It does not extend Theorem 7.1's lower-semicontinuous hypothesis by assertion; it first obtains equality of two finite measures on a generating algebra and then identifies those measures on their full Borel sigma-algebra. + +#### 9.4 Apply the extension to the elder/type mark + +Parent Section 8 represents the maximin connection event using countably many strict rational polygonal-path tests in finitely many torus charts. Hence + + (t,phi) -> 1{ d_phi(x) = phi(y) } + +is Borel on the Morse distinct-critical-value locus, with the essential-maximum convention preserved. Hessian index indicators are Borel functions of the 2-jet. Their product with the elder mark is therefore a bounded Borel weight on D x E, so (9.1) applies. + +No local cap event is substituted for the global mark. The cap theorem is used later only to estimate the probability that this global elder mark fails. + +Disintegrate the two endpoint heights using their nondegenerate joint Gaussian density conditional on G(t)=0. At prescribed heights + + f(M)=b, f(S)=b-k r^3, + +the Jacobian Delta = |det H_M det H_S| and the type indicators give exactly the parent endpoint weight W_r. Thus candidate pair intensity contains the full-pin density times E_Q W_r; the selected intensity contains the same factor times the Q^W global elder probability p_r. There is no second determinant factor. + +Finally exhaust D toward the diagonal. The parent Section 10 bound is locally r dr, hence integrable at zero; nonnegative monotone convergence gives the near-diagonal formula. This last step imports the parent Sections 3/5/10 estimates and does not re-prove or accept them here. + +## 3. What this repair does and does not close + +Repaired if a nonauthor reviewer accepts this successor: +- exact theorem attribution for the continuous-cylinder base; +- regular conditional law on C^2; +- finite-measure formulation; +- functional monotone-class hypotheses; +- Borel product-space status of the elder/type mark. + +Still separate: +- contact covariance; +- residual-field independence in parent Section 4; +- typed boundary integration; +- full normalizer; +- Theorem A; +- off-diagonal/unbounded-mark domination; +- coefficient (15.2); +- numerical constants; +- RN/24-jet certificates; +- every authoritative scientific-status gate. + +## 4. Review request + +Review this exact file, not the parent prose by memory. Return: +- C1: primary-source theorem attribution; +- C2: C^2-valued regular conditional Gaussian kernel; +- C3: finiteness and equality of mu,nu on continuous cylinders; +- C4: Borel generation / functional monotone class; +- C5: elder/type mark measurability and determinant accounting; +- C6: scope boundary. + +Disposition per interface: ACCEPT / AMEND / COUNTEREXAMPLE / BLOCKED. The reviewer should explicitly state whether the original PR112 theorem-number objection is withdrawn, retained with a primary-source citation, or replaced by a narrower hypothesis objection. + +## 5. Source-integrity note + +The immediately preceding v1 commit on this branch was created through a string-serialization path that stripped several backslashes from LaTeX notation. It is preserved in Git provenance but is not a valid review target. This v1.1 file deliberately uses plain-text equations to remove that serialization ambiguity. Review only the exact current head and digest. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md b/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md new file mode 100644 index 000000000..0633aca3c --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/CAP.md @@ -0,0 +1,230 @@ +# A weaker separating-cylinder criterion, dimension-independent geometry, and an improved SIDE24 cubic bound + +Object: **MARKED-CYLINDER-CAP-20260924-v1**. Author: OpenAI / ChatGPT, 24 September 2026. +Disposition: author-side proof candidate. Ordinary mathematical argument below; exact finite controls accompany it. No independent acceptance or formal verification is claimed. +Tasks: CAP-01 and CAP-02, GitHub campaign #61. This is an additive successor to #57/#58, not a change to their historical statements. + +## 1. Results and exact division of scope + +There are two results, not one unqualified higher-dimensional Gaussian theorem. + +**Deterministic theorem.** Let d>=2 and use coordinates (x,y) in R x R^(d-1) on an embedded product cylinder + + D=[-2r,2r] x closed_ball(0,2r), r>0. + +Suppose f is C4 on a neighborhood of D, has critical pins M=(-r/2,0), S=(r/2,0), with f(M)=b and f(S)=b-kappa*r^3, kappa>0. For j=3,4 define the partial-block norm + + M_j=max_{a+c=j} sup_D || partial_x^a D_y^c f ||_op. + +For c=0 this is absolute value. For c>=1 the operator norm is the supremum on c transverse Euclidean unit vectors, independently chosen. In dimension two this is exactly the maximum of the coordinate partial derivatives used by #57, not the larger full-tensor operator norm. Put + + lambda=lambda_min(-D_y^2 f(M)). + +The explicit sufficient conditions are + + lambda > (4/(3*kappa))*r*M_3^2, + r*M_4 <= 3*kappa/10. (1) + +They give a unique transverse-critical ridge on D and a closed cap through S separating M from every older point. On a compact manifold, for a global Morse extension with distinct critical values, the ordinary superlevel elder death partner of M is S. Exactly one ascending unstable branch of S ends at M, for a smooth Riemannian gradient. No Morse-Smale assumption and no identification of the ridge with a flow trajectory are used. + +**Probabilistic corollary, narrower scope.** Retain EXACTLY the fixed-x-axis 2D normalized SIDE24 six-pin law of #57/#58: b=6/5, kappa=1/6, 08*r*m^2, r*n<=1/20. (4) + +The exact critical pins give the Hermite identity + + f(S)-f(M)=-(1/2) integral_{-r/2}^{r/2} + (x+r/2)(r/2-x) f_xxx(x,0) dx. + +The positive kernel has integral r^3/6, so its weighted average of f_xxx is 2. In particular m>=2 and some x0 in the pin interval satisfies f_xxx(x0,0)=2. This forced lower bound on m is useful; replacing m by the larger 1+m is unnecessary. + +The partial-block norms control increments by |Delta x|+||Delta y||, with no factor depending on dimension. Every point of D is within this product distance <5r of (x0,0), and likewise <5r from M. Thus + + f_xxx >= 2-5rn >= 7/4, + D_y^2 f <= -(lambda-5rm) I = -delta I, (5) + delta>rm(8m-5), 8m-5>=11. + +The Hessian inequality is in the quadratic-form order throughout D. A scalar lower bound on a diagonal entry cannot substitute for this matrix inequality when d>2. + +## 3. A vector ridge exists globally across the cylinder + +Let w(x)=grad_y f(x,0). It vanishes at the two pins and ||w''||<=m. Applying the scalar two-node interpolation remainder to every unit-vector projection of w gives, including outside the pin interval, + + ||w(x)|| <= (m/2)|x^2-r^2/4| <= (15/8)mr^2 <2mr^2. (6) + +The bound outside the nodes follows from the interpolation remainder on the convex hull of the three points, not from an unjustified inside-only estimate. + +Also integral_{-r/2}^{r/2} w'(t)dt=0. Subtract this average from w'(x), use the Lipschitz constant m, and integrate |x-t| to obtain + + ||w'(x)|| <= (m/r) integral_{-r/2}^{r/2}|x-t|dt + <=2mr, |x|<=2r. (7) + +This vector average argument does not posit a common point at which every component of w' vanishes. + +For fixed x, f(x,.) is delta-strongly concave on the transverse ball. On its boundary ||y||=2r, + + grad_y f(x,y).y <= ||w(x)|| ||y|| -delta||y||^2 <0. + +The maximum on the compact ball is therefore interior (moving inward increases the value at a boundary point). Strict concavity makes it unique. Call it h(x). It solves grad_y f(x,h(x))=0, has h(+-r/2)=0 and is C3 by the implicit function theorem. Strong monotonicity gives + + ||h(x)|| <= ||w(x)||/delta <2r/k, k=8m-5>=11. (8) + +Its derivative satisfies + + h'=-(D_y^2 f)^(-1) partial_x grad_y f. + +Equations (7),(8) and the mixed derivative bound give + + ||h'|| < (2rm+m||h||)/delta + <2/k+2/k^2=:u, u<=24/121. (9) + +Since m=(k+5)/8, + + m*u=(1+6/k+5/k^2)/4 <=48/121. (10) + +All these inequalities use transverse operator norms; there is no hidden dimension-dependent coordinate sum. + +## 4. The correct ridge derivative and all cap boundary faces + +Set g(x)=f(x,h(x)), F=g'=f_x(x,h(x)). With v=(1,h'), differentiating the transverse critical equation gives (H_f v)_y=0. In the third derivative of g, every term involving h'' is therefore zero. The exact identity is + + F''=f_xxx +3 f_xxy[h'] +3 f_xyy[h',h'] + + f_yyy[h',h',h']. (11) + +This is valid for a vector h. It does not assert that the ridge is an integral curve of the gradient. + +Using (9),(10), the adverse part is at most + + (48/121)[3+3*(24/121)+(24/121)^2] + =2554128/1771561. + +Consequently + + F'' > 7/4-2554128/1771561 + =2184415/7086244 >1/4. (12) + +The bound is weaker than #58's F''>1, but it is already sufficient for an older connecting endpoint. This is the second source of the improved depth condition. + +F has the prescribed zeros a=-r/2,c=r/2. Strict convexity makes them its only zeros, with F negative between and positive outside. Every critical point in D lies on the ridge, hence D contains precisely M and S. The Schur complement of D_y^2 f at these points is F'(a)<0 and F'(c)>0, proving that M is a maximum and S has exactly one positive Hessian direction. + +Define + + C=[-2r,r/2] x closed_ball(0,2r). + +Strong concavity gives f(x,y)<=g(x) and g(x)<=b on C. To control the longitudinal faces, subtract (x-a)(x-c)/8 from F. The difference is convex with zero values at a,c, hence nonnegative outside [a,c]. Integrating on either exterior interval gives + + g(-2r) <= b-(9/32)r^3 = s-(11/96)r^3, + g( 2r) >= s+(9/32)r^3 = b+(11/96)r^3, (13) + +where s=b-r^3/6. These are lower/upper inequalities at the actual ridge endpoints, not Taylor fits. + +For a transverse boundary point ||y||=2r, (5),(8) imply + + f(x,y) <=g(x)-(delta/2)||y-h(x)||^2 + rm(8m-5)>=22r and ||y-h||>20r/11. This covers the entire curved side of the cylinder, including its intersections with the end faces; in dimension two it covers both horizontal edges and their corners. + +On the right cap face x=c, h(c)=0, so + + f(c,y) <= s-(delta/2)||y||^2, + +with equality at S only. On the left face use f<=g and (13). Thus every boundary exit from C has value <=s. + +Finally, the ridge segment a<=x<=2r starts at M, passes through S and ends at z=(2r,h(2r)), where f(z)>b by (13). Its minimum is exactly s. A path through S at level s is not a preemption path staying strictly above s. + +## 5. Global elder pairing, actual branches, and mark scaling + +Every point with value >b lies outside C. Every continuous path from M to such a point must first intersect the boundary of C, so its minimum is <=s regardless of later remote excursions or reentry. The preceding ridge path has minimum exactly s and endpoint >b. Hence the maximin connection level to an older point is s. + +For a compact Morse function with distinct critical values, ordinary superlevel-component persistence identifies this maximin value with M's elder death level; the older endpoint excludes the essential/global-maximum case. The unique critical value s identifies S. This is the same separating-cap criterion established in #58, now with all its boundary premises re-proved under the weaker event. + +At S the Hessian is negative definite on the transverse hyperplane. The unstable eigenvector for gradient ascent has nonzero x component: its Hessian quadratic form is positive, whereas that hyperplane is strictly negative. This argument also holds for a smooth positive-definite Riemannian metric. One half-branch enters the cap at height >s; the other enters outside. The first cannot exit because its height increases and every boundary value is <=s. Its omega-limit is a critical point in the compact cap. Gradient monotonicity and finiteness of the critical set make this limit a single point; it cannot be S, so it is M. The other branch cannot enter C or tend to its interior point M. Exactly one branch ends at M. + +For general kappa>0 apply the normalized theorem to f/(6*kappa). Its pins have gap r^3/6 and arbitrary birth b/(6*kappa). Its derivative norms and transverse curvature are divided by 6*kappa. Condition (4) becomes exactly (1). The geometric ridge is unchanged. The reduced third derivative is >3*kappa/2, the longitudinal drop/rise is at least (27/16)*kappa*r^3, and the excess over the specified gap is (11/16)*kappa*r^3. The radial drop exceeds (26400/121)*kappa*r^3. + +An arbitrary orthonormal frame can be used in this pathwise argument as long as its product chart is embedded. Scaling or rotating the function here is not a proof that SIDE24's conditional covariance, pin energy or normalizer are invariant under that operation. + +## 6. What is imported for the 2D probability calculation + +Use the fixed-axis Q,Q^W law from Section 1 and the same random-variable coupling as #57 Sections 5 and 7. The following are retained premises, not newly rerun numerical programs: + +* Z>=(17/10)r^2, six-pin regression energy<8, and ||det H_M det H_S||_2/Z<23/5. +* q=f_yy(M) has density <=1/3 and ||q||_8<5. On typed support q<0. +* There is T>=1, JOINTLY independent of q, with ||T||_8<320 and M_3<=T+8|q|; also ||M_4||_8<340. +* In the SAME coupling, at q=-ru, the full typed weight satisfies the pointwise bound + + W<=r^4[T^2*u^2+3*T^3*u], u>=0. (15) + + Internal correlations among the residual field, axial/mixed variables and transverse innovation are retained. Independence is only from q. The coefficient 3 includes the saddle's mixed-square contribution. +* The pinned-genericity candidate #49 supplies the full-probability Morse/distinct-value locus for each fixed r. No simultaneous null-set statement over all parameters is imported. + +These facts are read from exact source bytes and their explicit chain. Hash verification does not establish their truth; H3 and LPW interval programs were not rerun by this delivery. The stronger geometry does not depend on #56's endpoint spectral candidate. + +## 7. The smaller exceptional layer and its retained far branch + +Set G_new={-q>8r M_3^2, rM_4<=1/20}. Section 2-5 give global pairing on G_new. Unlike an inference from a local-event probability, this uses a direct bound on G_new's complement. + +Put lambda=-q>0 on typed support. Failure of the depth condition implies + + lambda<=8r(T+8lambda)^2 + <=16rT^2+1024r lambda^2. + +Thus the failure lies in + + {01/(2048r)}. (16) + +The large-lambda branch is essential to the quadratic inequality and has NOT been discarded. + +For the near event, disintegrate on q independent of T. Use its density cap 1/3, dq=r du, and (15): + + E[W 1_near] <= (r^5/3) E integral_0^(32T^2) + [T^2 u^2+3T^3 u]du + = (r^5/3)[(32^3/3)E T^8+(3*32^2/2)E T^7]. (17) + +Division by the full Z and the moment bounds gives C3_new*r^3. Weighted Cauchy-Schwarz, INCLUDING its square root, and eighth-moment Markov give + + Q^W{lambda>1/(2048r)} < (23/5)(10240r)^4, + Q^W{M_4>1/(20r)} < (23/5)(6800r)^4. (18) + +Together (17),(18) prove (2),(3) at the imported premises. The same single complement controls nonadjacency and the event that both unstable branches end at M; no independence or equality of these events off G_new is asserted. + +The earlier LPW lower premise, when separately consumed, is unaffected: c_LPW=13/468574870385996070912000 and r<=1/356352 still compose with this upper bound for two-sided cubic order in the one fixed-axis law. This delivery does not re-establish that lower premise or give a limiting coefficient. + +## 8. Exact comparison and remaining work + +The predecessor constants from #57/#58 are + + C3_old=(10/51)[(256^3/3)320^8+(3*256^2/2)320^7], + C4_old=(23/5)[81920^4+6800^4]. + +Exact arithmetic checks C3_old>500*C3_new, C4_old>500*C4_new and + + C3_new+C4_new/20 <2.4*10^23. + +Their absorbed-constant ratio is about 511.803 (display only; the comparison >500 is rational). The tighter good event is strictly weaker than the old sufficient event; neither claims to characterize all successful pairs. + +For the three-dimensional probability program the deterministic matrix/ridge obstacle now has a concrete candidate solution, but the probability of a small transverse eigenvalue and intersections of matrix boundary strata remains to be proved under the correct eight-pin law. Full-annulus expected counts, 24-jet modulus certificates, all-angle/all-mark estimates and lifetime integration remain distinct targets. Small failure probability alone does not bound a count on the failure event. + +Source identities and external reconnaissance are in SOURCE_OBSERVATIONS.json and RECONNAISSANCE.md. Tests check exact algebra, finite examples and deliberate mutations, not arbitrary continuous fields or the analytic imported chain. Review should focus on the vector average in (7), all partial-block norms, the h'' cancellation, entire radial side, marked scaling, and the independence/density/normalizer powers in (17). diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md b/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md new file mode 100644 index 000000000..213594d6c --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/ERR.md @@ -0,0 +1,44 @@ +# Additive erratum — endpoint Hessian congruence in UNIFORM_MATRIX_CAP_AND_LIFETIME + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1-ERRATUM-1 +**Scientific effect:** correction of a displayed linear-algebra sentence only; no theorem promotion. + +Source object: `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, raw SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. + +## Correction + +In Section 5, immediately after the convergence of endpoint Hessians, the source says: + +> Here the congruence uses `diag(sqrt(r),I)`, so its off-diagonal entries are `sqrt(r) beta_i`. + +Given the source's own definition + + H_i = [[r alpha_i, r beta_i^T], + [r beta_i, A_i]], + +that displayed congruence factor is incorrect. The matrix used by the argument is + + D_r = diag(r^(-1/2), I), + +because + + D_r H_i D_r + = [[alpha_i, sqrt(r) beta_i^T], + [sqrt(r) beta_i, A_i]]. + +It also satisfies + + det(D_r H_i D_r) = det(H_i)/r, + +which is exactly the determinant normalization used in Sections 5–7. + +With `alpha_M -> -6k`, `alpha_S -> 6k`, bounded `beta_i`, and `A_i -> A_0`, this corrected congruence gives + + D_r H_M D_r -> diag(-6k,A_0), + D_r H_S D_r -> diag( 6k,A_0). + +No formula (5.3), (5.4), (5.5), or the subsequent soft-factor/eigenvalue integration is changed by this correction. The original file is retained byte-for-byte for provenance; consumers of its Section 5 type-convergence argument must read this erratum with it. + +## Review requirement + +Because this correction touches the load-bearing full-normalizer/type-convergence interface A3, the pending nonauthor A1–A7 review on main #63 should review the source **together with this erratum**. A green test or obvious intended meaning is not acceptance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md new file mode 100644 index 000000000..ef2aa5795 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_2.md @@ -0,0 +1,433 @@ +# Planar cubic elder mark: a compact-bar transfer theorem + +Object: OA-PLANAR-LOCAL-ELDER-20260930-v1. +Authors/contributors: OpenAI root, elder_bridge_audit and its local_topology child, foreground continuation, 30 September 2026. The companion scale analysis has an additional OpenAI contributor, mixed_geometry_audit. The proof is AUTHOR_SIDE / HOLD pending qualified source-bound mathematical review. All these agents have the same provider and earn zero organizational-independence credit. Existing source headers are not treated as live acceptance decisions. Publication, source hashes and engineering checks do not establish mathematical acceptance or promote any parent claim. + +This is a new full proof candidate. Only this additive directory is written; no source body, count theorem, register, graph, numerical certificate or integration disposition is changed. Exact dependency interfaces, provenance and unresolved obligations appear below and in SOURCES.json. + +## 1. Exact source cut and estimands + +The main source cut is Math- commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751. Immutable source links and blob identities are in SOURCES.json. + +* [P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md, blob dfed3b8d318a3ab1950957f393307733a4bef3f2, §§1,8,10–14. Read with its congruence erratum (blob 213594d6ca6a86fb938110f4d166d9ce275a02d0), §9 Borel repair (blob fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a), and [REC] §1 reading rule/W1/embedding restriction. [REC] is reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md, blob 75da2597971510f843f8d90c743950cb8c177342. +* [CUB] frontiers/planar_cubic_cluster_20260929/PROOF.md, blob bb446d08db8a944537a743ad550b88c1c2ad5758, §2 (C1)–(C14), §3 Lemma S, §4 (G6)–(G10). +* [LOW] frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md, blob aaefc8da9668e819591dd9db0b34442394f2362e, §§1–4; and frontiers/elder_lower_all_d_20260929/PROOF.md, blob 7f41c9e315be0a7985d4c770a11a2cdf717bd46a, §§0,3–5. +* [TWO] frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md at PR166 head be9b7fa95de9b7e59a67347640e157ed3027f624, blob a32fd5f7d941bbe1fe943df045b1e0fbec8d691c, §§1–4, (M1)–(M2), Theorem L and (L3). +* [Q] frontiers/c6_palm_route_20260929/PROOF.md, blob 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5, §1.2 Theorem Q and §1.3 non-claims. +* [U] frontiers/unrestricted_selection_difference_20260929/PROOF.md, blob 5a55b179a974a90c65d257f3f93765cc6fb30bb8, §§1,3–5. + +For the actual pinned law Q_r^W, the pins are M_r=(-r/2,0), S_r=(r/2,0), heights b and b-kr^3, and both gradients zero. The weight is W_r=F_2(H_M)F_1(H_S), with exactly one FULL Z_r=E_Q W_r. + +N_r in [Q]/[TWO] counts **additional critical points of all indices in the open height window**, excluding the two pins. The ordered pair measure (L3) is a measure of two additional window witnesses. Its microscopic limit has two index-1 saddles. It is not a maximum/saddle persistence-pair measure. + +The elder selector in [P] is different: +d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t)), +and p_r=Q_r^W{d_f(M)=f(S)} on the Morse distinct-value locus. Its candidate maximum/saddle intensity is r A_r dr db dk d sigma with A_r=12 pi_r(Z_r/r^2); its selected intensity is r A_r p_r with the same measures. The lifetime pushforward ell=kr^3 supplies r dr/dell=ell^(-1/3)/(3k^(2/3)), [P] §§10–11. There is no adjacency normalization, no factor 1/2 and no second Z_r. + +Consequently, localization of (L3) alone does not supply the elder mark. The common leading candidate/elder lifetime density is already supplied by the reconciled D1 cap/selection/Kac–Rice chain, rather than by C6 localization. [LOW] identifies the compact-window rejected density order ell^(2/3); [U] separately shows a positive far rejected density and forbids importing that compact-window decay as an unrestricted difference rate. + +The following theorem adds an actual topological mark to the planar cubic sector. It uses neither a new count argument nor any remote witness theorem. + +## 2. Deterministic theorem + +Fix k>0 and a typed cubic from [CUB]: +P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3. +Write M=(-1/2,0), S=(1/2,0), so P(M)=0, P(S)=-k and both gradients vanish. With the determinant-one shear u=X+aZ/(12k), set +B=beta-a^2/(12k), +D=(c-a beta/(4k)+a^3/(72k^2))/2. +The typed domain is s<-|B|/2. + +Exclude the two algebraic sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0}. +Both are nonzero polynomial zero sets after clearing fixed positive k denominators, hence Lebesgue-null. The first is [CUB]'s window transition; the second is the only additional exclusion made here. + +Let C(P) be the additional critical points with -k0. Suppose their scaled functions + g_i(X,Z)=(f_i(r_iX,r_iZ)-b)/r_i^3 + converge to P in C^2 on every fixed compact disk, and retain the exact pins, gradients and values 0,-k. Then their global normalized death levels satisfy + (d_{f_i}(M_i)-b)/r_i^3 -> h_*(P). + No conditions are imposed outside the shrinking coordinate disks. +3. Continue the finitely many points of C(P) to local critical points of g_i by pinned C^2 root stability, and include the exact pinned saddle S. For every sufficiently large i the global death level is exactly the highest *actual* height in this finite continued set. On the global Morse distinct-value locus its unique highest point is the actual elder partner. If C(P) is empty, that point is exactly S_i; if C(P) is nonempty, it is a continued window saddle and S_i is not the partner. Thus the selector-failure indicator converges to 1{C(P) nonempty}. + +The theorem is planar and pointwise in the cubic and coupling realization. It makes no assertion uniform over marks, dimensions, radii or approaching exceptional sets. + +## 3. The critical-chord identity supplies the lower bound + +For any polynomial of degree at most three and any two critical points M,Y, put q(t)=P(M+t(Y-M)). Since q'(0)=q'(1)=0, interpolation gives exactly +q(t)=P(M)+(P(Y)-P(M))(3t^2-2t^3). +This also holds when the restriction has lower degree. + +Here P(M)=0 and h=P(Y)<0. On 0<=t<=2 the minimum of h(3t^2-2t^3) is h, attained at t=1; the endpoint is -4h>0. Every negative critical point therefore gives a finite path from M to a strictly higher point whose bottleneck is exactly its own critical height. + +Apply this to Y=S and to each window saddle. It follows that d_P(M)>=h_*. + +For g_i, the same fixed finite path has endpoint strictly above zero and minimum at least h_* minus an error tending to zero. Hence the global normalized death has liminf at least h_*. This path is a path in the actual surface after rescaling; no claim about its being a separatrix or an ascending branch is made. + +When C(P) is nonempty, h_*>-k, so the strict margin alone proves eventual failure of selection of S_i. + +## 4. No finite-height escape at infinity + +In the sheared coordinates the leading homogeneous cubic is +H(u,Z)=2ku^3+(B/2)uZ^2+(D/3)Z^3. +Its derivatives are +H_u=6ku^2+(B/2)Z^2, +H_Z=Z(Bu+DZ). + +A nonzero common zero requires Z nonzero. If B=0 it exists exactly when D=0; if B is nonzero it exists exactly when 12kD^2+B^3=0. Thus off Delta_k, grad H has no zero on the unit circle. Compactness and homogeneity give |grad H(z)|>=c|z|^2, and lower-order terms then give +|grad P(z)|>=c_0|z|^2 for |z|>=R_0 +after increasing R_0. The invertible shear preserves such a bound up to constants. + +There are no critical points of P with value in (h_*,0), by the definition of h_* and [CUB]'s complete classification of all window points. Near M a sufficiently high negative regular level t_0 has a small closed level oval enclosing a disk K_0. The disk contains only the strict maximum M. + +For any h in (h_*,t_0), choose a smooth height cutoff chi with 0<=chi<=1, equal to one on [h,t_0], supported in a slightly larger critical-free strip inside (h_*,0), and zero near zero. The vector field +V=-chi(P) grad P/|grad P|^2 +is smooth and globally bounded after setting it to zero outside that strip. The gradient-growth bound controls it at infinity, and compactness controls it elsewhere. Its flow is defined for the finite time t_0-h; it fixes M. The initial boundary stays where chi=1, so its P-value decreases at speed exactly one and ends at P=h. The image K_h is compact, contains M and has boundary P=h. + +Moreover P is nonpositive throughout K_h: the initial disk has P<=0 and along the flow P never increases. Thus every path from M to a point with P>0 exits K_h and has minimum at most h. Letting h decrease to h_* proves d_P(M)<=h_*. Together with §3 this proves part 1. + +This is the missing compact-bar/escape argument. Merely having finitely many window roots would not exclude a finite asymptotic critical level; the Delta_k exclusion and the gradient estimate do. + +## 5. Transfer of the upper bound to the actual global field + +Fix h with h_*0, the conic forces |u|<1/2, so this is negative. If B<0, either u>=1/2, when it is below -k, or u<-1/2; (C9) and 2s/B>1 again make it negative. For B=0, the two extra possible heights are s^3/(6D^2) and -k+s^3/(6D^2), both negative. Thus M is the only critical point at value zero or above. A second interior zero maximum of K_h would be another such critical point. Boundary h is negative. + +The exact pin M and C^2 convergence imply g_i<=0 on K_h for all sufficiently large i: in a fixed small ball around M its Hessian remains negative definite and M remains critical with value zero; outside that ball P has a strictly negative maximum on the remaining compact set, so C^0 convergence suffices. Boundary values of g_i are at most h+epsilon for all large i. + +Any actual global path to a point above birth must leave the embedded r_i K_h because there is no such point inside. It therefore crosses a boundary at value at most b+r_i^3(h+epsilon). This proves the normalized limsup is at most h+epsilon. Let epsilon->0 and h->h_* to prove part 2. + +This uses a local compact boundary to control the global maximin, so exterior connectivity cannot invalidate the bound. + +## 6. Exact selection of the highest actual continued local saddle + +Convergence of death heights alone would not identify the location or exact partner. The following finite-index strengthening supplies both. + +Let the finite set be C(P) union {S}. For each Y in it, let Y_i be the unique continued critical point, using exact S_i=S for the pinned saddle. All their values are negative for large i. They lie in the window except S, whose value is exactly -k. Define + + h_i=max{g_i(Y_i): Y in C(P) union {S}}. + +Fix t_0<0 close enough to zero to have a small maximum-level oval and to satisfy t_0>h_*. Then h_i 2k/(c_0 R_0^2)+1. + +For all large i, C^1 convergence gives |grad g_i|>=c_0|z|^2/2 on R_0<=|z|<=R+1. By [CUB] Lemma S, using exact pins and excluding Sigma_k, sufficiently small C^2 perturbations on B_(R+1) have exactly the continued roots in (-k,0), together with the exact pins at 0,-k. Small disjoint neighborhoods give one root each; the rest of the compact enlarged height band has a positive gradient floor. Thus there is no critical point in (h_i,0) inside B_(R+1). The same sufficiently large index satisfies all these statements, independently of the lower height h chosen next. + +### Exact upper bound by a common finite-disk collar + +Negative-definite Hessians and exact pinning give a small maximum-level oval for g_i at t_0 inside B_(R_0), enclosing M in a disk with g_i<=0. For each h with h_ih_i>=-k. A first passage from B_(R_0) to that boundary traverses the annulus, where speed is at most 2/(c_0 R_0^2). The chosen width makes this take more than k. Thus all boundary trajectories remain in B_R and reach height h at time t_0-h. They form a Jordan curve, and its enclosed image disk is inside B_R: the complement of B_R is connected and lies in the curve's unbounded component. The cap contains M and g_i<=0 throughout since the cutoff flow never increases height. + +Every older path therefore crosses a boundary of height h. For this SAME sufficiently large i the conclusion holds for every h in (h_i,t_0). Taking h down to h_i gives exact normalized global death <=h_i, without exchanging i and h limits and regardless of the exterior geometry. + +### Exact lower bound through each continued critical point + +For any one Y, consider the moving straight critical chord + + q_i(t)=g_i(M+t(Y_i-M)), 0<=t<=2. + +The actual exact criticality of M and Y_i gives q_i'(0)=q_i'(1)=0. C^2 convergence and Y_i->Y give C^2 convergence of these one-dimensional restrictions to + + q(t)=P(Y)(3t^2-2t^3). + +Writing h_Y=P(Y)<0, the limiting derivative is negative on (0,1), positive on (1,2), and + + q''(0)=6h_Y<0, q''(1)=-6h_Y>0. + +C^2 convergence preserves the second-derivative signs near 0 and 1. Together with the *exact* derivative zeros, this preserves the required derivative signs on those endpoint neighborhoods. C^1 convergence preserves them on the remaining compact subintervals. Hence q_i decreases strictly from t=0 to t=1, then increases strictly to t=2. Its exact minimum is g_i(Y_i), and its endpoint is positive since its limit is -4h_Y>0. + +There are finitely many roots, so the same sufficiently large index works for all these paths, including S. Selecting the root attaining h_i provides an actual older path of exact minimum h_i. Thus normalized global death >=h_i. With the collar bound it equals h_i exactly. Even when two limiting saddle heights tie and the actual winning root switches, the finite collection of paths proves the same result. + +On the actual global Morse distinct-value locus, the unique critical value h_i identifies the highest continued local saddle as the ordinary elder partner. Empty sectors select S exactly; nonempty sectors select one of the continued window saddles and preempt S. This proves Theorem E part3 and identifies the actual partner, not only its limiting height. + +### A unique limiting position for almost every jet + +For the position mark only, exclude D=0 as well. This is another nonzero polynomial zero set in the physical jet coordinates, hence Lebesgue null. When B!=0, (C9) gives + + (d/du)(P/k)=-1-(4s/B)u. + +On the B>0 window conic this derivative is strictly negative (at its largest allowed u=u_w it is 1+2s/B<0); on the B<0 window conic it is strictly positive because u<-1/2 and s/B>1/2. Equal window heights therefore force the same u. If D!=0, the line s+Bu+DZ=0 then forces the same Z, so distinct window roots have distinct heights. For B=0 there is at most one window root. Accordingly outside this extra null set there is a unique highest cubic window point Y_*(theta) when n>0, and the actual elder partner's scaled position converges to it. Use S as the default position when n=0. This position exclusion is unnecessary for the exact finite-index death and partner conclusion above. + +## 7. A compact-sector marked selection law under the actual tilt + +This corollary uses only the planar compact-sector disintegration already displayed in [CUB] §4; it does not modify its count theorem. + +For a compact Borel C in the typed domain, define the finite measure +dM_C(theta)=1_C(theta) z_0^(-1) w(theta) h_0(0,a,beta,c) dtheta, +with the full z_0 of [CUB] (G3), and define lambda(theta)=-h_*(P_theta)/k in (0,1]. + +Under the regression coupling [CUB] (G6)–(G8), Theorem E applies pathwise for almost every theta and almost every coupling realization. Define I_r=1{the global partner of M is not S} and D_r=(d_f(M)-b)/r^3. For every bounded measurable psi(theta), the marked identity +r^(-3) E_QW[psi(Theta_r) I_r; Theta_r in C] + -> integral_C psi(theta) 1{n(theta)>0} dM_C(theta) +holds. + +For every bounded continuous phi(theta,t), +r^(-3) E_QW[phi(Theta_r,D_r); Theta_r in C] + -> integral_C phi(theta,h_*(theta)) dM_C(theta). +On the exceptional essential-class event define phi arbitrarily at D_r=-infinity; each good coupling eventually has an explicit local path to an older point, so that arbitrary choice disappears from the limit. + +Proof: replace the count indicator in [CUB] (G10) by the Borel elder mark, or by the indicated bounded functional of the Borel maximin. The multiplier is +1_C(theta) (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4) times mark]. +Theorem E supplies pointwise convergence of the mark. Equations (G7)–(G9) supply the same integrable domination as before. Dominated convergence proves both statements. The first works for every bounded measurable psi because theta itself is not moved by the coupling and the failure indicator converges pointwise; this also yields total variation for the joint theta/Boolean measure on the compact sector. The real-valued death-height mark is asserted weakly, not in total variation. + +The marked corollary gives a selection law on every fixed compact soft-jet sector. Compact convergence alone does not justify a full-global coefficient: it needs tightness of the rescaled *elder-failure measure*, rather than of the nonempty witness measure. The next section proves that tightness by retaining the rare scalar layer in the parent's cap-failure integral and explicitly converting endpoint control to the actual midpoint coordinates. + +## 8. Full planar elder-failure tightness and global marked selection law + +### 8.1. Fixed model and immutable interfaces + +Fix the planar periodized Gaussian model, L>0, b in R, k>0 and one frame. Write Q_r for the exact six-pin regression at M_r=(-r/2,0), S_r=(r/2,0), heights b,b-kr^3 and zero gradients. Set +W_r=F_2(H_M)F_1(H_S), Z_r=E_Q W_r, dQ_r^W=(W_r/Z_r)dQ_r. +Let +F_r={the global ordinary superlevel elder partner of M_r is not S_r}, +Theta_r=(f_zz(0)/r,f_xxz(0),f_xzz(0),f_zzz(0))=(s,a,beta,c), +D_r=(d_f(M_r)-b)/r^3. +The essential class uses d_f=-infinity. All event definitions are Borel by parent §8 and its reconciled §9 whole-field interface. + +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at main commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751, blob dfed3b8d318a3ab1950957f393307733a4bef3f2. It is read WITH the congruence erratum, §9 Borel replacement, and reconciliation/W1/embedding rule in SOURCES.json. The consumed equations here are (3.5), (4.2)–(4.3), (5.5), (6.2), (7.5)–(7.7), and the good-cap implication in §8. + +[CUB] is frontiers/planar_cubic_cluster_20260929/PROOF.md at the same main cut, blob bb446d08db8a944537a743ad550b88c1c2ad5758. The consumed equations are (C1)–(C14), Lemma S, (G1)–(G3), regression (G6)–(G9), physical midpoint disintegration (G10), and coefficients (G11) with finiteness (G13). + +For theta=(s,a,beta,c), let P_theta be its pinned cubic, B=beta-a^2/(12k), D=(c-a beta/(4k)+a^3/(72k^2))/2, and D_k={s<-|B|/2}. Let n(theta) be the existing cubic classifier and +h_*(theta)=max({-k} union {P_theta(Y):Y is an additional window critical point}). +The function h_* is Borel: by Theorem E it is the cubic maximin, whose strict upper-level events have the same countable finite polygonal-path representation as [P] §8; each path stays in a compact plane region. +Extend it by a fixed value on the excluded polynomial sets and outside D_k when using an ambient test function. Those Borel sets have null or zero-weight contribution to the limiting failure measure, so the extension changes none of the identities. +Write +w_+(theta)=9k^2 [B-2s]_+ [-B-2s]_+. +This equals w(theta)=9k^2(4s^2-B^2) on D_k and is zero elsewhere, including its boundary. + +Define the finite candidate limit measure +dmu_fail(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{theta in D_k,n(theta)>0} dtheta. (S1) +Here z_0 is the FULL normalizer limit [CUB] (G3), and h_0 is the actual raw midpoint Gaussian jet density under the exact contact law. No GOE substitution, adjacency normalization or extra pin-density factor appears. + +**Theorem S (fixed planar selection law).** Conditional on Theorem E and the exact source interfaces above, +r^(-3) Q_r^W(Theta_r in ·,F_r) -> mu_fail +in total variation as finite measures on R^4. Consequently, +r^(-3)(1-p_r) -> alpha_1+alpha_2, (S2) +with alpha_j exactly [CUB] (G11). + +Moreover, for every bounded continuous phi on R^4 x R, +r^(-3) E_QW[phi(Theta_r,D_r);F_r] + -> integral phi(theta,h_*(theta)) dmu_fail(theta). (S3) +Give the test an arbitrary bounded value on the essential-class event; that event has o(r^3) tilted mass. Thus the height-mark assertion is weak convergence, not total variation. After division by alpha_1+alpha_2>0 it is the actual conditional law given elder failure. + +There is no claim about the unmarked full r^(-3) joint theta measure. Its typical s-coordinate escapes and its total mass diverges. + +### 8.2. Good cap excludes failure + +Let G_r be the parent's good cap. On the almost-sure Morse distinct-value locus, [P] §8 proves that the global ordinary partner is S on G_r. Hence +F_r subset G_r^c (S4) +up to a Q_r^W-null set. This includes essential maxima: the good cap supplies a finite older endpoint, so the essential class cannot occur there. + +This implication concerns the same actual global selector p_r as [P], not a new local adjacency event. It is the only place where the cap is used to contain the failure event. + +### 8.3. Rare-layer tail bound from the independent endpoint residual + +In the plane set lambda=-f_zz(M)>0 on typed support. Under [P] (4.2), the residual field g_r is independent of the scalar endpoint transverse Hessian. Write +J_r=1+||g_r||_(C4). +Its moments of every finite order are bounded along small-r sequences by (4.3). The bounded C4 regression coefficients and bounded mean in (4.2) give the pathwise inequality +||f||_(C4)<=C(J_r+lambda). (S5) +The mean and E_Q f_zz(M) terms are absorbed into C using the bounded fixed-parameter pin regression and J_r>=1. + +Let D_cap>0 be the fixed scalar depth constant from [P] (7.5). By that split, depth failure is contained in +near: 01/(4D_cap r). +On the near branch, r<=1 implies +||f||_(C4)<=C'J_r^2, M3<=C'J_r^2. (S6) + +The m=1 typed weight bound (6.2) gives +W_r<=C r^2 J_r^4 lambda(lambda+C rJ_r^2) +on this branch. The density of lambda is bounded by (3.5), and J_r is independent of lambda. Retaining a tail restriction J_r>A in the SAME integration yields +E_Q[W_r;near,J_r>A] + <= C r^2 E[ J_r^4 1{J_r>A} + integral_0^(4D_cap rJ_r^2) lambda(lambda+C rJ_r^2) d lambda ] + <= C r^5 E[J_r^10;J_r>A]. +Divide by the FULL floor Z_r>=z_*r^2 from (5.5): +r^(-3) Q_r^W(near,J_r>A) + <= C E[J_r^10;J_r>A]. (S7) +Higher moments of J_r imply the right side tends to zero as A increases, uniformly along small-r sequences. + +The far branch (7.6) and the fourth-derivative exception (7.7) each have tilted mass O(r^4), hence o(r^3). Their union with the near branch covers G_r^c on typed support. Therefore the rare-layer tail estimate applies to F_r via (S4). + +An unconditional C4 tail estimate followed by Cauchy–Schwarz would lose the rare factor. The retained scalar integral above supplies r^5 before division by the one full Z_r; this is why the tilted r^3 scale survives. + +### 8.4. Explicit endpoint-to-midpoint control + +Between M and the midpoint, +|f_zz(0)|/r <= lambda/r + M3/2 +by the mean-value derivative bound. On the near branch, +lambda/r<=4D_cap J_r^2. +The other midpoint coordinates a,beta,c are third derivatives and are bounded by ||f||_(C3). Equations (S6) therefore imply +||Theta_r||<=C J_r^2. (S8) + +No density-coordinate replacement occurs here: (S8) is a deterministic control of the actual midpoint target by the scalar endpoint residual. Together with (S7) and the o(r^3) exceptions, it proves +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||Theta_r||>T)=0. (S9) +The same proof gives +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||f||_(C4)>T)=0. (S10) + +Thus the leading failed population is ambient-midpoint-soft in the plane. Since there is only one transverse direction, bounded residual sectors force f_zz(M), f_zz(0), f_zz(S)=O(r), the last by (5.1). The axial and mixed endpoint entries are O(r) by (5.1)–(5.2), so both endpoints have the r^4 determinant-product scale. At finite r the far and fourth-derivative branches still exist; their smaller mass is retained above. + +### 8.5. Compact-box disintegration, including the typed boundary + +Fix an ambient compact box K_T in R^4. Although [CUB] states its count theorem on compact subsets of D_k, its actual regression construction (G6), moment bound (G7), Taylor convergence (G8), and determinant bound (G9) apply to ANY bounded midpoint target. They rely on the nonsingular joint covariance and bounded target, not on endpoint typing. No count convergence is invoked outside D_k. + +Under this coupling, the endpoint Hessians divided by r converge to +H_M(P)=[[-6k,-a/2],[-a/2,s-beta/2]], +H_S(P)=[[6k,a/2],[a/2,s+beta/2]]. +Their determinants are 3k(B-2s) and 3k(B+2s). Since the axial signs are fixed, the limiting typed weight is exactly +W_r/r^4 -> F_2(H_M(P))F_1(H_S(P))=w_+(theta). (S11) +Each F_j is continuous even at a degenerate matrix: its determinant factor tends to zero there. Hence (S11) holds on the typed boundary as well, with zero weight. On the whole box the integrable envelope +W_r/r^4 <= C_T(1+||F||_(C4))^4 (S12) +follows from the same endpoint Taylor bound as (G9). + +The fixed-k algebraic exceptional sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0} +are nonzero polynomial zero sets, hence null for the absolutely continuous midpoint density. No distinct-height exclusion is needed between the two window saddles. + +For almost every theta in D_k and coupling realization, Theorem E gives +1_F_r -> 1{n(theta)>0}, +D_r -> h_*(theta). +For theta outside D_k, including its boundary, no convergence of the selector is required: its bounded multiplier is killed by (S11). Thus [CUB]'s physical midpoint disintegration (G10), with the count mark replaced by the Borel global failure mark, gives the density +u_r(theta) + = (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4)1_F_r]. (S13) +The one factor r in (S13) comes from f_zz(0)=rs. It is not a spectral/endpoint change of variables. The prefactor is r^(-3)*r*r^4/Z_r=r^2/Z_r. + +By (S11)–(S12), the full normalizer limit, local boundedness of h_r, and dominated convergence in theta and F, +integral_(K_T) |u_r(theta)-u_0(theta)| dtheta -> 0, +u_0(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{n(theta)>0}. (S14) +Here define the final indicator arbitrarily off D_k, where w_+=0. + +This simultaneously handles thin typed-boundary neighborhoods and the exceptional polynomial sets; neither can carry an unidentified atom of the leading failure measure. + +### 8.6. Exhaustion gives global total variation and the coefficient + +The limiting density u_0 is integrable. It is exactly the sum of the two nonempty coefficient densities (G11), whose integrability follows from [CUB] (G13) and Gaussian polynomial tails. Its total mass is alpha_1+alpha_2, finite and strictly positive. + +Choose T large. On K_T, the L1 error tends to zero by (S14). Outside K_T, the actual density's mass is controlled by (S9), and the limiting density's mass tends to zero by integrability. Therefore +integral_(R^4) |u_r-u_0| dtheta -> 0. +This is total-variation convergence of the finite failure measures (under either conventional factor in the TV norm), proving Theorem S and (S2). + +The proof exhausts only the marked failure measure. It never integrates the divergent n=0/unmarked total soft-jet measure and never asserts an uncentered r^(-3) probability law. + +### 8.7. The normalized death mark on actual failure + +For a bounded continuous phi(theta,t), use (S13) with multiplier +1_F_r phi(theta,D_r). +On every ambient compact box the same domination applies. Theorem E gives pointwise convergence on typed generic theta to +1{n(theta)>0} phi(theta,h_*(theta)); +outside the typed domain the zero weight suffices. Dominated convergence gives the compact-box marked limit. + +The bounded test's omitted actual mass is at most ||phi||_infinity times the failure tail in (S9). The limiting omitted mass is controlled by the integrable u_0. Exhausting boxes proves (S3). + +The essential-class mark has zero scaled limit: on a good typed cubic coupling Theorem E's finite critical-chord path supplies a local point above the birth for all small r, and outside the typed domain the weight limit is zero. The compact-box domination and (S9) again exhaust the actual essential-failure mass. Thus it is o(r^3). + +Normalizing (S3) by (S2) identifies the weak actual conditional failure-height law. This is not a density result for persistence lifetimes, not a rate, and not a spatial or real-height TV assertion. + +### 8.8. Actual replacement location and the lifetime mark + +Let T_r be the actual elder partner's coordinates divided by r in a fixed embedded midpoint chart. Give it a sentinel value when the partner is outside the chart or the class is essential. Theorem E part3 proves that on every generic typed coupling the partner is eventually a continued local window saddle on failure. Outside the additional null set D=0, it is the continuation of the unique highest root Y_*(theta), so + + T_r -> Y_*(theta), D_r -> h_*(theta). + +This is an actual partner assertion; no witness factorial pair is relabelled. The functions Y_* and h_* are Borel on the generic typed domain: the finitely many nondegenerate roots continue continuously locally, and strict height comparison selects a unique root. Arbitrary Borel extensions on the excluded sets have zero limit mass. + +For every bounded continuous test phi(theta,y,t), the same compact-box domination as S13, followed by the actual failure-tail bound S9, gives + + r^(-3) E_QW[phi(Theta_r,T_r,D_r);F_r] + -> integral phi(theta,Y_*(theta),h_*(theta)) dmu_fail(theta). + +Give bounded tests arbitrary values at either sentinel. On compact good couplings those sentinel events eventually vanish, and S9 exhausts their omitted failure mass. In particular their tilted mass is o(r^3). The joint location/death assertion is weak convergence. Dividing by the positive total mass alpha_1+alpha_2 identifies the genuine conditional replacement-partner law under the original pair-Palm law given failure. + +The corresponding normalized actual lifetime on failure is + + (b-d_f(M))/r^3 -> -h_*(theta) in (0,k), + lambda(theta)=-h_*(theta)/k in (0,1). + +Its law is the pushforward of mu_fail/(alpha_1+alpha_2); thus the local geometry determines which saddle pairs and its lifetime under this specified law. This is not an intensity obtained by duplicating rejected candidates attached to the same maximum. + +The weak finite replacement-location measures have converging total masses and a finite limit on R^2, hence they are tight. Therefore for every deterministic cutoff delta_r with delta_r/r->infinity, + + Q_r^W(F_r, dist(partner(M),midpoint)>delta_r)=o(r^3). + +For the proof, first fix a large scaled radius A, use tightness outside A, and then use delta_r/r>A for all sufficiently small r; let A increase. The o(r^3) sentinel mass is included. No growing-radius uniform root enclosure is asserted or needed. This localizes the **actual failed elder pairing**; it neither needs nor promotes the contextual factorial-count localization results. + +## 9. A compact rejected-candidate lifetime coefficient + +The local-to-elder bridge is now supplied by Theorem E and the full marked failure limit S2, rather than postulated from a count theorem. One precise lifetime consequence follows by consuming the parent's *existing* compact-parameter domination; no new uniform convergence theorem is needed. + +Retain exactly [P] §§10–12's populations: candidate maximum/saddle pairs and the selected subset, per unit midpoint volume, with birth b in a compact interval B and positive gap mark k in a compact interval K, both intervals of positive lengths and K bounded away from zero, and directed orientation u in S^1 with ordinary arc measure. Write + + A_r(b,k,u)=12 pi_r(u;v_r) Z_r(b,k,u)/r^2, + A_0=12 pi_0(u;v_0) z_0(b,k,u), + a_fail(b,k,u)=alpha_1(b,k,u)+alpha_2(b,k,u)>0. + +For every fixed (b,k,u), S2 identifies + + (1-p_r(b,k,u))/r^3 -> a_fail(b,k,u). + +The source [P] (7.8), with its compact parameter hypotheses, bounds this ratio by a common finite constant for all sufficiently small r, b,k and frames in these compacts. Equations (10.2)–(10.3) bound A_r uniformly and identify its limit. These are already-displayed source bounds, not uniformity deduced from our pointwise cubic topology theorem. The conditional a_fail is Borel as the limit of Borel quantities, and is bounded by that same cap constant. + +Let nu_rej^{B,K}(ell)=nu_cand^{B,K}(ell)-nu_eld^{B,K}(ell) denote the density of the **nonselected candidate counting measure**, in the parent's canonical Kac–Rice versions. With r=(ell/k)^(1/3), [P] (11.1)–(11.2) give exactly + + ell^(-2/3) nu_rej^{B,K}(ell) + = integral_(B x K x S^1) + [A_r(b,k,u)/(3k^(5/3))] + [(1-p_r(b,k,u))/r^3] db dk d sigma(u). + +The common compact bounds dominate this integrand by an integrable constant, because k is bounded below and the integration domain has finite measure. Pointwise convergence and dominated convergence prove + + nu_rej^{B,K}(ell) ~ C_fail^{B,K} ell^(2/3), + C_fail^{B,K}=integral_(B x K x S^1) + A_0 a_fail/(3k^(5/3)) db dk d sigma(u). + +The coefficient is finite and is strictly positive when B and K have positive lengths. It is an identified Gaussian integral, not a numerical estimate. The original full z_0 cancels if the integrand is written as + + A_0 a_fail/(3k^(5/3)) + = 4 pi_0(u;v_0) k^(-5/3) + integral_(D_k) w(theta) h_0(0,a,beta,c) + 1{n(theta)>0} dtheta. + +There is still only one original pin density, one determinant weight and one full normalizer. By direct integration, + + E N_rej^{B,K}(0,t] ~ (3/5) C_fail^{B,K} t^(5/3). + +This is a sharp coefficient for the selected-versus-candidate **difference** on the exact compact-mark population. It is not a separate second-order expansion of nu_eld: the convergence of A_r itself has not been given an expansion here. It is also not a density of replacement elder bars obtained by counting every rejected candidate. One actual maximum can participate in several rejected candidate pairs; doing that pushforward would require the appropriate de-biasing/selection identity. Theorem S3 instead identifies the actual death-height law under the specified candidate pair-Palm law conditioned on failure, which is a different, well-defined estimand. + +No unrestricted version of this difference asymptotic is inferred. [U] already proves a positive far rejected density and excludes such an inference. No k down to zero limit, new parameter uniformity, higher-dimensional lift or numerical coefficient is added. + +## 10. Hypotheses, exclusions and falsifiers + +The deterministic Theorem E assumes the exact displayed planar cubic, k>0, strict endpoint typing, Sigma_k and Delta_k excluded, pinned C^2 convergence on every fixed disk, and embedded shrinking charts. Exact partner statements additionally assume the actual global Morse distinct-value locus. Only the unique limiting position mark additionally excludes D=0; exact finite-index partner identification permits tied limiting heights. It does not need Morse–Smale or a separatrix identification. + +The Gaussian statements consume the exact original six-pin regression, full Z_r, nonsingular finite contact-jet covariance, full-field conditional C^4 moment coupling, and the reviewed cap implication/integration interfaces of [P] with all three reading rules. The algebraic exclusions are Lebesgue null, not discarded positive-probability events. The tightness proof includes the scalar large-lambda and fourth-derivative branches as o(r^3), handles the typed boundary by continuous determinant weight, and keeps all random-field correlations. Theorem S concerns fixed planar marks. Section9 uses only the parent's already uniform compact-mark bound for its explicitly restricted lifetime difference. + +The following would falsify the corresponding new step: + +* A generic typed displayed cubic whose maximin level differs from h_*, or a pinned local C^2 sequence on compact surfaces whose actual death fails to approach h_*, falsifies Theorem E. A sequence with no cubic window saddle but eventual preemption of S would specifically falsify its exact empty-sector argument. +* A failure of the cap's global older-path implication, or of the endpoint-independent residual and rare scalar integration with the original W_r/Z_r, would invalidate S4–S9 and reopen full failure exhaustion. +* A leading failure mass escaping every bounded midpoint-jet box would contradict S9; a nonzero leading typed-boundary mass would contradict S11–S14. These are explicit measure claims, not inferences from tests or finite samples. +* An erroneous Jacobian/full normalizer in G10 or the parent's radial lifetime ledger would invalidate the corresponding marked/lifetime identities. Unrestricted or replacement-bar uses of Section9's compact rejected-candidate coefficient are outside its scope. + +The new scientific candidate is the planar local-to-global actual elder partner, full rescaled selector-failure law, joint conditional replacement location/death/lifetime law, and compact rejected-candidate lifetime coefficient. Mixed factorial pairs remain two additional saddle witnesses rather than max/saddle persistence pairs. Higher-dimensional replacement geometry and any unrestricted refined lifetime law remain unproved by this note. Source-bound review and publication do not change any controlling status, graph, prize, premise or parent acceptance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md new file mode 100644 index 000000000..923d3236b --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/MARK_D.md @@ -0,0 +1,468 @@ +# Hard-negative fibres identify the actual elder law in every fixed dimension + +Author-side successor by OpenAI, 30 September 2026. Scientific effect NONE. +Disposition: AUTHOR_SIDE / HOLD pending qualified source-bound review. +The root, concave_fibre_geometry and higher_d_source_interface are contributors; +their source audits are not nonauthor acceptance. All source conditions below +are retained. No count moment or remote-count theorem is a premise. + +The complete deterministic lift, including its counterexample, is proved in +[HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). The detailed analytic source +interface and retained matrix integral are in +[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md). [SOURCES.json](SOURCES.json) +binds the complete public sources by immutable commit/path/blob/SHA256. +Together these files expose the full new argument; source hypotheses are not +accepted merely by being linked or having verified hashes. + +## 1. Model and the actual pairing estimand + +Fix an integer d>=3, torus side L>0, birth b in R, positive gap mark k, and an +orthonormal axial/transverse frame R. Use the exact variance-one periodized +Bargmann–Fock Gaussian field of [P], with strictly positive Fourier weights. +At M=(-r/2,0), S=(r/2,0), impose the original observations + + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r is this original 2(d+1)-observation Gaussian regression. Set + + W_r=F_d(H_M) F_(d-1)(H_S), Z_r=E_(Q_r) W_r, + Q_r^W=(W_r/Z_r) Q_r. + +F_j is absolute Hessian determinant restricted to j negative eigenvalues. +There is one original weight and one FULL normalizer. No adjacency restriction, +extra pin density or good-event normalizer is inserted. [P] gives +Z_r/r^2 -> z0>0, with its congruence erratum [ERR] and convergence-in-law reading +rule [REC]. The ordinary global elder death is the superlevel maximin + + d_f(M)=sup_(gamma(0)=M, f(gamma(1))>b) min_t f(gamma(t)). + +An essential maximum has d_f(M)=-infinity. On the almost-sure global Morse +distinct-value locus of [P] §8, d_f(M) is the value of the actual elder merge +saddle when finite. Define + + F_r={actual ordinary elder partner of M is not S}, + p_r=Q_r^W(F_r^c), D_r=(d_f(M)-b)/r^3. + +The event is Borel by [P] §8; whole-field marked Kac–Rice uses the exact [BOREL] +repair, not an assumption that elder selection is a continuous local mark. +These are candidate pair-Palm estimands; counting every rejected candidate +does not count each replacement elder bar once. + +## 2. Original spectral coordinates and the soft cubic + +Put m=d-1. Write the signed ordered eigenvalues of -D_y^2 f(0) as +mu1<...0, -k P in C2 + +on every fixed soft disk. Write M0=(-1/2,0), S0=(1/2,0) for the normalized +soft pins, keeping M,S for the physical original pins. The graph retains +EXACT critical pins and values G_r(M0)=0, G_r(S0)=-k, since the original hard +gradients vanish there and the hard stationary point is unique. The proof of the supplied C2 interface, +including its derivative powers, is displayed in SOURCE_INTERFACE §2. + +The missing geometry is a full-dimensional barrier, not another root count. +For a fixed soft disk containing the planar collar and doubled critical chords, +let eta=min_j h_j>0. Taylor and the fixed C4 bound give + + f_ww <= -(eta/2) I + +on that soft disk at physical r scale and on a hard ball of radius c0 r, for +all sufficiently small r. This is a pointwise fixed-target bound; constants may +depend on eta. Twice integrating along a straight hard fibre gives + + f(t,zeta_r(t)+v) <= f(t,zeta_r(t))-(eta/4)|v|^2. + +Choose fixed rho with D0=eta rho^2/4>k. The centred hard tube +|v|<=rho r^(3/2) fits inside the c0 r ball, since zeta_r=O(r^2) and +r^(3/2)=o(r). On its hard boundary the normalized drop is at least D0. +This is the quantitative hard-depth hypothesis of HARD_FIBRE_LEMMA. The r^2 +scale of the critical graph alone would give a drop of order r^4 and would +not block escape across a death window of order r^3. + +For each continued soft window root Y_r, and the exact normalized S0, put + + h_r=max({-k} union {G_r(Y_r):Y in C(P)}). + +Theorem H (actual highest partner). Under the displayed fixed-target hypotheses, +for every sufficiently small r, + + d_f(M)=b+r^3 h_r. (H1) + +On the actual global Morse distinct-value locus, the unique highest continued +graph saddle is the actual ordinary elder partner. Its full Hessian has one +positive and d-1 negative directions. If n=0 the partner is EXACTLY S; if n>0 +a continued local window saddle preempts S. Limiting ties do not obstruct H1; +the comparison is between the finitely many ACTUAL heights at that r. + +Proof. [LOCAL] §§3–6 construct, in the reduced profile, compact disks K_(r,h) +for every h_rh_* pointwise in this coupling. When D!=0, the highest soft root +Y_*(Theta) is unique and the actual partner divided by r tends, in the original +physical axial frame, to + + T_*(Theta)=R(X_*, O(Z_*,0,...,0)). (H2) + +The hard graph coordinates divided by r vanish. Sign changes of an eigenframe +change Z_*,tau and O together, leaving H2 invariant. No separatrix or +Morse–Smale premise is needed. HARD_FIBRE_LEMMA §8 gives an explicit smooth +counterexample if the contained hard-depth tube is omitted, even when graph +convergence is exact and the physical hard Hessian is uniformly negative. + +## 4. Full failure-measure exhaustion from the endpoint matrix + +H1 is pointwise in fixed jets. It does not alone identify the full rare failure +law. This section supplies the required exhaustion from the actual endpoint +matrix, independently of any count or near-occurrence event. + +On typed support put B_M=-D_y^2 f(M)>0, with ordered eigenvalues + + 0=2, + v=m(m-1)/2. + +[P] (4.2)–(4.3) regress the whole field on the ENTIRE endpoint matrix and give +an independent residual g_r. Put J=1+||g_r||C4, U=J+Lambda. All fixed J moments +are uniformly bounded, J is independent of the whole matrix, and + + K=1+||f||C4 <= C(J+||B_M||F) <= C U. (E1) + +Here Lambda is among the retained eigenvalues lambda2,...,lambdam; its being +independent of lambda1 for this conditional integration is why m>=2 matters. +No independence among matrix entries, derivative suprema or internal residual +evaluations is asserted. + +The exact global good-cap implication [CAP]/[P] §8 gives F_r subset G_r^c, + + G_r={lambda1>[4/(3k)] r M3^2, r M4<=3k/10}. + +[P] (7.7), divided by the FULL Z floor once, bounds the M4 exception by O(r^4), +which is o(r^3). On the remaining depth branch, + + lambda1<=D r U^2, + W_r<=C r^2 U^(2m) lambda1(lambda1+C r U). (E2) + +The second factor is essential: it retains the saddle mixed-square term in +[P] (6.2). The endpoint matrix density has Gaussian bound [P] (3.5); its +spectral Jacobian is at most C Lambda^v. For every retained event E depending +only on J and the hard eigenvalues, the SAME integral gives + + E_Q[W_r;depth,E] + <= C r^2 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m) 1_E + integral_0^(D r U^2) lambda1(lambda1+C rU) d lambda1 + <= C r^5 E_J integral_hard exp(-c sum_hard lambda_j^2) + Lambda^v U^(2m+6) 1_E d lambda_hard. (E3) + +The extension of the lambda1 interval beyond its actual ordered interval is +an upper bound of a nonnegative majorant. U and E are retained while integrating +lambda1; its Gaussian factor may be dropped. Equation E3 is the retained-mark +version of [P] (7.4), not a new independence or cutoff inversion. + +Taking E={U>A}, using arbitrarily high J moments and Gaussian polynomial +integrability, and dividing once by Z_r>=z_*r^2, gives, for every fixed p>0, + + limsup_r r^-3 Q_r^W(F_r,U>A) <= C_p A^-p. (E4) + +Taking E={lambda2<=epsilon} in E3 gives + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,lambda2<=epsilon)=0. (E5) + +Indeed integration of the hard Gaussian polynomial over this strip tends to +zero; for epsilon<=1 a constant times epsilon suffices after integrating all +other hard variables and the J moments. All higher-corank intersections are +included, with no inverse-hard-eigenvalue or inverse-gap factor. + +Weyl's eigenvalue inequality and Hessian Lipschitz control give + + |muj-lambdaj| <= r M3/2 <= C rU. + +On the depth branch this implies + + |s|<=D U^2+C U, |h|+|tau|+K<=C U. (E6) + +Consequently E4 is tightness of the FULL failure measure in the actual midpoint +jets and derivative norm, not just a compact-near count estimate. On U<=A, +midpoint h2<=epsilon implies endpoint lambda2<=2epsilon for sufficiently small +r at fixed epsilon,A. Equations E4–E5 and the M4 exception yield + + lim_epsilon->0 limsup_r + r^-3 Q_r^W(F_r,h2<=epsilon)=0. (E7) + +Midpoint hard eigenvalues of either sign near zero can carry no lost leading +failure mass. Inverse hard eigenvalues are used only at fixed positive targets +for graph convergence; they never enter E3. The complete detailed derivation +is SOURCE_INTERFACE §§4–5. No scalar shortcut is used: for d=2 one must retain +[LOCAL] §8's separate scalar near/far split, since Lambda=lambda1 there. + +## 5. The identified full elder-failure measure + +Let the limiting hard domain be 0=2) h_j^3 + product_(2<=i0} + ds dh dO d tau. (S1) + +It is extended by zero off the stated hard/typed domain. The deterministic +classifier bound [SC] (19) proves integrability: if x=-s>0 and n>0, then +x<=2|B|+(48kD^2)^(1/3), and + + integral_s w_+ 1{n>0} <=384k^2 |B|^3+2304k^3 D^2. + +The remaining hard spectral factor is a polynomial, while h0 has uniform +Gaussian decay in the ORIGINAL orthogonally rotated raw variables. All angular +variables remain inside the integral. Positive Gaussian density on open typed +nonempty sectors gives mu_fail(total)>0. In [SC] (20)'s notation its mass is + + a_fail=a1+a2. (S2) + +This consumes the displayed deterministic integral, not [SC]'s global count +limit. Its remote singleton coefficient beta_far does not enter S1 or S2. + +Theorem F (full fixed-dimensional selector-failure law). Under the exact +source interfaces and H1, in the extended spectral law of §2, + + r^-3 Q_r^W(Theta_r in dot, F_r) -> mu_fail in total variation, + (1-p_r)/r^3 -> a1+a2>0. (S3) + +Proof. On a fixed bounded spectral box with hard eigenvalues bounded below, +[SC] §2 disintegrates the actual additional full jet. Its soft eigenvalue +change has Jacobian r c_m J_r, where + + J_r=product_(j>=2)(h_j+rs) product_(2<=i product h_j^2 w_+. Its limiting +weight is zero off the typed domain and ON the typed boundary; selector +convergence is unnecessary there. For a typed generic target H1 gives the +actual indicator I=1_F eventually equal to 1{n>0}. The excluded algebraic +sets are null, not discarded positive mass. + +The independent complete-jet coupling [SC] (5)–(6) gives +K<=C_box(1+||F||C4). Its two-short-column bound [SC] (15) bounds W_r/r^4 by an +integrable fixed polynomial in that coupling norm on the box. Thus pointwise +convergence, the exact weight limit, and dominated convergence give compact +L1 convergence of S4 to S1, including near the typed boundary. Repeated hard +spectra are null; no uniform inverse-gap estimate is required on the box. +E4–E7 exhaust the ACTUAL failure measure, and S1 is integrable. First exhaust +bounded boxes and then let the positive hard lower bound decrease to zero. +The limsup of both excluded tails tends to zero. Compact L1 plus those two +tails is global L1, which is precisely the total variation statement S3. +Taking total masses proves its coefficient. QED. + +This establishes the pairing relevance of the microscopic soft geometry. +It does not equate two extra saddle witnesses with one maximum/saddle pair. +Neither a factorial moment nor fixed-remote mixing is used to infer selection. +When the fixed-r almost-sure global Morse/distinct-value locus is used in this +conditional coupling, take a countable intersection along an arbitrary fixed +sequence r_j->0. Dominated convergence along every such sequence proves the +stated limit. No common probability-one locus for uncountably many pinned laws +is assumed. + +## 6. Actual replacement location, death and lifetime + +On F_r let T_r be the actual elder partner's displacement from the midpoint, +in the local torus chart, divided by r. Use an isolated sentinel if the partner +is essential or outside that chart. For the same reason use a sentinel in D_r +when d_f(M) is not finite. These exceptions have r^-3 mass tending to zero: +on each regular compact jet sector H1 provides a local older path and actual +partner for all sufficiently small r; E4–E7 exhaust the remaining failure mass. + +Theorem M (actual marked law). The finite measures + + r^-3 Q_r^W((Theta_r,T_r,D_r) in dot, F_r) + +converge WEAKLY to the pushforward of S1 under + + Theta -> (Theta,T_*(Theta),h_*(Theta)). (M1) + +The conditional-on-failure law is M1 divided by a_fail. The lifetime mark +(b-d_f(M))/r^3 converges in that law to -h_* in (0,k), and its fraction of the +candidate lifetime converges to -h_*/k in (0,1). + +Proof. At a generic positive hard target H1 identifies the exact actual +winning continued root. Off the additional null set D=0, [LOCAL] §6's winning +soft root is unique; continuation and the vanishing r-scaled hard graph give +H2 and D_r->h_*. Insert any bounded continuous joint test function in S4. +The same compact dominated-convergence argument proves convergence of its +integral. E4–E7 remove the tails; sentinel mass has already been excluded. +No continuity across the polynomial transition sets is assumed. QED. + +Weak, not total variation, convergence is claimed for these real-valued marks. +Tightness of M1 also gives, for EVERY deterministic delta_r->0 with +delta_r/r->infinity, + + Q_r^W(F_r, dist(actual elder partner,midpoint)>delta_r)=o(r^3). (M2) + +For a given tolerance choose a fixed large radius with arbitrarily small +limiting marked tail, use weak tightness including the sentinel, then use +delta_r/r eventually larger than that radius. This is actual failed-partner +localization, rather than localization of a count-defined witness. + +## 7. Exact compact lifetime-density consequence + +Retain [P] §§10–12's exact compact population: birth b in an interval B, gap k +in an interval K, both of positive length and K bounded away from zero, axial +orientation u in S^(d-1) with ordinary area measure, per unit midpoint volume. +Choose a Borel transverse completion R(u) as in the source. The coefficient +a_fail is independent of that completion: the original selection probability +is physical and independent of a transverse coordinate choice, and S3 is its +unique limit. This does not assert torus rotational invariance or birth-factor +cancellation in higher dimensions. + +Use the source's ACTUAL radial ledger, in every fixed d: + + A_r=12 pi_r(R(u);v_r) Z_r/r^2, + nu_cand(ell)=ell^-1/3 integral_(B x K x S^(d-1)) + A_((ell/k)^(1/3))/(3k^(2/3)). + +The selected density has the additional p_r. The dimension powers have already +cancelled in [P] §10; no new spatial Jacobian or dimension-dependent lifetime +power may be inserted. [P] (7.8) supplies a common compact-mark bound on +(1-p_r)/r^3, and [P] (10.2)–(10.3) supply common bounds and A_r->A0. These are +existing domination interfaces, not uniformity inferred from pointwise H1. + +With r=(ell/k)^(1/3), the exact nonselected candidate density therefore has + + ell^-2/3 [nu_cand^(B,K)(ell)-nu_eld^(B,K)(ell)] + = integral_(B x K x S^(d-1)) + A_r/(3k^(5/3)) [(1-p_r)/r^3]. + +Dominated convergence and S3 identify + + nu_rej^(B,K)(ell) ~ C_fail^(B,K) ell^(2/3), + C_fail^(B,K)=integral_(B x K x S^(d-1)) + A0(b,k,u) a_fail(b,k,u)/(3k^(5/3)) db dk d sigma(u). (L1) + +C_fail is finite and strictly positive. The full z0 cancels if A0=12pi0z0 is +substituted into S1; there is still one original pin density and no additional +normalization. + +L1 is the compact candidate-minus-selected counting measure. It is not a +replacement-bar intensity obtained by counting all rejected candidates, nor a +standalone second-order selected-density expansion. An unrestricted refined +difference does not follow; its far rejected population is different. No new +k->0, d->infinity, volume, or escaping-mark uniformity is claimed. + +## 8. Hypotheses, explicit falsifiers and review boundary + +H1 assumes a generic typed cubic and its pinned reduced C2 convergence, an +embedded fixed soft collar, and a CONTAINED negative hard tube with drop D0>k. +The graph and tube are proved from [SC]'s actual smooth field on fixed positive +hard targets. The exact finite-r partner additionally uses the source's actual +global Morse distinct-value locus. A unique limiting position additionally +excludes D=0, a null set. The deterministic lemma does not require probability, +Morse–Smale, a separatrix assertion or control of the exterior. + +The stochastic statements require the original positive-spectrum field, +nonsingular contact/full-jet regression, independent endpoint-matrix residual +with all C4 moments, actual cap implication, double soft-factor weight bound, +full normalizer floor, and the original spectral density/coupling interfaces. +They are source-bound conditions, not consequences of tests or publication. +All inverse-hard constants are pointwise only. The actual rare failure tail +and multiple-soft exclusion are proved by E3–E7, rather than borrowed from a +near-count event. The full typed boundary is treated by its continuous zero +weight. The Haar-gauge extension is part of the precise measure statement. + +Falsifiers: + +- A full path bypassing a contained hard-depth cap at height above h_r, or a + graph-lifted moving chord whose exact minimum is not the continued root, + invalidates H1. The thin-tube counterexample in HARD_FIBRE_LEMMA §8 explains + why graph convergence alone is not a substitute for that hypothesis. +- A leading failure mass escaping every bounded midpoint jet set, or staying + at a nonvanishing fraction on h2<=epsilon as epsilon->0, contradicts E4/E7. + A defect in the parent cap implication, original residual independence or + retained matrix soft integral would reopen that analytic step. +- A dropped saddle mixed-square factor, extra restricted normalizer, omitted + hard/sign multiplicity or GOE substitution changes S1 and invalidates S3. +- A positive leading essential/sentinel mass contradicts M1. Total variation + for real-valued location/lifetime marks is outside the claim. +- Replacement-bar, unrestricted refined-density, new parameter-uniform or + numerical uses of L1 are outside this proof's interface. + +This is a new fixed-dimensional actual elder/failure/lifetime candidate, with +full proofs and source identities exposed. d=2 remains the separate [LOCAL] +scalar proof. No controlling acceptance, status, graph, prize, Boolean, parent +proof or source history is altered. Qualified nonauthor review must bind this +successor itself; acceptance of a count packet or of [LOCAL] alone is not an +acceptance of this higher-dimensional lift. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/P.md b/frontiers/lifetime_moment_boundary_20260930/sources/P.md new file mode 100644 index 000000000..dfed3b8d3 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/P.md @@ -0,0 +1,494 @@ +# Uniform matrix-cap selection and unrestricted short-lifetime asymptotics + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1. +**Author:** OpenAI / ChatGPT, 24 September 2026. +**Disposition:** author-side mathematical candidate; no nonauthor acceptance, formal verification, numerical finite-band certificate, or program-level closure is asserted. + +This is an additive continuation of MARKED-CYLINDER-CAP-20260924-v1, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`, reproduced unchanged in `sources/`. It addresses the matrix-probability, angular-uniformity, compact-mark, and lifetime-composition interfaces of campaign #61. It does **not** extend the predecessor's numerical constants to new parameters. Instead it derives new, non-numerically-evaluated constants by a uniform Gaussian argument. + +## 1. Model, convention, and the three theorems + +Fix a dimension d >= 2 and a side length L > 0. On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance + + K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2) + / sum_{n in Z^d} exp(-|Ln|^2/2). + +In particular d=3, L=24 is an actual three-dimensional SIDE24 application. No replacement by a nonperiodic covariance is made. Fix compact intervals B=[b_-,b_+] and K=[k_-,k_+] with 00, use the embedded local cylinder centered at the origin with pins + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=s=b-k r^3, + grad f(M)=grad f(S)=0. + +Q_{r,b,k,R} is the continuous Gaussian regression law at these 2(d+1) observations, not a conditioning on a positive-probability event. A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. Set + + W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}, + Z_r = E_Q W_r, dQ^W = (W_r/Z_r)dQ. + +The index counts negative eigenvalues. The FULL normalizer Z_r has no extra pin-density, coordinate Jacobian, or adjacency condition. Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. + +**Theorem A (uniform selection candidate).** There exist r_*>0 and C=z_* r^2 and the full transverse-matrix boundary layer, including simultaneous small transverse eigenvalues. It does not presume isotropy under arbitrary rotations of the torus. + +**Theorem B (compact-mark density candidate).** Suppose B and K have positive lengths. Count ordered maximum/saddle pairs with 00, + 0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3), (1.2) + +as ell decreases to zero. An exact finite-dimensional Gaussian integral for c_{B,K} appears in Section 11. + +Theorem B is a COMPACT MARK WINDOW result, especially its quantitative difference estimate. Theorem C below separately removes the mark and distance restrictions for the LEADING asymptotic. There is no stated numerical value of C, r_*, z_*, or c_{B,K}, and no claim that r_*=1/20. The new asymptotic normalizer proof does not replace or re-certify the historical H3 finite-band floor. + +**Theorem C (unrestricted finite-lifetime leading density candidate).** For the same fixed torus and dimension, let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. There is a positive finite constant c_{d,L} such that + + nu_cand^all(ell) ~ c_{d,L} ell^(-1/3), + nu_eld^all(ell) ~ c_{d,L} ell^(-1/3). (1.3) + +All birth heights, all positive scaled-gap marks and all spatial separations on the torus are included in (1.3). The essential global-maximum class is excluded from the finite-lifetime count. The constant is an explicitly identified finite Gaussian/angular integral, with the birth and gap integrals analytically eliminated in Section 15. No elementary closed form or numerical enclosure is claimed. Crucially, the compact-window O(ell^(2/3)) DIFFERENCE estimate in (1.2) is not asserted for the unrestricted densities. Theorem C has a separate domination and off-diagonal proof, not a change of label on Theorem B. + +## 2. Positive Fourier spectrum and finite-jet rank + +Poisson summation gives + + K_L(z)=sum_{n in Z^d} a_n exp(2 pi i n.z/L), + a_n=exp(-2 pi^2 |n|^2/L^2) / sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +Thus every Fourier mode has positive variance, and sum_n sqrt(a_n)(1+|n|)^q is finite for every q. A real sine/cosine expansion therefore converges in every C^q norm in every finite L^p norm. In particular the field has a smooth version and all finite moments of its global derivative suprema exist. + +Any finite list of distinct derivative evaluation functionals at distinct sites is linearly independent. Indeed a vanishing-variance linear combination must annihilate every Fourier mode because every a_n is positive. As a distribution it is a finite sum of derivatives of point masses with every Fourier coefficient zero, so it is zero. Smooth test functions supported at one site and with arbitrarily prescribed finite jets show that every coefficient is zero. At one site in a rotated frame the same assertion also follows from a polynomial P(R^T n) vanishing on all n in Z^d: the invertible rotation reduces this to a polynomial vanishing on the full integer lattice, hence the polynomial is zero. + +Only lists without repeated derivative functionals are called distinct. A Hessian is parametrized by its independent symmetric entries, not by d^2 allegedly independent entries. + +These arguments justify positive covariance for the original pins at each pair of distinct points and for every residual jet list modulo the pins used below. They also explain why finitely many observed modes or a numerically positive sampled covariance would not suffice for this proof. + +## 3. An exact nonsingular contact frame for all orientations + +Write x for the axial coordinate and y_1,...,y_m for the transverse coordinates, m=d-1. On the line y=0 set a=-r/2, c=r/2. Order the original observations as + + O_r=(f(a), f_x(a), f(c), f_x(c), + f_y1(a),f_y1(c), ..., f_ym(a),f_ym(c)). + +This is a permutation and orthogonal re-expression of the specified physical pins. Define U_r=T_r O_r by four axial rows + + U0 = (f(a)+f(c))/2, + U1 = (f(c)-f(a))/r, + U2 = (f_x(c)-f_x(a))/r, + U3 = (6/r^2)[f_x(a)+f_x(c)-2(f(c)-f(a))/r], (3.1) + +and, for every transverse direction j, two rows + + Vj0=(f_yj(a)+f_yj(c))/2, + Vj1=(f_yj(c)-f_yj(a))/r. + +The transformation is invertible for r>0 and has exact absolute determinant + + |det T_r| = (12/r^4)(1/r)^m = 12 r^(-(d+3)). (3.2) + +The prescribed target becomes exactly + + v_r=(b-k r^3/2, -k r^2, 0, 12k, 0,...,0). (3.3) + +This is not an approximate transformation, and the factor 12 in (3.2) is retained in the lifetime ledger. + +Taylor formulas with integral remainders give mean-square convergence, uniformly over frames, of U_r to the contact jet + + U0*=(f, f_x, f_xx, f_xxx, + f_y1,f_xy1,...,f_ym,f_xym) at 0. (3.4) + +The potentially dangerous U3 cancellation is an averaged third derivative, not a subtraction of uncontrolled random values. Uniform convergence of its covariances, and of cross-covariances with any fixed finite derivative order, follows either from those integral formulas or from the rapidly summable Fourier series. The target v_r converges uniformly on B x K to v_0=(b,0,0,12k,0,...,0). + +The functionals in (3.4) are distinct. Hence their covariance Sigma_0(R) is positive definite by Section 2. Its entries are continuous in R. The orthogonal group is compact, so its least eigenvalue has a positive uniform minimum. Consequently Sigma_r(R) remains uniformly positive definite for all sufficiently small r; Sigma_r and Sigma_r^{-1} converge uniformly. + +This compactness argument establishes existence, not an evaluated interval enclosure. It never equates Sigma_r at different orientations. Full torus rotational invariance is neither assumed nor true in general. + +Let A_r=D_y^2 f(M), represented by its m(m+1)/2 independent entries. Append these entries to U_r. At r=0 the extra functionals are f_yi_yj, i<=j; none duplicates a functional in (3.4). The enlarged covariance is therefore uniformly positive definite at contact and for small r. Its Schur complement shows that the Q-law of A_r has covariance uniformly bounded above and uniformly bounded away from zero. Its mean is uniformly bounded. In symmetric-matrix Lebesgue coordinates it has a density satisfying + + density_Q(A) <= C_0 exp(-c_0 ||A||_F^2), (3.5) + +for fixed positive C_0,c_0. This remains valid after any change of eigenvector orientation. Orthogonal invariance of the density is not required. + +## 4. Uniform conditional derivative moments and the independent residual + +Because Sigma_r^{-1} and v_r are uniformly bounded, the regression energy v_r^T Sigma_r^{-1}v_r is uniformly bounded. For each unconditioned standardized Fourier coefficient xi, covariance Cauchy-Schwarz bounds its conditional mean by the square root of this energy, and its conditional centered variance by one. Minkowski's inequality and the summable Fourier coefficients then give, for every finite p and derivative order q, + + sup_{r,b,k,R} || ||f||_{C^q(X)} ||_{L^p(Q)} < infinity. (4.1) + +The same proof applies to a centered residual after further conditioning on A_r. Write, in one full-field Gaussian coupling, + + f(z)=mu_r(z)+B_r(z).(A_r-E_Q A_r)+g_r(z), (4.2) + +where g_r is jointly independent of the entire symmetric matrix A_r. The regression coefficient B_r and its derivatives through order four are uniformly bounded, by the uniform inverse covariance from Section 3 and bounded derivative cross-covariances. The residual has uniformly bounded C^4 moments of every finite order. + +Independence in (4.2) is not asserted separately for Hessian entries, endpoint variables, or sampled derivative maxima. All residual evaluations are orthogonal to A_r; Gaussian independence on a countable dense collection and continuity give independence of the random smooth field. The residual's internal correlations are retained. + +Let J_r=1+||g_r||_{C^4(X)}, with a fixed coordinate norm. The partial-block operator norms of the cap theorem differ from this coordinate norm by finite constants depending only on d. Hence, for M3,M4 on the local cylinder, + + M3 <= K_0(J_r+||A_r||_F), + M4 <= K_0(J_r+||A_r||_F), + J_r >= 1, J_r independent of A_r, + sup ||J_r||_p < infinity for every finite p. (4.3) + +No inverse Hessian moment is used. Gaussian pin covariance and random field Hessians are different objects. + +## 5. Endpoint identities and the actual full normalizer + +For either pin i in {M,S}, put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, + A_i=D_y^2 f(i), + H_i=[[r alpha_i, r beta_i^T], [r beta_i,A_i]]. + +The two gradient pins imply the vector and scalar average bounds + + |alpha_i| <= M3/2, ||beta_i|| <= M3/2, + ||A_S-A_M||_op <= r M3. (5.1) + +For example integral_a^c f_xx=0; subtract this average at an endpoint and integrate the M3-Lipschitz bound. For beta use the average of grad_y f_x and its vector derivative. This does not invoke a simultaneous vector Rolle zero. + +The exact height identity gives a weighted mean of f_xxx equal to 12k on [a,c]. Variation of f_xxx over that interval is at most rM4. The same average calculation yields + + |alpha_M+6k| <= r M4/2, + |alpha_S-6k| <= r M4/2. (5.2) + +The block determinant identity, valid even when alpha_i or A_i is zero, is + + det H_i/r = alpha_i det A_i + -r beta_i^T adj(A_i) beta_i. (5.3) + +By (4.1),(5.1), det H_i/r and W_r/r^2 have uniformly bounded moments of every finite order. + +Let A_0 be D_y^2 f(0) under regression on U0*=v_0. Its law is a full-support nondegenerate Gaussian on symmetric matrices. Finite-dimensional Gaussian regression gives uniform convergence of the means and covariances of A_M to those of A_0. Equations (5.1),(5.2) imply that the congruence-scaled endpoint Hessians converge, in probability uniformly over parameters, to + + diag(-6k,A_0) at M, diag(6k,A_0) at S. + +Here the congruence uses diag(sqrt(r),I), so its off-diagonal entries are sqrt(r) beta_i. The limiting matrices are nonsingular almost surely. This follows from the full matrix density and k>=k_->0. Thus the type indicator converges to 1{A_0<0}. Uniform integrability from (5.3) gives + + Z_r/r^2 -> z_0(b,k,R) + := (6k)^2 E[det(A_0)^2 1{A_0<0}], (5.4) + +uniformly on the compact parameter set. + +For completeness, uniformity can be checked by a subsequence argument: any allegedly bad sequence of parameters has a convergent subsequence; continuous Gaussian means/covariances, (5.2), and uniform integrability give the stated limit on that subsequence. The discontinuity set of the type indicator has Gaussian probability zero. Equation (3.5) supplies a common integrable majorant. + +The right side of (5.4) is continuous and strictly positive: the Gaussian law has positive density on an open negative-definite matrix ball. Compactness therefore yields + + 00, with eigenvalues + + 00. + +Determinant monotonicity on positive-definite matrices therefore gives + + |det H_M| <= r (h/2) det B. (6.1) + +This is a canceled-pivot bound: no estimate of 1/alpha_M is taken. + +From ||A_S+B||<=rh, the ordered singular values of A_S are at most lambda_j+rh. The adjugate norm is at most the product of the largest m-1 such bounds. With the empty product equal to one when m=1, (5.3) gives + + |det H_S| <= r(h/2) [lambda_1+(3/2)rh] + product_{j=2}^m(lambda_j+rh). + +Hence the actual typed weight satisfies + + W_r <= (r^2 h^2/4) lambda_1[lambda_1+(3/2)rh] + product_{j=2}^m lambda_j(lambda_j+rh). (6.2) + +The second soft factor includes the saddle mixed-square term. Omitting it, or replacing it by an unrelated order-one factor, changes the radius power. No Gaussian-entry independence is needed for (6.2). + +## 7. Full matrix boundary-layer integration, including corank intersections + +By the predecessor's deterministic marked-cylinder theorem, pairing and the one-branch conclusion hold on + + G_r={lambda_min(-A_M)>[4/(3k)]rM3^2, + rM4<=3k/10}. + +On typed support A_M<0 automatically. It suffices to control G_r^c directly, not infer a converse from a local path-event bound. + +First suppose m>=2. In (4.3) absorb sqrt(m) into K and set U=J_r+Lambda>=1. Then h<=KU. Failure of the depth condition is contained in + + 00 and use h<=K(J_r+lambda). Depth failure implies + + lambda<=Dr(J_r+lambda)^2 + <=2DrJ_r^2+2Dr lambda^2. + +It is contained in the union + + 01/(4Dr). (7.5) + +The large-lambda branch is retained. On the near branch, h<=K(1+4D)J_r^2 for r<=1, J_r>=1. The bounded scalar density, (6.2), and direct integration in lambda give a numerator <=C r^5 E J_r^10. On the far branch, direct fourth-moment Markov with the actual weight gives + + E_Q[(W_r/r^2)1_far] + <=(4Dr)^4 E_Q[(W_r/r^2)lambda^4] <=C r^4. (7.6) + +Thus the scalar numerator is <=C r^5+C' r^6. This separate argument is why the all-dimension statement does not silently drop the scalar far branch. + +Finally, for every dimension, the fourth-derivative exception is bounded directly by + + E_Q[(W_r/r^2)1{rM4>3k_-/10}] + <=[10r/(3k_-)]^4 E_Q[(W_r/r^2)M4^4] <=C r^4. (7.7) + +This is a pointwise Markov bound followed by a joint moment estimate. It is not Cauchy-Schwarz with a missing square root. All joint moments in (7.6),(7.7) are finite by Sections 4-5. + +Divide the numerators by the FULL lower bound (5.5). The result is + + Q^W(G_r^c)<=C_3 r^3+C_4 r^4<=C r^3. (7.8) + +## 8. Pinned genericity and global event measurability + +The following argument applies at every fixed positive r and parameter, not on an asserted common probability-one set for uncountably many laws. + +Away from the pins, consider (grad f(x),Hf(x)v), with ||v||=1. Its range has dimension 2d; its parameter manifold has dimension 2d-1. Modulo the pin observations, these functionals have full covariance rank: gradient and Hessian orders are distinct, and the map H -> Hv from symmetric matrices onto R^d is surjective. For two distinct unpinned points, the map + + (grad f(x),grad f(y),f(x)-f(y)) + +has range dimension 2d+1 and parameter dimension 2d. For a tie to a pinned height use (grad f(x),f(x)-b), or the analogous s map, with dimensions d+1 and d. + +Exhaust the relevant domains by compact sets separated from pins and from coincident points; use finitely many sphere charts when v occurs. Covariance positivity from Section 2 and compactness give bounded Gaussian densities on each such set. An elementary overdetermined-zero lemma applies: on a mesh of size epsilon, a random C^1 map from dimension q to dimension p>q, with derivative norm bounded by N, could have a zero only if one of O(epsilon^-q) mesh values has norm O(N epsilon). The density bound makes the probability O(N^p epsilon^(p-q)), tending to zero. First fix N, then let N increase; the random derivative norm is almost surely finite. This proves absence of all these zeros. + +Pinned Hessians have a full density on their symmetric entries for fixed distinct pins, so their determinants are nonzero almost surely. The pinned heights differ because k,r>0. Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. The transfer to Q^W follows by absolute continuity and 0f(M)} min_t f(gamma(t)), + +with supremum of an empty set equal to -infinity. The event d_f(M)>h can be represented by a countable union of polygonal-path tests, with rational interior vertices and endpoints, all staying strictly above h and with endpoint value above f(M). Strict clearances allow approximation by such paths. For variable M, use countably many coordinate charts and an initial segment depending continuously on M. These are Borel tests in (M,f). Thus equality d_f(M)=f(S) is Borel. On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. + +On G_r the unchanged deterministic cap theorem controls every boundary exit and supplies a connection through S to a point higher than b. Therefore the GLOBAL ordinary elder partner is S, not merely a local adjacency candidate. Equations (7.8) and the full-probability locus prove Theorem A. The cap theorem also gives exactly one actual ascending branch to M. It does not identify that branch with the transverse ridge. + +## 9. A source-bound weighted Kac-Rice interface + +For clarity, this section specifies why a global elder mark is admissible; a formula for local Hessian index alone would not suffice. + +Apply Kac-Rice first on compact spatial pair domains separated from the diagonal, to the Gaussian vector field (grad f(x),grad f(y)). It is C^1 and has positive definite point covariance for x!=y by Section 2. Its derivative is block diagonal with Jacobian |det H_x det H_y|. Conditional laws of the whole smooth field depend continuously on the imposed gradient values by finite-dimensional Gaussian regression. Derivative moments are locally bounded. + +The expected-integral extension in Armentano-Azais-Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1, gives the identity for bounded nonnegative continuous weights in the location and an additional Gaussian field mark. It is enough first to use cylinder weights depending on finitely many fixed evaluations of f and its derivatives through order two. Their conditional distributions depend continuously on the imposed gradients by Gaussian regression, so the paper's continuity requirements hold. The field is a random element of the separable C^2 space on the torus. On each compact pair domain the two measures on location x C^2 are finite, by the unweighted formula. Cylinder evaluations on a countable dense set, together with location coordinates, generate this Borel sigma-algebra. Equality on their bounded continuous test-function algebra gives equality of the Borel measures by the monotone-class theorem. This extends the identity to bounded Borel global marks, rather than pretending the elder indicator is continuous or lower semicontinuous. No independence between the mark and gradient field is assumed. + +Use the Borel elder mark from Section 8 and the type indicators. Disintegrate the two heights using their nondegenerate joint Gaussian density conditional on the gradients. This gives the full-pin density times E_Q[W] for candidates, and the same quantity times p_r for selected pairs. The continuous Gaussian regression kernel defines the canonical density version; the underlying measure identity is unaffected by height-null-set choices. + +Exhaust the off-diagonal spatial domain toward r=0. The r dr bound proved in the next section makes the expectations locally integrable, and nonnegative monotone convergence completes the near-diagonal formula. No finite-grid approximation of the continuum field is substituted. + +## 10. Exact radial ledger: the dimension cancels + +Let pi_r(R;v_r) denote the density of U_r at (3.3). Since U_r=T_r O_r, + + density_{O_r}(specified pins) + =12 r^(-(d+3)) pi_r(R;v_r). (10.1) + +Use midpoint z, directed separation h=r u, birth b, and gap mark k. The change from the ordered pair (M,S) to (z,h) has absolute determinant one. Polar separation contributes r^(d-1) dr d sigma(u), where sigma is ordinary, NOT probability-normalized, surface area on S^(d-1). The height change (f(M),f(S))=(b,b-k r^3), at fixed r, contributes r^3 db dk. There is no factor 1/2: maximum/saddle pairs are ordered by their different roles. + +Consequently the candidate intensity per unit midpoint volume is + + r A_r(b,k,u) dr db dk d sigma(u), + A_r=12 pi_r(R;v_r) (Z_r/r^2). (10.2) + +The powers are + + spatial (d-1) + height-mark 3 + pin -(d+3) + + determinant-normalizer 2 = 1. + +By Sections 3 and 5, + + A_r -> A_0=12 pi_0(R;v_0) z_0(b,k,R)>0 (10.3) + +uniformly and with a uniform finite upper bound. Although frames were used in the proof, (10.2) is invariant under a rotation of the transverse frame: the original zero-gradient law and determinant weight are unchanged by that orthogonal coordinate change. Thus it is a function of u alone. Uniformity was proved on the compact orthogonal group and descends to the sphere; no nonexistent global continuous transverse frame on the sphere is assumed. + +The selected intensity is r A_r p_r with the same measures. This supplies the domination needed below, rather than assuming a fixed-axis statement can be integrated. + +## 11. Lifetime pushforward and the exact compact coefficient + +For ell=k_->0. This proves (1.2). It is an O(ell^(2/3)) difference, not a claim that either density separately has a second-order expansion with that remainder. Convergence of A_r to A_0 was not assigned a numerical rate. + +## 12. Short bars, moments, and the exact remaining boundary + +Integrating the density versions gives the following expected per-unit-volume results for the same spatial/mark-restricted populations: + + E N_eld(0,t] ~ (3/2)c_{B,K} t^(2/3), + 0<=E[N_cand(0,t]-N_eld(0,t]]<=C t^(5/3). (12.1) + +For any real q>-2/3, + + E sum_{selected, ell<=t} ell^q + ~ c_{B,K} t^(q+2/3)/(q+2/3). (12.2) + +The same leading expression holds for candidates. For q>-5/3 the nonselected difference alone satisfies + + E sum_{candidate but not selected, ell<=t} ell^q + <= C_q t^(q+5/3). (12.3) + +For q<=-2/3 the two individual expectations in (12.2) are infinite for every sufficiently small positive t, because their densities have a positive ell^(-1/3) leading term. Equation (12.3) concerns the nonselected counting measure directly, not subtraction of two infinities. In particular the compact-window inverse-lifetime threshold is p<2/3 for integrability of ell^(-p), with logarithmic divergence at p=2/3. This is unrelated to the inverse-Hessian/Schur thresholds in issue #56. + +The corresponding unrestricted cumulative and individual-moment corollaries follow from Theorem C with c_{d,L} in place of c_{B,K}, by the same integrable-power argument. The compact-window nonselected difference rates (12.1),(12.3) are not automatically unrestricted rates. + +## 13. Removing all birth and gap cutoffs by a direct intensity majorant + +The covariance Sigma_r of U_r depends on radius and frame, NOT on the imposed target. Section 3 therefore supplies a fixed r_0>0, independent of b and k, on which its eigenvalues are bounded above and away from zero. Reduce r_0 below the torus injectivity scale and below one. Do NOT require a globally uniform lower bound for Z_r or a globally uniform constant in (1.1). + +For every b in R, k>0 and 0= c(b^2+k^2), (13.1) + +with fixed c,C>0. For the second inequality it suffices to use its two coordinates b-k r^3/2 and 12k: if r<=1, then b^2<=2(b-k r^3/2)^2+k^2/2. Thus the invertible covariance bound yields + + pi_r(R;v_r) <= C exp[-c(b^2+k^2)]. (13.2) + +Conditional Fourier-coefficient means are bounded by C|v_r| and their centered variances by one. The same summability/Minkowski proof as Section 4 now gives the TARGET-GROWTH bound + + E_Q (1+||f||_{C^3})^p <= C_p(1+|b|+k)^p (13.3) + +for every fixed finite p, uniformly over r and frames on this r_0 band. This is not the compact-target bound (4.1) applied outside its scope; it is its explicit polynomial-growth version. + +By (5.1),(5.3), for r<=1, + + |det H_i|/r <= C (1+||f||_{C^3})^d, + 0<=W_r/r^2<=C(1+||f||_{C^3})^(2d). + +These inequalities remain true when k is arbitrarily small; no division by k or by Z_r has occurred. Combining with (13.2),(13.3) gives the full unnormalized-intensity majorant + + 0<=A_r(b,k,u)=12 pi_r(R;v_r) Z_r/r^2 + <= C(1+|b|+k)^(2d) exp[-c(b^2+k^2)] =: H(b,k). (13.4) + +For every fixed positive r and every b,k>0, Z_r is positive and finite: append BOTH endpoint Hessians to the distinct-site pins; the full residual Gaussian density is positive on a product of an open maximum cone and an open index-(d-1) cone. Thus p_r is defined, even though no globally uniform lower floor has been used. For every fixed b,k>0, Theorem A on a compact neighborhood implies p_r->1 as r->0. + +At a fixed lifetime ell, the near-pair restriction r<=r_0 is exactly k>=ell/r_0^3. After the exact change (11.1), + + ell^(1/3) nu_cand^near(ell) + = integral_{R x (0,infinity) x S^(d-1)} + 1{k>=ell/r_0^3} A_{(ell/k)^(1/3)}(b,k,u) + /(3 k^(2/3)) db dk d sigma, (13.5) + +with the integrand defined to be zero where the indicator is zero. The selected near density has the additional factor p_r. Both integrands are dominated by H(b,k)/(3 k^(2/3)). This majorant is integrable: the k exponent -2/3 is strictly greater than -1 at zero, and Gaussian tails absorb the polynomial at infinity and in b. This explicitly handles the shrinking-k boundary where a compact-window selection constant can blow up. + +For each fixed b,k>0,u, the indicator tends to one, A_r->A_0 and p_r->1. Dominated convergence therefore gives the same limit for the two near densities: + + c_{d,L}=4 integral_{R x (0,infinity) x S^(d-1)} + k^(-2/3) pi_0(R;v_0) z_0(b,k,R) + db dk d sigma(u), (13.6) + 00, + D_u=E[(det A_u)^2 1{A_u<0} | V_u=0]. (15.1) + +Here A_u conditional on V_u=0 is an actual centered nondegenerate Gaussian symmetric matrix, not automatically GOE. Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and D_u>0. All are continuous in the frame and invariant under transverse-frame changes. + +The contact density and cone moment factor as + + pi_0(R;v_0)=p_(f,V_u)(b,0) p_G(0) phi_tau_u(12k), + z_0=(6k)^2 E[(det A_u)^2 1{A_u<0} | f=b,V_u=0], + +where phi_tau is the centered scalar Gaussian density of standard deviation tau. Integrating b by ordinary Gaussian disintegration gives p_Vu(0) D_u. Thus (13.6) becomes + + c_{d,L}=144 integral_{S^(d-1)} p_G(0) p_Vu(0) D_u + [integral_0^infinity k^(4/3) + phi_tau_u(12k) dk] d sigma. + +The scalar integral can be evaluated exactly. With t=12k and the elementary gamma integral, + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + = Gamma(7/6) tau^(4/3) / [24^(1/3) sqrt(pi)]. + +Consequently the full leading coefficient is + + c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] + * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) + D_u d sigma(u). (15.2) + +This eliminates both unbounded mark integrals. The remaining angular and negative-definite-cone expectation are finite, strictly positive and specified by derivatives of the EXACT periodized covariance. Formula (15.2) is an explicit finite-dimensional Gaussian representation, not a reported elementary closed form or numerical enclosure. The factor24 is a numerical Jacobian/gamma factor, NOT the side length L; it is the same factor for every L. Changing the field amplitude by a>0 scales the coefficient by a^(-2/3), consistent with nu_(af)(ell)=a^(-1)nu_f(ell/a); this is an additional normalization check. + +## 16. Supplied interfaces, retained limits, and review + +**Interfaces supplied by this candidate:** the actual transverse matrix boundary argument (including d=3); a uniform compact-mark asymptotic full normalizer; all orientations without false isotropy; the global Borel marked Kac-Rice interface; exact pin/radial/lifetime factors; compact-window leading density and quantitative selection difference; a separate integrable unnormalized Gaussian majorant removing ALL birth/gap cutoffs; an off-diagonal bound completing ALL spatial separations; the unrestricted finite-bar leading density and its cumulative/moment consequences; and analytic elimination of birth and gap in its coefficient. + +**Not supplied:** a numerical r_* or C on a prescribed band; a globally uniform O(r^3) probability constant over unbounded b,k; the compact O(ell^(2/3)) difference as an unrestricted remainder; an elementary closed form or numerical enclosure of (15.2); a rigorously evaluated lower coefficient for pairing failure; the original numerical H3/LPW replay; a shrinking-annulus RN certificate; a24-jet displacement enclosure; formal proof verification; nonauthor mathematical acceptance; or unrestricted P15 prize closure. No historical scientific status is changed by this author-side proof. + +The exact-arithmetic program checks pin transforms/targets, block determinants, finite typed-matrix envelopes, spectral Jacobians, radius/lifetime exponents, explicit small-eigenvalue integral models, global target coercivity and coefficient scaling. Semantic mutations challenge the load-bearing factors. These are finite algebra/implementation controls, NOT Gaussian simulation or proof of continuum estimates. + +Priority nonauthor review: (i) desingularized full residual covariance; (ii) uniform all-field moments and independent-matrix regression; (iii) both soft determinant factors; (iv) eigenvalue integration retaining the random largest-eigenvalue width and all corank strata; (v) scalar far branch; (vi) full-normalizer type convergence; (vii) continuous-to-Borel marked Kac-Rice extension; (viii) every factor in the pin/lifetime ledger; (ix) target-growth bound (13.3) and integrability at k=0 in (13.5); (x) distinct-site compact covariance in the far domain; and (xi) parity, b-disintegration and the gamma factor in (15.2). + +### Primary-source attribution + +Armentano, Azais and Leon, arXiv:2304.07424v3, Sections 2 and 7, supply established Gaussian Kac-Rice/expected-integral background. The precise full-field/Borel and pair application is explained in Section 9, not attributed wholesale to that paper. Curry, *The Fiber of the Persistence Map for Functions on the Interval*, arXiv:1706.06059v2, Section 3, supplies elder-rule background; the all-dimensional separating cap is the explicitly identified in-project predecessor. Beliaev, Cammarota and Wigman, *No repulsion between critical points for planar Gaussian random fields*, arXiv:1911.03455, is neighboring two-point/index-asymptotic work, not a source of our elder-selection conclusion. No priority or novelty claim follows from this bounded reconnaissance. diff --git a/frontiers/lifetime_moment_boundary_20260930/sources/REC.md b/frontiers/lifetime_moment_boundary_20260930/sources/REC.md new file mode 100644 index 000000000..75da25979 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/sources/REC.md @@ -0,0 +1,309 @@ +# D1 Theorem A selection chain — source-bound reconciliation + +**Object:** D1-CHAIN-RECONCILIATION-20260928-v1. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. +**Effect:** register reconciliation. This record introduces **no new mathematical claim** about the parent. It binds +existing nonauthor reviews to exact source bytes, discharges every residual item they left open, reviews the two +parent sections that no earlier review covered (§§11–12), and states the resulting scoped status. The register edits +it justifies are listed in §9. `lemma_closed`, prizes and premises are untouched. + +## 1. The reconciled object and its reading rule + +The D1 parent is OpenAI/ChatGPT's UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, with the imported deterministic +MARKED-CYLINDER-CAP-20260924-v1. The parent bytes are immutable. The object reconciled here is the parent **read +together with** the three amendments in the table below. Every consumer must cite this identity set, not the parent +alone. + +| Role | Path (Math-) | Git blob | Bytes | SHA256 | +|---|---|---|---|---| +| Parent (P) | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | 40261 | `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7` | +| Cap import (CAP) | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` | `0633aca3c2a2882b0de4399da0a75d64c2e6b2e1` | 15160 | `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc` | +| Amendment E1: congruence erratum | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` | `213594d6ca6a86fb938110f4d166d9ce275a02d0` | 1782 | `bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028` | +| Amendment E2: §9 replacement v1.1 | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` | `fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a` | 9062 | `845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f` | +| Amendment E3: wording W1 and embedding radius | this record, §5 | — | — | — | + +**Reading rule.** + +- Read P §5 with E1. The congruence factor is `D_r = diag(r^{-1/2}, I)`, not the displayed `diag(√r, I)`. +- Read P §9 as replaced by E2. +- Read P l. 165 with W1 (§5 below). +- Read P §1's "sufficiently small r" as including the explicit embedding bound `r < L/(4√2)` (§5 below). + +No formula of P changes. + +**Machine binding.** Each component above is also its own source node in the hard-gate graph +(`frontiers/downstream_gate_20260925/GRAPH.json`), required by the D1 node: E1, E2, CAP (whole file; only §§1–5 +are consumed) and this record's directory, which carries E3 and `LEDGER.json`. A byte change or deletion of any +component therefore proposes D1, D2 and D3 for revalidation even when P's bytes are unchanged (§9, §10). + +**Name collision.** "Theorem A" here always means P (1.1). The fixed-remote mean-measure theorem in +`frontiers/remote_window_20260924/PROOF.md` is also called "Theorem A". That is a different theorem (D4), and +nothing in this record concerns it. + +## 2. Reconciled statements + +The model is P §1: + +- the exact L-periodized Gaussian field on `R^d/(LZ^d)`, for fixed `d >= 2` and `L > 0`; +- the six pins, generally `2(d+1)` observations; +- the typed weight `W_r` and the FULL normalizer `Z_r = E_Q W_r`; +- `p_r(b,k,R)`, the `Q^W`-probability that the global ordinary superlevel elder partner of `M` is `S`. + +The three statements below are accepted at existential scope. There is no numerical `C`, `r_*`, `z_*`, `c_{B,K}` +or `c_{d,L}`. + +- **Theorem A (P 1.1).** For compact `B` and `K ⊂ (0,∞)`, there are `C` and `r_* > 0` such that + `0 <= 1 - p_r <= C r^3`. This holds uniformly over `b ∈ B`, `k ∈ K`, all orthonormal frames and `0 < r <= r_*`. +- **Theorem B (P 1.2).** Let `B` and `K ⊂ (0,∞)` be compact intervals **of positive length** (P l. 37; a singleton + interval gives zero mark-window intensity). The compact-window densities satisfy + `ν_cand ~ ν_eld ~ c_{B,K} ℓ^{-1/3}` with `c_{B,K} > 0`. Their difference satisfies `0 <= ν_cand - ν_eld <= C ℓ^{2/3}`. +- **Theorem C (P 1.3).** All births, all positive gaps and all separations are included, and the essential class is + excluded. Then `ν_cand^all ~ ν_eld^all ~ c_{d,L} ℓ^{-1/3}` with `0 < c_{d,L} < ∞` given by (15.2). This is the + leading term only: no unrestricted remainder or difference rate is included. +- **§12 corollaries.** These are the cumulative and moment consequences (12.1)–(12.3) and their unrestricted leading + forms. + +**Sharpness in d = 2 (composition, separately sourced).** In the plane, the direct lower bound +`1 - p_r >= c r^3` combines with Theorem A to give `1 - p_r = Θ(r^3)` and `ν_cand - ν_eld = Θ(ℓ^{2/3})` on compact +marks. The lower bound is (E2) of `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md`, +blob `aaefc8da…`, reviewed in `reviews/d1_elder_lower_claude_20260928/`. **No lower bound is asserted for `d >= 3`.** + +## 3. The four components of the chain + +| Component | Sections | Output | Interfaces | +|---|---|---|---| +| (I) Matrix-cap normalizer | P §§2–5 with E1 and W1 | `Z_r/r^2 -> z_0 > 0` uniformly; floor `z_* r^2 <= Z_r <= z^* r^2` (5.4)–(5.5) | A1, A2, A4, A3 | +| (II) Cubic pairing-defect bound | P §§6–7 | `E_Q[W_r 1_{G_r^c}] <= C r^5`, hence `Q^W(G_r^c) <= C r^3` (7.8) | A5, A6, A7 | +| (III) Global elder selection | CAP §§2–5 and P §8 | on `G_r` ∩ {Morse, distinct values}, the global elder partner of `M` is `S`; hence `1-p_r <= Q^W(G_r^c)` | CAP, D1-A | +| (IV) Marked Kac–Rice and lifetime | E2 (§9), P §§10–15 | intensity identities, radial ledger, pushforward, majorant, off-diagonal bound, coefficient (15.2) | D1-B/E2, D1-C, §11, §12, D1-D, D1-E | + +Theorem A = (I) + (II) + (III). Theorem B = Theorem A + (IV) through §11. Theorem C = (IV) + Theorem A, which gives +`p_r -> 1` pointwise in §13. + +## 4. Interface ledger + +**Providers.** OpenAI is the author of P, CAP, E1 and E2. The nonauthor providers are xAI/Grok and Anthropic/Claude. +All lanes share one GitHub account, so no organizational independence is claimed anywhere. The full machine-readable +ledger, with every review locator, is `LEDGER.json`. + +**Review records.** + +| ID | Provider and session | Record | Verdicts | +|---|---|---|---| +| G1 | xAI Grok 4.7 via Cursor, `bc-ce4bf0bc` | [main#63 comment 5841570965](https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5841570965) | D1-A–E ACCEPT | +| C1 | Anthropic Claude, this session | `reviews/d1_theorem_a_nonauthor_20260928/REVIEW.md` (blob `bc11369c…`, Math-#106 v2) and [main#63 comment 5873873903](https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5873873903) | A1–A7, CAP, Theorem A and E2 C1–C6 ACCEPT; W1 | +| G2 | xAI Grok, Lucas lane | `reviews/d1_section9_borel_grok_lucas_20260928/REVIEW.md` (blob `48ea9ad5…`) | E2 C1–C6 ACCEPT. Narrative record; its runner is absent. | +| G3 | xAI Grok via Cursor, replay rounds 3–9 | main `incoming/grok-session-20260926-replay/` | Partial slices; see the table below | +| G4 | xAI Grok 4.7 via Cursor, `bc-81faa729` and `bc-dfbce246` | [main#67 5841270276](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841270276), [5841782206](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206) | While reviewing D2, re-derived P (6.2), the depth-failure integral, the m=1 split, and the far and M4 Markov pieces. Also checked the (11.3) = (13.6) coefficient identity. | +| C2 | Anthropic Claude, this record | §6 below | §11 and §12 ACCEPT; spot re-derivations of (13.1), (13.4) and (15.2) | +| O1 | OpenAI technical review of Math-#106, [5341593068](https://github.com/d6g8k5htny-coder/Math-/pull/106#pullrequestreview-5341593068) | — | Confirms W1 and the A5/A6 fixtures. **Author provider, so non-discharge.** | + +**Interfaces.** "Full" means a nonauthor provider re-derived the whole interface. "Partial" means a nonauthor provider +checked part of it; partial checks never discharge on their own. `reconciliation_check.py` computes the full-depth +counts from `LEDGER.json`. + +| Interface | P lines | Claim | Full-depth nonauthor ACCEPT | Partial second-provider check | +|---|---|---|---|---| +| S1 §1 | 9–53 | model, conventions, statements of A–C | G1, C1 (xAI, Anthropic) | — | +| A1 §2 | 54–68 | positive spectrum, jet rank, smooth version | C1 | G3 "qualitative positivity OK" | +| A2 §3 | 69–114 | contact frame, `12 r^{-(d+3)}`, uniform gap, (3.5) | C1 (including the gap proof) | G3 certified `det T_r` | +| A4 §4 | 115–137 | (4.1)–(4.3), independent residual | C1 | — | +| A3 §5 | 138–183 | (5.1)–(5.5), type convergence, UI, floor | C1 (O1 agrees but is author-provider) | G3 UI-majorant structure and `A_M -> A_0` | +| A3-E1 | erratum | `D_r = diag(r^{-1/2}, I)` | C1, G3 | — | +| A5 (6.1) | 184–197 | canceled-pivot bound, all m | C1 | G3 (m=1) | +| A6 (6.2) | 198–211 | two soft factors, including the 3/2 saddle term | C1, G4 | — | +| A7 §7 | 212–275 | `r^5` numerators, corank strata, m=1 far branch, M4, (7.8) | C1 | G4 numerator structure | +| CAP §§1–5 | cap 1–168 | deterministic separating cap, global maximin | C1, G3 | — | +| D1-A §8 | 276–297 | pinned genericity, Borel elder event, `G_r =>` partner S | G1, C1 | — | +| D1-B §9 | 298–309 | Borel marked Kac–Rice as written | G1 | — | +| D1-B-E2 | E2 | §9 replacement v1.1, C1–C6 | C1, G2 | — | +| D1-C §10 | 310–336 | radial ledger `r A_r dr db dk dσ` | G1 | C1 (#123 E9 check) | +| §11 | 337–364 | pushforward, (11.3), difference (1.2) | C2 | G4 ((11.1) and the (11.3) identity) | +| §12 | 365–385 | cumulative and moment corollaries | C2 | — | +| D1-D §§13–14 | 386–446 | target-growth majorant, `k -> 0`, off-diagonal (14.1) | G1 | C2 spot checks | +| D1-E §15 | 447–481 | parity, disintegration, (15.2) | G1 | C2 gamma factor | + +Every interface has a full-depth ACCEPT from at least one nonauthor provider. Six have full-depth ACCEPTs from both +nonauthor providers: S1, A3-E1, A6, CAP, D1-A and D1-B-E2. Nine more have a partial check by the other provider. Three +have a single provider at any depth: A4, §12, and D1-B. D1-B is the original §9 text, which the reading rule replaces by +D1-B-E2 (two providers), so it needs no second check. The project's established reconciliations (D2, D3, D4, D6 in STATUS) +each rest on one provider-distinct nonauthor review, so this meets the existing bar. §8 invites second-provider checks, +prioritized by that gap, but they are not a precondition. + +## 5. Residual items and their discharge + +Every item that an earlier record left open, AMEND or "not closed" is listed below with where it is discharged. + +| Open item | Raised in | Discharge | +|---|---|---| +| A3: wrong congruence factor `diag(√r,I)` | main#63 5841947606; G3 `A3_TYPE_CONVERGENCE` C1 | E1, landed by Math-#64 at `d8f5505`. Algebra confirmed by G3 and C1. | +| A3: UI of `W_r/r^2` | G3 C4 | C1 A3 item 4: `sup E(W_r/r^2)^2 < ∞` from (5.3), (5.1), `‖A_S‖ <= ‖A_M‖ + rM3` and A4. G3 `A3_UI_MAJORANT` gives the same majorant. | +| A3: type-indicator convergence under `D_r` | G3 | C1 W1: Slutsky joint convergence in law. The discontinuity set lies in `{det A_0 = 0}`, which is null (C1 item 5). | +| A3: `z_* > 0` | G3 C3 | C1: `A_0` is a nondegenerate Gaussian on `Sym_m`, so it charges the open cone `{A < 0}`. Continuity plus compactness gives (5.5). | +| A3: `A_M -> A_0` uniformly | G3 `A_M_TO_A0` (conditional on A1–A2) | C1 A3 item 3 and A1–A2 below. | +| A2: "single missing estimate", the uniform Gramian gap | G3 `A2_RESIDUAL_GAP` | C1 A2. Each coordinate of `U_r ⊕ vec A_r` is `∫ ∂^α f(tu) dμ_r(t)` with `μ_r -> δ_0`, so the covariance is jointly continuous on `[0,r_0] × O(d)`. It is positive definite at `r=0` by A1. Compactness gives `λ_min >= c_* > 0`. This is existential, which is all Theorem A claims. | +| A5 for `m >= 2` | G3 table | C1 A5: `det H_M = rα det(B' )` with `B' = A - (r/α)ββ^T`, and `0 < -B' <= B` on typed support. Symbolic for m=1,2,3; probed on 1,962 exact instances. | +| A7 `r^5` integrals; far branch | G3 table; C1 v1 | C1 A7, and G4. The v2 correction is that `λ >= 1/(Dr)` lies in the **majorant** event, not necessarily in depth failure. The far branch is kept in the upper bound. | +| Cap pairing: embedded chart and Morse/§8 | G3 `EMBEDDED_CHART_AND_MORSE`, `CAP_PAIRING_IDENTITIES` | §8 by G1 and C1. The embedding radius is made explicit below. | +| Embedding radius not displayed | G3 `EMBEDDED_CHART_AND_MORSE` ("AMEND the slogan") | Below: `r < L/(4√2)`. | +| §9 write-up gap (monotone-class passage) | Math-PR112 review, recorded in E2 | E2 v1.1, reviewed by C1 and G2. C1 withdrew the theorem-number objection on primary-source grounds (arXiv:2304.07424v3, Theorems 2.1, 2.2, 7.1, Remarks 7–8). | +| §§11–12 never reviewed | This reconciliation (line audit) | §6 below. | +| "Second non-Claude review of W1 and A2 before integration" | C1 request | W1 is wording only and is confirmed by O1 (non-discharge) and G3's in-law route. For A2 see the provider note in §4. The request stands as §8's open invitation. It is not a blocker under the existing bar. | + +**W1 (wording; proof unchanged).** P l. 165 says the scaled endpoint Hessians converge "in probability" to +`diag(∓6k, A_0)`. Correctly: + +- `D_r H_M D_r - diag(-6k, A_M)` and `D_r H_S D_r - diag(6k, A_M)` tend to 0 in probability, by (5.1), (5.2) and + `‖A_S - A_M‖ <= rM3`; +- `A_M -> A_0` in law. + +Slutsky then gives joint convergence in law, and continuous mapping applies off the null set `{det A_0 = 0}`. P's +subsequence paragraph (l. 176) already argues in law. + +**Embedding radius.** Let `D = [-2r, 2r] × B̄(0, 2r)`, the cap cylinder, which contains all of `G_r`'s geometry. Then +`D ⊂ B̄(0, 2√2 r)`. On `R^d/(LZ^d)` the exponential chart at any point is injective on the open ball of radius +`L/2`. So `D` is embedded whenever `2√2 r < L/2`, that is `r < L/(4√2)`. Theorem A's existential `r_*` may be, and is, +taken below this. This makes explicit what P §1 says with "for sufficiently small r". + +## 6. Review of the previously unreviewed §§11–12 (C2) + +The line audit found that no earlier record reviewed P lines 337–384. G1 lists 215–217, 276–296, 298–308, +88–94 and 310–335, 386–445, and 447–480. C1 assumed that D1-A–E covered "§§10–15", which it did not. Both sections +were reviewed here from the bytes. + +**§11 — ACCEPT.** + +- (11.1): `r = (ℓ/k)^{1/3}` gives `r dr/dℓ = (1/3) k^{-2/3} ℓ^{-1/3}`. +- Restriction: for `ℓ < k_- r_*^3`, every `k ∈ K` has `r < r_*`, so the whole pushforward stays inside Theorem A's + range. +- (11.2): at fixed `(b,k,u)`, pushing `r A_r dr` forward to `ℓ` gives `ℓ^{-1/3} A_{(ℓ/k)^{1/3}}/(3k^{2/3}) dℓ`. + Integrating over `B × K × S^{d-1}` gives the density version; `ν_eld` carries the extra factor `p_r`. +- (11.3): as `ℓ -> 0`, `A_r -> A_0` uniformly with a uniform bound (10.3), and `p_r -> 1` uniformly by Theorem A. + Dominated convergence on the compact domain gives `ℓ^{1/3} ν -> ∫ A_0/(3k^{2/3}) = 4∫ k^{-2/3} π_0 z_0 = c_{B,K}`, + using `A_0 = 12 π_0 z_0`. Positivity follows from `π_0, z_0 > 0` on a set of positive measure. G4 verified the same + identity in (13.6)'s form. +- Difference: `ν_cand - ν_eld = ℓ^{-1/3}∫ A_r(1-p_r)/(3k^{2/3})`. Theorem A gives `<= ℓ^{-1/3} ∫ A^* C r^3/(3k^{2/3})` + with `r^3 = ℓ/k`, so the bound is `C ℓ^{2/3} ∫ A^*/(3k^{5/3})`. This is finite because `k >= k_- > 0`, which proves + (1.2). The exponent `ℓ^{2/3} k^{-5/3}` is checked exactly. + +**§12 — ACCEPT.** + +- (12.1): `∫_0^t c ℓ^{-1/3}(1+o(1)) dℓ = (3/2)c t^{2/3}(1+o(1))`, and `∫_0^t C ℓ^{2/3} = (3/5)C t^{5/3}`. +- (12.2): for `q > -2/3` the integrand `ℓ^{q-1/3}` is integrable at 0, giving `c t^{q+2/3}/(q+2/3)`. +- (12.3): for `q > -5/3`, `∫_0^t ℓ^q C ℓ^{2/3} = C t^{q+5/3}/(q+5/3)`. +- For `q <= -2/3`, each individual expectation diverges because `c > 0`. (12.3) is stated for the nonselected measure + directly, not as a difference of infinities. +- The unrestricted leading forms follow from Theorem C in the same way. The compact difference rates are correctly + **not** extended. + +**Spot re-derivations for §§13–15.** These do not replace G1; they are a second provider's check. + +- (13.1): `(b - kr^3/2)^2 + 144k^2 >= (b^2+k^2)/2` for `r <= 1`. Checked exactly at sample rationals and by the + inequality `b^2 <= 2(b - kr^3/2)^2 + k^2/2`. +- (13.4): `W_r/r^2 <= C(1+‖f‖_{C^3})^{2d}` follows from (5.1) and (5.3) with no division by `k`. +- (15.2): with `t = 12k` and `w = s^2/2`, `144∫_0^∞ k^{4/3}φ_τ(12k)dk = Γ(7/6)τ^{4/3}/(24^{1/3}√π)`. Checked + numerically to relative `1e-9` in `reconciliation_check.py`, and exactly for the power bookkeeping. + +## 7. Consumption contract + +| Consumer | Consumes from D1 | After this reconciliation | +|---|---|---| +| D2 Theorem R, `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf…`) | The §8 comparison `1-p_r <= Q^W(G_r^c)`, §9 (read as E2), §10, §14 (14.1), and (15.2) as `c`. **Not** (1.1). | Every consumed interface is reconciled. `claims` moves FAIL_CLOSED -> REVIEWED_SCOPED at Theorem R's stated `O(1)` scope. | +| D3 SIDE24 coefficient, `coefficients/side24_v1/PROOF.md` (blob `44b66f04…`) | (15.2) as the leading coefficient of Theorem C | The persistence interpretation is reconciled. `claims` moves HOLD_WITH_DOMAIN -> REVIEWED_SCOPED. The arithmetic acceptance (#65) is unchanged. | +| Direct lower bound, `ELDER_LOWER_AND_DENSITY_GAP.md` (E4)/(E11) | Theorem A (1.1) upper half, §§9–11 | The two-sided planar composition is now fully sourced. (E2) itself stays as reviewed in #123. | +| D4 fixed-remote (RM), RC (#110), IW (#107), #120 | P's model and observation **conventions** only, plus RM's own "Theorem A" | Unaffected. No D1 theorem is consumed. | +| Any new consumer | — | Cite the §1 identity set, this record, and a scope no larger than §2. | + +**Still not consumable.** + +- Numerical `C`, `r_*`, `z_*`, `c_{B,K}` or `c_{d,L}`. +- (1.2)'s difference rate outside compact marks. +- Any lower bound for `d >= 3`. +- A globally uniform `O(r^3)` constant over unbounded marks. +- RN, 24-jet, JETMOD or H3/LPW numerical certificates. +- `lemma_closed`, any prize, or "unconditional acceptance of the lifetime parent", which is a forbidden global + inference in `claims/LANDING_CLAIMS.json`. This acceptance is scoped. + +## 8. Independence, lineage and requests + +**Dual role.** The reconciler is also reviewer C1 of A1–A7, CAP and E2, and reviewer C2 of §§11–12. This is disclosed +rather than hidden. + +**What the reconciliation relies on, and what it does not.** Nothing here is accepted on the strength of a green test, +a hash, a same-provider review or O1. Every interface rests on a written nonauthor derivation. + +**Merge.** The register PR carrying this record must be merged by a non-Claude lane after reading it. Claude does not +self-merge. + +**Invitation.** A second-provider check is invited in this priority order: + +1. A4 and §12, which have one provider at any depth; +2. the full-depth gaps behind partial checks: A2's uniform gap, A5 for `m >= 2`, A7's division by the floor, and + §11's difference bound; +3. the reverse direction for D1-C, D1-D and D1-E, where xAI is full-depth and Claude is partial. + +Any lane may return COUNTEREXAMPLE with an exact equation. That would reopen only the affected interface and its +dependents, and the graph's reverse-impact rule would mark them REVALIDATION_REQUIRED. + +## 9. Register transitions justified by this record + +Math-, in the same PR: + +- `PROOF_INDEX.md`: D1 moves to "Reviewed scoped results" with the §2 scope. The §9 repair is recorded as reviewed by + C1 and G2. The D2 entry is updated to name its reconciled imports. +- `claims/LANDING_CLAIMS.json`: + - `lifetime-remainder` FAIL_CLOSED -> REVIEWED_SCOPED; + - `side24-coefficient` HOLD_WITH_DOMAIN -> REVIEWED_SCOPED; + - dependency reviews are bound to local review files. +- `frontiers/downstream_gate_20260925/GRAPH.json`: + - `math.uniform-matrix-cap-lifetime` AUTHOR_SIDE_CANDIDATE -> PROVED_REVIEWED, with this record as evidence and + the §2 scope. It is the only classification that satisfies a still-required premise, and D2 and D3 require it. + - D2 and D3 gain `review_disposition: ACCEPT` metadata under the existing convention (commit `cc3a991`). + - Four reading-rule component nodes (E1, E2, CAP and this record's directory) are added as `PROVED_REVIEWED` + source nodes with required edges from D1. Each carries the ledger reviews its role needs: C1 and G3 for E1 and + CAP, C1 and G2 for E2. This record is an Anthropic, nonauthor record; its W1 and radius items were raised in C1 + and G3 and its §6 is C2. OpenAI's technical check of this record (review 5345687857) is listed as author-provider + non-discharge. This closes the gap that review found: before it, E1, E2 and W1 were named only in metadata, so + editing them left D1's source snapshot unchanged. + - No node becomes `controlling`. Satisfying D2's and D3's required edge to D1 is not promotion: their own + classifications stay author-side, so `promotion_allowed` still refuses both. + - The fail-closed tests are kept on a pre-reconciliation fixture, so they still prove that an unreviewed parent + blocks both dependents. New tests make real two-commit edits and deletions of each component with P unchanged, + and require D1, D2 and D3 to be proposed `REVALIDATION_REQUIRED`. An unrelated edit must propose nothing, and + the pre-binding graph is shown to miss the component edit. + +main, in a companion PR pinned to this record's merged commit: + +- `STATUS.md`: D1 moves from AMEND/open to ACCEPT — scoped. +- The glossary's D1 row is updated. +- The museum and shop projections are regenerated. + +## 10. Checks + +`reconciliation_check.py` (Python standard library): + +1. Binds every §1 source and every local review record by Git blob, bytes and SHA256. +2. Cross-checks the hard-gate graph against `LEDGER.json`: + - D1 is bound to P's exact bytes; + - E1, E2, CAP and this directory are D1 source nodes behind required edges; + - each component node carries exactly the ledger reviews of its interfaces; + - the record's author provider differs from P's; + - two negative controls (a dropped edge; an author-only review basis) are rejected. +3. Validates `LEDGER.json`: + - every interface has at least one nonauthor ACCEPT; + - author-provider reviews are non-discharge; + - P's §§1–16 line ranges are fully covered; + - every residual item has a discharge pointer. +4. Re-verifies the reading-rule mathematics exactly: + - E1 congruence versus the displayed factor; + - (5.3), (6.1) and (6.2) on exact rational instances; + - (7.3) and the (7.5) split; + - the cap constants and κ-scaling; + - the embedding radius, the radial and lifetime ledgers, (11.1)–(11.3), the difference exponent and the §12 + integrals; + - the (13.1) coercivity bound; + - the (15.2) gamma factor, numerically. +5. Rejects nine semantic mutants. + +A green run is evidence that the bookkeeping and finite identities hold. It is not the analytic review, which lives in +C1, G1, G2, G4 and §6. diff --git a/frontiers/lifetime_moment_boundary_20260930/test_verify.py b/frontiers/lifetime_moment_boundary_20260930/test_verify.py new file mode 100644 index 000000000..4024f8783 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/test_verify.py @@ -0,0 +1,111 @@ +"""Real CLI controls for receipt selection and immutable local source copies.""" +from pathlib import Path +import hashlib +import json +import shutil +import subprocess +import sys +import tempfile +import unittest + +HERE = Path(__file__).resolve().parent +REPO = HERE.parents[1] +LIFETIME = Path('frontiers/lifetime_moment_boundary_20260930') +RADIAL = Path('reviews/radial_moment_corollaries_20260930') + + +class PublicationVerifierTests(unittest.TestCase): + def setUp(self): + temporary = tempfile.TemporaryDirectory() + self.addCleanup(temporary.cleanup) + self.root = Path(temporary.name) / 'repo' + for packet in (LIFETIME, RADIAL): + shutil.copytree(REPO / packet, self.root / packet, + ignore=shutil.ignore_patterns('__pycache__')) + + def replay(self): + flags = ['-O'] if sys.flags.optimize else [] + return subprocess.run( + [sys.executable, '-B', *flags, '-S', + str(self.root / LIFETIME / 'verify.py')], + cwd=self.root, capture_output=True, text=True, timeout=30) + + def test_clean_packet_replays_both_modes(self): + result = self.replay() + self.assertEqual(result.returncode, 0, result.stderr) + report = json.loads(result.stdout) + self.assertTrue(report['passed']) + self.assertEqual([row['mode'] for row in report['modes']], + ['normal', 'optimized']) + + def test_changed_optimized_lifetime_receipt_is_rejected(self): + path = self.root / LIFETIME / 'CHECK_RECEIPT_OPTIMIZED.json' + data = json.loads(path.read_text()) + data['good_quartic_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('lifetime replay differs', result.stderr) + + def test_changed_optimized_loss_receipt_is_rejected(self): + path = self.root / LIFETIME / 'LOSS_CHECK_RECEIPT_OPTIMIZED.json' + data = json.loads(path.read_text()) + data['rational_probability_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('loss replay differs', result.stderr) + + def test_changed_normal_receipt_remains_rejected(self): + path = self.root / LIFETIME / 'CHECK_RECEIPT.json' + data = json.loads(path.read_text()) + data['good_quartic_cases'] += 1 + path.write_text(json.dumps(data)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('lifetime replay differs', result.stderr) + + def test_changed_optimized_radial_receipt_is_rejected_after_custody_check(self): + path = self.root / RADIAL / 'moment-checks-optimized.json' + data = json.loads(path.read_text()) + data['tests'] += 1 + path.write_text(json.dumps(data)) + # A self-consistent custody record must not replace actual replay. + manifest = self.root / RADIAL / 'COROLLARY_CUSTODY.json' + custody = json.loads(manifest.read_text()) + for row in custody['artifacts']: + if row['path'] == path.name: + row['bytes'] = path.stat().st_size + row['sha256'] = hashlib.sha256(path.read_bytes()).hexdigest() + manifest.write_text(json.dumps(custody)) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('radial replay differs', result.stderr) + + def check_alias_rejected(self, relative): + path = self.root / relative + target = self.root / 'byte-identical-target' + path.rename(target) + path.symlink_to(target, target_is_directory=target.is_dir()) + result = self.replay() + self.assertNotEqual(result.returncode, 0) + self.assertIn('symlink', result.stderr.lower()) + + def test_byte_identical_lifetime_source_symlink_is_rejected(self): + self.check_alias_rejected(LIFETIME / 'sources/P.md') + + def test_byte_identical_radial_source_symlink_is_rejected(self): + self.check_alias_rejected(RADIAL / 'P.md') + + def test_lifetime_source_directory_symlink_is_rejected(self): + self.check_alias_rejected(LIFETIME / 'sources') + + def test_lifetime_parent_directory_symlink_is_rejected(self): + self.check_alias_rejected(Path('frontiers')) + + def test_radial_parent_directory_symlink_is_rejected(self): + self.check_alias_rejected(Path('reviews')) + + +if __name__ == '__main__': + unittest.main(verbosity=2) diff --git a/frontiers/lifetime_moment_boundary_20260930/verify.py b/frontiers/lifetime_moment_boundary_20260930/verify.py new file mode 100644 index 000000000..af57cbc79 --- /dev/null +++ b/frontiers/lifetime_moment_boundary_20260930/verify.py @@ -0,0 +1,90 @@ +"""Replay finite controls and exact packet custody; not theorem acceptance.""" +from pathlib import Path +import hashlib +import json +import stat +import subprocess +import sys + +HERE = Path(__file__).absolute().parent +REPO = HERE.parents[1] +RADIAL = REPO / 'reviews/radial_moment_corollaries_20260930' + + +def require(ok, message): + if not ok: + raise RuntimeError(message) + + +def read_regular(root, name): + part = Path(name) + require(part.parts and not part.is_absolute() and '..' not in part.parts, + 'unsafe source path') + path = root / part + cursor = REPO + require(not cursor.is_symlink() and cursor.is_dir(), 'invalid repository root') + for component in path.relative_to(REPO).parts: + cursor = cursor / component + mode = cursor.lstat().st_mode + require(not stat.S_ISLNK(mode), 'symlink source path: ' + str(cursor)) + require(stat.S_ISREG(mode) if cursor == path else stat.S_ISDIR(mode), + 'non-regular source path: ' + str(cursor)) + return path.read_bytes() + + +def verify_lifetime_paths(): + sources = json.loads(read_regular(HERE, 'SOURCES.json')) + for row in sources['sources']: + read_regular(HERE, row['local_path']) + for name in ('PROOF.md', 'check_exact.py', 'LOSS_COROLLARY.md', 'check_loss_exact.py'): + read_regular(HERE, name) + + +def verify_radial(root): + custody = json.loads(read_regular(root, 'COROLLARY_CUSTODY.json')) + for row in custody['artifacts'] + custody['sources']: + name = row.get('path', row.get('local_path')) + raw = read_regular(root, name) + require(len(raw) == row['bytes'], 'size drift: ' + name) + require(hashlib.sha256(raw).hexdigest() == row['sha256'], 'SHA256 drift: ' + name) + if 'git_blob' in row: + blob = hashlib.sha1(f'blob {len(raw)}\0'.encode() + raw).hexdigest() + require(blob == row['git_blob'], 'Git blob drift: ' + name) + return len(custody['artifacts']) + len(custody['sources']) + + +def main(): + require(sys.version_info[:2] == (3, 11), 'Python 3.11 required') + verify_lifetime_paths() + count = verify_radial(RADIAL) + rows = [] + for flags in ([], ['-O']): + mode = 'optimized' if flags else 'normal' + suffix = '_OPTIMIZED' if flags else '' + lifetime = subprocess.run([sys.executable, '-B', *flags, '-S', str(HERE/'check_exact.py')], + check=True, capture_output=True, text=True) + current = json.loads(lifetime.stdout) + expected = json.loads(read_regular(HERE, f'CHECK_RECEIPT{suffix}.json')) + require(current == expected, 'lifetime replay differs from exact receipt') + radial = subprocess.run([sys.executable, '-B', *flags, '-S', str(RADIAL/'moment_consequence_checks.py')], + check=True, capture_output=True, text=True) + current_radial = json.loads(radial.stdout) + require(current_radial == json.loads(read_regular(RADIAL, f'moment-checks-{mode}.json')), + 'radial replay differs from exact receipt') + loss = subprocess.run([sys.executable, '-B', *flags, '-S', str(HERE/'check_loss_exact.py')], + check=True, capture_output=True, text=True) + current_loss = json.loads(loss.stdout) + require(current_loss == json.loads(read_regular(HERE, f'LOSS_CHECK_RECEIPT{suffix}.json')), + 'loss replay differs from exact receipt') + rows.append({'mode': mode, + 'lifetime_quartics': current['good_quartic_cases'], + 'lifetime_negative_controls': current['outside_hypothesis_axial_negative_controls'], + 'normalization_exponents': current['normalization_exponent_cases'], + 'lifetime_sources': len(current['source_identity_checks']), + 'radial_tests': current_radial['tests'], 'loss_controls': current_loss}) + print(json.dumps({'passed': True, 'modes': rows, 'radial_custody_records': count, + 'scientific_effect': 'NONE'}, sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/frontiers/local_elder_geometry_20260930/MIXED_SCALES.md b/frontiers/local_elder_geometry_20260930/MIXED_SCALES.md new file mode 100644 index 000000000..57c24c49b --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/MIXED_SCALES.md @@ -0,0 +1,248 @@ +# Mixed window geometry: the scale distinction and a fixed-scaled falsifier + +Read-only source-bound mathematical note, 30 September 2026. +Disposition: AUTHOR_SIDE / HOLD. This note is not a new repository verdict, +count/rate packet, independent-provider review, or register transition. +Scientific effect NONE. No GitHub, STATUS, GRAPH, navigation, or source edits. + +## Model and exact source boundary + +Fix the variance-one periodized Gaussian field on a torus of side L>0, +an orthonormal frame, birth b in R, and positive gap mark k>0. The pins are +M=(-r/2,0), S=(r/2,0), their values are b,b-kr^3, and both gradients vanish. +Use the original endpoint Gaussian regression Q_r, endpoint determinant mark +W_r, and FULL endpoint-only normalizer Z_r=E_Qr W_r. Write P_r=Q_r^W. +N_r counts all additional critical points in I_r=(b-kr^3,b), excluding M,S. +In d=2, W_r=F_2(H_M)F_1(H_S). No adjacency or persistence selection is conditioned. + +Immutable reads: main 98fdb54954a2d9ac0b8ba8ca03e279701f70d751; +PR162 head 382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf; +PR166 head be9b7fa95de9b7e59a67347640e157ed3027f624. +Full proofs and source manifests were fetched using GitHub tools. The proof +copies used below have matching Git blob hashes. Applications keep their source +hypotheses and review boundaries; publication does not remove those conditions. + +| Tag | Exact path and Git blob | Interface used | +|---|---|---| +| MAIN-MIX | frontiers/c6_cluster_law_20260929/PROOF.md; ba492c8e62e58bfc055fc8254c790e893d346ba5 | Current repaired main proof, §5 Lemmas5.1–5.2, (5.3a)–(5.4) | +| C6 | frontiers/c6_palm_route_20260929/PROOF.md; 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5 | §1 Theorem Q, total factorial upper bounds; ordinary fourth moment follows with the first moment | +| DL | frontiers/d5_dimension_lift_20260929/PROOF.md; 9d82c707fdb17d3072a8930f26dabedf59e456fc | Theorem I_d, (1.5)–(1.6), height-window shell and annulus bounds | +| CUB | frontiers/planar_cubic_cluster_20260929/PROOF.md; bb446d08db8a944537a743ad550b88c1c2ad5758 | §2 (C3)–(C11), exact planar classifier; §4 (G3), (G6)–(G10), compact-sector regression and rare density | +| SC | frontiers/spectral_cluster_closure_20260929/PROOF.md; 16c56821b52fd76b0be791622b9c3809eafde75a | PR162, §7 (23)–(24), fixed-origin-jet alternative mixed proof | +| TS | frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md; a32fd5f7d941bbe1fe943df045b1e0fbec8d691c | PR166, §4 Theorem L, (L1)–(L3) | + +MAIN-MIX's actual successor review is OpenAI5360192822 at +f7c33955d005c03d1010835f52f500c4b853a847, bound to the blob above. It accepts +the repaired conditional theorem and specifically its near/far expectation. +It is source-exposed and provides zero organizational-independence credit. +This current 70131-byte proof differs from the historical 51313-byte source +1ce3769e76c540ab03b42f2deb6f2d1124459a81 criticized by SC. +MAIN-MIX SOURCE_MAP.json is blob ef51c582d0b1c874aabb01fce3b35c10631f7fc6. + +SC's separate §7–8 ACCEPT is OpenAI5359879627, same-provider nonauthor scrutiny. +TS's actual conditional Slice A ACCEPT is Anthropic5360227991 at mathematical +head 7c82252533c3fe14ee262f7f87c0549c10968592, bound to the same TS blob. +TS SOURCES.json, blob59d11113e208965fefeaadd1bd5866585e77dc82, pins its SC/CUB/ +RM/C6/P interfaces. That consumer review does not revalidate its parents. +The falsifier below consumes CUB's Gaussian (G6)–(G10) interface as a conditional +source premise; it does not infer acceptance of that interface from the separate +deterministic-classifier review. This note itself remains AUTHOR_SIDE / HOLD. + +## What the existing scale statements establish + +For fixed finite A>=4 and fixed positive physical rho, MAIN-MIX (5.4) gives + + P_r(N_A>0,N_far^rho>0) + <= E_r[N_A 1{N_far^rho>0}] <= C_(A,rho) r^(9/2), + +where N_A counts points in B(0,Ar) and N_far^rho those at physical distance +at least rho. Its remote covariance floor is proved by five-jet whitening, +with the complete raw collar frame and the residualized pin gradient frame. +All observation targets and the original W_r/Z_r are retained. Constants are +not uniform as A grows or rho shrinks. + +This existing event estimate also implies the counted mixed little-o. Let +E={N_A>0,N_far^rho>0}. C6 and the first moment give E_r N_r^4<=Cr^3, and +N_A N_far^rho<=N_r^2 1_E. Hence + + r^-3 E_r[N_A N_far^rho] + <= (r^-3 E_r N_r^4)^(1/2) (r^-3 P_r(E))^(1/2) -> 0. + +SC has a different proof: its r^(9/2) estimate is only on a fixed compact +origin-jet box. Exhaustion gives its full o_(A,rho)(r^3) without claiming that +global polynomial rate. TS (L2) explicitly converts that result to counted +mixed pair mass. The two arguments and their review bindings are not interchangeable. + +TS (L1) gives the stronger localization statement for ANY deterministic cutoff + + delta_r -> 0, delta_r/r -> infinity: + +for every fixed q>=2, with N_in counting points inside delta_r, + + r^-3 E_r[(N_r)_q-(N_in)_q] -> 0. + +Thus all leading second-factorial mass is inside every such shrinking physical +ball. Its coefficient is 2a2 and its ordered-pair limit is TS (L3). This is a +conditional PR166 result, not a main-register update. Its proof takes r->0 at +fixed near and remote cutoffs, then the near cutoff to infinity and the physical +remote cutoff to zero. No convergence rate or spatial-moment interchange is supplied. + +DL's height-window intermediate estimate is uniform in A0 and rho: + + E_r N({A0 r<=|x|<=rho}) <= C r^3(A0^-2+rho^2), + A0>=4, A0 rinfinity and +s->0 is genuinely intermediate. In contrast, fixed A0 and fixed rho leave a +possibly nonzero r^3 coefficient. The following exact sector proves why a +fixed large multiple of r cannot be treated as a fixed physical remote cutoff. + +## Exact fixed-scaled mixed sector + +This construction is planar, d=2. Fix k>0. Pick an integer n>=4 and a fixed z>0. +Define + + alpha = (1/2) sqrt(1+3/n^2), + lambda = 9k/(n^2 z^2), + t = (n alpha+1)/(n+1), + a=0, beta=B=-lambda, s=-lambda t, + D=lambda(1-alpha)/((n+1)z), c=2D. + +Since 1/20. + +The two proposed extra roots are + + (u1,Z1)=(-alpha,z), (u2,Z2)=(-1,-nz). + +Their stationarity is exact. CUB (C7) requires + + Q=6k(u^2-1/4)+(B/2)Z^2=0, + s+Bu+DZ=0 when Z!=0. + +At the first point, alpha^2-1/4=3/(4n^2), giving + + Q1=9k/(2n^2)-lambda z^2/2=0, + s+B u1+D Z1 + =lambda[-t+alpha+(1-alpha)/(n+1)]=0. + +At the second point, + + Q2=9k/2-lambda n^2 z^2/2=0, + s+B u2+D Z2 + =lambda[1-t-n(1-alpha)/(n+1)]=0. + +Both Z coordinates are nonzero. For B<0, CUB (C11) puts a root in the strict +height window precisely when u_w0, + eta2=(3t-1)/2 in (1/4,1). + +For n>=4, alpha<=sqrt(19)/8<11/20 and t<2/3. Therefore eta1<1/50<1, +so both height inequalities can also be checked without the classifier. +Their Hessian determinants, by CUB (C8), are + + det Hess P at root1 = -3k lambda(4t alpha-1)<0, + det Hess P at root2 = -3k lambda(4t-1)<0. + +They are strict, nondegenerate saddles. The cubic has exactly these two extra +in-window points by Theorem C. They remain distinct and away from the pins. + +## Pin and collar locations in physical coordinates + +For z=1/10, the first root's physical distance from M is + + r sqrt((alpha-1/2)^2+1/100). + +Using alpha-1/2<1/20, this is less than (sqrt(5)/20)r (n/10)r. + +For any prescribed fixed A, choose the finite integer n>10A. The first point +is near the maximum pin and the second is outside the fixed scaled ball Ar. +Both distances still tend to zero physically with r. + +For a compact-collar example instead take z=1. Then both distances from the +first root to the pins exceed r/4, and its midpoint radius is at most +(sqrt(83)/8)r<1.2r. It lies inside the fixed collar C(1/4,4). The second root's +radius exceeds nr; choose a finite n>A. These are two variants of the same +construction, not a change of the conditioned endpoint law. + +Because n and z are fixed before r->0, a sufficiently large FIXED scaled radius +R contains both roots with strict spatial margins. For sufficiently small r, +the Rr chart embeds in the fixed torus. No limit n=n(r)->infinity is used. + +## Positive Gaussian coefficient and the order on fixed scaled sets + +Choose disjoint small bounded open spatial neighborhoods U,V of the two roots. +For z=1/10, ensure U lies in the punctured radius-1/4 disk about (-1/2,0) and +V lies outside the radius-A midpoint ball. For z=1, ensure U lies in C(1/4,4) +and V lies outside radius A. Both closures fit inside one fixed radius R ball. + +The strict typing, heights, nondegenerate roots and spatial margins persist +on a sufficiently small compact four-dimensional jet box C of positive volume +around theta0=(s,a,beta,c). Allow a to vary in this box and evaluate positions +in the ORIGINAL coordinates X=u-aZ/(12k). Root continuity preserves the same +U,V inclusions after reducing the box. This avoids treating the measure-zero +slice a=0 itself as a positive Gaussian event. + +On C, the two stable extra roots consist of exactly one point in U and one in V. +CUB (G6)–(G9) couple the actual endpoint-conditioned field on C, give C^2 +convergence to this cubic in the fixed R ball, and yield + + W_r/r^4 -> w(theta), Z_r/r^2 -> z0>0. + +The same inverse-function/root-stability argument makes both spatial indicators +eventually one for almost every coupled realization and theta in C. Retaining +those indicators in CUB's actual disintegration (G10), then using its compact +domination, gives + + r^-3 P_r{Theta_r in C, N_(rU)>=1, N_(rV)>=1} + -> m_C, + m_C=z0^-1 integral_C w(theta) h0(0,a,beta,c) dtheta > 0. + +Here h0 is the strictly positive contact Gaussian density from CUB (G2)–(G3). +On the compact box w and h0 have positive minima, so + + m_C >= (min_C w)(min_C h0) |C|/z0 > 0. + +The power is the original rare-density ledger r*r^4/r^2=r^3: one curvature +coordinate A=rs, two soft endpoint determinants, and the FULL original normalizer. +No normalized soft-sector or witness-conditioned normalizer is introduced. + +Consequently E_r[N_(rU)N_(rV)] has a positive order-r^3 lower bound. +Since U and V are disjoint, + + 2 N_(rU)N_(rV) <= (N_r)_2, + +and C6 Theorem Q supplies the order-r^3 upper bound. Conditional on those exact +source interfaces, this fixed-scaled mixed raw pair mass is therefore Theta(r^3). +The coefficient can become small as the fixed A and chosen n grow; it remains +strictly positive for every finite choice. This is consistent with all iterated +or shrinking-cutoff localization statements above. + +Both counted extra points in this sector are saddles. Their factorial pair is +not a maximum–saddle persistence pair, and no elder matching of either point has +been asserted. That separate height-mark/selection problem is unchanged. diff --git a/frontiers/local_elder_geometry_20260930/MIXED_SCALES_REVIEW.md b/frontiers/local_elder_geometry_20260930/MIXED_SCALES_REVIEW.md new file mode 100644 index 000000000..5184de0c8 --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/MIXED_SCALES_REVIEW.md @@ -0,0 +1,64 @@ +# Noncontributor review of the fixed-scaled mixed geometry note + +Reviewer: OpenAI, elder_mark_nonauthor_review, 30 September 2026. This reviewer and the read-only child reviewer did not author or amend MIXED_SCALES.md. Same provider as its contributors; zero organizational-independence credit. No proof body, register, source, remote review or scientific status was mutated. + +Reviewed object: mixed_geometry_audit/MIXED_SCALES.md, SHA256 +d5892abc8a1d35e8bbf10aa14092ea4cd2458662953de22b6f9422e89f525b3b. + +The publication copy local_elder_geometry_20260930/MIXED_SCALES.md has the same exact SHA256. The published dependency map local_elder_geometry_20260930/SOURCES.json is bound at SHA256 aefaf14b9177870028dce73a50906d06e472a21d92543f8974c10fc0726fa30a; its 14 source entries independently match their cached exact bytes and distinguish the elder core from the companion-only count interfaces. + +**Verdict: ACCEPT, conditional on the exact named source interfaces.** No amendment is required for the fixed-scaled falsifier, the compact Gaussian sector argument, or the conclusions separating fixed scaled and fixed physical distances. + +## Source boundary + +I inspected the actual local source interfaces and independently checked their Git blob identities: + +| Interface | Git blob | +|---|---| +| MAIN-MIX §5, Lemmas 5.1–5.2, (5.3a)–(5.4) | ba492c8e62e58bfc055fc8254c790e893d346ba5 | +| C6 §1.2 Theorem Q | 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5 | +| DL Theorem I_d, (1.5)–(1.6), and first-moment bound | 9d82c707fdb17d3072a8930f26dabedf59e456fc | +| CUB classifier and (G6)–(G10) | bb446d08db8a944537a743ad550b88c1c2ad5758 | +| SC §7, (23)–(24) | 16c56821b52fd76b0be791622b9c3809eafde75a | +| TS §4 Theorem L, (L1)–(L3) | a32fd5f7d941bbe1fe943df045b1e0fbec8d691c | + +All six matched the note's exact mathematical source identities. This review is a consumer review of those interfaces; it does not independently revalidate the entire 70,131-byte MAIN-MIX proof, the full Gaussian regional count closure, or every remote review record. The note's source premises remain conditional, including CUB's Gaussian compact-sector theorem separately from its deterministic classifier. + +## Existing scale conclusions + +The event estimate cited from current MAIN-MIX is the displayed \(O_{A,\rho}(r^{9/2})\) result, with fixed \(A\) and physical \(\rho\). The note uses the repaired source, not SC's historical target. MAIN-MIX conditions on the full raw collar observations, and residualizes the pin gradient frame before whitening in the endpoint regime. The remote covariance comparison and its limiting floor are the source's actual statements. + +The conversion to a counted mixed estimate is valid. The source count bounds give +\[ +N^4=(N)_4+6(N)_3+7(N)_2+N,\qquad \mathbb E N^4=O(r^3). +\] +Together with \(N_A N_{\rm far}\le N^2 1_E\), Cauchy–Schwarz gives the stated rescaled little-o and, from the MAIN-MIX exponent, a bound of order \(r^{15/4}\) for fixed cutoffs. No independence of the counts is assumed. + +SC only asserts the polynomial mixed rate on a fixed jet box; its full mixed result is little-o after exhaustion. TS explicitly supplies the counted statement and localization for every cutoff \(\delta_r\to0\), \(\delta_r/r\to\infty\). DL's annulus bound is uniform in the displayed cutoffs but leaves a potentially nonzero \(r^3\) coefficient when both are fixed. These statements have correctly different orders of limits. TS's factor \(2a_2\) counts two orderings of two additional saddle witnesses, not an elder maximum/saddle pair. + +## Exact two-root construction + +The definitions give +\[ +\frac12<\alpha1\) and \(4t>1\). Thus both are distinct nondegenerate saddles strictly inside the window. The complete classifier proves there are exactly two additional window roots. + +All location margins check. At \(z=1/10\), the first root is in the punctured radius-\(1/4\) disk about the maximum pin, and the second has midpoint radius exceeding \(n/10\). At \(z=1\), the first root is in the fixed collar, away from both pin disks, with midpoint radius below \(1.2\), while the second has radius exceeding \(n\). Choosing a finite \(n\) before \(r\to0\) places the second root outside any prescribed finite scaled radius \(A\); it still lies at a physical distance tending to zero. + +## Positive Gaussian mass + +The proof correctly thickens the central \(a=0\) slice to a compact four-dimensional jet box of positive volume. The root locations are evaluated in the original coordinates under the inverse shear, so perturbing \(a\) preserves the chosen spatial sets after reducing the box. Strict root, height, type and spatial margins permit the actual local-root stability argument. + +Under CUB's conditional coupling, the two spatial occurrence indicators are eventually one for almost every target and realization, and its integrable endpoint-weight envelope applies. Inserting these bounded indicators in G10 therefore proves the stated positive \(m_C\), with the exact ledger \(r\cdot r^4/Z_r\sim r^3/z_0\). The full original normalizer is used once. This is not an event supported only on the null slice \(a=0\). + +The event supplies a positive order-\(r^3\) lower bound for \(\mathbb E[N_{rU}N_{rV}]\). Disjointness gives \(2N_{rU}N_{rV}\le(N_r)_2\), and C6's exact Theorem Q supplies the matching upper bound. Consequently the fixed-scaled mixed raw pair mass is \(\Theta(r^3)\), conditionally on those source interfaces. + +There is no contradiction with shrinking-cutoff localization: for any fixed finite construction both roots eventually lie inside \(\delta_r\) whenever \(\delta_r/r\to\infty\). No sequence \(n=n(r)\), growing-cutoff uniformity, or actual elder pair is claimed. The fixed-scaled counterexample therefore refutes only the improper identification of a large fixed multiple of \(r\) with a fixed positive physical remote distance. diff --git a/frontiers/local_elder_geometry_20260930/NONAUTHOR_REVIEW.md b/frontiers/local_elder_geometry_20260930/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..8ed018591 --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/NONAUTHOR_REVIEW.md @@ -0,0 +1,152 @@ +# Source-bound noncontributor review of the planar cubic elder mark + +Reviewer: OpenAI, elder_mark_nonauthor_review, 30 September 2026. + +This reviewer did not author or amend the proof. A read-only child reviewer, also OpenAI, checked the shear and critical-point algebra. All lanes use the same provider. This review earns zero organizational-independence credit and makes no source, scientific-register, claim-status, promotion, or publication decision. + +Reviewed object: local_elder_geometry_20260930/PROOF.md, SHA256 +f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b. + +**Verdict: ACCEPT, source-scoped, of the complete integrated mathematical candidate at those exact bytes:** Theorem E, §§2–6, including exact identification of the highest actual continued local saddle; the compact marked law, §7; the full fixed-planar failure-measure and joint conditional partner-location/death/lifetime law, §8; and the compact rejected-candidate lifetime difference coefficient, §9. The estimand, normalization and limitations in §§1,10 agree with the sources and the proved results. This review does not accept a higher-dimensional extension, an unrestricted refined lifetime law, an actual replacement-bar intensity obtained by duplicating rejected candidates, numerical constants, or any later successor bytes. + +## Source scope + +The published dependency map reviewed here is local_elder_geometry_20260930/SOURCES.json, SHA256 aefaf14b9177870028dce73a50906d06e472a21d92543f8974c10fc0726fa30a. It has 14 immutable source entries with required-core and contextual/companion-only roles. I independently matched all 14 entries against the exact cached source files for byte length, SHA256 and Git blob identity; all matched. The source refs include the stated main cut and the explicit SC/TS branch cuts. No parent interface is accepted merely by being listed or by having a matching hash. + +Earlier in the audit, I also checked the broader historical 19-file cache map elder_bridge_audit/SOURCES.json against those same identity fields; all matched. That historical cache count is not the published map's entry count. Both checks bind source identity, not mathematical acceptance. + +For the new mathematics, the load-bearing source interfaces inspected were: + +- [CUB], blob bb446d08db8a944537a743ad550b88c1c2ad5758: (C1)–(C14), Lemma S, and (G1)–(G10). The algebraic classifier, its fixed-disk pinned stability, full regression coupling, full normalizer, rare jet Jacobian, and conditional domination are the actual inputs. +- [P], blob dfed3b8d318a3ab1950957f393307733a4bef3f2: model/rank/regression interfaces in §§1–5, genericity and Borel maximin in §8, and the pair/lifetime ledger in §§10–11. I read the congruence erratum, blob 213594d6ca6a86fb938110f4d166d9ce275a02d0, the Borel repair, blob fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a, and the reconciliation reading rule/W1/embedding restriction, blob 75da2597971510f843f8d90c743950cb8c177342. +- [Q], blob 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5, §1.2–1.3, and [TWO], blob a32fd5f7d941bbe1fe943df045b1e0fbec8d691c, §§1–4, for the additional-window-critical-point estimand and its separation from the elder selector. Neither count theorem supplies or is used to prove the new topological mark. +- [LOW], blob aaefc8da9668e819591dd9db0b34442394f2362e, §§1–4, and [U], blob 5a55b179a974a90c65d257f3f93765cc6fb30bb8, §§1,3–5, for the compact-window density loss and the obstruction to importing it as an unrestricted rate. + +This review consumes those explicitly stated source interfaces and respects the published map's required versus context-only distinction. It does not revalidate every imported theorem in the broader historical source closure, including the complete global C6 proof or the complete D1 matrix-cap integration chain. Their existing review provenance is not new acceptance by this reviewer. + +## Deterministic theorem + +### Critical chord and global lower bound + +I checked the cubic interpolation identity independently. A degree-at-most-three restriction \(q(t)\) with \(q'(0)=q'(1)=0\) and endpoint values \(q(0)=0,q(1)=h\) has derivative \(q'(t)=6h\,t(1-t)\), hence \(q(t)=h(3t^2-2t^3)\). For \(h<0\), its minimum on \([0,2]\) is exactly \(h\), and its endpoint value is \(-4h>0\). Thus every negative critical point, including the pinned saddle, supplies an actual finite older path. This is sufficient for the maximin lower bound and does not assume that an arbitrary saddle is incident to a particular gradient branch. + +Taking the highest window critical height proves \(d_P(M)\ge h_*\). Its strict margin above \(-k\) when the window is nonempty persists under local \(C^0\) convergence, proving eventual preemption of \(S_i\) without any global counting inference. + +### No escape at infinity and compact upper barriers + +The shear and homogeneous gradient criterion are correct. When \(B\ne0\), a nonzero common gradient zero must have \(Z\ne0\) and \(u=-DZ/B\); substituting into \(H_u\) gives \(Z^2(12kD^2+B^3)/(2B^2)\). When \(B=0\), a nonzero common zero exists exactly for \(D=0\). Off \(\Delta_k\), continuity on the unit circle and degree-two homogeneity therefore give the asserted quadratic lower bound for \(|\nabla P|\), including after the invertible shear. + +In the critical-free strip between \(h_*\) and zero, the height-cutoff normalized gradient field is smooth. It is bounded on compact sets because its active support avoids critical points, and bounded at infinity by the quadratic gradient estimate. Finite-time completeness follows from boundedness. Its flow maps the initial closed maximum disk to a compact disk, fixes \(M\), carries its boundary to the prescribed height, and never increases height inside. A path with endpoint value above zero must leave this disk and cross that boundary. This rules out both distant connectivity and finite-height asymptotic escape, and proves the matching global upper bound. + +### Transfer to a completely unrestricted exterior + +For each fixed \(h>h_*\), §5 now chooses the initial oval at \(t_0>h\), so the preceding flow constructs the required barrier. The negative-definite Hessian near the exactly pinned maximum ensures \(g_i\le0\) there. Away from that neighborhood, \(P<0\) on the remaining compact part of the cap, giving the same statement by \(C^0\) convergence. Consequently an actual global older path must cross the embedded cap boundary, regardless of the geometry elsewhere on the surface. The two inequalities establish convergence of the actual global death height. + +The assertion that the only zero in the cap is \(M\) is valid. The displayed extra-critical-point height algebra proves it, and it also follows from the fact that the initial disk has only \(M\) at zero while descent preserves negativity of every other initial point. No global Morse assumption on the cubic below the window is used. + +### Exact, eventual selection of the highest actual continued saddle + +I specifically checked the repaired quantifiers and strengthened vector-field construction in §6. Lemma S is applied on \(B_{R+1}\), preserving the exact values and gradients of both pins; it gives exactly the finite continued window-root set for the same sufficiently large index. Let \(h_i\) be the maximum of those actual heights and the exact pinned height \(-k\). There are no critical points in \((h_i,0)\) on the disk, \(h_i\ge-k\), and \(h_i0\). Thus \(C^2\) convergence preserves the derivative signs close to the stationary endpoints, while \(C^1\) convergence preserves the strict signs on the intervening compact intervals and up to \(t=2\). The chord's exact minimum is \(g_i(Y_i)\), and its endpoint remains strictly above birth. + +The convergence disk contains every doubled chord. Since there are finitely many candidates, the same large index works for all of them, even if tied limiting heights cause the highest actual continuation to switch. The winning chord proves the exact lower bound \(h_i\). Together with the collar it identifies the actual global death as \(h_i\) and, on the global Morse distinct-value locus, the unique highest continued local saddle as the actual elder partner. Empty sectors select \(S\) exactly; nonempty sectors select a continued window saddle. + +For a unique limiting position, the additional null exclusion \(D=0\) is sufficient and is correctly separate from the finite-index theorem. By (C9), +\[ + (P/k)'(u)=-1-(4s/B)u=-\frac{B+4su}{B}. +\] +The classifier gives \(B+4su>0\) throughout the window, so height is strictly monotone in \(u\) for \(B\ne0\). For \(D\ne0\), the line fixes \(Z\) uniquely from \(u\); distinct window roots therefore have distinct heights. The \(B=0\) case has at most one window root. At \(D=0\), a typed two-root sector can have equal limiting heights and opposite \(Z\), so a unique limiting position would not be justified without that exclusion. The proof uses the original coordinates under the inverse shear. A read-only noncontributor child review independently checked this strengthening and tie criterion. + +I found no counterexample under the stated hypotheses. The exclusions matter as stated: the window transition must be avoided for the exact empty-sector stability, and the additional leading-homogeneous degeneracy set is excluded for the chosen no-escape proof. The theorem remains pointwise and planar; no uniformity as these sets are approached is proved. + +## Compact Gaussian marked law and normalization + +The extra-conditioned regression coupling has the original endpoint observations and the physical midpoint jets. Its contact covariance is the full ten-dimensional planar degree-three jet covariance, and the source supplies a uniform \(C^4\) bound on each compact target sector. This is the correct order of conditioning for a rare soft-layer event; subtracting an unconditional tail from a sector of size \(r\) would not be justified. + +For each good target jet and coupling realization, Theorem E gives eventual equality of the Boolean elder-failure mark to \(1_{n>0}\), and convergence of the normalized death mark to \(h_*\). The local endpoint Hessians give \(W_r/r^4\to w\), with the source's integrable conditional majorant. Thus replacing the count indicator in (G10) by either mark is legitimate. + +The power ledger is exact: + +\[ + r^{-3}\cdot r\cdot r^4/Z_r=r^2/Z_r\longrightarrow1/z_0. +\] + +The \(r\) factor is the physical jet change \(A=rs\). The \(r^4\) factor is the two soft endpoint determinants. The full normalizer scales as \(r^2z_0\); it is used once. The slice density \(h_0(0,a,\beta,c)\) and the weight \(w=9k^2(4s^2-B^2)\) retain the original jet coordinates. No adjacency restriction, extra determinant, second normalization, or factor \(1/2\) occurs. + +The joint target/Boolean law converges in total variation because its two densities converge pointwise and in \(L^1\) under the same domination. For the real death-height mark, bounded continuous tests converge by dominated convergence; this is correctly a weak assertion. A total-variation assertion for the moving real point mass would not follow. The treatment of the essential class is harmless: every good coupling eventually has a strictly older local endpoint, so \(D_r=-\infty\) ceases to occur in the pointwise argument. + +The source's fixed-\(r\) genericity statement suffices by considering arbitrary countable sequences \(r_i\to0\) and applying Fubini/disintegration; no common probability-one set for uncountably many conditional laws is required. The Borel maximin and elder selector come from [P] §8, read with its Borel repair, rather than being inferred from local roots. + +The compact law alone does not justify removal of the target sector. Finiteness or localization of the additional-critical-point measure would not supply that step. The integrated §8 supplies the required domination/tightness of the actual rescaled elder-failure measure and explicitly identifies the endpoint cap coordinates with the midpoint jets, as checked below. + +## Full fixed-planar failure-measure exhaustion + +The source premise \(F_r\subset G_r^c\) is the parent's actual global older-path cap implication on typed support. It includes the essential class, which cannot occur on the good cap. The proof correctly estimates this marked failure measure rather than trying to exhaust the divergent unmarked \(r^{-3}\) population. + +In the plane, the endpoint transverse Hessian is the one scalar \(-\lambda\), and [P] (4.2) supplies a residual field independent of it. The bounded regression coefficient functions imply \(\|f\|_{C^4}\le C(J_r+\lambda)\). The m=1 depth split in [P] (7.5) must keep both its near branch \(\lambda\le C rJ_r^2\) and its far branch \(\lambda>C'/r\); the integrated proof keeps both. On the near branch, the typed determinant bound gives +\[ + W_r\le C r^2J_r^4\lambda(\lambda+CrJ_r^2). +\] +The endpoint scalar density is bounded and independent of \(J_r\). Retaining \(J_r>A\) inside the same scalar integration, the two terms each integrate to \(O(r^3J_r^6)\). The numerator is therefore +\[ + O\!\left(r^5\mathbb E[J_r^{10};J_r>A]\right). +\] +Division by the one full normalizer \(Z_r\ge z_*r^2\) leaves the required \(r^3\) factor. Higher residual moments make the remaining tail small uniformly for small \(r\). [P] (7.6)–(7.7) give the far and fourth-derivative exceptions \(O(r^4)\) tilted mass, hence negligible after \(r^{-3}\) scaling. This checks the critical rare-layer estimate without an invalid unconditional-tail subtraction or independence assertion about the full field. + +The endpoint-to-midpoint control is explicit and valid: +\[ + |f_{zz}(0)|/r\le\lambda/r+M_3/2,\qquad + \|\Theta_r\|\le CJ_r^2 +\] +on the near branch. The remaining three target coordinates are third derivatives. Thus the retained residual tail genuinely exhausts the actual ambient midpoint target in the failed population. There is no unproved density-coordinate equivalence and, in the plane, no unintegrated hard-eigenvalue or corank intersection. + +The bounded-target regression, Taylor convergence and endpoint determinant envelope extend to every ambient compact box; their hypotheses are covariance nonsingularity and bounded targets, not endpoint typing. I independently checked the two limiting endpoint Hessians. Their fixed opposite axial signs make the typed limiting determinant product precisely +\[ + w_+(\theta)=9k^2[B-2s]_+[-B-2s]_+. +\] +Every \(F_j\) is continuous at a degenerate matrix because its determinant vanishes there; thus this product is zero outside the typed domain and on its boundary. No convergence of the selector outside the typed domain is needed. The ambient density convergence therefore handles arbitrarily thin typed-boundary sectors, rather than silently restricting to compacts separated from the boundary. + +Within the generic typed domain, Theorem E gives the actual selector convergence. The algebraic exceptions are null; the integrated proof now explicitly provides a Borel maximin representative and an arbitrary null-set extension of \(h_*\). Compact dominated convergence proves the \(L^1\) density limit. Actual failure-target tightness and the source's integrability of the nonempty cubic density then prove full \(L^1\), hence total-variation, convergence of the finite target/failure measures on \(\mathbb R^4\). + +The limiting total mass is \(\alpha_1+\alpha_2\), with no factor two: the mark is Boolean failure, not a factorial count. The same bounded-test domination proves weak convergence of the actual normalized death height under failure. The essential-failure mass is \(o(r^3)\): generic typed couplings eventually have a finite older local endpoint, off-typed target weights vanish, and the failure-target tails exhaust the measure. Dividing by the positive limiting failure mass gives the stated actual conditional failure-height law. This does not assert total variation in the moving real death-height coordinate. + +### Actual replacement position and localization + +The strengthened finite-index theorem gives the actual partner, rather than a root chosen only by a limiting count. Off \(D=0\), it is eventually the continuation of the unique highest limiting root; this selector is locally continuous on the generic typed domain by nondegenerate root stability and the strict height gap. It is therefore Borel, with the stated arbitrary null-set extension. + +For bounded continuous joint position/death tests, the same compact-box endpoint-weight domination and the actual failure-target tail apply. The chart/essential sentinel events eventually vanish on every good typed coupling; elsewhere the limiting typed weight is zero. Their full scaled failed mass is consequently \(o(1)\), or unscaled tilted mass \(o(r^3)\), after compact exhaustion. This proves the stated joint weak conditional partner-location and death-height law. The failure lifetime is its pushforward, with \(-h_*\in(0,k)\) and \(-h_*/k\in(0,1)\). + +The finite actual partner-location measures converge weakly with converging total mass and are tight. For any positive deterministic \(\delta_r\) with \(\delta_r/r\to\infty\), fix a large scaled radius \(A\), take \(r\) small enough that \(\delta_r/r>A\), and then let \(A\) grow. Tightness and the \(o(r^3)\) sentinel mass give the claimed \(o(r^3)\) mass of failed partners beyond \(\delta_r\). This is a valid conclusion about the actual partner under the specified pair-Palm law. It does not import factorial-count localization or supply a growing-radius root estimate, a spatial moment, a rate, or position total variation. + +I found no gap in §8. A read-only noncontributor child review separately reached the same conclusion for the tail and height law. Its same-provider status adds no organizational-independence credit. + +## Compact rejected-candidate lifetime coefficient + +The new §9 integrates a fixed-parameter result using the parent's existing compact-parameter bounds. This is legitimate without uniform convergence of the new cubic theorem: [P] (7.8) uniformly bounds \((1-p_r)/r^3\), while (10.2)–(10.3) uniformly bound \(A_r\), over compact birth/gap/orientation parameters with \(k\) bounded away from zero. + +I independently rechecked the pushforward. For \(r=(\ell/k)^{1/3}\), the original ledger gives +\[ + \ell^{-2/3}\nu_{\rm rej}^{B,K}(\ell) + =\int_{B\times K\times S^1} + \frac{A_r}{3k^{5/3}}\frac{1-p_r}{r^3} + \,db\,dk\,d\sigma. +\] +The exponent follows from \(\ell^{-1/3}r^3=\ell^{2/3}/k\). The displayed integrand is dominated on the finite compact integration domain. Pointwise Theorem S and dominated convergence therefore identify the exact coefficient \(\int A_0(\alpha_1+\alpha_2)/(3k^{5/3})\). It is finite, and positive for positive-length birth and gap intervals. Borel measurability follows from the canonical field regression and maximin mark, or by taking their pointwise limit as the proof does. + +Substituting \(A_0=12\pi_0z_0\) and \(\alpha_1+\alpha_2=z_0^{-1}\int w h_0 1_{n>0}\) yields the stated \(4\pi_0k^{-5/3}\) formula; the full normalizer cancels once. Direct integration of \(\ell^{2/3}\) gives the cumulative factor \(3/5\). + +The population is correctly the nonselected candidate counting measure. A rejected candidate pair cannot be turned into a count of replacement elder bars without accounting for repeated candidates attached to a maximum. The text explicitly avoids that inference. It also correctly avoids a standalone second-order expansion of the selected lifetime density, since no sufficient expansion of \(A_r\) was supplied, and avoids an unrestricted difference asymptotic contradicted by the source's positive far rejection density. + +## Earlier version and final scope + +The predecessor elder_bridge_audit/PLANAR_ELDER_MARK.md, SHA256 +5fbdafcea3a5866888beaa6509aeba12710075c59d83094e61076298dfce4de9, was AMEND as written: §5 did not explicitly choose \(t_0>h\), and §6 spoke of a finite-disk normalized flow without giving its smooth global extension. I notified the root reviewer, who authored the repairs. The current spatial cutoff, enlarged-disk stability, annulus bound, and common-index statement resolve those presentation gaps. They did not require changing the theorem. + +I first accepted only the repaired compact theorem at SHA256 b048ed2e9e64bd7c0913319768bed1e882c49ef12de34bbc034210fbb3c69157. I then reviewed the new tightness note at SHA256 ec4824c154e66082923549f0775a0e15330d5eebac6d81439992775b588d2fd5 and the integrated pre-strengthening candidate at SHA256 49ad8e443570a89fad4b69e46c0a3307182e989ca25d7715ae46ce49cc90a3ef, accepting its full tail/death/difference-coefficient mathematics under positive-length birth and gap intervals. I raised the need to state those positive-length assumptions explicitly for literal asymptotic equivalence to a positive coefficient; the root authored that wording repair. + +Finally I read the complete strengthened successor, with targeted rereading of the tool-output truncation, and checked its exact partner and joint position/localization arguments. The current ACCEPT binds only the full integrated SHA256 at the top of this review. Later changes require a new source-bound review. No remote mutation was made by this reviewer. diff --git a/frontiers/local_elder_geometry_20260930/PROOF.md b/frontiers/local_elder_geometry_20260930/PROOF.md new file mode 100644 index 000000000..ef2aa5795 --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/PROOF.md @@ -0,0 +1,433 @@ +# Planar cubic elder mark: a compact-bar transfer theorem + +Object: OA-PLANAR-LOCAL-ELDER-20260930-v1. +Authors/contributors: OpenAI root, elder_bridge_audit and its local_topology child, foreground continuation, 30 September 2026. The companion scale analysis has an additional OpenAI contributor, mixed_geometry_audit. The proof is AUTHOR_SIDE / HOLD pending qualified source-bound mathematical review. All these agents have the same provider and earn zero organizational-independence credit. Existing source headers are not treated as live acceptance decisions. Publication, source hashes and engineering checks do not establish mathematical acceptance or promote any parent claim. + +This is a new full proof candidate. Only this additive directory is written; no source body, count theorem, register, graph, numerical certificate or integration disposition is changed. Exact dependency interfaces, provenance and unresolved obligations appear below and in SOURCES.json. + +## 1. Exact source cut and estimands + +The main source cut is Math- commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751. Immutable source links and blob identities are in SOURCES.json. + +* [P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md, blob dfed3b8d318a3ab1950957f393307733a4bef3f2, §§1,8,10–14. Read with its congruence erratum (blob 213594d6ca6a86fb938110f4d166d9ce275a02d0), §9 Borel repair (blob fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a), and [REC] §1 reading rule/W1/embedding restriction. [REC] is reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md, blob 75da2597971510f843f8d90c743950cb8c177342. +* [CUB] frontiers/planar_cubic_cluster_20260929/PROOF.md, blob bb446d08db8a944537a743ad550b88c1c2ad5758, §2 (C1)–(C14), §3 Lemma S, §4 (G6)–(G10). +* [LOW] frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md, blob aaefc8da9668e819591dd9db0b34442394f2362e, §§1–4; and frontiers/elder_lower_all_d_20260929/PROOF.md, blob 7f41c9e315be0a7985d4c770a11a2cdf717bd46a, §§0,3–5. +* [TWO] frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md at PR166 head be9b7fa95de9b7e59a67347640e157ed3027f624, blob a32fd5f7d941bbe1fe943df045b1e0fbec8d691c, §§1–4, (M1)–(M2), Theorem L and (L3). +* [Q] frontiers/c6_palm_route_20260929/PROOF.md, blob 89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5, §1.2 Theorem Q and §1.3 non-claims. +* [U] frontiers/unrestricted_selection_difference_20260929/PROOF.md, blob 5a55b179a974a90c65d257f3f93765cc6fb30bb8, §§1,3–5. + +For the actual pinned law Q_r^W, the pins are M_r=(-r/2,0), S_r=(r/2,0), heights b and b-kr^3, and both gradients zero. The weight is W_r=F_2(H_M)F_1(H_S), with exactly one FULL Z_r=E_Q W_r. + +N_r in [Q]/[TWO] counts **additional critical points of all indices in the open height window**, excluding the two pins. The ordered pair measure (L3) is a measure of two additional window witnesses. Its microscopic limit has two index-1 saddles. It is not a maximum/saddle persistence-pair measure. + +The elder selector in [P] is different: +d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t)), +and p_r=Q_r^W{d_f(M)=f(S)} on the Morse distinct-value locus. Its candidate maximum/saddle intensity is r A_r dr db dk d sigma with A_r=12 pi_r(Z_r/r^2); its selected intensity is r A_r p_r with the same measures. The lifetime pushforward ell=kr^3 supplies r dr/dell=ell^(-1/3)/(3k^(2/3)), [P] §§10–11. There is no adjacency normalization, no factor 1/2 and no second Z_r. + +Consequently, localization of (L3) alone does not supply the elder mark. The common leading candidate/elder lifetime density is already supplied by the reconciled D1 cap/selection/Kac–Rice chain, rather than by C6 localization. [LOW] identifies the compact-window rejected density order ell^(2/3); [U] separately shows a positive far rejected density and forbids importing that compact-window decay as an unrestricted difference rate. + +The following theorem adds an actual topological mark to the planar cubic sector. It uses neither a new count argument nor any remote witness theorem. + +## 2. Deterministic theorem + +Fix k>0 and a typed cubic from [CUB]: +P(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3. +Write M=(-1/2,0), S=(1/2,0), so P(M)=0, P(S)=-k and both gradients vanish. With the determinant-one shear u=X+aZ/(12k), set +B=beta-a^2/(12k), +D=(c-a beta/(4k)+a^3/(72k^2))/2. +The typed domain is s<-|B|/2. + +Exclude the two algebraic sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0}. +Both are nonzero polynomial zero sets after clearing fixed positive k denominators, hence Lebesgue-null. The first is [CUB]'s window transition; the second is the only additional exclusion made here. + +Let C(P) be the additional critical points with -k0. Suppose their scaled functions + g_i(X,Z)=(f_i(r_iX,r_iZ)-b)/r_i^3 + converge to P in C^2 on every fixed compact disk, and retain the exact pins, gradients and values 0,-k. Then their global normalized death levels satisfy + (d_{f_i}(M_i)-b)/r_i^3 -> h_*(P). + No conditions are imposed outside the shrinking coordinate disks. +3. Continue the finitely many points of C(P) to local critical points of g_i by pinned C^2 root stability, and include the exact pinned saddle S. For every sufficiently large i the global death level is exactly the highest *actual* height in this finite continued set. On the global Morse distinct-value locus its unique highest point is the actual elder partner. If C(P) is empty, that point is exactly S_i; if C(P) is nonempty, it is a continued window saddle and S_i is not the partner. Thus the selector-failure indicator converges to 1{C(P) nonempty}. + +The theorem is planar and pointwise in the cubic and coupling realization. It makes no assertion uniform over marks, dimensions, radii or approaching exceptional sets. + +## 3. The critical-chord identity supplies the lower bound + +For any polynomial of degree at most three and any two critical points M,Y, put q(t)=P(M+t(Y-M)). Since q'(0)=q'(1)=0, interpolation gives exactly +q(t)=P(M)+(P(Y)-P(M))(3t^2-2t^3). +This also holds when the restriction has lower degree. + +Here P(M)=0 and h=P(Y)<0. On 0<=t<=2 the minimum of h(3t^2-2t^3) is h, attained at t=1; the endpoint is -4h>0. Every negative critical point therefore gives a finite path from M to a strictly higher point whose bottleneck is exactly its own critical height. + +Apply this to Y=S and to each window saddle. It follows that d_P(M)>=h_*. + +For g_i, the same fixed finite path has endpoint strictly above zero and minimum at least h_* minus an error tending to zero. Hence the global normalized death has liminf at least h_*. This path is a path in the actual surface after rescaling; no claim about its being a separatrix or an ascending branch is made. + +When C(P) is nonempty, h_*>-k, so the strict margin alone proves eventual failure of selection of S_i. + +## 4. No finite-height escape at infinity + +In the sheared coordinates the leading homogeneous cubic is +H(u,Z)=2ku^3+(B/2)uZ^2+(D/3)Z^3. +Its derivatives are +H_u=6ku^2+(B/2)Z^2, +H_Z=Z(Bu+DZ). + +A nonzero common zero requires Z nonzero. If B=0 it exists exactly when D=0; if B is nonzero it exists exactly when 12kD^2+B^3=0. Thus off Delta_k, grad H has no zero on the unit circle. Compactness and homogeneity give |grad H(z)|>=c|z|^2, and lower-order terms then give +|grad P(z)|>=c_0|z|^2 for |z|>=R_0 +after increasing R_0. The invertible shear preserves such a bound up to constants. + +There are no critical points of P with value in (h_*,0), by the definition of h_* and [CUB]'s complete classification of all window points. Near M a sufficiently high negative regular level t_0 has a small closed level oval enclosing a disk K_0. The disk contains only the strict maximum M. + +For any h in (h_*,t_0), choose a smooth height cutoff chi with 0<=chi<=1, equal to one on [h,t_0], supported in a slightly larger critical-free strip inside (h_*,0), and zero near zero. The vector field +V=-chi(P) grad P/|grad P|^2 +is smooth and globally bounded after setting it to zero outside that strip. The gradient-growth bound controls it at infinity, and compactness controls it elsewhere. Its flow is defined for the finite time t_0-h; it fixes M. The initial boundary stays where chi=1, so its P-value decreases at speed exactly one and ends at P=h. The image K_h is compact, contains M and has boundary P=h. + +Moreover P is nonpositive throughout K_h: the initial disk has P<=0 and along the flow P never increases. Thus every path from M to a point with P>0 exits K_h and has minimum at most h. Letting h decrease to h_* proves d_P(M)<=h_*. Together with §3 this proves part 1. + +This is the missing compact-bar/escape argument. Merely having finitely many window roots would not exclude a finite asymptotic critical level; the Delta_k exclusion and the gradient estimate do. + +## 5. Transfer of the upper bound to the actual global field + +Fix h with h_*0, the conic forces |u|<1/2, so this is negative. If B<0, either u>=1/2, when it is below -k, or u<-1/2; (C9) and 2s/B>1 again make it negative. For B=0, the two extra possible heights are s^3/(6D^2) and -k+s^3/(6D^2), both negative. Thus M is the only critical point at value zero or above. A second interior zero maximum of K_h would be another such critical point. Boundary h is negative. + +The exact pin M and C^2 convergence imply g_i<=0 on K_h for all sufficiently large i: in a fixed small ball around M its Hessian remains negative definite and M remains critical with value zero; outside that ball P has a strictly negative maximum on the remaining compact set, so C^0 convergence suffices. Boundary values of g_i are at most h+epsilon for all large i. + +Any actual global path to a point above birth must leave the embedded r_i K_h because there is no such point inside. It therefore crosses a boundary at value at most b+r_i^3(h+epsilon). This proves the normalized limsup is at most h+epsilon. Let epsilon->0 and h->h_* to prove part 2. + +This uses a local compact boundary to control the global maximin, so exterior connectivity cannot invalidate the bound. + +## 6. Exact selection of the highest actual continued local saddle + +Convergence of death heights alone would not identify the location or exact partner. The following finite-index strengthening supplies both. + +Let the finite set be C(P) union {S}. For each Y in it, let Y_i be the unique continued critical point, using exact S_i=S for the pinned saddle. All their values are negative for large i. They lie in the window except S, whose value is exactly -k. Define + + h_i=max{g_i(Y_i): Y in C(P) union {S}}. + +Fix t_0<0 close enough to zero to have a small maximum-level oval and to satisfy t_0>h_*. Then h_i 2k/(c_0 R_0^2)+1. + +For all large i, C^1 convergence gives |grad g_i|>=c_0|z|^2/2 on R_0<=|z|<=R+1. By [CUB] Lemma S, using exact pins and excluding Sigma_k, sufficiently small C^2 perturbations on B_(R+1) have exactly the continued roots in (-k,0), together with the exact pins at 0,-k. Small disjoint neighborhoods give one root each; the rest of the compact enlarged height band has a positive gradient floor. Thus there is no critical point in (h_i,0) inside B_(R+1). The same sufficiently large index satisfies all these statements, independently of the lower height h chosen next. + +### Exact upper bound by a common finite-disk collar + +Negative-definite Hessians and exact pinning give a small maximum-level oval for g_i at t_0 inside B_(R_0), enclosing M in a disk with g_i<=0. For each h with h_ih_i>=-k. A first passage from B_(R_0) to that boundary traverses the annulus, where speed is at most 2/(c_0 R_0^2). The chosen width makes this take more than k. Thus all boundary trajectories remain in B_R and reach height h at time t_0-h. They form a Jordan curve, and its enclosed image disk is inside B_R: the complement of B_R is connected and lies in the curve's unbounded component. The cap contains M and g_i<=0 throughout since the cutoff flow never increases height. + +Every older path therefore crosses a boundary of height h. For this SAME sufficiently large i the conclusion holds for every h in (h_i,t_0). Taking h down to h_i gives exact normalized global death <=h_i, without exchanging i and h limits and regardless of the exterior geometry. + +### Exact lower bound through each continued critical point + +For any one Y, consider the moving straight critical chord + + q_i(t)=g_i(M+t(Y_i-M)), 0<=t<=2. + +The actual exact criticality of M and Y_i gives q_i'(0)=q_i'(1)=0. C^2 convergence and Y_i->Y give C^2 convergence of these one-dimensional restrictions to + + q(t)=P(Y)(3t^2-2t^3). + +Writing h_Y=P(Y)<0, the limiting derivative is negative on (0,1), positive on (1,2), and + + q''(0)=6h_Y<0, q''(1)=-6h_Y>0. + +C^2 convergence preserves the second-derivative signs near 0 and 1. Together with the *exact* derivative zeros, this preserves the required derivative signs on those endpoint neighborhoods. C^1 convergence preserves them on the remaining compact subintervals. Hence q_i decreases strictly from t=0 to t=1, then increases strictly to t=2. Its exact minimum is g_i(Y_i), and its endpoint is positive since its limit is -4h_Y>0. + +There are finitely many roots, so the same sufficiently large index works for all these paths, including S. Selecting the root attaining h_i provides an actual older path of exact minimum h_i. Thus normalized global death >=h_i. With the collar bound it equals h_i exactly. Even when two limiting saddle heights tie and the actual winning root switches, the finite collection of paths proves the same result. + +On the actual global Morse distinct-value locus, the unique critical value h_i identifies the highest continued local saddle as the ordinary elder partner. Empty sectors select S exactly; nonempty sectors select one of the continued window saddles and preempt S. This proves Theorem E part3 and identifies the actual partner, not only its limiting height. + +### A unique limiting position for almost every jet + +For the position mark only, exclude D=0 as well. This is another nonzero polynomial zero set in the physical jet coordinates, hence Lebesgue null. When B!=0, (C9) gives + + (d/du)(P/k)=-1-(4s/B)u. + +On the B>0 window conic this derivative is strictly negative (at its largest allowed u=u_w it is 1+2s/B<0); on the B<0 window conic it is strictly positive because u<-1/2 and s/B>1/2. Equal window heights therefore force the same u. If D!=0, the line s+Bu+DZ=0 then forces the same Z, so distinct window roots have distinct heights. For B=0 there is at most one window root. Accordingly outside this extra null set there is a unique highest cubic window point Y_*(theta) when n>0, and the actual elder partner's scaled position converges to it. Use S as the default position when n=0. This position exclusion is unnecessary for the exact finite-index death and partner conclusion above. + +## 7. A compact-sector marked selection law under the actual tilt + +This corollary uses only the planar compact-sector disintegration already displayed in [CUB] §4; it does not modify its count theorem. + +For a compact Borel C in the typed domain, define the finite measure +dM_C(theta)=1_C(theta) z_0^(-1) w(theta) h_0(0,a,beta,c) dtheta, +with the full z_0 of [CUB] (G3), and define lambda(theta)=-h_*(P_theta)/k in (0,1]. + +Under the regression coupling [CUB] (G6)–(G8), Theorem E applies pathwise for almost every theta and almost every coupling realization. Define I_r=1{the global partner of M is not S} and D_r=(d_f(M)-b)/r^3. For every bounded measurable psi(theta), the marked identity +r^(-3) E_QW[psi(Theta_r) I_r; Theta_r in C] + -> integral_C psi(theta) 1{n(theta)>0} dM_C(theta) +holds. + +For every bounded continuous phi(theta,t), +r^(-3) E_QW[phi(Theta_r,D_r); Theta_r in C] + -> integral_C phi(theta,h_*(theta)) dM_C(theta). +On the exceptional essential-class event define phi arbitrarily at D_r=-infinity; each good coupling eventually has an explicit local path to an older point, so that arbitrary choice disappears from the limit. + +Proof: replace the count indicator in [CUB] (G10) by the Borel elder mark, or by the indicated bounded functional of the Borel maximin. The multiplier is +1_C(theta) (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4) times mark]. +Theorem E supplies pointwise convergence of the mark. Equations (G7)–(G9) supply the same integrable domination as before. Dominated convergence proves both statements. The first works for every bounded measurable psi because theta itself is not moved by the coupling and the failure indicator converges pointwise; this also yields total variation for the joint theta/Boolean measure on the compact sector. The real-valued death-height mark is asserted weakly, not in total variation. + +The marked corollary gives a selection law on every fixed compact soft-jet sector. Compact convergence alone does not justify a full-global coefficient: it needs tightness of the rescaled *elder-failure measure*, rather than of the nonempty witness measure. The next section proves that tightness by retaining the rare scalar layer in the parent's cap-failure integral and explicitly converting endpoint control to the actual midpoint coordinates. + +## 8. Full planar elder-failure tightness and global marked selection law + +### 8.1. Fixed model and immutable interfaces + +Fix the planar periodized Gaussian model, L>0, b in R, k>0 and one frame. Write Q_r for the exact six-pin regression at M_r=(-r/2,0), S_r=(r/2,0), heights b,b-kr^3 and zero gradients. Set +W_r=F_2(H_M)F_1(H_S), Z_r=E_Q W_r, dQ_r^W=(W_r/Z_r)dQ_r. +Let +F_r={the global ordinary superlevel elder partner of M_r is not S_r}, +Theta_r=(f_zz(0)/r,f_xxz(0),f_xzz(0),f_zzz(0))=(s,a,beta,c), +D_r=(d_f(M_r)-b)/r^3. +The essential class uses d_f=-infinity. All event definitions are Borel by parent §8 and its reconciled §9 whole-field interface. + +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at main commit 98fdb54954a2d9ac0b8ba8ca03e279701f70d751, blob dfed3b8d318a3ab1950957f393307733a4bef3f2. It is read WITH the congruence erratum, §9 Borel replacement, and reconciliation/W1/embedding rule in SOURCES.json. The consumed equations here are (3.5), (4.2)–(4.3), (5.5), (6.2), (7.5)–(7.7), and the good-cap implication in §8. + +[CUB] is frontiers/planar_cubic_cluster_20260929/PROOF.md at the same main cut, blob bb446d08db8a944537a743ad550b88c1c2ad5758. The consumed equations are (C1)–(C14), Lemma S, (G1)–(G3), regression (G6)–(G9), physical midpoint disintegration (G10), and coefficients (G11) with finiteness (G13). + +For theta=(s,a,beta,c), let P_theta be its pinned cubic, B=beta-a^2/(12k), D=(c-a beta/(4k)+a^3/(72k^2))/2, and D_k={s<-|B|/2}. Let n(theta) be the existing cubic classifier and +h_*(theta)=max({-k} union {P_theta(Y):Y is an additional window critical point}). +The function h_* is Borel: by Theorem E it is the cubic maximin, whose strict upper-level events have the same countable finite polygonal-path representation as [P] §8; each path stays in a compact plane region. +Extend it by a fixed value on the excluded polynomial sets and outside D_k when using an ambient test function. Those Borel sets have null or zero-weight contribution to the limiting failure measure, so the extension changes none of the identities. +Write +w_+(theta)=9k^2 [B-2s]_+ [-B-2s]_+. +This equals w(theta)=9k^2(4s^2-B^2) on D_k and is zero elsewhere, including its boundary. + +Define the finite candidate limit measure +dmu_fail(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{theta in D_k,n(theta)>0} dtheta. (S1) +Here z_0 is the FULL normalizer limit [CUB] (G3), and h_0 is the actual raw midpoint Gaussian jet density under the exact contact law. No GOE substitution, adjacency normalization or extra pin-density factor appears. + +**Theorem S (fixed planar selection law).** Conditional on Theorem E and the exact source interfaces above, +r^(-3) Q_r^W(Theta_r in ·,F_r) -> mu_fail +in total variation as finite measures on R^4. Consequently, +r^(-3)(1-p_r) -> alpha_1+alpha_2, (S2) +with alpha_j exactly [CUB] (G11). + +Moreover, for every bounded continuous phi on R^4 x R, +r^(-3) E_QW[phi(Theta_r,D_r);F_r] + -> integral phi(theta,h_*(theta)) dmu_fail(theta). (S3) +Give the test an arbitrary bounded value on the essential-class event; that event has o(r^3) tilted mass. Thus the height-mark assertion is weak convergence, not total variation. After division by alpha_1+alpha_2>0 it is the actual conditional law given elder failure. + +There is no claim about the unmarked full r^(-3) joint theta measure. Its typical s-coordinate escapes and its total mass diverges. + +### 8.2. Good cap excludes failure + +Let G_r be the parent's good cap. On the almost-sure Morse distinct-value locus, [P] §8 proves that the global ordinary partner is S on G_r. Hence +F_r subset G_r^c (S4) +up to a Q_r^W-null set. This includes essential maxima: the good cap supplies a finite older endpoint, so the essential class cannot occur there. + +This implication concerns the same actual global selector p_r as [P], not a new local adjacency event. It is the only place where the cap is used to contain the failure event. + +### 8.3. Rare-layer tail bound from the independent endpoint residual + +In the plane set lambda=-f_zz(M)>0 on typed support. Under [P] (4.2), the residual field g_r is independent of the scalar endpoint transverse Hessian. Write +J_r=1+||g_r||_(C4). +Its moments of every finite order are bounded along small-r sequences by (4.3). The bounded C4 regression coefficients and bounded mean in (4.2) give the pathwise inequality +||f||_(C4)<=C(J_r+lambda). (S5) +The mean and E_Q f_zz(M) terms are absorbed into C using the bounded fixed-parameter pin regression and J_r>=1. + +Let D_cap>0 be the fixed scalar depth constant from [P] (7.5). By that split, depth failure is contained in +near: 01/(4D_cap r). +On the near branch, r<=1 implies +||f||_(C4)<=C'J_r^2, M3<=C'J_r^2. (S6) + +The m=1 typed weight bound (6.2) gives +W_r<=C r^2 J_r^4 lambda(lambda+C rJ_r^2) +on this branch. The density of lambda is bounded by (3.5), and J_r is independent of lambda. Retaining a tail restriction J_r>A in the SAME integration yields +E_Q[W_r;near,J_r>A] + <= C r^2 E[ J_r^4 1{J_r>A} + integral_0^(4D_cap rJ_r^2) lambda(lambda+C rJ_r^2) d lambda ] + <= C r^5 E[J_r^10;J_r>A]. +Divide by the FULL floor Z_r>=z_*r^2 from (5.5): +r^(-3) Q_r^W(near,J_r>A) + <= C E[J_r^10;J_r>A]. (S7) +Higher moments of J_r imply the right side tends to zero as A increases, uniformly along small-r sequences. + +The far branch (7.6) and the fourth-derivative exception (7.7) each have tilted mass O(r^4), hence o(r^3). Their union with the near branch covers G_r^c on typed support. Therefore the rare-layer tail estimate applies to F_r via (S4). + +An unconditional C4 tail estimate followed by Cauchy–Schwarz would lose the rare factor. The retained scalar integral above supplies r^5 before division by the one full Z_r; this is why the tilted r^3 scale survives. + +### 8.4. Explicit endpoint-to-midpoint control + +Between M and the midpoint, +|f_zz(0)|/r <= lambda/r + M3/2 +by the mean-value derivative bound. On the near branch, +lambda/r<=4D_cap J_r^2. +The other midpoint coordinates a,beta,c are third derivatives and are bounded by ||f||_(C3). Equations (S6) therefore imply +||Theta_r||<=C J_r^2. (S8) + +No density-coordinate replacement occurs here: (S8) is a deterministic control of the actual midpoint target by the scalar endpoint residual. Together with (S7) and the o(r^3) exceptions, it proves +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||Theta_r||>T)=0. (S9) +The same proof gives +lim_(T->infinity) limsup_(r->0) + r^(-3) Q_r^W(F_r,||f||_(C4)>T)=0. (S10) + +Thus the leading failed population is ambient-midpoint-soft in the plane. Since there is only one transverse direction, bounded residual sectors force f_zz(M), f_zz(0), f_zz(S)=O(r), the last by (5.1). The axial and mixed endpoint entries are O(r) by (5.1)–(5.2), so both endpoints have the r^4 determinant-product scale. At finite r the far and fourth-derivative branches still exist; their smaller mass is retained above. + +### 8.5. Compact-box disintegration, including the typed boundary + +Fix an ambient compact box K_T in R^4. Although [CUB] states its count theorem on compact subsets of D_k, its actual regression construction (G6), moment bound (G7), Taylor convergence (G8), and determinant bound (G9) apply to ANY bounded midpoint target. They rely on the nonsingular joint covariance and bounded target, not on endpoint typing. No count convergence is invoked outside D_k. + +Under this coupling, the endpoint Hessians divided by r converge to +H_M(P)=[[-6k,-a/2],[-a/2,s-beta/2]], +H_S(P)=[[6k,a/2],[a/2,s+beta/2]]. +Their determinants are 3k(B-2s) and 3k(B+2s). Since the axial signs are fixed, the limiting typed weight is exactly +W_r/r^4 -> F_2(H_M(P))F_1(H_S(P))=w_+(theta). (S11) +Each F_j is continuous even at a degenerate matrix: its determinant factor tends to zero there. Hence (S11) holds on the typed boundary as well, with zero weight. On the whole box the integrable envelope +W_r/r^4 <= C_T(1+||F||_(C4))^4 (S12) +follows from the same endpoint Taylor bound as (G9). + +The fixed-k algebraic exceptional sets +Sigma_k={12kD^2+(s-B)^2(B+2s)=0}, +Delta_k={12kD^2+B^3=0} +are nonzero polynomial zero sets, hence null for the absolutely continuous midpoint density. No distinct-height exclusion is needed between the two window saddles. + +For almost every theta in D_k and coupling realization, Theorem E gives +1_F_r -> 1{n(theta)>0}, +D_r -> h_*(theta). +For theta outside D_k, including its boundary, no convergence of the selector is required: its bounded multiplier is killed by (S11). Thus [CUB]'s physical midpoint disintegration (G10), with the count mark replaced by the Borel global failure mark, gives the density +u_r(theta) + = (r^2/Z_r) h_r(rs,a,beta,c) + E[(W_r/r^4)1_F_r]. (S13) +The one factor r in (S13) comes from f_zz(0)=rs. It is not a spectral/endpoint change of variables. The prefactor is r^(-3)*r*r^4/Z_r=r^2/Z_r. + +By (S11)–(S12), the full normalizer limit, local boundedness of h_r, and dominated convergence in theta and F, +integral_(K_T) |u_r(theta)-u_0(theta)| dtheta -> 0, +u_0(theta)=z_0^(-1) w_+(theta) h_0(0,a,beta,c) + 1{n(theta)>0}. (S14) +Here define the final indicator arbitrarily off D_k, where w_+=0. + +This simultaneously handles thin typed-boundary neighborhoods and the exceptional polynomial sets; neither can carry an unidentified atom of the leading failure measure. + +### 8.6. Exhaustion gives global total variation and the coefficient + +The limiting density u_0 is integrable. It is exactly the sum of the two nonempty coefficient densities (G11), whose integrability follows from [CUB] (G13) and Gaussian polynomial tails. Its total mass is alpha_1+alpha_2, finite and strictly positive. + +Choose T large. On K_T, the L1 error tends to zero by (S14). Outside K_T, the actual density's mass is controlled by (S9), and the limiting density's mass tends to zero by integrability. Therefore +integral_(R^4) |u_r-u_0| dtheta -> 0. +This is total-variation convergence of the finite failure measures (under either conventional factor in the TV norm), proving Theorem S and (S2). + +The proof exhausts only the marked failure measure. It never integrates the divergent n=0/unmarked total soft-jet measure and never asserts an uncentered r^(-3) probability law. + +### 8.7. The normalized death mark on actual failure + +For a bounded continuous phi(theta,t), use (S13) with multiplier +1_F_r phi(theta,D_r). +On every ambient compact box the same domination applies. Theorem E gives pointwise convergence on typed generic theta to +1{n(theta)>0} phi(theta,h_*(theta)); +outside the typed domain the zero weight suffices. Dominated convergence gives the compact-box marked limit. + +The bounded test's omitted actual mass is at most ||phi||_infinity times the failure tail in (S9). The limiting omitted mass is controlled by the integrable u_0. Exhausting boxes proves (S3). + +The essential-class mark has zero scaled limit: on a good typed cubic coupling Theorem E's finite critical-chord path supplies a local point above the birth for all small r, and outside the typed domain the weight limit is zero. The compact-box domination and (S9) again exhaust the actual essential-failure mass. Thus it is o(r^3). + +Normalizing (S3) by (S2) identifies the weak actual conditional failure-height law. This is not a density result for persistence lifetimes, not a rate, and not a spatial or real-height TV assertion. + +### 8.8. Actual replacement location and the lifetime mark + +Let T_r be the actual elder partner's coordinates divided by r in a fixed embedded midpoint chart. Give it a sentinel value when the partner is outside the chart or the class is essential. Theorem E part3 proves that on every generic typed coupling the partner is eventually a continued local window saddle on failure. Outside the additional null set D=0, it is the continuation of the unique highest root Y_*(theta), so + + T_r -> Y_*(theta), D_r -> h_*(theta). + +This is an actual partner assertion; no witness factorial pair is relabelled. The functions Y_* and h_* are Borel on the generic typed domain: the finitely many nondegenerate roots continue continuously locally, and strict height comparison selects a unique root. Arbitrary Borel extensions on the excluded sets have zero limit mass. + +For every bounded continuous test phi(theta,y,t), the same compact-box domination as S13, followed by the actual failure-tail bound S9, gives + + r^(-3) E_QW[phi(Theta_r,T_r,D_r);F_r] + -> integral phi(theta,Y_*(theta),h_*(theta)) dmu_fail(theta). + +Give bounded tests arbitrary values at either sentinel. On compact good couplings those sentinel events eventually vanish, and S9 exhausts their omitted failure mass. In particular their tilted mass is o(r^3). The joint location/death assertion is weak convergence. Dividing by the positive total mass alpha_1+alpha_2 identifies the genuine conditional replacement-partner law under the original pair-Palm law given failure. + +The corresponding normalized actual lifetime on failure is + + (b-d_f(M))/r^3 -> -h_*(theta) in (0,k), + lambda(theta)=-h_*(theta)/k in (0,1). + +Its law is the pushforward of mu_fail/(alpha_1+alpha_2); thus the local geometry determines which saddle pairs and its lifetime under this specified law. This is not an intensity obtained by duplicating rejected candidates attached to the same maximum. + +The weak finite replacement-location measures have converging total masses and a finite limit on R^2, hence they are tight. Therefore for every deterministic cutoff delta_r with delta_r/r->infinity, + + Q_r^W(F_r, dist(partner(M),midpoint)>delta_r)=o(r^3). + +For the proof, first fix a large scaled radius A, use tightness outside A, and then use delta_r/r>A for all sufficiently small r; let A increase. The o(r^3) sentinel mass is included. No growing-radius uniform root enclosure is asserted or needed. This localizes the **actual failed elder pairing**; it neither needs nor promotes the contextual factorial-count localization results. + +## 9. A compact rejected-candidate lifetime coefficient + +The local-to-elder bridge is now supplied by Theorem E and the full marked failure limit S2, rather than postulated from a count theorem. One precise lifetime consequence follows by consuming the parent's *existing* compact-parameter domination; no new uniform convergence theorem is needed. + +Retain exactly [P] §§10–12's populations: candidate maximum/saddle pairs and the selected subset, per unit midpoint volume, with birth b in a compact interval B and positive gap mark k in a compact interval K, both intervals of positive lengths and K bounded away from zero, and directed orientation u in S^1 with ordinary arc measure. Write + + A_r(b,k,u)=12 pi_r(u;v_r) Z_r(b,k,u)/r^2, + A_0=12 pi_0(u;v_0) z_0(b,k,u), + a_fail(b,k,u)=alpha_1(b,k,u)+alpha_2(b,k,u)>0. + +For every fixed (b,k,u), S2 identifies + + (1-p_r(b,k,u))/r^3 -> a_fail(b,k,u). + +The source [P] (7.8), with its compact parameter hypotheses, bounds this ratio by a common finite constant for all sufficiently small r, b,k and frames in these compacts. Equations (10.2)–(10.3) bound A_r uniformly and identify its limit. These are already-displayed source bounds, not uniformity deduced from our pointwise cubic topology theorem. The conditional a_fail is Borel as the limit of Borel quantities, and is bounded by that same cap constant. + +Let nu_rej^{B,K}(ell)=nu_cand^{B,K}(ell)-nu_eld^{B,K}(ell) denote the density of the **nonselected candidate counting measure**, in the parent's canonical Kac–Rice versions. With r=(ell/k)^(1/3), [P] (11.1)–(11.2) give exactly + + ell^(-2/3) nu_rej^{B,K}(ell) + = integral_(B x K x S^1) + [A_r(b,k,u)/(3k^(5/3))] + [(1-p_r(b,k,u))/r^3] db dk d sigma(u). + +The common compact bounds dominate this integrand by an integrable constant, because k is bounded below and the integration domain has finite measure. Pointwise convergence and dominated convergence prove + + nu_rej^{B,K}(ell) ~ C_fail^{B,K} ell^(2/3), + C_fail^{B,K}=integral_(B x K x S^1) + A_0 a_fail/(3k^(5/3)) db dk d sigma(u). + +The coefficient is finite and is strictly positive when B and K have positive lengths. It is an identified Gaussian integral, not a numerical estimate. The original full z_0 cancels if the integrand is written as + + A_0 a_fail/(3k^(5/3)) + = 4 pi_0(u;v_0) k^(-5/3) + integral_(D_k) w(theta) h_0(0,a,beta,c) + 1{n(theta)>0} dtheta. + +There is still only one original pin density, one determinant weight and one full normalizer. By direct integration, + + E N_rej^{B,K}(0,t] ~ (3/5) C_fail^{B,K} t^(5/3). + +This is a sharp coefficient for the selected-versus-candidate **difference** on the exact compact-mark population. It is not a separate second-order expansion of nu_eld: the convergence of A_r itself has not been given an expansion here. It is also not a density of replacement elder bars obtained by counting every rejected candidate. One actual maximum can participate in several rejected candidate pairs; doing that pushforward would require the appropriate de-biasing/selection identity. Theorem S3 instead identifies the actual death-height law under the specified candidate pair-Palm law conditioned on failure, which is a different, well-defined estimand. + +No unrestricted version of this difference asymptotic is inferred. [U] already proves a positive far rejected density and excludes such an inference. No k down to zero limit, new parameter uniformity, higher-dimensional lift or numerical coefficient is added. + +## 10. Hypotheses, exclusions and falsifiers + +The deterministic Theorem E assumes the exact displayed planar cubic, k>0, strict endpoint typing, Sigma_k and Delta_k excluded, pinned C^2 convergence on every fixed disk, and embedded shrinking charts. Exact partner statements additionally assume the actual global Morse distinct-value locus. Only the unique limiting position mark additionally excludes D=0; exact finite-index partner identification permits tied limiting heights. It does not need Morse–Smale or a separatrix identification. + +The Gaussian statements consume the exact original six-pin regression, full Z_r, nonsingular finite contact-jet covariance, full-field conditional C^4 moment coupling, and the reviewed cap implication/integration interfaces of [P] with all three reading rules. The algebraic exclusions are Lebesgue null, not discarded positive-probability events. The tightness proof includes the scalar large-lambda and fourth-derivative branches as o(r^3), handles the typed boundary by continuous determinant weight, and keeps all random-field correlations. Theorem S concerns fixed planar marks. Section9 uses only the parent's already uniform compact-mark bound for its explicitly restricted lifetime difference. + +The following would falsify the corresponding new step: + +* A generic typed displayed cubic whose maximin level differs from h_*, or a pinned local C^2 sequence on compact surfaces whose actual death fails to approach h_*, falsifies Theorem E. A sequence with no cubic window saddle but eventual preemption of S would specifically falsify its exact empty-sector argument. +* A failure of the cap's global older-path implication, or of the endpoint-independent residual and rare scalar integration with the original W_r/Z_r, would invalidate S4–S9 and reopen full failure exhaustion. +* A leading failure mass escaping every bounded midpoint-jet box would contradict S9; a nonzero leading typed-boundary mass would contradict S11–S14. These are explicit measure claims, not inferences from tests or finite samples. +* An erroneous Jacobian/full normalizer in G10 or the parent's radial lifetime ledger would invalidate the corresponding marked/lifetime identities. Unrestricted or replacement-bar uses of Section9's compact rejected-candidate coefficient are outside its scope. + +The new scientific candidate is the planar local-to-global actual elder partner, full rescaled selector-failure law, joint conditional replacement location/death/lifetime law, and compact rejected-candidate lifetime coefficient. Mixed factorial pairs remain two additional saddle witnesses rather than max/saddle persistence pairs. Higher-dimensional replacement geometry and any unrestricted refined lifetime law remain unproved by this note. Source-bound review and publication do not change any controlling status, graph, prize, premise or parent acceptance. diff --git a/frontiers/local_elder_geometry_20260930/README.md b/frontiers/local_elder_geometry_20260930/README.md new file mode 100644 index 000000000..96d5d4a55 --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/README.md @@ -0,0 +1,19 @@ +# Planar local geometry determines the elder mark + +The full proof is [PROOF.md](PROOF.md). The geometry companion is [MIXED_SCALES.md](MIXED_SCALES.md). This is an additive mathematical candidate, AUTHOR_SIDE / HOLD pending qualified source-bound review; no controlling status, graph, premise, prize or parent theorem is promoted. + +For the exact planar six-pin Gaussian law at fixed positive gap mark k, the proposed bridge has three parts: + +1. A generic typed cubic has elder death height equal to its highest window saddle height, with the pinned saddle height -k as the default. Finite critical-chord paths and compact isolating boundaries transfer this to the actual global field under pinned local C^2 convergence. The actual partner is eventually the highest continued local saddle; empty sectors select the pinned saddle exactly. +2. The parent's existing rare weighted cap-failure integration controls escaped midpoint jets. Combining it with the topology gives the full selector-failure coefficient + \[ + r^{-3}(1-p_r)\longrightarrow\alpha_1+\alpha_2, + \] + and the weak joint actual replacement-location/death/lifetime law conditioned on failure. The finite failure measure on the midpoint jets converges in total variation, and actual failed elder partners localize inside every cutoff delta_r with delta_r/r tending to infinity. This does not turn a pair of extra saddles into a persistence pair. +3. On the parent's exact compact birth/gap population, its existing Kac–Rice pushforward gives an identified positive Gaussian coefficient for the rejected-candidate density, of order lifetime^(2/3). This is a candidate-minus-selected difference, not a separate second-order expansion of the selected density or a duplicated replacement-bar count. + +The mixed companion proves the necessary scale distinction. Fixed physical remote and genuinely shrinking intermediate contributions are negligible according to the exact current sources. Beyond any *fixed scaled* radius Ar, however, an open positive-weight cubic sector can contain a second saddle while another lies near a pin or in the collar, with order-r^3 mass. Both points remain at the microscopic r scale. Those are extra-witness factorial pairs, rather than maximum/saddle elder pairs. + +[SOURCES.json](SOURCES.json) links every consumed public full source by immutable commit/path/blob/SHA256 and separates the elder theorem's required inputs from contextual count interfaces. [RECONNAISSANCE.md](RECONNAISSANCE.md) explains why global diagram stability alone does not give this labelled local-to-global result. [NONAUTHOR_REVIEW.md](NONAUTHOR_REVIEW.md) and [MIXED_SCALES_REVIEW.md](MIXED_SCALES_REVIEW.md) preserve the actual noncontributor review reasoning and its exact prose cut. All participating authors/reviewers in this packet are OpenAI; these reviews have zero organizational-independence credit. + +The complete hypotheses, null exclusions, cap-tail and typed-boundary treatment, normalizer/Jacobian ledger, mathematical falsifiers and remaining exclusions are in PROOF §§2–10. No higher-dimensional replacement law, unrestricted refined lifetime difference, new small-gap uniformity or numerical coefficient is claimed. There is no new count script, historical certificate replay, workflow or register change. diff --git a/frontiers/local_elder_geometry_20260930/RECONNAISSANCE.md b/frontiers/local_elder_geometry_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..330298793 --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/RECONNAISSANCE.md @@ -0,0 +1,11 @@ +# Local elder geometry: reconnaissance and mathematical scope + +This note accompanies a self-contained topology proof, rather than claiming novelty or importing a diagram-stability theorem as a missing proof. The exact public workspace sources are in SOURCES.json; all source assertions are restricted to their displayed hypotheses. + +The retrieved primary reference is David Cohen-Steiner, Herbert Edelsbrunner and John Harer, *Stability of Persistence Diagrams*, SCG 2005, full text: https://math.uchicago.edu/~shmuel/AAT-readings/Data%20Analysis%20/Stability.pdf. Its main theorem bounds bottleneck distance between diagrams for tame continuous functions on a common triangulable space by their global sup-norm distance. That theorem concerns the whole diagrams and a global norm. It does not, by itself, identify a specified maximum's partner from convergence on shrinking local disks when the exterior field is arbitrary. + +The explicit substitute used here has two parts. A cubic critical-chord interpolation supplies a finite connection to an older point at the candidate saddle height. A compact isolating level boundary gives the reverse bound on *all* global older paths, independent of how they connect outside the disk. The nonvanishing leading homogeneous gradient rules out loss of that boundary at finite height. Exact pinning and a uniformly bounded finite-disk flow then prove exact selection in the empty sector, rather than infer it from convergence of death heights alone. + +D1 already supplies the global Borel maximin event, the original typed Gaussian law, a cap theorem guaranteeing selection on its good event, its rare weighted cap-failure integration, and the Kac–Rice lifetime pushforward. The new proof uses those interfaces with the published planar cubic classifier. It does not use C6 factorial localization as a substitute for elder pairing. + +No priority claim is made. The cubic interpolation, regular-level isotopy and weighted Gaussian disintegration are proved in the accompanying text. The new result remains a source-scoped author-side candidate until reviewed on its exact prose and dependency identities. 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Local paths omitted. Required interfaces remain conditional on their displayed hypotheses and source-bound review scope.", + "scientific_effect": "NONE" +} diff --git a/frontiers/radial_rate_20260930/MANIFEST.json b/frontiers/radial_rate_20260930/MANIFEST.json new file mode 100644 index 000000000..b06df6fa3 --- /dev/null +++ b/frontiers/radial_rate_20260930/MANIFEST.json @@ -0,0 +1,73 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "NONAUTHOR_REVIEW_OPEN", + "files": [ + { + "path": "PLAN.md", + "bytes": 1772, + "sha256": "3a436b2f58a3b79bcb5471474d94fff7c07f5f4dce838525fed117cbb5d446db", + "git_blob": "db61274a4ce7f2e9e6c4b75303a285230f44d5db" + }, + { + "path": "PROOF.md", + "bytes": 15755, + "sha256": "a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710", + "git_blob": "65f24b1f9a7b80522a2ee059f5f6877c9f970f0d" + }, + { + "path": "README.md", + "bytes": 3186, + "sha256": "b31b309916639ce111f28f4c81fea061a62ac777c6a9f241185bcfd0f59655b8", + "git_blob": "cc0e2f5a45c76aa60a3782002321f0c47d0f3b17" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 2002, + "sha256": "54118b87d9a0dbe78ed9f2162ad413d1a9bfc1c1122c53855274b300751ede2b", + "git_blob": "3eb935b8d3fa2c10e8220c71c5243488beb25e31" + }, + { + "path": "RESULTS.json", + "bytes": 438, + "sha256": "c9cdb842befa92b8e26eb02831a1d20838cc75465f1db7840eba1c0eb1c70b4f", + "git_blob": "c7494046e449f30a6346572b2760cca6c1c62afb" + }, + { + "path": "SOURCES.json", + "bytes": 1396, + "sha256": "22b3064a278faff1d915dfb658e9d7e41ec5c4ee69f31e944d47a699a82f8a07", + "git_blob": "37f3dc7b5583bf2f4f0a8d1bb1fd5bd39ff12838" + }, + { + "path": "VALIDATION.md", + "bytes": 2898, + "sha256": "b14b2fb0576c7c7fd1a03baf3d7f479d901f186d0ef0a49dac5156805908409d", + "git_blob": "5a4a11b112723aaf9a67c1aeaafa3a216e5f17e8" + }, + { + "path": "rate.py", + "bytes": 6344, + "sha256": "e96be67c1703780dcd48113fa55809d98b87f0e498da132b162c9640d300577e", + "git_blob": "d213a2762e10a07f3290f9ae1b4dd9f8f73e5667" + }, + { + "path": "test_rate.py", + "bytes": 7286, + "sha256": "f2a096b91de1978dbaa6c1301ac1013d454cadcf96aa705fb66df3c43f53324f", + "git_blob": "4074549e593ad0f02f6931c6df211e25a02d84ea" + }, + { + "path": "test_verify.py", + "bytes": 5792, + "sha256": "1e8f2790b5861b520463ab1d44b87c29dd43ba46b652c46992889ecd001bd4c7", + "git_blob": "258288f8efa9039782432daf6dde77e2b594ef39" + }, + { + "path": "verify.py", + "bytes": 5169, + "sha256": "7f7006a1f3ba74964a00023dd418624612980e19f8e40a3506245ae43966119c", + "git_blob": "d5789cbe7b94b6243b55a62491ecd19bdeb9cbe0" + } + ] +} diff --git a/frontiers/radial_rate_20260930/PLAN.md b/frontiers/radial_rate_20260930/PLAN.md new file mode 100644 index 000000000..db61274a4 --- /dev/null +++ b/frontiers/radial_rate_20260930/PLAN.md @@ -0,0 +1,29 @@ +# Inverse-radius expansion: execution plan + +Goal: derive quantitative radius-tail and mark-law results from the exact R11 +integral of the already-formed limiting microscopic POINT intensity. The original +Gaussian-field limit r->0 is taken first. No whole-cluster maximum or finite-r +rate is to be inferred. + +Design: reciprocal radius z=1/Z makes the physical cutoff an interval with +explicit endpoints. Map it to [-1,1], prove a smooth even integrated function of +1/t using Gaussian derivative domination, and compute its first two coefficients. +Deduce radial and fixed latent-mark total-variation rates, keeping the sign- +retaining law separate. Preserve the complete original anisotropic covariance, +Haar factors and full normalizer. No original source or peer branch is edited. + +Execution: isolated new runtime directory and new additive GitHub branch. +Dylan's repeated autonomy instruction authorizes foreground work; no timers or +background agents are restarted. No subagent tool is available here. + +1. Read source R11-R13, exact Gaussian/spectral interface and current peer claims. +2. Write and observe failing tests for exact shape integrals and endpoint series. +3. Implement standard-library rational controls; test both signs and parity. +4. Write complete domination and coefficient proof, plus all scope exclusions. +5. Verify exact historical source identities, packet inventory and both Python + modes; include hostile Git fixtures and semantic mutants. +6. Publish a new source-bound candidate and request nonauthor analytic review. + +Separate supporting task: bounded nonauthor review of PR173 Route C; accept the +conditional count implication and repair its inadmissible fixed-R substitution +by a diagonal argument. No graph execution or parent revalidation. diff --git a/frontiers/radial_rate_20260930/PROOF.md b/frontiers/radial_rate_20260930/PROOF.md new file mode 100644 index 000000000..65f24b1f9 --- /dev/null +++ b/frontiers/radial_rate_20260930/PROOF.md @@ -0,0 +1,334 @@ +# Smooth inverse-radius expansion and sharp rates for microscopic extreme marks + +Object: OA-RADIAL-RATE-20260930-v1. Author: OpenAI / GPT-6 Astra Pro. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No parent proof, register, Boolean, prize, or earlier +review is changed. This conversation authored SC/P and contributed to the +related cluster line; no independent revalidation of those premises is implied. + +## 1. Exact object and scope + +Fix d>=2, L>0, b real, k>0 and an orthonormal frame, under the original model +and full endpoint normalizer. This paper concerns the ALREADY-FORMED limiting +microscopic POINT intensity of [R]. First the original field radius r tends to +zero; only then does the microscopic distance threshold t tend to infinity. +No finite-r convergence rate is proved. Whole-cluster maximum selection and +its moving tie boundaries are not consumed or treated here. + +Exact source identities are in SOURCES.json. [R] is PR166 RADIAL_TAIL.md at +80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4: +R7-R13 supply the root-resolved integral, Gaussian domination and positive leading +coefficient. [SC] is PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the nondegenerate raw Gaussian contact +jet, orthogonal spectral convention and finite nonempty microscopic measure. +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at +7a1cb09a9d58d179393e6d29146252f714b9c9bf, blobdfed3b8d318a3ab1950957f393307733a4bef3f2: +only the exact periodized covariance and contact conditioning, used in Section7. +Neither PR169's whole-cluster candidate, its companion-ratio addendum, nor any +numerical coefficient note is a premise. Source reviews are not reviews here. + +Use [R]'s variables lambda=(a,h,O,xi), with ordered positive hard curvatures h, +normalized Haar O, and remaining cubic entries xi. Put + + A=a/(12k), gamma=sqrt(1+A^2), + beta0=a^2/(12k), c0=a^3/(144k^2), + A0=O diag(0,-h) O^T, + H(h)=product h_j^3 product_(i0 on S. + +For y=1/Z define + + g_lambda,u,v(y)=G_lambda(beta0+k b0 y^2, + c0+(a b0/4)y^2+2k(v-b0u)y^3). (2) + +[R, R11] says exactly that the tail F(t)=integral sum_i 1{|z_i|>t}dM is + + F(t)=K0 integral_(S,lambda) Q H + integral_R |Z|^-12 g(1/Z) + 1{(u-AZ)^2+Z^2>t^2}dZ du dv d lambda. (3) + +The multiplicity of roots is already correct in this formula. No extra factor +for a double cluster, pin density, or endpoint normalizer is introduced. + +## 2. A fixed-domain smooth even representation + +**Theorem 1.** There is a smooth even function C(delta), defined for +|delta|<2/3 and with every fixed derivative bounded on |delta|<=1/3, such that + + F(t)=t^-11 C(1/t), t>=3. (4) + +Smoothness also holds in L1 for the underlying UNSIGNED shape/parameter measures. +The claim is C-infinity, not analyticity or convergence of an infinite Taylor +series. Consequently for every fixed N>=0 there are finite coefficients C_(2j) +with + + F(t)=sum_(j=0..N) C_(2j) t^(-11-2j)+O_N(t^(-13-2N)), + -F'(t)=sum_(j=0..N)(11+2j)C_(2j)t^(-12-2j) + +O_N(t^(-14-2N)). (5) + +C_(0)>0 is [R]'s C_star. Constants are at the fixed source parameters; no growing +dimension, volume, vanishing-gap, or parameter-uniform claim is made. + +**Proof.** The reciprocal substitution in (3) gives |Z|^-12 dZ=|y|^10 dy on +both signs. For t>|u| the physical radius condition is + + (t^2-u^2)y^2+2uAy-gamma^2<0. + +Set delta=1/t and y=delta w. Its two endpoints in w are + + w_plus/minus(delta)=m(delta) +/- d(delta), + m(delta)=-uA delta/(1-u^2 delta^2), + d(delta)=sqrt(gamma^2-u^2 delta^2)/(1-u^2 delta^2). (6) + +Thus (4) holds with + + J_delta(u,v,lambda)=d(delta) integral_(-1)^1 + [m(delta)+d(delta)x]^10 + g(delta[m(delta)+d(delta)x]) dx, + C(delta)=K0 integral Q H J_delta du dv d lambda. (7) + +There is no moving integration boundary left. On |delta|<=delta0<2/3 the +denominator is uniformly positive and the square-root radicand is bounded +below by a positive constant, since |u|<=3/2 and gamma>=1. Each fixed derivative +of m,d is bounded by a polynomial in1+|a| with constants depending on delta0,k. + +For every fixed j the nondegenerate Gaussian contact density satisfies + + sup_(beta,c) |partial_(beta,c)^j G_lambda(beta,c)| + <= C_j exp[-c_j(|h|^2+a^2+|xi|^2)] (8) + +uniformly in O. Indeed each derivative is a polynomial times the same Gaussian; +absorb that polynomial into a weaker Gaussian exponent and then discard beta,c +from its quadratic lower bound. A bounded mean changes the constants only. +Uniform tensor norm equivalence under O supplies the displayed retained variables. +No inverse hard eigenvalue is used. This is an elementary derivative extension +of [R]'s raw Gaussian bound, not an extra stochastic assumption. + +The chain rule in (7), for any fixed number of delta derivatives, now gives an +integrable majorant of the form + + C Q H (1+|a|)^M exp[-c(|h|^2+a^2+|xi|^2)], (9) + +on the bounded shape domain and x in[-1,1]. The actual shifted beta,c values +may be large: (8) is a supremum over them, not an invalid compactness assumption. +This proves differentiation under all integrals and smoothness in L1 as well +as for the total mass. The exact identities m(-delta)=-m(delta), +d(-delta)=d(delta), followed by x->-x, show J_(-delta)=J_delta. This cancellation +holds even when g(y) is not even; its cubic y^3 term is retained. + +Taylor's theorem with the integrated derivative bounds proves (5), including +its differentiated remainder. It is NOT obtained by formally differentiating +an unspecified big-O estimate. C(0) is the positive source coefficient. QED. + +## 3. Explicit second coefficient + +Write C_0=C_(0), C_2=C_(2), to distinguish coefficients from their tail powers11,13. +Set the differential operator acting on G_lambda at fixed a,h,O,xi + + D_a=k partial_beta+(a/4)partial_c. + +The pointwise density has expansion g(y)=G0+g2 y^2+g3 y^3+O(y^4), with + + g2=b0 D_a G0. (10) + +There is no linear y term. Expanding the exact endpoints, or differentiating +(7), gives + + j0=(2/11)gamma^11 G0, + j2=u^2 gamma^9(1+12A^2)G0+(2/13)b0 gamma^13 D_a G0, + J_delta=j0+j2 delta^2+O_L1(delta^4). (11) + +For the geometric part, write w_+=gamma-uA delta+ +[u^2 gamma-u^2/(2gamma)]delta^2+O(delta^3), with the corresponding reflected +negative endpoint. Then (w_+^11-w_-^11)/11 equals +2gamma^11/11+u^2 gamma^9(1+12A^2)delta^2+O(delta^4). +The g2 term contributes2g2 gamma^13 delta^2/13. The odd g3 term first contributes +at order delta^4 after the two physical sides are summed; it is not dropped +from the all-order argument. + +The necessary exact shape integrals are + + I=integral_S Q=246528/35, + U2=integral_S u^2 Q=240192/35, + B=integral_S b0 Q=-428544/7=3I-12U2, + Uabs=integral_S |u|Q=5898627/880. (12) + +They follow by integrating the polynomial Q in v and then on the strips cut +at u=-1/2 and u=0. The companion computes them in exact Fraction arithmetic. +In particular + + C_0=K0*(2I/11) integral H gamma^11 G0 d lambda, + C_2=K0 integral H [U2 gamma^9(1+12A^2)G0 + +(2B/13)gamma^13 D_a G0]d lambda. (13) + +Both integrals are finite by (8). C_0>0. No general sign or nonzero assertion +for C_2 is made in arbitrary dimension/frame; Section7 supplies an exact +noncancellation result at a narrower source scope. No numerical evaluation of +either Gaussian integral is claimed. + +## 4. Rates for the radial ratio and unsigned marks + +Let P_t be point-INTENSITY sampling conditioned on physical radius R>t. Define +q=R/t>=1 and retain Y=(u,v,lambda), but FORGET sign Z. These are the root-resolved +latent coordinates of [R], not a configuration-uniform selection. Define the +finite measures with densities + + dmu0=K0 Q H j0 dY, dmu2=K0 Q H j2 dY, + c=C_2/C_0. (14) + +mu0 has mass C_0 and mu2 mass C_2. The joint law of (q,Y) has the L1 expansion + + dP_t(q,Y)=11q^-12 dq dmu0(Y)/C_0 + +t^-2 [13q^-14 dq dmu2(Y)/C_0 + -11c q^-12 dq dmu0(Y)/C_0]+O_L1(t^-4). (15) + +In particular its total-variation distance to the stated independent limiting +product is O(t^-2). Projection to Y, or to the downward height +eta=u+1/2+v/6, retains this rate. This supplies an O(t^-2) TV rate for the +intensity-selected height law H_eta/I in [R, H1-H2]. No spatial-configuration +TV claim is made: the map to actual positions depends on t. + +Proof: for fixed Y, the radius tail density is t^-11 K0 QH J_(1/t). +Differentiate the smooth expression already justified in Section2; at tq its +radial density times t equals + +t^-11 K0 QH q^-12[11J_(1/(tq))+(1/(tq))J'_(1/(tq))]. + +Divide by F(t)=t^-11 C(1/t) and use the L1 Taylor estimates. The remainder is +bounded after integration by a constant times t^-4 q^-12, integrable on q>=1. +This proves (15) without differentiating a merely pointwise asymptotic. QED. + +Let Pareto_11 have density11q^-12 on q>=1. For the scalar radial ratio, (15) gives + + p_t(q)=11q^-12[1+c t^-2((13/11)q^-2-1)]+O_L1(t^-4). + +**Sharp scalar coefficient:** + + dTV(Law_(P_t)(R/t),Pareto_11) + =kappa |c| t^-2+O(t^-4), + kappa=(2/13)(11/13)^(11/2). (16) + +The same leading coefficient holds for the supremum distance of their CDFs. +The correction changes sign only at q0=sqrt(13/11). Its positive integral is +q0^-11-q0^-13=kappa; its total integral is zero. Half its L1 norm is therefore +kappa, proving (16). The uniform CDF remainder follows from the L1 remainder; +the maximum of q^-11(1-q^-2) is also kappa. If C_2=0 the stated error is O(t^-4), +not an unjustified claim of a nonzero second-order term. + +As another quantitative consequence, uniformly over q>=1, + + F(tq)/F(t)=q^-11[1+c t^-2(q^-2-1)]+O(t^-4 q^-11). (17) + +This is a source-specific second-order regular-variation statement. Generic +second-order tail theory is not claimed novel. + +## 5. Retaining the sign has a different, strictly slower sharp rate + +The cancellation in Section2 used both signs. Keep epsilon=sign Z and the +same Y. The two half integrals are exactly + + J_delta,epsilon=integral_0^(d+epsilon m) w^10 g(epsilon delta w)dw, + J_delta,epsilon=j0/2+epsilon h1 delta+O_L1(delta^2), + h1=-uA gamma^10 G0. (18) + +The linear term comes from the physical cutoff shift m. g'(0)=0, so no density +term cancels it. Applying the differentiated density argument to each half yields + + dTV(Law_(P_t)(epsilon,R/t,Y), + UniformSign x Pareto_11 x mu0/C_0) + =B_sign t^-1+O(t^-2), + B_sign=(K0/C_0) Uabs integral H |A|gamma^10 G0 d lambda>0. (19) + +Indeed the first signed correction has density +12 epsilon h1 q^-13 K0 QH/C_0; its half-L1 norm, summed over the two signs and +integrated over q, is the displayed B_sign because integral_1^infinity12q^-13dq=1. +Finiteness follows from (8); strict positivity follows from the full positive +Gaussian density on open sets with a!=0 and positive hard spectrum. The gauge +convention for O is exactly that in [R]; (19) is a result for those retained +latent variables, not a statement that a sign alone has a nonzero marginal bias. +Marginalizing can cancel a correction, and specifically forgetting epsilon +cancels this one. Thus signed and unsigned rates must not be conflated. + +## 6. A finite part for the critical eleventh moment + +Let M11(t)=integral sum_i min(R_i,t)^11 dM. The prior theorem gave its leading +logarithm. There is now a finite constant L11 such that + + M11(t)=11 C_0 log t+L11-(11/2)C_2 t^-2+O(t^-4). (20) + +One exact definition is + + L11=11 integral_0^1 x^10 F(x)dx + +11 integral_1^infinity[x^10 F(x)-C_0/x]dx. + +The first integral is finite by finite total point mass. The second converges +by (5) at N=1. Tonelli gives M11(t)=11 integral_0^t x^10 F(x)dx; subtracting +its limiting finite part and integrating the controlled remainder proves (20). +No numerical value for L11 is asserted. This is again after the r limit only. + +## 7. Exact positive second correction in the aligned planar periodic model + +**Theorem 2.** In d=2, with the axial coordinate aligned with one of the lattice +axes of [P]'s EXACT periodized kernel, C_2>0 for every fixed L,k>0 and b real. +Consequently the radial TV and CDF errors in (16) are genuinely Theta(t^-2), +with the positive coefficient in (16); not merely an upper bound. + +Proof: [P]'s kernel factors as K_L(x,z)=k_L(x)k_L(z). Let m2,m4,m6 be the even +spectral moments of the one-dimensional variance-one factor; m2>0 and the full +positive lattice spectrum gives m4-m2^2>0 and m6-m4^2/m2>0. At contact condition +on (f,fx,fxx,fxxx,fz,fxz)=(b,0,0,12k,0,0). + +Even-total-order jets and odd-total-order jets are independent. For the odd +soft cubic jets (a,beta,c)=(fxxz,fxzz,fzzz), separability and reflection in each +axis give a centered DIAGONAL conditional Gaussian with variances + + va=vbeta=m2(m4-m2^2), vc=m6-m4^2/m2. (21) + +For transparency: Cov(beta,(fx,fxxx))=(-m2^2,m2 m4) is -m2 times the first row +of Cov(fx,fxxx), so its conditional mean is -m2 fx=0 even though fxxx=12k. +a and c correlate only with fz in that odd conditioning block, with covariances +-m2^2 and -m4; their raw covariance m2 m4 is removed by that regression. Their +conditional covariance with beta vanishes by coordinate reflection. The diagonal +variances follow by subtracting these rank-one regressions. The even soft +curvature has its own strictly positive conditional density at zero and merely +multiplies this cubic Gaussian by a positive b-dependent scalar. + +Therefore on the cusp + + D_a G0=-[a^2/(12vbeta)+a^4/(576k^2 vc)]G0<=0. (22) + +In (13), U2>0, B<0 and G0>0. The geometric term is strictly positive and the +second term is nonnegative. Thus C_2>0, without replacing finite L by a continuum +covariance or numerically approximating a lattice sum. The same common even +b-dependent factor cancels from c=C_2/C_0 and B_sign at this planar aligned +scope. We do not extend this diagonal regression or noncancellation assertion +to arbitrary rotations or d>=3 without a separate argument. QED. + +## 8. Verification and review boundary + +A: reciprocal physical cutoff, fixed-domain map, C-infinity L1 Gaussian derivative +domination, exact parity and differentiated all-order asymptotic expansion. +B: j0/j2 coefficients, all four exact shape integrals, signed/unsigned TV and CDF +constants, finite-part moment, and aligned planar regression/noncancellation. + +The stdlib companion checks shape integrals, formal endpoint equations, exact +Taylor coefficients, cancellation and signed bias, independent scalar controls +and deliberately false variants. These are finite checks, not a formal proof +of any Gaussian input or differentiation under an infinite-dimensional integral. +No Lean formalization, global status change, coefficient enclosure or self-merge. + +No rate for r->0, no uniform simultaneous t(r)->infinity statement, no claim +about the remote singleton population, no convergence rate for whole-cluster +maximum selectors across ties, and no spatial independence/elder pairing is +added by this packet. Sign-forgetting is part of (15), not an optional omission. diff --git a/frontiers/radial_rate_20260930/README.md b/frontiers/radial_rate_20260930/README.md new file mode 100644 index 000000000..cc0e2f5a4 --- /dev/null +++ b/frontiers/radial_rate_20260930/README.md @@ -0,0 +1,61 @@ +# Microscopic radial rates and an even inverse-radius expansion + +OpenAI / GPT-6 Astra Pro. Source-bound conditional candidate; nonauthor review +OPEN. Scientific effect NONE. No self-merge or parent/status/graph modification. + +The earlier point-intensity result F(t)~C0 t^-11 is sharpened here to a smooth +even expansion in delta=1/t, with controlled derivatives to every fixed order. +The original field limit r->0 is taken FIRST. This is not a rate for that limit. + +Main manuscript PROOF.md: +- F(t)=C0 t^-11+C2 t^-13+O(t^-15), with an explicit finite Gaussian-derivative + formula for C2 and all-even fixed-order extensions. +- The radial ratio's Pareto(11) TV error has coefficient + (2/13)(11/13)^(11/2)|C2/C0| t^-2+O(t^-4). +- The unsigned shape/parameter/radial joint law has an L1 second-order expansion; + its height marginal consequently converges in TV at O(t^-2). +- RETAINING sign Z gives a strictly positive first-order joint TV coefficient, + Bsign t^-1+O(t^-2). Forgetting sign cancels that term. +- The capped eleventh moment has a finite renormalized limit and an explicit + t^-2 correction, not merely a leading logarithm. +- For the exact aligned PLANAR periodized kernel, Gaussian separability proves + C2>0 at every fixed L,k>0 and b real. Thus the radial error is genuinely + Theta(t^-2) at that narrower scope. No isotropic replacement is used. + +No numerical Gaussian coefficient, finite-r spatial moment/rate, simultaneous +radius limit, whole-cluster maximum rate, spatial TV convergence or independence +between pin pairs is asserted. All-rate statements concern the root-resolved +POINT measure. The parent whole-cluster and elder-geometry lanes are untouched. + +## Replay + +From a full checkout with the three pinned historical source objects: + + python -B -S frontiers/radial_rate_20260930/verify.py + +Standalone download: + + python -B -S verify.py --local-only + +The latter reports source_pins_checked=false. Real local Git fixtures test +custody behavior, not the actual project sources. The full workflow verifies +all source identities and the flat packet before replaying both Python modes, +deterministic results and eight semantic negative controls. No third-party +Python library is required by the shipped scripts. + +Forty local tests passed in each normal/optimized mode at author validation. +They include exact shape integrals, endpoint identities, signed/unsigned formal +series, a high-precision scalar diagnostic, planar Gaussian Schur complements, +and real Git replacement/noncommit/path/inventory attacks. Tests do not prove +the Gaussian domination, differentiation, or the new probability limits. + +## Review requests + +A: Sections2-3, exact physical-cutoff inversion, fixed-domain representation, +C-infinity L1 domination, parity, derivatives and explicit second coefficient. +B: Sections4-7, normalized joint/radial TV rates, signed first-order bias, finite +critical-moment part and exact planar regression/noncancellation. + +The author is exposed to SC/P and related source packets. New reviews must state +that exposure and bind the exact mathematical bytes; existing source verdicts +or green workflows are not automatic acceptance of these new deductions. diff --git a/frontiers/radial_rate_20260930/RECONNAISSANCE.md b/frontiers/radial_rate_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..3eb935b8d --- /dev/null +++ b/frontiers/radial_rate_20260930/RECONNAISSANCE.md @@ -0,0 +1,33 @@ +# Narrow primary-source reconnaissance — 30 September 2026 UTC + +Exact searches actually executed: +1. second order regular variation Pareto approximation total variation Hall class tail expansion paper +2. site.arxiv.org Palm theory extremes anchored tail processes Planinic 2104.03810 + +Primary records consulted: +- Hrvoje Planinic, Palm theory for extremes of stationary regularly varying time + series and random fields, arXiv:2104.03810v2 (14 March2023). + https://arxiv.org/abs/2104.03810 + Direct abstract/metadata open succeeded. It distinguishes typical exceedances + from typical clusters; no generic sampling/Palm principle is claimed new here. +- Second-order regular variation, convolution and the central limit theorem, + Stochastic Processes and their Applications69(2),139-159(1997), + DOI10.1016/S0304-4149(97)00042-2. + https://www.sciencedirect.com/science/article/pii/S0304414997000422 + The search response supplied the abstract describing second-order tail theory. + Direct open returned403. No full-text/theorem-level audit is claimed. +- L. de Haan and L. Peng, Rates of Convergence for Bivariate Extremes, + Journal of Multivariate Analysis61(2),195-230(1997), DOI10.1006/jmva.1997.1669. + https://doi.org/10.1006/jmva.1997.1669 + Search abstract states rates in total variation and uniform metrics under a + second-order regular-variation condition. Direct open returned403. + +This new packet derives its particular inverse-radius expansion from exact +GitHub source R11, not by importing an unchecked second-order tail theorem. +Generic second-order regular variation and metric convergence rates are prior +frameworks. No exhaustive novelty, priority, or absence claim is made. + +Consensus was not invoked in this pass: the user requested GitHub/Superpowers, +and prior actual repeated attempts had returned the quota error. No changed +quota status is asserted. All external access depth above is limited to the +actual returned abstracts/metadata. No PDF or full paper was inspected. diff --git a/frontiers/radial_rate_20260930/RESULTS.json b/frontiers/radial_rate_20260930/RESULTS.json new file mode 100644 index 000000000..c7494046e --- /dev/null +++ b/frontiers/radial_rate_20260930/RESULTS.json @@ -0,0 +1,21 @@ +{ + "mathematical_acceptance": false, + "passed": true, + "scientific_effect": "NONE", + "scope": "finite rational coefficient and parity controls only", + "series_cases": 12, + "shape_integrals": { + "ABS_U": "5898627/880", + "B": "-428544/7", + "I": "246528/35", + "U2": "240192/35" + }, + "signed_joint_rate": 1, + "tail_exponents": [ + 11, + 13, + 15 + ], + "tv_crossing_squared": "13/11", + "unsigned_radial_rate": 2 +} diff --git a/frontiers/radial_rate_20260930/SOURCES.json b/frontiers/radial_rate_20260930/SOURCES.json new file mode 100644 index 000000000..37f3dc7b5 --- /dev/null +++ b/frontiers/radial_rate_20260930/SOURCES.json @@ -0,0 +1,33 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "id": "R", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "use": "R7-R13 exact point-intensity integral/majorant; H1-H2 limiting height law." + }, + { + "id": "SC", + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "use": "Raw nondegenerate contact Gaussian, orthogonal spectral conventions, finite microscopic measure. Author exposure." + }, + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "use": "Exact product-periodized covariance and contact pin vector; no new parent acceptance." + } + ] +} diff --git a/frontiers/radial_rate_20260930/VALIDATION.md b/frontiers/radial_rate_20260930/VALIDATION.md new file mode 100644 index 000000000..5a4a11b11 --- /dev/null +++ b/frontiers/radial_rate_20260930/VALIDATION.md @@ -0,0 +1,45 @@ +# Author validation and limits + +The initial eight-test mathematics scaffold returned seven failing assertions +(the zero odd-series test already passed on the zero scaffold). After exact +implementation all eight passed. Expanded mathematics tests total21. + +The nineteen-test custody scaffold returned25 failing assertions including +subtests. The implemented verifier passed all19. It rejects changed identities, +missing/duplicate/unexpected source IDs, unsafe or missing paths, noncommit tree +and tag objects, Git symlinks, replacement refs, and altered packet inventories. +The replacement fixture verifies that ordinary Git reads really see the forged +bytes before confirming that the protected verifier still authenticates only +the actual original source. This is a test fixture, not observed source tampering. + +Complete local result:40 tests PASS in each normal and optimized Python mode; +12 exact formal-series parameter cases; eight explicit semantic mutants rejected +with rc1 in each mode; unknown mutant rc2; deterministic baseline equal to +RESULTS.json. The source/packet verifier checks its inventory before and after +execution. No temporary output is placed in the packet during replay. + +Exact controls cover all four shape integrals, formal physical-cutoff equations, +odd-term cancellation, the signed first-order term, the density multiplier13, +the TV crossing at sqrt(13/11), and exact planar Schur regressions. A Decimal +scalar diagnostic checks the constant-density endpoint remainder at three +thresholds; it is a finite diagnostic, not a Gaussian coefficient enclosure. +Exploratory SymPy integration reproduced the shape constants, but SymPy is not +a shipped dependency or a substitute for the analytic derivation. + +Local runtime Python3.13.5, Git2.47.3. The workflow requests Python3.11.16, as in +the actual upstream workflows. All shipped syntax uses standard-library3.11 +features. The local raw.githubusercontent.com request failed with DNS resolution +error, so no complete upstream checkout was obtained. Local-only replay states +source_pins_checked=false. The real three-source authentication remains a +HOSTED requirement, not inferred from local Git fixtures or manifest strings. +No hosted pass is anticipated in this author-time record. + +The full Gaussian derivative domination and L1 smoothness are written analytic +obligations. There is no Lean formalization of this new packet. No numerical +value of C0,C2,Bsign or the critical-moment finite part is evaluated. The retained +sign and the order r->0 before t->infinity are load-bearing qualifications. + +Peer support actually delivered: native review5360751376 accepts PR173 §3 Route C +at its conditional count-estimate scope and requests an explicit diagonal repair +of the alternate fixed-R/moving-cutoff reading. It does not authorize graph +execution or count as an independent review of this conversation's source proofs. diff --git a/frontiers/radial_rate_20260930/rate.py b/frontiers/radial_rate_20260930/rate.py new file mode 100644 index 000000000..d213a2762 --- /dev/null +++ b/frontiers/radial_rate_20260930/rate.py @@ -0,0 +1,170 @@ +"""Exact finite controls for the inverse-radius asymptotic candidate. + +Standard library only. Series calculations are formal algebra, not authentication +of Gaussian premises or a proof of domination/total-variation convergence. +""" +import argparse +from fractions import Fraction as F +import json +from math import comb + +MUTANTS=('omit-shift','omit-density-term','drop-a-shear','odd-term-survives', + 'unsigned-is-signed','wrong-b-integral','wrong-density-factor','wrong-tv-root') + + +def rational(x): + if type(x) not in (int,F): raise TypeError('exact int/Fraction required') + return F(x) + + +def add(a,b): + out=dict(a) + for i,x in b.items():out[i]=out.get(i,F(0))+x + return {i:x for i,x in out.items() if x} + + +def scale(a,c): return {i:x*c for i,x in a.items() if x*c} + + +def mul(a,b): + out={} + for i,x in a.items(): + for j,y in b.items():out[i+j]=out.get(i+j,F(0))+x*y + return {i:x for i,x in out.items() if x} + + +def power(a,n): + out={0:F(1)} + for _ in range(n):out=mul(out,a) + return out + + +def integral(a,lo,hi): + return sum((x*(hi**(i+1)-lo**(i+1))/F(i+1) for i,x in a.items()),F(0)) + + +def shape_integrals(): + b={0:F(3),2:F(-12)};u={1:F(1)};hi={0:F(3),1:F(-6)} + # Q=(4v^2-b^2)(b-4uv); dictionary index is v power. + Q={3:scale(u,-16),2:scale(b,4),1:scale(mul(u,power(b,2)),4),0:scale(power(b,3),-1)} + ans={key:F(0) for key in ('I','U2','B','ABS_U')} + for lo_u,hi_u,sgn_b,sgn_u in ((F(-3,2),F(-1,2),-1,-1),(F(-1,2),F(0),1,-1),(F(0),F(1,2),1,1)): + low=scale(b,F(sgn_b,2));vint={} + for j,c in Q.items(): + difference=add(power(hi,j+1),scale(power(low,j+1),-1)) + vint=add(vint,scale(mul(c,difference),F(1,j+1))) + for key,weight in [('I',{0:F(1)}),('U2',{2:F(1)}),('B',b),('ABS_U',{1:F(sgn_u)})]: + ans[key]+=integral(mul(vint,weight),lo_u,hi_u) + return ans + + +def smul(a,b,n): + out=[F(0)]*(n+1) + for i,x in enumerate(a[:n+1]): + for j,y in enumerate(b[:n+1-i]):out[i+j]+=x*y + return out + + +def spow(a,m,n): + out=[F(1)]+[F(0)]*n + for _ in range(m):out=smul(out,a,n) + return out + + +def endpoints(u,A,g,n): + u,A,g=map(rational,(u,A,g)) + if type(n) is not int or not 0<=n<=12: raise ValueError('series order 0..12 required') + if g<=0 or g*g!=1+A*A: raise ValueError('positive exact gamma^2=1+A^2 required') + inv=[F(0)]*(n+1);root=[F(0)]*(n+1) + binom=F(1) + for j in range(n//2+1): + inv[2*j]=u**(2*j) + if j:binom*= (F(1,2)-(j-1))/j + root[2*j]=g*binom*(-u*u/(g*g))**j + d=smul(inv,root,n) + m=[F(0)]+[-u*A*x for x in inv[:n]] + plus=[x+y for x,y in zip(m,d)];minus=[x-y for x,y in zip(m,d)] + return plus,minus + + +def validate_phi(phi,n): + if not isinstance(phi,dict) or any(type(j) is not int or j<0 for j in phi): + raise ValueError('nonnegative polynomial orders required') + return {j:rational(x) for j,x in phi.items() if j<=n} + + +def tail_series(u,A,g,phi,n): + plus,minus=endpoints(u,A,g,n);phi=validate_phi(phi,n) + out=[F(0)]*(n+1) + for j,c in phi.items(): + p=spow(plus,11+j,n-j);m=spow(minus,11+j,n-j) + for i in range(n-j+1):out[i+j]+=c*(p[i]-m[i])/F(11+j) + return out + + +def half_series(u,A,g,phi,n,sign): + if type(sign) is not int or sign not in (-1,1): raise ValueError('sign must be +/-1') + plus,minus=endpoints(u,A,g,n);phi=validate_phi(phi,n) + endpoint=plus if sign==1 else [-x for x in minus] + out=[F(0)]*(n+1) + for j,c in phi.items(): + p=spow(endpoint,11+j,n-j) + for i in range(n-j+1):out[i+j]+=c*sign**j*p[i]/F(11+j) + return out + + +def density_series(coeffs):return [(11+j)*x for j,x in enumerate(coeffs)] + + +def predicted_second(u,A,g,phi0,phi2): + return u*u*g**9*(1+12*A*A)*phi0+F(2,13)*g**13*phi2 + + +def check(mutant=None): + vals=shape_integrals() + expected={'I':F(246528,35),'U2':F(240192,35),'B':F(-428544,7),'ABS_U':F(5898627,880)} + if mutant=='wrong-b-integral': vals['B']=-vals['B'] + if vals!=expected: raise ValueError('shape integral mismatch') + cases=0 + for u in (F(-5,4),F(-3,4),F(-1,4),F(1,4)): + for p in (F(-1,2),F(0),F(1,3)): + A=2*p/(1-p*p);g=(1+p*p)/(1-p*p) + phi={0:F(2),2:F(-3),3:F(5),4:F(7)} + got=tail_series(u,A,g,phi,6) + if got[0]!=2*g**11*phi[0]/11:raise ValueError('leading coefficient') + guess=predicted_second(u,A,g,phi[0],phi[2]) + if mutant=='omit-shift':guess-=10*u*u*A*A*g**9*phi[0] + if mutant=='omit-density-term':guess-=F(2,13)*g**13*phi[2] + if mutant=='drop-a-shear':guess=predicted_second(u,F(0),F(1),phi[0],phi[2]) + if got[2]!=guess:raise ValueError('second coefficient mismatch') + odd=[got[i] for i in (1,3,5)] + if mutant=='odd-term-survives':odd[0]=1 + if any(odd):raise ValueError('odd inverse powers do not cancel') + hplus=half_series(u,A,g,phi,6,1);hminus=half_series(u,A,g,phi,6,-1) + if [a+b for a,b in zip(hplus,hminus)]!=got:raise ValueError('half sum') + h1=-u*A*g**10*phi[0] + if mutant=='unsigned-is-signed':h1=F(0) + if hplus[1]!=h1:raise ValueError('signed first-order bias') + den=density_series(got) + expected_den=13*got[2] + if mutant=='wrong-density-factor':expected_den=11*got[2] + if den[2]!=expected_den:raise ValueError('radial density coefficient') + cases+=1 + crossing2=F(13,11) if mutant!='wrong-tv-root' else F(11,13) + if F(13,11)/crossing2!=1:raise ValueError('TV crossing root') + return {'passed':True,'mathematical_acceptance':False,'scientific_effect':'NONE', + 'scope':'finite rational coefficient and parity controls only', + 'shape_integrals':{k:str(v) for k,v in expected.items()}, + 'series_cases':cases,'tail_exponents':[11,13,15], + 'unsigned_radial_rate':2,'signed_joint_rate':1,'tv_crossing_squared':str(crossing2)} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args() + try:result=check(args.mutant) + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True));raise SystemExit(1) + print(json.dumps(result,sort_keys=True,indent=2)) + +if __name__=='__main__':main() diff --git a/frontiers/radial_rate_20260930/test_rate.py b/frontiers/radial_rate_20260930/test_rate.py new file mode 100644 index 000000000..4074549e5 --- /dev/null +++ b/frontiers/radial_rate_20260930/test_rate.py @@ -0,0 +1,137 @@ +from fractions import Fraction as F +import unittest +import rate + +class InitialTests(unittest.TestCase): + def test_shape_mass(self): self.assertEqual(rate.shape_integrals()['I'],F(246528,35)) + def test_shape_u2(self): self.assertEqual(rate.shape_integrals()['U2'],F(240192,35)) + def test_shape_b(self): self.assertEqual(rate.shape_integrals()['B'],F(-428544,7)) + def test_shape_abs_u(self): self.assertEqual(rate.shape_integrals()['ABS_U'],F(5898627,880)) + def test_leading_boundary(self): + c=rate.tail_series(F(-3,4),F(3,4),F(5,4),{0:F(1)},4) + self.assertEqual(c[0],2*F(5,4)**11/11) + def test_second_boundary(self): + u,A,g=F(-3,4),F(3,4),F(5,4) + c=rate.tail_series(u,A,g,{0:F(1),2:F(2)},4) + self.assertEqual(c[2],u*u*g**9*(1+12*A*A)+F(4,13)*g**13) + def test_even_series(self): + c=rate.tail_series(F(-3,4),F(3,4),F(5,4),{0:F(1),2:F(2),3:F(7)},6) + self.assertEqual([c[i] for i in (1,3,5)],[0,0,0]) + def test_signed_first_coefficient(self): + u,A,g=F(-3,4),F(3,4),F(5,4) + c=rate.half_series(u,A,g,{0:F(1),2:F(2),3:F(7)},4,1) + self.assertEqual(c[1],-u*A*g**10) + + +class ExtendedTests(unittest.TestCase): + def test_endpoint_equation(self): + for u in (F(-5,4),F(-1,2),F(1,4)): + for p in (F(-1,2),F(0),F(1,3)): + A=2*p/(1-p*p);g=(1+p*p)/(1-p*p);n=8 + for B in rate.endpoints(u,A,g,n): + square=rate.smul(B,B,n) + for j in range(n+1): + value=square[j]-(u*u*square[j-2] if j>=2 else 0) + value+=2*u*A*B[j-1] if j>=1 else 0 + value-=g*g if j==0 else 0 + self.assertEqual(value,0) + def test_general_polynomial_parity(self): + phi={0:F(1),1:F(3),2:F(-5),3:F(7),4:F(2),5:F(-1)} + got=rate.tail_series(F(-4,3),F(3,4),F(5,4),phi,10) + self.assertTrue(all(got[j]==0 for j in (1,3,5,7,9))) + def test_half_reflection(self): + u,A,g=F(-3,4),F(3,4),F(5,4);phi={0:F(2),2:F(-3),3:F(5)} + plus=rate.half_series(u,A,g,phi,8,1);minus=rate.half_series(u,A,g,phi,8,-1) + self.assertEqual(minus,[x*(-1)**j for j,x in enumerate(plus)]) + def test_cubic_density_term_first_enters_at_four(self): + args=F(-3,4),F(3,4),F(5,4) + x=rate.tail_series(*args,{0:F(1)},6);y=rate.tail_series(*args,{0:F(1),3:F(5)},6) + self.assertEqual(x[:4],y[:4]);self.assertNotEqual(x[4],y[4]) + def test_shear_zero_removes_first_signed_bias(self): + got=rate.half_series(F(-3,4),F(0),F(1),{0:F(1),2:F(3),3:F(5)},6,1) + self.assertEqual(got[1],0);self.assertNotEqual(got[3],0) + def test_density_multiplier(self): + coeff=[F(2),0,F(3),0,F(5)] + self.assertEqual(rate.density_series(coeff),[F(22),0,F(39),0,F(75)]) + def test_b_integral_relation(self): + c=rate.shape_integrals();self.assertEqual(c['B'],3*c['I']-12*c['U2']) + def test_constant_density_decimal_remainder(self): + from decimal import Decimal,localcontext + with localcontext() as ctx: + ctx.prec=70;D=Decimal;u=D('-0.75');A=D('0.75');g=D('1.25') + c0=2*g**11/11;c2=u*u*g**9*(1+12*A*A) + vals=[] + for t in (D(40),D(80),D(160)): + delta=1/t;disc=(g*g-u*u*delta*delta).sqrt();den=1-u*u*delta*delta + bp=(-u*A*delta+disc)/den;bm=(-u*A*delta-disc)/den + J=(bp**11-bm**11)/11 + vals.append(abs((J-c0-c2*delta*delta)/delta**4)) + self.assertTrue(all(x0 for x in expected)) + def test_second_correction_positive_in_aligned_case(self): + vals=rate.shape_integrals() + for a in (F(0),F(1),F(-3)): + for k in (F(1,2),F(1),F(2)): + vbeta,vc=F(2),F(6);da=-(a*a/(12*vbeta)+a**4/(576*k*k*vc)) + self.assertGreater(vals['U2'],0) + self.assertGreaterEqual(vals['B']*da,0) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/radial_rate_20260930/test_verify.py b/frontiers/radial_rate_20260930/test_verify.py new file mode 100644 index 000000000..258288f8e --- /dev/null +++ b/frontiers/radial_rate_20260930/test_verify.py @@ -0,0 +1,102 @@ +"""Custody regressions on real local Git objects, never on invented upstream data.""" +import copy +import hashlib +import json +import os +from pathlib import Path +import subprocess +import tempfile +import unittest +import verify as v + + +def entry(path,data,**kw): + return dict(path=path,bytes=len(data),sha256=hashlib.sha256(data).hexdigest(), + git_blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest(),**kw) + +class SourceTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.repo=Path(t.name) + self.root=self.repo/'frontiers'/'packet';self.root.mkdir(parents=True) + self.git('init','-q');self.data=b'fixture\n' + for name in ('R','SC','P'):(self.repo/(name+'.md')).write_bytes(self.data) + self.git('add','R.md','SC.md','P.md');self.git('commit','-qm','fixture') + self.commit=self.git('rev-parse','HEAD').decode().strip() + self.es=[entry(x+'.md',self.data,id=x,commit=self.commit) for x in ('R','SC','P')];self.write() + def git(self,*args): + env=os.environ.copy();env.pop('GIT_NO_REPLACE_OBJECTS',None) + return subprocess.run(['git','-c','user.name=Fixture','-c','user.email=fixture@example.invalid',*args], + cwd=self.repo,env=env,check=True,capture_output=True,timeout=15).stdout + def write(self,es=None): + (self.root/'SOURCES.json').write_text(json.dumps({'sources':self.es if es is None else es})) + def test_valid_nested(self):self.assertEqual(v.sources(self.root),3) + def test_identity_fields(self): + for k,val in (('bytes',0),('bytes',True),('sha256','0'*64),('git_blob','0'*40)): + es=copy.deepcopy(self.es);es[0][k]=val;self.write(es) + with self.subTest(k=k,val=val),self.assertRaises(ValueError):v.sources(self.root) + def test_missing_and_duplicate_inventory(self): + for es in ([],self.es[:-1],self.es+[self.es[0]]): + self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_wrong_id(self): + es=copy.deepcopy(self.es);es[0]['id']='X';self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_unsafe_paths(self): + for p in ('../R.md','/R.md','./R.md','x//R.md','x/../R.md',''): + es=copy.deepcopy(self.es);es[0]['path']=p;self.write(es) + with self.subTest(path=p),self.assertRaises(ValueError):v.sources(self.root) + def test_missing_file(self): + es=copy.deepcopy(self.es);es[0]['path']='missing.md';self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_noncommit_tree(self): + es=copy.deepcopy(self.es);es[0]['commit']=self.git('rev-parse','HEAD^{tree}').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_annotated_tag(self): + self.git('tag','-a','tag','-m','tag');es=copy.deepcopy(self.es) + es[0]['commit']=self.git('rev-parse','tag').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_missing_commit(self): + es=copy.deepcopy(self.es);es[0]['commit']='0'*40;self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_leaf_symlink(self): + (self.repo/'R.md').unlink();(self.repo/'R.md').symlink_to('SC.md');self.git('add','R.md');self.git('commit','-qm','link') + es=copy.deepcopy(self.es);es[0]['commit']=self.git('rev-parse','HEAD').decode().strip();self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + def test_replacements_disabled(self): + changed=b'forged bytes\n';(self.repo/'R.md').write_bytes(changed);self.git('add','R.md');self.git('commit','-qm','changed') + newer=self.git('rev-parse','HEAD').decode().strip();self.git('replace',self.commit,newer) + self.assertEqual(self.git('show',self.commit+':R.md'),changed) + self.assertEqual(v.sources(self.root),3) + es=copy.deepcopy(self.es);es[0].update(entry('R.md',changed));self.write(es) + with self.assertRaises(ValueError):v.sources(self.root) + +class PacketTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory();self.addCleanup(t.cleanup);self.root=Path(t.name);self.data=b'payload\n' + (self.root/'proof.md').write_bytes(self.data) + self.entries=[entry('proof.md',self.data)];self.write() + def write(self): (self.root/'MANIFEST.json').write_text(json.dumps({'files':self.entries})) + def test_valid(self):self.assertEqual(v.inventory(self.root),1) + def test_changed(self): + (self.root/'proof.md').write_bytes(b'changed') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_extra(self): + (self.root/'extra').write_bytes(b'x') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_missing(self): + (self.root/'proof.md').unlink() + with self.assertRaises(ValueError):v.inventory(self.root) + def test_duplicate(self): + self.entries+=self.entries;self.write() + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_payload(self): + (self.root/'proof.md').unlink();(self.root/'proof.md').symlink_to('MANIFEST.json') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_symlink_manifest(self): + p=self.root/'MANIFEST.json';p.rename(self.root/'real');p.symlink_to('real') + with self.assertRaises(ValueError):v.inventory(self.root) + def test_json_duplicate(self): + p=self.root/'MANIFEST.json';p.write_text('{"files":[],"files":[]}') + with self.assertRaises(ValueError):v.inventory(self.root) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/radial_rate_20260930/verify.py b/frontiers/radial_rate_20260930/verify.py new file mode 100644 index 000000000..d5789cbe7 --- /dev/null +++ b/frontiers/radial_rate_20260930/verify.py @@ -0,0 +1,104 @@ +"""Exact packet replay; --local-only does not authenticate upstream Git sources.""" +import argparse +import hashlib +import json +import os +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + +IDS={'R','SC','P'} +MUTANTS=('omit-shift','omit-density-term','drop-a-shear','odd-term-survives', + 'unsigned-is-signed','wrong-b-integral','wrong-density-factor','wrong-tv-root') + + +def read_json(path): + if path.is_symlink() or not path.is_file():raise ValueError('regular JSON file required') + def unique(pairs): + ans={} + for k,v in pairs: + if k in ans:raise ValueError('duplicate JSON key') + ans[k]=v + return ans + ans=json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + if not isinstance(ans,dict):raise ValueError('JSON object required') + return ans + + +def path_parts(path): + if not isinstance(path,str) or not path:raise ValueError('nonempty path required') + p=PurePosixPath(path) + if p.is_absolute() or str(p)!=path or any(x in ('.','..') or not re.fullmatch('[A-Za-z0-9_.-]+',x) for x in p.parts): + raise ValueError('unsafe path') + return p.parts + + +def identity(data,e): + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if type(e['bytes']) is not int or len(data)!=e['bytes'] or hashlib.sha256(data).hexdigest()!=e['sha256'] or blob!=e['git_blob']: + raise ValueError('identity mismatch: '+e['path']) + + +def git(root,*args): + run=subprocess.run(['git','--no-replace-objects',*args],cwd=root, + env=dict(os.environ,GIT_NO_REPLACE_OBJECTS='1'),capture_output=True,timeout=45) + if run.returncode:raise ValueError('Git object read failed: '+run.stderr.decode(errors='replace')[:200]) + return run.stdout + + +def sources(root): + es=read_json(root/'SOURCES.json')['sources'] + if not isinstance(es,list) or len(es)!=len(IDS) or {e['id'] for e in es}!=IDS: + raise ValueError('missing, duplicate or unexpected source IDs') + seen=set() + for e in es: + parts=path_parts(e['path']);commit=e['commit'] + if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit):raise ValueError('exact commit required') + pair=(commit,e['path']) + if pair in seen:raise ValueError('duplicate source binding') + seen.add(pair) + if git(root,'cat-file','-t',commit).strip()!=b'commit':raise ValueError('not a commit object') + for j in range(1,len(parts)+1): + prefix='/'.join(parts[:j]);records=git(root,'ls-tree','--full-tree','-z',commit,'--',prefix).split(b'\0') + records=[x for x in records if x] + if len(records)!=1:raise ValueError('missing or ambiguous source path') + meta,name=records[0].split(b'\t',1);mode,kind,blob=meta.decode().split() + if name.decode()!=prefix:raise ValueError('path mismatch') + if jt} dM. (R1) + +This is point-intensity sampling: a cluster with two qualifying points contributes +two, not one. Every assertion below is a statement about the limiting microscopic +measure. In an application to the Gaussian field, r->0 is taken before t->infinity. +No finite-r uniform spatial moment or rate is inferred. + +**Theorem R.** There is an explicit finite positive C_star such that + + F(t) ~ C_star t^-11. (R2) + +For p>=0 the microscopic intensity moment integral sum_i R_i^p dM is finite +exactly when p<11. The same threshold holds for the maximum radius of a nonempty +cluster under its normalized limiting law. At the boundary, the capped intensity +moment satisfies + + integral sum_i min(R_i,t)^11 dM ~ 11 C_star log t. (R3) + +A full formula for C_star is given in (R13). No numerical evaluation of its +Gaussian integral is claimed. + +## 2. Resolve one stationary point by an exact change of variables + +Retain the canonical shear invariants of CUB: + + u=X+aZ/(12k), B=beta-a^2/(12k), + D=(c-a beta/(4k)+a^3/(72k^2))/2, x=-s>0. + +At fixed a the inverse raw-jet map is + + beta=a^2/(12k)+B, + c=a^3/(144k^2)+aB/(4k)+2D, + |d(beta,c)/d(B,D)|=2. (R4) + +The factor 2 is essential. This is a change in the JET coordinates, separate +from the determinant-one spatial shear. The remaining third-jet entries, denoted +xi, are unchanged. + +Let + + b0(u)=3-12u^2, v=xZ^2/k. + +For an extra root, the canonical conic and line give the inverse parametrization + + x=kv/Z^2, + B=k b0(u)/Z^2, + D=k(v-b0(u)u)/Z^3, Z!=0. (R5) + +Conversely, every parameter in the region defined below with Z!=0 gives a strict +window root at (X,Z)=(u-aZ/(12k),Z). The normalized DOWNWARD height is + + eta=-P/k=u+1/2+v/6. (R6) + +Typed endpoints require v>|b0(u)|/2. The open height window requires 0-6u-3 is automatic on R7: +for -1/2<=u<1/2 its right side is nonpositive; for -3/2=0. The upper height inequality and v>|b0|/2 are +compatible exactly on the displayed u interval. Boundaries are excluded. + +An explicit determinant computation gives the ORIENTED Jacobian + + det d(x,B,D)/d(v,u,Z) + =6k^3(12u^2+4uv-3)/Z^8 + =-6k^3[b0(u)-4uv]/Z^8. (R8) + +On R7 both b0(u)-4uv and 4v^2-b0(u)^2 are strictly positive. The first is also +the saddle determinant sign from CUB; it can be checked directly: for u<-1/2, +use v>|b0|/2, and for u>=0 use v<3-6u, giving a lower bound +12(u-1/2)^2, with the remaining interval -1/2<=u<0 immediate. The second follows +from v>|b0|/2. Thus the Jacobian is nonzero in the open region. + +For each fixed (x,B,D) in the typed domain there are at most two extra window +roots. The usual change of variables with multiplicity is legitimate here: +cover R7 times (R minus {0}) by countably many neighborhoods where R5 is a +diffeomorphism, make them disjoint measurably, apply the ordinary change of +variables to each, and sum the resulting nonnegative integrals. The multiplicity +is exactly the number of counted roots, rather than an omitted or extra factor +of two. Degenerate/boundary jets are algebraically null and not in the strict +window. This justifies applying R5 to a point-intensity sum. + +Define + + Q(u,v)=[4v^2-b0(u)^2][b0(u)-4uv]. (R9) + +The endpoint weight is w=9k^4[4v^2-b0(u)^2]/Z^4. Combining this with the absolute +Jacobian R8 and the raw-jet Jacobian 2 of R4 gives exactly + + w ds d beta dc =108 k^7 Q(u,v)|Z|^-12 du dv dZ, (R10) + +in the point-counted integral, with a and all other spectral/tensor variables +retained. This identity incorporates the root multiplicity through the map, +not by an additional constant. The two signs of Z will be integrated next. + +## 3. Tail scaling, domination and the complete coefficient + +Let h,O,xi be the remaining variables of M, and let tau(a,beta,c,xi) denote the +full rotated cubic tensor in those coordinates. Its raw contact Gaussian density +remains evaluated at + + h0(A0,Rot_O(tau)), A0=O diag(0,-h)O^t. + +It is NOT replaced by an isotropic density. Combining M1 and R10 gives the exact +representation of F(t) as the integral over R7 and Z!=0 of + + (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12 + h0(A0,Rot_O tau(a, a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4 Z^2)+2k(v-b0 u)/Z^3, xi)) + *1{ (u-aZ/(12k))^2+Z^2 > t^2 }, (R11) + +with measure du dv dZ da dxi dh dO, ordered positive h, and the empty-product +conventions for d=2. The coefficient a b0/(4Z^2) in c includes the k cancellation +in aB/(4k). Formula R11 is often where confusing the spatial and jet Jacobians, +or integrating out a before testing the actual radius, gives a wrong exponent +or coefficient. + +Put Z=t z and gamma(a)=sqrt(1+(a/(12k))^2). Then |Z|^-12 dZ=t^-11 |z|^-12 dz. +At each fixed nonzero z the raw jet tends to the cusp slice + + beta_cusp=a^2/(12k), c_cusp=a^3/(144k^2), + +and the radius indicator tends almost everywhere to 1{gamma(a)|z|>1}. +For t>=3 and |u|<=3/2, the indicator in R11 implies + + |z|>1/[2gamma(a)] + +by the triangle inequality t<|u|+gamma|Z|. Orthogonal norm equivalence and SC's +Gaussian density bound imply, uniformly in t,z,u,v,O, + + h0(A0,Rot_O tau(a,beta,c,xi)) + <= C exp[-c(|h|^2+a^2+|xi|^2)]. + +Dropping beta and c from this upper bound is legitimate; it never introduces +an inverse hard eigenvalue. Integration in z over |z|>1/(2gamma) costs at most +a constant times gamma^11. Q is bounded on the bounded region R7, H(h) is a +nonnegative polynomial, and Gaussian integration in h,a,xi dominates gamma^11. +Dominated convergence therefore applies to t^11 F(t). + +Let + + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) + da dxi dh dO. (R12) + +This is finite by the same majorant and positive by positivity of the Gaussian +density and H on open sets. Integrating both signs of z gives +integral_(gamma|z|>1)|z|^-12 dz=2 gamma^11/11. Finally, + + I_shape=integral_Dshape Q(u,v) du dv=246528/35, + + C_star=(216/11) k^7 (c_m/z0) I_shape J_cusp + =(53250048/385) k^7 (c_m/z0) J_cusp. (R13) + +For an exact check of I_shape, integrate v first using + + (4b0/3)v^3 -4u v^4 -b0^3 v +2u b0^2 v^2. + +The strips -3/2=2 under the source interface. The exponent 11 +is dimension-independent here because the hard directions affect only the +finite integral J_cusp; the soft jet weights are s,B of degree -2, D of degree -3, +and endpoint weight of degree -4, for total 2+2+3+4=11. This explanatory homogeneity +is not substituted for the dominated-convergence proof. + +For p>0, Tonelli gives the moment identity integral R^p=p integral_0^infinity +t^(p-1)F(t)dt. Finite total point mass handles t<=1 and R2 handles infinity. +This proves the threshold in Theorem R and the boundary formula R3. If R_max +is the maximum radius of a nonempty cluster, its unnormalized tail lies between +F(t)/2 and F(t), since there are at most two points. Its normalized law therefore +has the same exact moment threshold, although no exact maximum-tail coefficient +is claimed. QED. + +## 4. A universal law for an intensity-selected far local point + +Normalize the point-intensity measure restricted to R_i>t. For that selected +point retain its canonical shape variables (u,v), downward height eta, radial +ratio q=R_i/t, sign epsilon=sign Z, and the remaining variables (a,h,O,xi). + +**Theorem H (extreme local marks).** As t->infinity, the joint law converges +weakly to the product of: + +1. shape density Q(u,v)/I_shape on R7; +2. a Pareto radial ratio q>=1 with density 11q^-12; +3. a symmetric sign epsilon in {-1,1}; +4. the parameter measure with density H(h) gamma(a)^11 times the cusp Gaussian + in R12, divided by J_cusp. + +In particular eta=u+1/2+v/6 is independent of the limiting radial ratio and the +remaining parameter measure. Its probability density on 01. The limiting Gaussian depends only on (a,h,O,xi), while +Q depends only on (u,v). Division by R13 gives exactly the four factors above. +No claim of spatial independence in the original field is involved. + +To obtain H1, substitute v=6e-6u-3, with dv=6de. Conditions R7 become + + -1/2-sqrt(e) < u < 1/2-sqrt(1-e), 0=0 a reproducible rational formula before dividing by I_shape is + + integral e^j H_eta(e) de + =(110592/7)[1/(j+9/2)-B(j+1,9/2)+1/(j+1)] + +13824[1/(j+5)-2/(j+4)+5/(j+3)-4/(j+2)], (H4) + +with B(j+1,9/2)=j!/product_(l=0..j)(9/2+l). No special-function numeric library +is needed to check these moments. Seven such moments are checked independently +against exact double polynomial integration on R7. QED. + +## 5. Evidence boundary and useful falsifiers + +The finite companion independently checks the oriented root Jacobian, the inverse +raw-jet map and its factor 2, stationary heights and gradients at 648 rational +parameter choices, the two exact shape integrals, H4 versus direct integration, +and the k^7/radius-power ledgers. Eight explicit false variants must fail. +These controls do not prove the Gaussian bound, change-of-variables multiplicity, +weak convergence or domination; those are the written analytic obligations above. + +The result would fail if the contact slice lost full positive Gaussian density, +if the physical radius were incorrectly replaced by |Z| before retaining a, +or if the invariant density were assumed isotropic rather than kept under Haar. +A finite-r spatial-moment convergence claim, a global tail including remote +singletons, an ordinary configuration-selected version of H1, or a numerical +Gaussian value of C_star would require new work and is not supplied here. diff --git a/frontiers/two_scale_cluster_geometry_20260929/README.md b/frontiers/two_scale_cluster_geometry_20260929/README.md new file mode 100644 index 000000000..4036b28db --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/README.md @@ -0,0 +1,96 @@ +# Two-scale cluster geometry and an exact eleven-power tail + +OpenAI / GPT-6 Astra Pro. Author-side conditional mathematical packet, nonauthor +review COMPLETE at the stated conditional scope: Claude5360227991 (A), +5360178611 (B), and5360216551 (C). See REVIEW_RECORD.md. Scientific effect NONE; +no source proof or global status is changed. +No self-merge or organizational-independence claim. + +TWO_SCALE_LAW.md is a conditional consumer of the exact spectral proof in +Math-#162. It upgrades the count interface to a weak law of whole two-scale +configurations: microscopic roots are scaled by 1/r, remote singletons keep +physical torus positions. It proves localization of all leading ordered pair +mass, describes remote height marks, and shows why spatial total variation of +double clusters is impossible in d>=3 even though weak convergence holds. + +RADIAL_TAIL.md computes the tail of the resulting limiting MICROSCOPIC point +intensity. The tail is C_star t^-11 with a finite positive Gaussian coefficient; +its geometric factor is exactly 53250048/385 times k^7. The radial moment threshold +is p<11. An intensity-selected point far out in this limiting local law has an +independent Pareto(11) radial ratio and a universal downward-height density, with +mean 5771/7062. This is NOT a finite-r uniform tail or a global law including +remote singletons; the order is r->0 first, radius->infinity second. + +The exact scalar classifier of #158 is consumed only at its algebraic scope. +The #159 competitor is not a premise, and the #157 exponential-moment theorem +is not consumed. Source-condition failures block the corresponding applications. +A green check does not decide any mathematical review. + +## Replay + + python -B -S verify.py + +Run inside a full Git checkout with the immutable commits in SOURCES.json +available. This verifies all six named source objects (including ancestors' +Git modes), the flat packet inventory, 52 unit tests in each Python mode, +deterministic output, eight semantic negative controls and unknown-label refusal. +The workflow fetches the two non-base source commits explicitly. + + python -B -S verify.py --local-only + +This explicitly excludes project source-object authentication and reports +source_pins_checked=false. Local isolated-Git tests still exercise the checker; +they do not establish the actual project source identities. The new arguments +are not Lean-formalized; the repository's existing formal gate is separate. + +Exact controls include 648 rational stationary-point substitutions, the signed +three-variable Jacobian, raw-jet inverse, two shape-strip integrals, seven height +moments computed in two independent ways, localization inequalities and source +custody regressions. No Gaussian coefficient is numerically estimated. + +## Completed scoped reviews + +A. Two-scale configuration convergence, arbitrary cutoff, remote marks, counted +pair localization and the d>=3 spatial-TV obstruction. +B. Root change of variables with multiplicity, all Jacobian factors, Gaussian +majorant with the shear parameter retained, tail coefficient and moment boundary. +C. Extreme intensity-selected mark law, universal height density and exact +rational companion. All three reviews bind the unchanged mathematical files +at7c82252533c3fe14ee262f7f87c0549c10968592. One Claude session reviewed these +three disjoint scopes; this is not three independent-provider votes. Current +head/base verification and nonauthor integration remain separate requirements. + +The custody successor disables Git replacement objects and requires the declared +source identity to be a commit object, not merely a forty-hex tree-ish name. Four +real-Git red/green regressions cover forged/legitimate replacement bindings and +tree/tag substitution. Both mathematical manuscripts and all mathematical inputs +remain byte-identical to the initial reviewed head. + +## Current proof-to-review custody replay + +The earlier 52-test count describes the preceding custody repair. The current +replay runs 61 unit tests in each Python mode, including nine additional test +methods for proof and review bindings. The verifier requires both manuscripts, +the source/result declarations, both review records, the executable checker and +verifier, and both test modules. Regenerating MANIFEST.json cannot silently omit +these files or move an old review to different proof bytes. + +REVIEW_RECORD.json records the actual A/B/C native IDs, COMMENTED states, raw +review-body byte counts and SHA256 digests, exact reviewed commit, proof mapping, +conditional scopes and exposure. The verifier anchors the entire frozen record +and the accepted manuscript/source identities outside the regenerable inventory. +Intentional future mathematical changes require reviewed updates to those +anchors and the review record. Both current manuscripts, SOURCES.json, +RESULTS.json and geometry.py remain unchanged. + +This is an offline consistency check. Its successful summary explicitly reports +remote_review_authenticity_checked=false and mathematical_acceptance=false. +The native bodies were actually fetched during integration preflight; the local +CLI does not fetch them or authenticate a remote reviewer. In --local-only mode, +the source declaration's frozen bytes are still checked, while the historical +Git objects remain excluded and source_pins_checked=false. + +The same workflow now also runs for main pushes affecting this packet or its own +workflow, providing a scoped post-merge replay route. Prior PR/manual coverage +was not itself a verifier defect. Hosted success must bind the actual candidate +or merge commit; this local note supplies no CI result or mathematical vote. diff --git a/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md b/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..d6e523e5f --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md @@ -0,0 +1,45 @@ +# External reconnaissance — foreground recovery, 29 September 2026 Central + +Scope: primary-source context for Gaussian critical-point geometry, not an +exhaustive novelty, priority or full-text literature audit. No external theorem +is imported as an unverified substitute for the source-bound arguments. + +Consensus.search was actually called for Gaussian random field persistence, +rare critical clusters, spatial scaling, cubic saddles and marked processes. +It returned the monthly 30-search quota error (reset October 1), not papers. + +Primary arXiv abstract/metadata pages actually opened: + +1. Diego Armentano, Jean-Marc Azais and Jose Rafael Leon, On a general Kac-Rice +formula for the measure of a level set, arXiv:2304.07424v3. +https://arxiv.org/abs/2304.07424v3 +The page distinguishes v3 (5 December 2023) from later v4 (22 April 2025). The +in-project sources cite v3; this memo does not silently replace that version. +Only metadata and abstract were read here, not an independent framework audit. + +2. Dmitry Beliaev, Valentina Cammarota and Igor Wigman, No repulsion between +critical points for planar Gaussian random fields, arXiv:1911.03455v3 +(27 November 2020). https://arxiv.org/abs/1911.03455 +The abstract concerns near-diagonal two-point correlations for isotropic +stationary fields and index-sensitive behavior. It is relevant context, not the +same determinant-tilted, coalescing-pinned law or the new limiting radial tail. +No priority claim follows from that distinction. + +3. Job Feldbrugge and coauthors, Stochastic Homology of Gaussian vs. non-Gaussian +Random Fields: Graphs towards Betti Numbers and Persistence Diagrams, +arXiv:1908.01619 (2019). https://arxiv.org/abs/1908.01619 +The abstract supplies random-field topology and persistence context. This pass +does not claim to have searched every occurrence of a cubic normal form in its +full text or the wider literature. + +The new calculation gives its stationary-root change of variables, integrable +Gaussian majorant, shape integrals and marked convergence arguments in full. +The sources in SOURCES.json keep their own review conditions. The generic +measure-theoretic tools are not claimed novel, and the exact radius/height +result is an author-side candidate pending separate nonauthor review. + +For the later custody repair, the official Git documentation was searched and +read at https://git-scm.com/docs/git and the primary git-replace manual at +https://www.kernel.org/pub/software/scm/git/docs/git-replace.html. They document +replacement refs and the --no-replace-objects override. This external check +supports the engineering repair only, not any mathematical claim. diff --git a/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json b/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json new file mode 100644 index 000000000..99678cd67 --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json @@ -0,0 +1,11 @@ +{ + "mathematical_acceptance": false, + "passed": true, + "radial_factor_at_k1": "53250048/385", + "scientific_effect": "NONE", + "scope": "exact finite identities, not Gaussian or limiting-law verification", + "shape_integral": "246528/35", + "stationary_map_cases": 648, + "tail_exponent": 11, + "universal_height_mean": "5771/7062" +} diff --git a/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json new file mode 100644 index 000000000..a4b0fb99c --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json @@ -0,0 +1,77 @@ +{ + "schema": "math166-scoped-review-binding-v1", + "scientific_effect": "NONE", + "mathematical_acceptance": false, + "remote_review_authenticity_checked": false, + "reviewed_commit": "7c82252533c3fe14ee262f7f87c0549c10968592", + "proofs": [ + { + "path": "TWO_SCALE_LAW.md", + "bytes": 17139, + "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "git_blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c" + }, + { + "path": "RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4" + } + ], + "source_manifest": { + "path": "SOURCES.json", + "bytes": 2923, + "sha256": "de09c2cf3c9082a09d168f451f78a1a60b150f9f55c0c6b9bfe94ac26652dfe0", + "git_blob": "59d11113e208965fefeaadd1bd5866585e77dc82" + }, + "reviews": [ + { + "slice": "A", + "native_review_id": 5360227991, + "native_state": "COMMENTED", + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360227991", + "submitted_at": "2026-09-30T00:52:00Z", + "reviewed_commit": "7c82252533c3fe14ee262f7f87c0549c10968592", + "proof_path": "TWO_SCALE_LAW.md", + "body_bytes": 7840, + "body_sha256": "c63efc292d7cf209e2770419cc159276897d0ba955c32ee1d2af88c8aa7f0ed8", + "reported_disposition": "ACCEPT at the stated conditional scope", + "scope": "Theorem T, Theorem L, ordered-pair law L3, sampling/order of limits and spatial-TV obstruction V1; conditional on exact imported source interfaces, without a rate or parent revalidation." + }, + { + "slice": "B", + "native_review_id": 5360178611, + "native_state": "COMMENTED", + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360178611", + "submitted_at": "2026-09-30T00:42:15Z", + "reviewed_commit": "7c82252533c3fe14ee262f7f87c0549c10968592", + "proof_path": "RADIAL_TAIL.md", + "body_bytes": 7631, + "body_sha256": "3042d958b769b0a9ada1359b6f8d9461c090060fe5057cff795c99b6c6c2894d", + "reported_disposition": "ACCEPT at the stated conditional scope", + "scope": "RADIAL_TAIL R4-R13: the already-formed limiting microscopic point-intensity tail, full coefficient and moment boundary; conditional on SC/CUB, with r tending to zero before radius tends to infinity." + }, + { + "slice": "C", + "native_review_id": 5360216551, + "native_state": "COMMENTED", + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360216551", + "submitted_at": "2026-09-30T00:49:42Z", + "reviewed_commit": "7c82252533c3fe14ee262f7f87c0549c10968592", + "proof_path": "RADIAL_TAIL.md", + "body_bytes": 4277, + "body_sha256": "9a2c89c6e0c122fc75cc96a2b6392b47ab33bc47feb072b9f2a56315c527df84", + "reported_disposition": "ACCEPT at the stated conditional scope", + "scope": "RADIAL_TAIL H1-H4: extreme marks under point-intensity sampling of the already-formed microscopic measure, not a uniformly sampled configuration or finite-r global law.", + "proof_binding_note": "The native C review refers to B's same full Git blob and an abbreviated SHA256. The integration preflight matched the complete proof bytes at the native reviewed commit; this record does not claim C printed a fresh full digest." + } + ], + "exposure": { + "reviewer_provider": "Anthropic", + "reviewer_model_attribution": "Claude", + "same_session_for_all_three": true, + "organizational_independence_credit": 0, + "notes": "Actual native review bodies were read back by the integration preflight. These are three disjoint scopes from one provider-distinct session, not three independent votes. Parent proofs remain imported; no new analytic review is supplied by this custody repair." + }, + "binding_policy": "The verifier pins this complete record and the accepted proof/source byte identities. Regenerating MANIFEST.json cannot transfer these old reviews to changed mathematics. An intentional future update requires reviewed changes to the bindings and verifier anchors. Offline replay does not contact GitHub or authenticate a remote reviewer." +} diff --git a/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md new file mode 100644 index 000000000..31896de23 --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md @@ -0,0 +1,117 @@ +# Scoped review and custody repair record + +Scientific effect NONE. This is a record of actual reviews, not a new scientific +status register or a self-issued acceptance. Same GitHub login gives no +organizational-independence credit. + +## Completed analytic Slice B + +Anthropic Claude review5360178611, submitted2026-09-30 00:42:15Z: +https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360178611 + +Binding: head7c82252533c3fe14ee262f7f87c0549c10968592, RADIAL_TAIL.md +blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4, SHA256 +250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973. + +ACCEPT scope: R4-R13, the tail F(t)~C_star*t^-11 of the already-formed limiting +microscopic point intensity, the complete coefficient, p<11 radial-moment +threshold and capped-eleventh-moment asymptotic, at fixed d,L,b,k,frame and +conditional on the exact SC/CUB interfaces. The reviewer did not author the +pinned sources; prior reviewer exposure to SC is disclosed. No source theorem +is independently revalidated by this consumer review. + +Four nonblocking clarifications are adopted without changing the proof bytes: +R8's sign is chart orientation and R10 uses its absolute value; root multiplicity +uses CUB Theorem C/C6; the density majorant is SC(10); dimension dependence stays +in c_m/z0,H,h0 and J_cusp, not the shape integral/exponent. + +At the initial B-only record, A and C remained open; the courtesy height check +was not counted as a verdict. The separate actual A/C reviews below now complete +the requested mathematical scope, without changing either manuscript. + +## Completed analytic Slice C + +Anthropic Claude review5360216551, submitted2026-09-30 00:49:42Z: +https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360216551 + +Binding: the same mathematical head and RADIAL_TAIL.md blob as B. ACCEPT of +Theorem H and H1-H4: the four-factor limit under point-intensity sampling, the +universal downward-height density, its exact normalization and moments. This +uses the already formed microscopic measure and then radius->infinity, not a +finite-r global distribution or uniformly sampled configuration. The parameter +and orientation factors need not be universal. Courtesy algebra checks in B +were not substituted for this full C review. + +Its two nonblocking notes are reading clarifications: the bounded test uses +q=R_i/t at finite t before passing to gamma(a)|z|, and the H4 moment index is a +nonnegative integer with the second bracket coming from the polynomial density. + +## Completed analytic Slice A + +Anthropic Claude review5360227991, submitted2026-09-30 00:52:00Z: +https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360227991 + +Binding: head7c82252533c3fe14ee262f7f87c0549c10968592, TWO_SCALE_LAW.md +bloba32fd5f7d941bbe1fe943df045b1e0fbec8d691c, SHA256 +e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769. + +ACCEPT of Theorem T, Theorem L, ordered-pair law L3, the sampling/order-of-limits +statements and spatial-TV obstruction V1, conditional on the exact SC/CUB/RM/C6/P +interfaces. The review checks entire-configuration functionals, both spatial +scales, arbitrary admissible cutoff, count-weighted tail control, uniform remote +height marks and the absolutely-continuous versus collinear-support argument. +It does not revalidate the consumed Gaussian parent proofs or supply a rate. + +Three nonblocking notes are retained as reading clarifications: near configuration +convergence holds for almost every (theta,F), with hard coordinates O(r^2)/r->0; +the configuration space is Polish with locally constant cardinality; separated +spatial exhaustion suffices for V1 since pins and coincidence diagonal are null, +and the latter contains no ordered distinct pair. + +## Combined mathematical disposition + +All A/B/C requests now have actual nonauthor ACCEPT at their stated conditional +scope. The same provider-distinct Claude session supplied all three; this is +not three independent votes or organizational independence. Both manuscripts +are unchanged throughout custody and metadata successors. No review merges the +packet, promotes a global register, removes parent conditions, evaluates a +numerical Gaussian coefficient or establishes finite-r spatial-moment rates. + +## Custody repair, separate from mathematical acceptance + +Author-side audit reproduced replacement-ref and tree/tag-substitution defects +with four failing real-Git regression assertions. Source lookup now explicitly +disables Git replacement objects and requires actual commit type. All52 tests +pass in each local Python mode; old tests and semantic mutants remain. The +changed verifier received Codex engineering comment5901897673 at repair commit +80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337: no major issues found. That comment +is not a mathematical verdict or an exhaustive security guarantee. Targeted +source run36652196578 and full downstream/formal run36652196938 both succeeded +at the repair commit. A later metadata successor still needs fresh current-head +checks; no old-head run is automatically rebound. The mathematical files, +geometry.py, source pins and RESULTS.json have not changed. + +## Current integration custody repair + +At b8b028ea0a3f45aa8d2640b339977a04154909a4, independent full-CLI fixtures showed +that changing either manuscript, or removing either manuscript, still passed +after honestly regenerating MANIFEST.json. The ordinary unresigned mutation +correctly failed. These were inventory/review-binding defects, not mathematical +counterexamples. The original reviewed manuscripts remain unchanged. + +The current repair requires the proof/review declarations and every executable +or test module used by the advertised replay. It binds both current manuscripts +and SOURCES.json to fixed accepted identities and pins the complete additive +REVIEW_RECORD.json, including actual native review body identities. A co-edit of +a manuscript, review metadata and manifest cannot rebind an old review merely by +re-signing the mutable inventory. Nine new regression methods cover these cases; +the current suite contains 61 tests per Python mode. Existing exact geometry, +eight semantic mutants and Git replacement/commit-type controls remain intact. + +The A/B/C records above retain their original conditional scope and exposure. +C's native proof identity cross-references B's full blob and gives an abbreviated +digest; actual bytes at the native reviewed commit were independently matched +in preflight. The machine-readable record makes that distinction explicit. +Offline validation authenticates local byte relationships only: it grants no +remote-review authentication, mathematical acceptance or independence credit. +Fresh engineering review, current-head CI and integration remain separate gates. diff --git a/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json b/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json new file mode 100644 index 000000000..59d11113e --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json @@ -0,0 +1,62 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "condition": "Source identity is not a new review or an automatic status transition. All applications retain the pinned source hypotheses.", + "sources": [ + { + "id": "SC", + "commit": "7a44130b41b827577424a73c4feac4cde20b5568", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "use": "Sections4-8: actual spectral/root limit, domination, radius and mixed-event estimates, full count limit. Conditional input; not revalidated." + }, + { + "id": "CUB", + "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "use": "Theorem C and exact shear/height/type identities only. Separate Gaussian sector theorem not consumed." + }, + { + "id": "RM", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "frontiers/remote_window_20260924/PROOF.md", + "bytes": 18355, + "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", + "use": "Sections2-5, complete height-disintegrated integrand12 and normalized kernel13; finite-r spatial Kac-Rice framework." + }, + { + "id": "C6", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "frontiers/c6_palm_route_20260929/PROOF.md", + "bytes": 53364, + "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "git_blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5", + "use": "Theorem Q for polynomial count uniform integrability, not a spatial moment bound." + }, + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "use": "Section2 distinct-site jet rank for finite-r spatial absolute continuity; original model conventions." + }, + { + "id": "D5", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "use": "Global first moment and uniform R-dependent shell interface, as composed in SC." + } + ] +} diff --git a/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md b/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md new file mode 100644 index 000000000..a32fd5f7d --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md @@ -0,0 +1,331 @@ +# Two-scale marked clusters and localization of all leading pair mass + +Object: OA-TWO-SCALE-CLUSTER-GEOMETRY-20260929-v1. +Author: OpenAI / GPT-6 Astra Pro, foreground recovery, 29 September 2026 (Central). +Disposition: AUTHOR-SIDE CONDITIONAL PROOF; NONAUTHOR REVIEW REQUIRED. +Scientific effect NONE. No source proof, prior verdict, global register, Boolean, +prize or integration disposition is changed. A shared GitHub account is not +organizational independence. + +## 1. Exact interface, not a silent revalidation + +Fix d>=2, L>0, k>0, b in R, and one orthonormal axial/transverse frame. Use the +original periodized Gaussian field, pins M=(-r/2,0), S=(r/2,0), values b,b-kr^3, +zero gradients, determinant weight W_r and FULL normalizer Z_r of [SC]. Write +P_r=Q_r^W. N_r counts all additional critical points in the open height window, +excluding M,S. No observation set or normalizer is replaced. + +All source identities are in SOURCES.json. This paper is a new conditional +consumer of [SC], not a review of it. In particular: + +[SC] frontiers/spectral_cluster_closure_20260929/PROOF.md, exact commit +7a44130b41b827577424a73c4feac4cde20b5568, blob +16c56821b52fd76b0be791622b9c3809eafde75a. + +[CUB] frontiers/planar_cubic_cluster_20260929/PROOF.md, exact commit +bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8, blob +bb446d08db8a944537a743ad550b88c1c2ad5758. Only its deterministic cubic classifier +is consumed; not its separate Gaussian compact-sector theorem. + +[RM] frontiers/remote_window_20260924/PROOF.md, Sections 2-5: the complete +height-disintegrated integrand in (12) is Lambda_j(x)+O_rho(r), uniformly at +fixed remote separation rho and throughout the window. + +[C6] frontiers/c6_palm_route_20260929/PROOF.md, Theorem Q, together with [SC]'s +first moment: every fixed ordinary count moment is O(r^3). + +[P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md, +Section 2: full distinct-site jet rank. Used only for the finite-r spatial +absolute-continuity statement in Section 6. + +RM, C6 and P are pinned at source cut 7a1cb09a9d58d179393e6d29146252f714b9c9bf. +A defect or unfulfilled condition in an imported result blocks the corresponding +application here. Publication, tests and a status string do not remove that +condition. No theorem from the competing #159 proof or the #157 Fourier packet +is imported. + +The following exact portions of SC are needed, not just its final count law: + +* (14)-(18): the integrable full-field/spectral near-occurrence majorant, + negligible exceptional branch, and pointwise convergence of the individual + roots, heights and indices in each fixed rescaled ball. +* (19)-(20): a finite spectral measure on nonempty cubic configurations, with + positive masses a1,a2 for one and two points. +* (21)-(25): beta_far=k integral Lambda is finite and positive, remote multiple + probability O_rho(r^5), mixed near/remote probability o_(R,rho)(r^3), and + middle-window mean <=C r^3(R^-2+rho^2), with the last constant independent + of R>=4 and small rho. +* (3), or its C6 tail argument: polynomially weighted count convergence and + E_r N_r^p=O_p(r^3) for every fixed positive integer p. + +Let m=d-1. In SC's coordinates theta=(s,h,O,tau), ordered hard eigenvalues are +0 0, delta_r/r -> infinity. (T1) + +For sufficiently small r the delta_r ball is embedded. For each additional +window critical point x put eta=(b-f(x))/(k r^3) in (0,1), and retain its index j. +Use the following two tags: + + micro: dist(x,0)=delta_r, with mark (x,eta,j). (T2) + +The micro position uses the unique local lift, expressed in the fixed original +frame. Remote positions remain actual torus points; they are not divided by r. +The state space Y is the disjoint topological union + + Y_micro=R^d x (0,1) x {0,...,d}, + Y_remote=(X minus {0}) x (0,1) x {0,...,d}. + +Let Conf_+(Y) be the disjoint union over n>=1 of the finite symmetric products +Y^n/S_n, with the quotient product topology. Multiplicities are allowed in this +ambient space; the particular laws below are simple almost surely. Cardinality +is a continuous, discrete-valued function. This space retains configurations, +not merely their one-point intensity. + +Let Xi_r be the tagged configuration (T2). Define the finite nonempty measure + + L_r(G)=r^-3 E_r[G(Xi_r);N_r>0]. (T3) + +For a spectral jet theta, define the micro-only configuration + + C_micro(theta)=sum_(X,Z in C(theta)) + delta_(micro, X e_axis+Z e_soft(O), -P_theta(X,Z)/k,d-1). + +Let L_micro be the pushforward of 1{n>=1}dM under this map. Let L_remote be the +singleton-configuration measure induced by + + k sum_(j=0..d) Lambda_j(x) dx d eta, + x in X minus {0}, 0 L=L_micro+L_remote weakly as finite measures (T5) + +on Conf_+(Y). The convergence also holds when testing G times any fixed +nonnegative integer power of the total count, for bounded continuous G. +The mass of L is a1+a2+beta_far=nu1+nu2. In particular, the probability law +of Xi_r conditioned on N_r>0 converges weakly to L/(nu1+nu2). + +This is weak convergence of spatially marked configurations. It is not stated +as spatial total-variation convergence; Section 6 shows that such a statement +would be false in general. Count and tag-count marginals, which have a discrete +state space, do converge in total variation. + +### Proof: fixed-radius limits, then the regional sandwich + +Fix R>=4 and a sufficiently small rho>0. For small r, Rr0} union {N_R>0,N_far>0} + union {N_far>=2} union {N_R>=3}. + +SC's exact bounds imply + + limsup_(r->0) r^-3 P_r(E_bad) <= C(R^-2+rho^2). (T6) + +Only the middle term survives this limsup. The mixed bound here is the full +o(r^3) result, not a uniform extension of the compact-jet O(r^(9/2)) estimate. + +For fixed R, SC's pointwise hard-direction slaving gives convergence of the +individual micro roots, their normalized heights and their indices for almost +every spectral jet and field coupling realization. Height and observation-circle +exceptional sets are null. Its integrable near-occurrence majorant (16), with the +exceptional branch handled by (14), therefore proves convergence of the entire +bounded continuous CONFIGURATION functional G, not only sums of one-point +functions, to the pushforward of 1{n_R>=1}dM. Here n_R counts cubic roots in the +R ball. All angular variables remain under the integral. No independent root +labelling is needed: local labellings from root stability induce convergence in +the symmetric-product topology. + +For the remote component, change height variable t to eta=(b-t)/(kr^3) in +RM (12). Its complete integrand differs uniformly from Lambda_j(x) by O_rho(r). +The rescaled first-moment measure on D_rho x (0,1) x indices thus converges in +full variation to k Lambda_j(x) dx d eta. On N_far>=2, the mean number of remote +points is at most E_r(N_far)_2=O_rho(r^5). Subtracting this multiple-point +intensity shows that the SINGLETON-configuration measure has the same limit. +This proves T4 first on D_rho. In particular the limiting eta is uniform on +(0,1), independent of the remote location and index under the normalized remote +component; this is a marked consequence of the uniform height disintegration, +not a field-independence assertion. + +On the complement of E_bad the global configuration is precisely either the +near configuration or a remote singleton, but never both. Testing a bounded G +and comparing these three finite measures gives an error bounded by a fixed +multiple of ||G||_infinity P_r(E_bad). Hence T6 bounds the scaled error. + +For almost every fixed theta, every cubic root is finite and n_R=n eventually +as R->infinity. The measure 1{n>=1}M is finite by SC (19), so dominated convergence +of bounded configuration functionals gives the full micro pushforward. Meanwhile +SC (21) bounds the missing remote mass by C rho^2. Let r->0 first at fixed R,rho, +then R->infinity and rho->0. These steps prove T5 for every bounded continuous G, +including G=1. They apply to any delta_r obeying T1 because only the inequalities +RrM] <= C_q/M, (T7) + +using the (q+1)-st ordinary moment, including q=0. The limiting count is at most +two. Let M increase after the finite truncation limit. This proves the weighted +claim. Total mass tends to the positive value nu1+nu2, so normalization is valid. +Discrete tag-count convergence follows from the coordinate probabilities and +the vanishing scaled mass outside the three possible limit sectors. QED. + +## 3. What singleton and double-cluster sampling mean + +Conditional on N_r=1, Theorem T gives a mixture of a micro singleton, with weight + + a1/(a1+beta_far), + +and a remote singleton, with weight beta_far/(a1+beta_far). The first is the +pushforward of 1{n=1}dM/a1; the second has density T4/beta_far. Conditional on +N_r=2, it gives the micro pair pushforward of 1{n=2}dM/a2, and both points have +index d-1. Positivity of a1,a2,beta_far was established by SC and remains an input. + +A law obtained by sampling a nonempty configuration is not the same as a law +obtained by selecting one point from the intensity measure. The latter weights +clusters by their sizes. RADIAL_TAIL.md uses explicitly the second convention. +Likewise a first-moment formula of the form E sum phi(point) does not, by itself, +identify the full configuration law; T5 uses bounded continuous functionals of +ALL points in each configuration. + +Plainly scaling ALL global positions by 1/r and seeking a finite Euclidean +configuration limit would lose a positive component. For fixed rho>0, the +remote singleton event has scaled probability tending to k integral_Drho Lambda, +while its rescaled distance is at least rho/r. Dividing by P_r(N>0)~(nu1+nu2)r^3 +shows that a fixed positive fraction of the conditional mass escapes every +compact spatial set. The appropriate remedy is the two-scale space T2, not +silently deleting the remote contribution. + +## 4. All leading ordered pair mass is microscopic + +Let N_in count the points in the delta_r ball, with any cutoff T1. For q>=2, +(N)_q-(N_in)_q counts ordered distinct q-tuples not wholly inside that ball. + +**Theorem L (factorial localization).** For every fixed integer q>=2, + + r^-3 E_r[(N_r)_q-(N_in)_q] -> 0. (L1) + +In particular all order-r^3 second-factorial mass is microscopic and its limiting +coefficient is 2a2. The assertion does not hold for q=1, whose omitted coefficient +is beta_far>0. + +**Proof.** On the good event used in T6, either all points belong to the near ball +and hence to the delta_r ball, or the only point is a remote singleton. For q>=2 +the difference in L1 is zero in both cases. Everywhere it is between zero and +N_r^q 1_Ebad. By Cauchy-Schwarz and C6, + + r^-3 E_r[N_r^q 1_Ebad] + <= (r^-3 E_r N_r^(2q))^(1/2) + (r^-3 P_r(E_bad))^(1/2). + +Take the small-r limsup, then R->infinity and rho->0 using T6. This proves L1. +The coefficient 2a2 follows from SC's count law and polynomial moment convergence. +The identical argument for fixed R,rho and their mixed event gives + + E_r[N_R N_far^rho]=o_(R,rho)(r^3), (L2) + +using N_R N_far<=N^2 and E N^4=O(r^3). Thus the probability localization also +controls the COUNTED mixed pair mass; an event bound alone without uniform +integrability would not have supplied that conclusion. QED. + +For a bounded continuous function of an ordered pair of micro position/height/ +index marks, the scaled ordered-pair measure converges weakly to + + integral_(n=2) [delta_(z1(theta),z2(theta)) + +delta_(z2(theta),z1(theta))] dM(theta). (L3) + +This is T5 with a count-quadratic functional and T7, followed by L1 to remove +nonmicroscopic pairs. No arbitrary root ordering enters L3. It gives a precise +conditional answer to localization of the leading pair mass, without changing +any existing scientific register or accepting a different proof of that problem. + +## 5. The order of limits remains a real restriction + +T5 concerns a two-scale weak limit as r->0. The sharp radius tail in the companion +paper concerns the tail of the already formed MICROSCOPIC limiting measure. +Neither result asserts a convergence rate or an interchange with a radius that +grows arbitrarily quickly at finite r. Count-polynomial uniform integrability +T7 is not spatial-moment uniform integrability. In particular an eleventh radial +moment conclusion for the limiting local law must not be advertised as an +asymptotic for the finite-r global eleventh moment, which also has remote points. + +## 6. A rigorous obstruction to spatial total-variation convergence in d>=3 + +In the micro double-cluster limit, both transverse displacement vectors are +multiples of the same soft eigenvector. Let E_col be the event that a two-point +configuration has both tags micro and linearly dependent transverse vectors. +The limiting probability conditional on N=2 is one. + +For any fixed r>0, the factorial spatial measure of two distinct additional +critical points under P_r is absolutely continuous away from the pins and the +coincidence diagonal. To see this, exhaust that domain by compact separated +sets; [P]'s distinct-site jet rank gives a nonsingular two-gradient covariance. +The Gaussian two-point Kac-Rice formula with the single endpoint weight W_r and +single Z_r gives a Lebesgue density there. Polynomial Gaussian moments make it +finite on each compact piece. A spatially null set therefore has zero factorial +measure. The conditional law on N=2 is dominated, as a null-set statement, by +this nonnegative factorial measure. No unproved uniform near-diagonal density +bound is needed for this absolute-continuity assertion. + +When d>=3, the transverse dimension m>=2. Linear dependence of two transverse +vectors is a Lebesgue-null algebraic condition: for example the zero set of the +sum of squares of their 2-by-2 minors, a nonzero polynomial. Hence P_r(E_col|N=2)=0 +whenever that conditioning is defined. SC implies it is defined for all small r +because P_r(N=2)~a2 r^3 and a2>0. The probability total-variation distance is thus + + dTV(Law(Xi_r|N_r=2), L_micro restricted to n=2 / a2)=1. (V1) + +Weak convergence is fully compatible with V1. Latent-jet/count total variation +from SC must not be pushed through an r-dependent spatial map as if the map were +fixed. No V1 claim is made in d=2. + +## 7. Review scope + +New obligations: the marked configuration-space sandwich (not merely intensity), +remote height disintegration, arbitrary-cutoff localization with counted tails, +and the finite-r versus limiting-support argument V1. The cubic, SC domination +and imported Gaussian estimates keep their own review obligations. Exact finite +controls verify exponent ledgers and count inequalities; they do not certify +these measure-theoretic or Gaussian steps. The radius/height calculation is +separate in RADIAL_TAIL.md and has its own nonauthor review request. diff --git a/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md b/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md new file mode 100644 index 000000000..33f00592e --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md @@ -0,0 +1,49 @@ +# Executed validation and scope + +Scientific effect NONE. Finite tests are not analytic acceptance. + +The mathematical companion was test-first: ten assertion failures were observed +against explicit missing-implementation stubs; implementation then passed those +ten tests and the expanded suite. A separate real-Git custody suite was written +against stubs: 19 tests produced 24 intended assertion failures (some tests had +multiple subcases), then all 19 passed after implementation. Logs are retained +in the conversation evidence bundle, not invented as upstream execution receipts. + +Initial standalone suite: 29 mathematical tests plus 19 custody tests = 48 tests +in each normal/optimized mode. The mathematical executable verifies 648 rational +stationary substitutions and exact root/weight/height constants. Eight semantic +mutants must each return1; an unknown mutant must return2. Normal and optimized +RESULTS.json outputs are byte-identical. + +The source checker uses git ls-tree --full-tree for repository-root paths while +running from the nested packet directory. Fixtures include changed size/SHA/blob, +missing path, malformed commit, path traversal, leaf and parent Git symlinks, +missing/duplicate/empty inventories, duplicate JSON, and packet/manifest symlinks. +All six declared sources are mandatory; no prior or auxiliary source is skipped. + +The local environment did not obtain a full network checkout of the project. +Therefore local-only replay explicitly reports source_pins_checked=false. +Actual project Git-object identities and the final packet inventory must pass +hosted replay at the published commit. Full downstream and existing Lean gates +remain separate and must be read at their current tested head/base. No local +full-repository run or Lean formalization of these new proofs is claimed. + +Mathematical limits not validated by the checker: Gaussian rank/regression, +spatial root convergence, change-of-variables multiplicity, domination, two-scale +configuration topology, singular-support TV obstruction and the continuum tail +asymptotics. These are the written proof obligations for nonauthor reviewers. + +## Replacement-object and object-type repair + +A later author-side audit reproduced four intended assertion failures in actual +local Git repositories: replacement refs could forge a commit/path binding, +replacement refs could incorrectly invalidate a legitimate original pin, and a +tree or annotated-tag object could pass as a declared forty-hex commit. These +were verifier weaknesses, not observations of altered project evidence. + +The repair uses git --no-replace-objects for every source-object lookup and +requires cat-file -t to return commit before tree traversal. Four new regressions +pass after the change. The complete suite is now 29 mathematical plus 23 custody +tests = 52 per Python mode, with all previous exact and mutant controls retained. +All mathematical and source-pin bytes remain unchanged. The final successor's +hosted replay must be checked separately from initial-head runs. diff --git a/frontiers/two_scale_cluster_geometry_20260929/geometry.py b/frontiers/two_scale_cluster_geometry_20260929/geometry.py new file mode 100644 index 000000000..f7fd5b699 --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/geometry.py @@ -0,0 +1,217 @@ +"""Exact finite controls for two-scale geometry and the limiting radial tail. + +Standard library only. No Gaussian integral is numerically approximated. +Sparse polynomials have keys (power_of_u, power_of_v) and Fraction values. +""" +import argparse +from fractions import Fraction as F +import json +import math + +MUTANTS = ('omit-jet-factor-two', 'omit-z-reflection', 'power-ten', + 'wrong-cusp', 'height-reversed', 'wrong-window', 'wrong-k-power', + 'half-root-jacobian') + + +def rational(x): + if type(x) not in (int, F): + raise ValueError('exact int or Fraction required') + return F(x) + + +def nonnegative_integer(q): + if type(q) is not int or q < 0: + raise ValueError('nonnegative integer required') + return q + + +def positive_k(k): + k = rational(k) + if k <= 0: + raise ValueError('positive k required') + return k + + +def add(p, q): + out = dict(p) + for powers, c in q.items(): + out[powers] = out.get(powers, F(0)) + c + return {powers:c for powers,c in out.items() if c} + + +def scale(p, c): + c = rational(c) + return {powers:c*a for powers,a in p.items() if c*a} + + +def mul(p, q): + out = {} + for (i,j), a in p.items(): + for (k,l), b in q.items(): + powers = (i+k,j+l) + out[powers] = out.get(powers,F(0))+a*b + return {powers:c for powers,c in out.items() if c} + + +def power(p, n): + nonnegative_integer(n) + out = {(0,0): F(1)} + for _ in range(n): + out = mul(out,p) + return out + + +def integrate_strip(poly, lower, upper, left, right): + """Exact integral: u in [left,right], polynomial lower(u)<=v<=upper(u).""" + left, right = map(rational,(left,right)) + if left > right or any(j for i,j in set(lower)|set(upper)): + raise ValueError('ordered endpoints and univariate bounds required') + out = {} + for (i,j), c in poly.items(): + v_integral = add(power(upper,j+1),scale(power(lower,j+1),-1)) + out = add(out,mul({(i,0): c/F(j+1)},v_integral)) + return sum((c*(right**(i+1)-left**(i+1))/F(i+1) + for (i,j),c in out.items()),F(0)) + + +def shape_polynomial(): + b = {(0,0):F(3),(2,0):F(-12)} + return mul(add({(0,2):F(4)},scale(mul(b,b),-1)), + add(b,{(1,1):F(-4)})) + + +def shape_integrals(height_order=0): + nonnegative_integer(height_order) + b = {(0,0):F(3),(2,0):F(-12)} + upper = {(0,0):F(3),(1,0):F(-6)} + height = {(0,0):F(1,2),(1,0):F(1),(0,1):F(1,6)} + poly = mul(shape_polynomial(),power(height,height_order)) + return (integrate_strip(poly,scale(b,F(-1,2)),upper,F(-3,2),F(-1,2)), + integrate_strip(poly,scale(b,F(1,2)),upper,F(-1,2),F(1,2))) + + +def radial_factor(k): + """Coefficient multiplying the remaining spectral/Gaussian cusp integral.""" + return F(216,11)*positive_k(k)**7*sum(shape_integrals()) + + +def tail_power(): + return 2+2+3+4 # ds,dB,dD and the typed determinant weight + + +def height_moment(q): + """Exact integral of eta^q H_eta(eta), without probability normalization.""" + nonnegative_integer(q) + beta = F(math.factorial(q),1) + for j in range(q+1): + beta /= F(9,2)+j + return (F(110592,7)*(1/(F(q)+F(9,2))-beta+F(1,q+1)) + +13824*(F(1,q+5)-F(2,q+4)+F(5,q+3)-F(4,q+2))) + + +def raw_jet(k,a,B,D): + k = positive_k(k) + a,B,D = map(rational,(a,B,D)) + return a*a/(12*k)+B, a**3/(144*k*k)+a*B/(4*k)+2*D + + +def point_map(k,u,v,z,a): + """Recover original s,a,beta,c from one canonical stationary point.""" + k = positive_k(k) + u,v,z,a = map(rational,(u,v,z,a)) + if z == 0: + raise ValueError('nonzero transverse coordinate required') + b = 3-12*u*u + s = -k*v/z**2 + B = k*b/z**2 + D = k*(v-b*u)/z**3 + beta,c = raw_jet(k,a,B,D) + return s,a,beta,c + + +def cubic_gradient(k,s,a,beta,c,X,Z): + k = positive_k(k) + s,a,beta,c,X,Z = map(rational,(s,a,beta,c,X,Z)) + return (6*k*X*X-F(3,2)*k+a*X*Z+beta*Z*Z/2, + s*Z+a*(X*X-F(1,4))/2+beta*X*Z+c*Z*Z/2) + + +def cubic_height(k,s,a,beta,c,X,Z): + k = positive_k(k) + s,a,beta,c,X,Z = map(rational,(s,a,beta,c,X,Z)) + return (2*k*X**3-F(3,2)*k*X-k/2+s*Z*Z/2 + +a*(X*X-F(1,4))*Z/2+beta*X*Z*Z/2+c*Z**3/6) + + +def root_jacobian(k,u,v,z): + k = positive_k(k) + u,v,z = map(rational,(u,v,z)) + if z == 0: + raise ValueError('nonzero transverse coordinate required') + return 6*k**3*(12*u*u+4*u*v-3)/z**8 + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def run_checks(mutant=None): + raw = raw_jet(1,6,0,0) + if mutant == 'wrong-cusp': raw = (raw[0],F(3)) + require(raw == (F(3),F(3,2)), 'cusp raw-coordinate inverse') + exponent = 10 if mutant == 'power-ten' else tail_power() + require(exponent == 11,'weighted radial homogeneity') + volume = sum(shape_integrals()) + if mutant == 'wrong-window': volume = shape_integrals()[1] + require(volume == F(246528,35),'both typed height-window strips') + factor = radial_factor(1) + if mutant in ('omit-jet-factor-two','omit-z-reflection','half-root-jacobian'): + factor /= 2 + require(factor == F(53250048,385),'raw-jet, root-Jacobian and sign factors') + value = radial_factor(2) + if mutant == 'wrong-k-power': value /= 2 + require(value == 128*factor,'k-to-the-seventh scaling') + mean = height_moment(1)/height_moment(0) + if mutant == 'height-reversed': mean = 1-mean + require(mean == F(5771,7062),'height measured downward from birth') + for q in range(7): + require(sum(shape_integrals(q)) == height_moment(q), + 'independent double-integral/height-density moment comparison') + cases = 0 + for k in (F(1,2),F(1),F(2)): + for u in (F(-5,4),F(-1),F(-3,4),F(-1,4),F(0),F(1,4)): + b = 3-12*u*u + for frac in (F(1,4),F(1,2),F(3,4)): + v = abs(b)/2+frac*(3-6*u-abs(b)/2) + for z in (F(-3),F(-1),F(1,2),F(2)): + for a in (F(-2),F(0),F(6)): + s,a,beta,c = point_map(k,u,v,z,a) + X = u-a*z/(12*k) + require(cubic_gradient(k,s,a,beta,c,X,z)==(0,0),'stationary map') + eta = u+F(1,2)+v/6 + require(0 s=-3/2,B=-9/4,D=-3/8. + self.assertEqual(g.point_map(1,-1,6,2,0), (F(-3,2),F(0),F(-9,4),F(-3,4))) + def test_cusp_curve(self): + self.assertEqual(g.raw_jet(1,6,0,0), (F(3),F(3,2))) + def test_exponent(self): + self.assertEqual(g.tail_power(),11) + + +class AdditionalTests(unittest.TestCase): + def test_sparse_algebra(self): + p={(1,0):F(1),(0,0):F(1)} + self.assertEqual(g.power(p,2),{(2,0):F(1),(1,0):F(2),(0,0):F(1)}) + self.assertEqual(g.add(p,g.scale(p,-1)),{}) + def test_exact_rectangular_integral(self): + self.assertEqual(g.integrate_strip({(2,1):F(1)},{},{(0,0):F(2)},0,3),18) + def test_constant_integral(self): + self.assertEqual(g.integrate_strip({(0,0):F(1)},{},{(0,0):F(1)},0,1),1) + def test_empty_polynomial(self): + self.assertEqual(g.integrate_strip({}, {}, {(0,0):F(1)},0,1),0) + def test_bad_order(self): + with self.assertRaises(ValueError): g.height_moment(-1) + def test_float_rejected(self): + with self.assertRaises(ValueError): g.radial_factor(1.0) + def test_bool_rejected(self): + with self.assertRaises(ValueError): g.radial_factor(True) + def test_zero_k_rejected(self): + with self.assertRaises(ValueError): g.radial_factor(0) + def test_zero_z_rejected(self): + with self.assertRaises(ValueError): g.point_map(1,0,1,0,0) + def test_invalid_strip(self): + with self.assertRaises(ValueError): g.integrate_strip({}, {}, {(0,1):F(1)},0,1) + def test_height_moments_independent(self): + for q in range(8): + self.assertEqual(sum(g.shape_integrals(q)),g.height_moment(q)) + def test_height_variance_positive(self): + mass=g.height_moment(0) + self.assertGreater(g.height_moment(2)/mass-(g.height_moment(1)/mass)**2,0) + def test_root_jacobian_independent(self): + def det(a): + return (a[0][0]*(a[1][1]*a[2][2]-a[1][2]*a[2][1]) + -a[0][1]*(a[1][0]*a[2][2]-a[1][2]*a[2][0]) + +a[0][2]*(a[1][0]*a[2][1]-a[1][1]*a[2][0])) + for k in (F(1,2),F(1),F(3)): + for u in (F(-5,4),F(-3,4),F(-1,4),F(1,4)): + b=3-12*u*u; bp=-24*u + v=(abs(b)/2+3-6*u)/2 + for z in (F(-3),F(1),F(4)): + rows=[[k/z**2,0,-2*k*v/z**3], + [0,k*bp/z**2,-2*k*b/z**3], + [k/z**3,-k*(bp*u+b)/z**3,-3*k*(v-b*u)/z**4]] + self.assertEqual(det(rows),g.root_jacobian(k,u,v,z)) + def test_raw_inverse(self): + for k in (F(1,2),F(1),F(3)): + for a in (F(-3),F(0),F(4)): + for B,D in ((F(-2),F(1,3)),(F(1),F(-1))): + beta,c=g.raw_jet(k,a,B,D) + self.assertEqual(beta-a*a/(12*k),B) + self.assertEqual((c-a*beta/(4*k)+a**3/(72*k*k))/2,D) + def test_root_homogeneity(self): + k,u,v,z,a=F(1),F(-3,4),F(3),F(2),F(3) + s,a,beta,c=g.point_map(k,u,v,z,a) + B=beta-a*a/(12*k); D=(c-a*beta/(4*k)+a**3/(72*k*k))/2 + for t in (F(2),F(5),F(10)): + st,at,betat,ct=g.point_map(k,u,v,t*z,a) + self.assertEqual(st,s/t**2) + self.assertEqual(betat-a*a/(12*k),B/t**2) + self.assertEqual((ct-a*betat/(4*k)+a**3/(72*k*k))/2,D/t**3) + def test_factorial_localization_inequality(self): + def falling(n,q): + out=1 + for j in range(q):out*=max(n-j,0) + return out + for near in range(5): + for mid in range(5): + for far in range(5): + n=near+mid+far + bad=bool(mid or (near and far) or far>=2 or near>=3) + for q in range(2,6): + loss=falling(n,q)-falling(near,q) + self.assertGreaterEqual(loss,0) + self.assertLessEqual(loss,n**q*bad) + def test_moment_uniform_integrability_bound(self): + for q in range(5): + for M in range(1,10): + for n in range(M+1,30): + self.assertLessEqual(F(n**q),F(n**(q+1),M)) + def test_counting_grid(self): + self.assertEqual(g.run_checks()['stationary_map_cases'],648) + def test_cli_modes_and_mutants(self): + import subprocess,sys + from pathlib import Path + script=str(Path(g.__file__).resolve()) + reference=None + for flags in (['-B','-S'],['-B','-O','-S']): + out=subprocess.run([sys.executable,*flags,script],capture_output=True,timeout=30) + self.assertEqual(out.returncode,0,out.stderr) + if reference is None: reference=out.stdout + self.assertEqual(out.stdout,reference) + for name in g.MUTANTS: + bad=subprocess.run([sys.executable,*flags,script,'--mutant',name],capture_output=True,timeout=30) + self.assertEqual(bad.returncode,1,(name,bad.stdout,bad.stderr)) + bad=subprocess.run([sys.executable,*flags,script,'--mutant','unknown'],capture_output=True,timeout=30) + self.assertEqual(bad.returncode,2) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/two_scale_cluster_geometry_20260929/verify.py b/frontiers/two_scale_cluster_geometry_20260929/verify.py new file mode 100644 index 000000000..6ecc60a58 --- /dev/null +++ b/frontiers/two_scale_cluster_geometry_20260929/verify.py @@ -0,0 +1,181 @@ +"""Fail-closed local packet and historical Git-source verification. + +Default replay verifies all six declared project source objects. --local-only +excludes those objects explicitly; real local Git fixtures still test the code. +""" +import argparse +import hashlib +import json +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + +SOURCE_IDS = frozenset(('SC','CUB','RM','C6','P','D5')) +REQUIRED_PACKET_FILES = frozenset(('TWO_SCALE_LAW.md','RADIAL_TAIL.md','SOURCES.json', + 'RESULTS.json','REVIEW_RECORD.md','REVIEW_RECORD.json', + 'geometry.py','verify.py','test_geometry.py','test_custody.py')) +# These anchors are deliberately outside the regenerable MANIFEST.json. Updating +# mathematics requires a reviewed change to these anchors and the native-review +# record; re-signing an inventory cannot transfer an old review to new bytes. +ACCEPTED_FILES = ( + {'path':'TWO_SCALE_LAW.md','bytes':17139, + 'sha256':'e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769', + 'git_blob':'a32fd5f7d941bbe1fe943df045b1e0fbec8d691c'}, + {'path':'RADIAL_TAIL.md','bytes':12533, + 'sha256':'250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973', + 'git_blob':'0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4'}, + {'path':'SOURCES.json','bytes':2923, + 'sha256':'de09c2cf3c9082a09d168f451f78a1a60b150f9f55c0c6b9bfe94ac26652dfe0', + 'git_blob':'59d11113e208965fefeaadd1bd5866585e77dc82'}, +) +# Pins the entire local record, including actual A/B/C IDs, native COMMENTED +# states, review-body hashes, proof mappings, conditional scopes and exposure. +# This authenticates a frozen local record, not GitHub or the reviewer's identity. +FROZEN_REVIEW_RECORD = { + 'path':'REVIEW_RECORD.json','bytes':4226, + 'sha256':'958d2e1ae8ea5d034d0d87efce6cde9be57ebb9d7c0b671e69d3d6598a6675ca', + 'git_blob':'a4b0fb99c24e2edf41973ea2a5e5176584602855', +} +MUTANTS = ('omit-jet-factor-two','omit-z-reflection','power-ten','wrong-cusp', + 'height-reversed','wrong-window','wrong-k-power','half-root-jacobian') + + +def read_json(path): + path=Path(path) + if path.is_symlink() or not path.is_file(): + raise ValueError('regular JSON source required: '+str(path)) + def unique(pairs): + result={} + for key,value in pairs: + if key in result:raise ValueError('duplicate JSON key: '+key) + result[key]=value + return result + return json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique) + + +def safe_path(path): + if not isinstance(path,str) or not path or '\\' in path or '\0' in path: + raise ValueError('unsafe path') + pure=PurePosixPath(path) + if pure.is_absolute() or str(pure)!=path or any(p in ('.','..') for p in path.split('/')): + raise ValueError('unsafe path: '+path) + return pure + + +def identity(data,entry): + expected_bytes=entry['bytes'] + if type(expected_bytes) is not int or expected_bytes<0: + raise ValueError('invalid source size') + sha=hashlib.sha256(data).hexdigest() + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if len(data)!=expected_bytes or sha!=entry['sha256'] or blob!=entry['git_blob']: + raise ValueError('identity mismatch: '+entry['path']) + + +def git(root,*args): + return subprocess.run(['git','--no-replace-objects',*args],cwd=root,check=True,capture_output=True, + timeout=60).stdout + + +def tree_entry(root,commit,path): + # --full-tree is essential: callers run from a NESTED packet directory. + data=git(root,'ls-tree','-z','--full-tree',commit,'--',path) + records=[r for r in data.split(b'\0') if r] + if len(records)!=1:raise ValueError('missing or ambiguous source: '+path) + header,found=records[0].split(b'\t',1) + mode,kind,blob=header.decode('ascii').split() + if found.decode('utf-8')!=path:raise ValueError('source path substitution') + return mode,kind,blob + + +def verify_source_entry(root,entry): + commit=entry['commit'];path=entry['path'];pure=safe_path(path) + if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit): + raise ValueError('immutable forty-hex commit required') + if git(root,'cat-file','-t',commit).strip()!=b'commit': + raise ValueError('source identity must name a commit object') + for length in range(1,len(pure.parts)): + prefix='/'.join(pure.parts[:length]) + mode,kind,blob=tree_entry(root,commit,prefix) + if mode!='040000' or kind!='tree':raise ValueError('non-tree source ancestor') + mode,kind,blob=tree_entry(root,commit,path) + if mode not in ('100644','100755') or kind!='blob': + raise ValueError('regular Git blob required: '+path) + if blob!=entry['git_blob']:raise ValueError('Git tree/blob identity mismatch: '+path) + identity(git(root,'cat-file','blob',blob),entry) + + +def verify_sources(root): + entries=read_json(root/'SOURCES.json')['sources'] + if not isinstance(entries,list) or len(entries)!=len(SOURCE_IDS): + raise ValueError('six source entries required') + ids=[e['id'] for e in entries] + if len(set(ids))!=len(ids) or set(ids)!=SOURCE_IDS: + raise ValueError('missing, duplicate or unexpected source identity') + for entry in entries:verify_source_entry(root,entry) + return len(entries) + + +def verify_review_bindings(root): + record=read_json(root/'REVIEW_RECORD.json') + identity((root/'REVIEW_RECORD.json').read_bytes(),FROZEN_REVIEW_RECORD) + if (record['proofs']!=list(ACCEPTED_FILES[:2]) or + record['source_manifest']!=ACCEPTED_FILES[2]): + raise ValueError('review proof/source mapping mismatch') + for entry in ACCEPTED_FILES: + path=root/entry['path'] + if path.is_symlink() or not path.is_file(): + raise ValueError('regular reviewed source required: '+entry['path']) + identity(path.read_bytes(),entry) + return len(record['reviews']) + + +def verify_inventory(root): + entries=read_json(root/'MANIFEST.json')['files'] + if not isinstance(entries,list) or not entries:raise ValueError('empty packet inventory') + names=[e['path'] for e in entries] + if len(set(names))!=len(names) or 'MANIFEST.json' in names: + raise ValueError('duplicate or self-referential packet inventory') + if not REQUIRED_PACKET_FILES.issubset(names): + raise ValueError('missing required proof or review packet leaf') + if sorted(p.name for p in root.iterdir())!=sorted(names+['MANIFEST.json']): + raise ValueError('packet inventory mismatch') + for entry in entries: + pure=safe_path(entry['path']) + path=root/entry['path'] + if len(pure.parts)!=1 or path.is_symlink() or not path.is_file(): + raise ValueError('flat regular packet leaf required') + identity(path.read_bytes(),entry) + verify_review_bindings(root) + return len(entries) + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--local-only',action='store_true') + args=parser.parse_args() + root=Path(__file__).resolve().parent + count=verify_inventory(root) + source_count=0 if args.local_only else verify_sources(root) + expected=(root/'RESULTS.json').read_bytes() + for flags in (['-B','-S'],['-B','-O','-S']): + subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'],cwd=root, + check=True,timeout=120) + command=[sys.executable,*flags,str(root/'geometry.py')] + result=subprocess.run(command,check=True,capture_output=True,timeout=60) + if result.stdout!=expected or result.stderr: + raise ValueError('deterministic checker output mismatch') + for mutant in MUTANTS+('unknown',): + result=subprocess.run(command+['--mutant',mutant],capture_output=True,timeout=60) + if result.returncode!=(2 if mutant=='unknown' else 1): + raise ValueError('negative control not rejected: '+mutant) + verify_inventory(root) + print(json.dumps({'passed':True,'packet_leaves':count,'source_count':source_count, + 'source_pins_checked':not args.local_only,'scientific_effect':'NONE', + 'review_bindings_checked':True,'review_count':3, + 'remote_review_authenticity_checked':False, + 'mathematical_acceptance':False},sort_keys=True)) + + +if __name__=='__main__':main() diff --git a/reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md b/reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md new file mode 100644 index 000000000..70d05080e --- /dev/null +++ b/reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md @@ -0,0 +1,99 @@ +# External reviews preserved for C6-RESIDUAL-CLOSURE-20260930-v1 + +**Object:** C6-RESIDUAL-CLOSURE-EXTERNAL-REVIEWS-20260930-v1 (component of the residual-closure record; source of the +proposed graph node `math.c6r-component.cluster-law-acceptance`; since v1.3 also the preserved copy of Math-#173 review +5360645884, the non-Claude read of this record's §3 on which `math.c6r-component.residual-closure-record` rests). +**What this is.** The nonauthor acceptance of the C6 cluster law (Math-#159, `frontiers/c6_cluster_law_20260929/PROOF.md`, +blob `ba492c8e62e58bfc055fc8254c790e893d346ba5`, the bytes now on `main`) is a GitHub pull-request review, not a file in +the repository. It is read back on 30 September 2026 through the session's GitHub connector and transcribed here with +its identity fields (review id, URL, posting account, provider named in the body, commit reviewed, `submitted_at`). +Everything is reproduced as returned. The in-repository pointer to the same acceptance is +`frontiers/c6_cluster_law_20260929/SOURCE_MAP.json` (`nonauthor_acceptance: true`; `revision_note`: "Nonauthor +acceptance recorded on 30 September 2026 (review 5360192822); proof bytes frozen"). +**Status of this evidence.** A pull-request review is a mutable external object: this file preserves what was read, it +does not make the review immutable, and an offline check cannot re-authenticate it (the packet checker is stdlib-only +and has no network). What the checker binds is this file's bytes and the presence of the quoted verdict lines in it +(`VERDICTS`). The reviewer states its exposure (author of earlier imported sources and of Math-#161) and that the read +is not organizational independence; nothing here adds a review or a provider. No Cursor agent was contacted or +restarted (owner stop of 27 September 2026). Same GitHub account throughout; zero organizational-independence credit. +The Math-#162 reviews (A 5359968447, B 5359845136, C 5359879627, D 5360082678) are recorded in the repository itself, +in `frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md`, and are not repeated here. + +## Index + +| # | Object | PR | Review | Posted by | Provider named | Commit reviewed | `submitted_at` | Verdict line | +|---|---|---|---|---|---|---|---|---| +| 1 | Math-#159 `frontiers/c6_cluster_law_20260929/PROOF.md`, blob `ba492c8e…` | Math-#159 | 5360192822 | `d6g8k5htny-coder` (owner account; the body names OpenAI / GPT-6 Astra Pro) | OpenAI / GPT-6 Astra Pro, source-exposed nonauthor | `f7c33955d005c03d1010835f52f500c4b853a847` | 2026-09-30T00:44:51Z | "VERDICT: ACCEPT Theorem N, Corollaries Lambda/S, and the explicitly sequential NEAR POINT-INTENSITY Corollary X at the stated fixed-d, fixed-torus, compact-positive-gap parameter scope, conditional only on the exact reviewed SOURCE_MAP inputs." | +| 2 | this record, `RECONCILIATION.md` §3 at head `b3fac79` ((D1), Route A, Route B; preserved in `REVIEWED_SECTION_3_b3fac79.md`) | Math-#173 | 5360645884 | `d6g8k5htny-coder` (owner account; the body names OpenAI / GPT-5.6 Sol) | OpenAI / GPT-5.6 Sol, source-exposed nonauthor (integrator of Math-#162/#166) | `b3fac79875f28bacd135c0aa5e65a47a41ae0fdf` | 2026-09-30T02:06:27Z | "MATHEMATICAL VERDICT ON §3: ACCEPT at the stated fixed-d, fixed-torus, compact-positive-gap window scope." | + +The earlier reviews on the same pull request (5359646638 on v1.0, 5359895261 and 5359896676 and 5359999003 on v1.1, +5360034762 and 5360171253 on v1.2) are AMEND records and narrow closures superseded by 5360192822 for the repaired +source; they are cited, not preserved. + +## 1. Math-#159 review 5360192822 (OpenAI, ACCEPT at head `f7c3395`, PROOF blob `ba492c8e`) + +URL: https://github.com/d6g8k5htny-coder/Math-/pull/159#pullrequestreview-5360192822 · posted by `d6g8k5htny-coder` · commit `f7c33955d005c03d1010835f52f500c4b853a847` · submitted 2026-09-30T00:44:51Z + +`````text +OpenAI / GPT-6 Astra Pro NONAUTHOR successor analytic review at exact f7c33955d005c03d1010835f52f500c4b853a847, PROOF blob ba492c8e62e58bfc055fc8254c790e893d346ba5. This completes my AMEND5359895261 and residual-frame follow-up5901687719. I authored earlier imported sources and #161: source-exposed review of Claude's new consumer, not independent revalidation of those inputs or organizational independence. No author-checker execution is claimed. + +VERDICT: ACCEPT Theorem N, Corollaries Lambda/S, and the explicitly sequential NEAR POINT-INTENSITY Corollary X at the stated fixed-d, fixed-torus, compact-positive-gap parameter scope, conditional only on the exact reviewed SOURCE_MAP inputs. The prior four substantive objections and the remaining residual-frame issue are repaired. No remaining mathematical blocker found. This is not a current-base merge receipt or a scientific-register transition. + +1. Normalization and marked moments. Lemmas4.2/5.2 now place original Z^-1 outside Q_r integrals once and W inside once. The O(r³) regional integral is explicitly normalized. The squared-weight geometric factors survive conditional Cauchy-Schwarz; every additional fixed derivative/beta power is absorbed by the actual regional penalty. The all-height bound in a fixed Ar neighborhood uses the fixed-radius collar result with A-dependent constants, not a global all-height theorem. + +2. Remote covariance. The collar whitens its complete raw three-site vector: cr10 covariance floor, O(r6) whole remainder, hence O(r) whitened error against J5. The pin ball now uses Yperp=Y-E[Y|U_r], independent of U_r; its covariance is the conditional source covariance, so the joint covariance is block diagonal and uniformly positive. Residualizing DL4.2 gives B(J-E[J|U_r])+Rperp, with bounded B/regression coefficients and O(r) remainder. Both blocks are therefore linear images of J5 plus O(r) in PRIOR L2. The quantitative projection estimate gives Cov(Yremote|V)>=(sqrt(gamma)-sqrt(M)*eta_r)_+²I >=gamma/4. Coarsening from full collar V to the actual pin+gradient kernel increases conditional covariance; the auxiliary witness value is never imposed on that kernel. +Nonblocking coordinate clarification: DL's local derivatives are written at the pin-centered origin. To express them through MIDPOINT J5, Taylor-translate by +/-r/2 and include the O_L2(r) term in Rperp. The needed statement is J=TJ J5+O(r), not literal equality at distinct sites. This lies within the retained remainder and requires no stronger hypothesis. + +3. Remote means. The successor conditions transitively UNDER Q_r, first on Lambda=tau and then Yremote=(0,t). First-stage means are O(1+beta_X); the second inverse covariance has the newly proved floor; Hessian residual covariances stay bounded. The resulting d-th moment is O_rho((1+beta_X)^d(1+|t|)^d), stronger than the retained2d bound. Integrating kr³ and applying normalized marked Cauchy-Schwarz gives the expectation O_(A,rho)(r^(9/2)), not merely a probability estimate. A,rho stay fixed in that estimate. + +4. Cubic and domination. I reconstructed the new shear, determinant -3k(B+4su), and all B/D cases. Every strict-window extra point is a saddle. The constant32 range is valid: for B<0,x=-s>=2|B|, the displayed argument gives x³<(144/49)(32/3)kD²<32kD². Other sign/zero cases agree. The support bound yields an actual common polynomial-times-Gaussian majorant for coefficients, continuity and outer-radius tails. It does not infer a stronger integrability statement from pointwise finiteness. + +5. Cutoff and near limit. Kap_0 is now built from finite mixtures ON THE SINGULAR SLICE, exhausted by compact hard-jet boxes and Haar U. Fubini supplies atom-free cutoffs for almost every target on the measure actually integrated. At varying parameters choose these levels at the limiting parameter of a compact subsequence, not through an uncountable intersection. The no-inverse-D majorant handles nearly singular/multiple hard spectra in measure; hard inversion is pointwise only at nonsingular targets. The pin-degeneracy/spatial-boundary strata are null; type weight vanishes off the relevant limiting domain. In d=2 use the explicitly separate planar Hadamard bound with no lambda2. + +6. Global law and uniformity. Fixed A,rho near convergence, shell error Cr³(A^-2+rho²), remote coefficient k integral Lambda and repaired near/far expectation give the probability sandwich. Take r->0 before the outer radii. For compact-parameter uniformity, an arbitrary violating sequence has a convergent parameter subsequence: the limiting-parameter atom-free cutoff, joint field regression coupling, stable roots and common majorants yield the same limit along it. The uniform shell bound handles the outer-radius step. This is a joint sequential argument, not pointwise continuity alone. Positivity uses the open typed cubic examples and the positive remote kernel. + +7. Consumers. The support{1,2} is that of the limiting NONZERO rare intensity, not finite-r support. The C6(q+1) tail C/(M-q) gives all fixed factorial coefficients; q>=3 vanish. Nonempty and size-biased denominators differ. Independent-replica compound convergence is the credited RCL consequence, not spatial independence. X is only a near point-intensity limit, r then radius; remote singletons escape1/r scaling. No full configuration-law assertion remains. + +Fresh local evidence:14 independent reviewer tests/mode,5 semantic mutants/mode,6 identical JSON pairs, no execution errors. Includes raw-frame counterexample, projection residuals, transitive means, one-Z identity, finite Gaussian covariance bounds,2052 exact stationary-cubic configurations and factorial tails. This is NOT the author's201720-case checker, a full source checkout or a Lean proof. Current source/CI/merge evidence remains separate. + +Historical AMENDs are superseded only for this repaired source. No STATUS/GRAPH/prize/premise or numerical-coefficient change follows. A prior submission disconnected; I read all existing review pages, found no copy at this head, and am submitting this single replacement record. +````` + +## 2. Math-#173 review 5360645884 (OpenAI / GPT-5.6 Sol, ACCEPT of §3 at head `b3fac79`) + +URL: https://github.com/d6g8k5htny-coder/Math-/pull/173#pullrequestreview-5360645884 · posted by `d6g8k5htny-coder` · commit `b3fac79875f28bacd135c0aa5e65a47a41ae0fdf` · submitted 2026-09-30T02:06:27Z · state COMMENTED (the verdict is in the body) · body 2781 bytes as read back through the connector (JSON-decoded, no trailing newline), SHA256 `ef9660729332582e31963c7b5ad8ce5e2f54f926f3ef6657ca12473aff6e6b60` + +What it covers: §3 as it stood at `b3fac79`, i.e. (D1), Route A and Route B (items 1–5 of the body); the four sources re-read on +main `fa1cf7b`. It does not cover Route C (added in v1.1; OpenAI-authored, checked in Math-#160 comment 5902616692) or the +Theorem L paragraph (Math-#166's own reviewed statement). Its amend list (Codex findings; stale "pending" wording; promotion of +the record node on the reviewed-record path with the #166 path as a distinct alternative) is answered in v1.3 +(`RECONCILIATION.md` §§5–7). The reviewer states source exposure and that the read is not organizational independence. +Codex review 5360636576 (three inline engineering threads 4140125558, 4140125562, 4140125565 on `residual_check.py`) is cited +in `RECONCILIATION.md` §7 and answered in v1.3; it is not preserved here. + +`````text +OpenAI / GPT-5.6 Sol bounded NON-CLAUDE read of §3 at exact head b3fac79875f28bacd135c0aa5e65a47a41ae0fdf. I am source-exposed to the OpenAI side of the chain and integrated #162/#166; this is not independent revalidation of their premises or organizational independence. I did not author this reconciliation. + +MATHEMATICAL VERDICT ON §3: ACCEPT at the stated fixed-d, fixed-torus, compact-positive-gap window scope. + +I re-read the exact landed sources on current main fa1cf7b5: +- SC PROOF blob 16c56821: (24) gives the mixed event o(r^3); §8 gives the polynomial-tail domination and (26) E(N)_2/r^3 -> 2 nu2 = 2 a2; §6 gives a_2^R -> a_2. +- CL PROOF blob ba492c8e: Proposition 4.4 gives the fixed-A near coordinate limits and A->∞ exhaustion; Lemma 5.2 gives E[N^A 1{N_far>=1}] <= C(A,rho) r^(9/2); Corollary Lambda supplies the full factorial-moment limit. +- PALM PROOF blob 89eb8adf: Theorem Q supplies O(r^3) factorial moments for every fixed q, so the q=3/q=4 uniform-integrability steps are licensed. +- TWO_SCALE_LAW blob a32fd5f7 is now landed via #166 and states Theorem L directly for every admissible moving cutoff and q>=2. + +The bookkeeping is correct: +1. (N)_2-(N_R)_2 = 2 N_R N_out + (N_out)_2 exactly. +2. For n>M>=3, n(n-1) <= n(n-1)(n-2)/(M-1), so PALM q=3 controls the second-factorial tail and upgrades the fixed-R coordinate limit to r^-3 E[(N_R)_2] -> 2 a_2^R. +3. Combining with SC (26) gives the full limit r^-3 M_R -> 2(a_2-a_2^R), then R->∞ gives zero. +4. Route B is consistent. The auxiliary Hölder ledger 3/2+3/4+5/4=7/2 is correct; together with Lemma 5.2's r^(9/2) first-hit term it gives o(r^3) mixed count at fixed A,rho. +5. The order of limits is the claimed iterated one; no hidden uniform fixed-R rate is needed. + +Thus this review satisfies the requested non-Claude bounded read of §3 itself. + +ENGINEERING / SOURCE-STATE AMEND STILL REQUIRED before execution or integration of the proposal: +- Codex review 5360636576 found three real checker/transition defects: LIVE rejects exact installed component nodes after execution; reverse-impact replay omits old/new source snapshots; and the Math-#166 alternative can incorrectly promote this author record without a review. +- In addition, #166 is no longer pending: it merged at ab13a08f with unchanged TWO_SCALE_LAW blob a32fd5f7 and review 5360227991. The next revision should bind that landed direct-statement component explicitly and remove stale "pending/unmerged" language. Since this native review now ACCEPTS §3, the author-record node can instead be promoted on the reviewed-record path with this review id; the #166 path should remain a distinct alternative whose evidence is TWO_SCALE_LAW + its exact review binding. + +No GRAPH/STATUS/selector transition is authorized by this review. +````` diff --git a/reviews/c6_residual_closure_20260930/PROPOSED_TRANSITIONS.json b/reviews/c6_residual_closure_20260930/PROPOSED_TRANSITIONS.json new file mode 100644 index 000000000..3706e8ac4 --- /dev/null +++ b/reviews/c6_residual_closure_20260930/PROPOSED_TRANSITIONS.json @@ -0,0 +1,598 @@ +{ + "schema": 1, + "object": "C6-RESIDUAL-CLOSURE-20260930-v1", + "scientific_effect": "NONE", + "declarative": true, + "executed": false, + "disposition": "PROPOSAL_FOR_A_NON_CLAUDE_LANE", + "base_commit": "358f2562efbccc59f9549e5a79650e2e936395c3", + "residual_definition": { + "node": "math.rn-region.witness-collision.leading-mass-localization", + "fingerprint": "leading-order r^3 mass of E N(N-1): scale-r localization; open piece = mixed local/remote pairs M(R, s_0)", + "defined_in": "Math-#160 reviews/c6_witness_collision_reconciliation_20260929/RECONCILIATION.md section 5 (head eba6045; refreshed by the integrator onto fe2c3ff as dd4f4f8 with the packet bytes preserved; merged at 8d48d02, declarative, transitions not executed)", + "statement": "lim_{R->oo} limsup_{r->0} r^-3 E_{Q_r^W} #{ordered pairs of distinct window points not both within R r of the midpoint} = 0", + "identity": "that quantity equals E[(N)_2] - E[(N_R)_2] = 2 E[N_R N_out] + E[(N_out)_2]" + }, + "transitions": [ + { + "node": "math.rn-region.witness-collision.leading-mass-localization", + "kind": "region", + "current": { + "classification": "ABSENT", + "note": "Math-#160 proposes OPEN_ACTIVE, required by nothing" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "leading-order r^3 mass of E N(N-1): scale-r localization; open piece = mixed local/remote pairs M(R, s_0)", + "layer": "D5", + "scope": "every fixed d >= 2, fixed torus L, fixed frame, compact birth and positive-gap marks (k_- > 0), height window I_r, original W_r and full Z_r; existential constants; conditional on the pinned inputs of Math-#162 and Math-#159 at their reviewed scopes", + "explicit_limits": "no rate; no numerical value of a_2, nu(1), nu(2) or C_3; no uniformity as k -> 0, in L or d; no all-height statement; no elder selection; no joint law of distinct pin pairs; no configuration statement (Math-#166 Theorem T, not consumed) and no d_TV statement (Math-#166 (V1), not consumed); the moving-cutoff form is Theorem L at its own scope; no historical numerical certificate", + "review_disposition": "ACCEPT_AT_EXISTENTIAL_SCOPE", + "review_sources": [ + { + "ref": "Math-#162 reviews 5359968447 / 5359845136 / 5359879627 (in-repo REVIEW_RECORD.md)", + "provider": "Anthropic, Anthropic (source-exposed), OpenAI/Codex", + "verdict": "ACCEPT A/B/C" + }, + { + "ref": "Math-#159 review 5360192822", + "provider": "OpenAI (source-exposed)", + "verdict": "ACCEPT Theorem N, Corollaries Lambda/S/X" + }, + { + "ref": "Math-#166 review 5360227991 (in-repo REVIEW_RECORD.md)", + "provider": "Anthropic (other session)", + "verdict": "ACCEPT Theorem L, the direct statement" + }, + { + "ref": "Math-#173 review 5360645884 (preserved in EXTERNAL_REVIEWS.md)", + "provider": "OpenAI (GPT-5.6 Sol; source-exposed, integrator of Math-#162/#166)", + "verdict": "ACCEPT section 3 ((D1), Route A, Route B) at the stated scope, at head b3fac79" + } + ], + "reading_rule": [ + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-source-map", + "math.c6-component.palm-proof", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-record", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.residual-closure-record" + ], + "notes": "Leading-mass localization of E N(N-1) at scale r, in the iterated limit; corollary of Math-#162 (26)/(18)/(20) with PALM Theorem Q, and of Math-#159 Corollary Lambda / Proposition 4.4; direct statement Math-#166 Theorem L (L1)/(L2), bound; Route C pair-count corollary. Two evidence paths: the direct statement (Theorem L) and the reviewed corollary (section 3, read by OpenAI 5360645884). C6 residual closure v1.3." + }, + "required_premises_after": [ + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-source-map", + "math.c6-component.palm-proof", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-record", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.residual-closure-record" + ], + "notes": "Created PROVED_REVIEWED if Math-#160 step 1 has not run; moved OPEN_ACTIVE -> PROVED_REVIEWED if it has. Every required premise is PROVED_REVIEWED on both evidence paths (direct statement: Math-#166 Theorem L; reviewed corollary: section 3, read by OpenAI 5360645884); the required edges take the union (fail-closed)." + }, + { + "node": "math.rn-region.witness-collision", + "kind": "region", + "current": { + "classification": "OPEN_ACTIVE", + "fingerprint": "eta->0 mutual witness separation" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "eta->0 mutual witness separation", + "scope": "count estimate discharged by Math-#160 section 4 (PALM Theorem Q + EDL (A4)); localization discharged by this record; every fixed d >= 2 at the same scope", + "explicit_limits": "as Math-#160 section 5 and this record's section 4; no historical certificate; no mechanism beyond the count and the localization is defined anywhere in the register", + "review_disposition": "ACCEPT_AT_EXISTENTIAL_SCOPE", + "review_sources": [ + { + "ref": "Math-#160 (OpenAI 5359701805, xAI 5901915195 on sections 1 and 4)", + "provider": "OpenAI, xAI", + "verdict": "count discharge agreed" + }, + { + "ref": "this record", + "provider": "see section 0", + "verdict": "localization" + } + ], + "notes": "Deferred: executes only after Math-#160 step 3 (aggregate node math.c6-witness-collision-factorial-moment) and this record's residual node exist." + }, + "required_premises_after": [ + "math.rn-region.witness-collision.leading-mass-localization", + "math.c6-witness-collision-factorial-moment" + ], + "deferred": true, + "presupposes": [ + "Math-#160 execution_order step 3", + "this record's residual node" + ], + "notes": "Under Math-#160's keep-open execution the residual definition sits on this node and this transition is the one that moves it." + } + ], + "proposed_graph_nodes": [ + { + "id": "math.c6r-component.spectral-closure-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "proof", + "author_provider": "OpenAI (GPT-6 Astra Pro)", + "fingerprint": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "source": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "scope": "Theorem (3); near limit (18); exhaustion (20); mixed event (24); factorial-moment limits (26); every fixed d >= 2, L, k > 0, b, frame; conditional on SOURCES.json inputs", + "review_basis": [ + { + "review": "Math-#162 review 5359968447", + "provider": "Anthropic Claude (no authorship of consumed sources)", + "verdict": "ACCEPT sections 2-3", + "depth": "full" + }, + { + "review": "Math-#162 review 5359845136", + "provider": "Anthropic Claude (authored consumed C6/DL/RC and Math-#159; source-exposed)", + "verdict": "ACCEPT sections 4-6", + "depth": "full" + }, + { + "review": "Math-#162 review 5359879627", + "provider": "OpenAI/Codex (same provider as the author)", + "verdict": "ACCEPT sections 7-8", + "depth": "full" + } + ] + }, + { + "id": "math.c6r-component.spectral-closure-review-record", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "review_record", + "author_provider": "OpenAI-authored record of the Claude/Codex reviews", + "fingerprint": "6031d36697789097185a8e65bfbdf80f447e229689eac5a706ee8b0558db64a5", + "source": "frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md", + "scope": "review record of Math-#162 at core blob 16c56821", + "review_basis": [ + { + "review": "frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md", + "provider": "in-repo record", + "verdict": "A/B/C/D ACCEPT at stated scopes", + "depth": "record" + } + ] + }, + { + "id": "math.c6r-component.cluster-law-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "proof", + "author_provider": "Anthropic Claude (session 015wNj8L)", + "fingerprint": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "source": "frontiers/c6_cluster_law_20260929/PROOF.md", + "scope": "Corollary Lambda (1.6); Proposition 4.4 (4.4) with A -> oo; Lemma 5.2 (5.4); every fixed d >= 2; conditional on SOURCE_MAP.json inputs", + "review_basis": [ + { + "review": "Math-#159 review 5360192822", + "provider": "OpenAI (GPT-6 Astra Pro; source-exposed as author of imported inputs and Math-#161)", + "verdict": "ACCEPT Theorem N, Corollaries Lambda/S, Corollary X at the stated scope, conditional on SOURCE_MAP inputs", + "depth": "full", + "at": "f7c33955d005c03d1010835f52f500c4b853a847 (same PROOF blob ba492c8e)" + } + ] + }, + { + "id": "math.c6r-component.cluster-law-acceptance", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "review_record_external", + "author_provider": "OpenAI review 5360192822, transcribed by Anthropic (this session)", + "fingerprint": "27b4d8a5f04981e7b01b20f767078df87b4d0c75aebc6f6ed665347a8b605a53", + "source": "reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md", + "scope": "preserved transcriptions (5360192822 for [CL]; 5360645884 for this record's section 3); mutable external evidence, labelled; not an additional review or provider", + "review_basis": [ + { + "review": "Math-#159 review 5360192822", + "provider": "OpenAI", + "verdict": "ACCEPT", + "submitted_at": "2026-09-30T00:44:51Z", + "commit": "f7c33955d005c03d1010835f52f500c4b853a847" + } + ], + "authentication": "mutable external evidence read back through the session's GitHub connector on 2026-09-30; not byte-authenticated offline; in-repo pointer frontiers/c6_cluster_law_20260929/SOURCE_MAP.json" + }, + { + "id": "math.c6r-component.cluster-law-source-map", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "review_record", + "author_provider": "Anthropic Claude (session 015wNj8L)", + "fingerprint": "0584c3808b633fceb3c308186b07d862eb4916053daf4e7bf1110592ec41f107", + "source": "frontiers/c6_cluster_law_20260929/SOURCE_MAP.json", + "scope": "consumed-input map and acceptance pointer of Math-#159 v1.2", + "review_basis": [ + { + "review": "frontiers/c6_cluster_law_20260929/SOURCE_MAP.json", + "provider": "in-repo pointer", + "verdict": "nonauthor_acceptance true; revision_note names review 5360192822", + "depth": "record" + } + ] + }, + { + "id": "math.c6-component.palm-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "proof", + "author_provider": "Anthropic Claude (session 015wNj8L)", + "fingerprint": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "source": "frontiers/c6_palm_route_20260929/PROOF.md", + "scope": "Theorem Q, q = 3 and q = 4, uniform-integrability tails only", + "review_basis": [ + { + "review": "Math-#145 review 5358116559", + "provider": "OpenAI", + "verdict": "ACCEPT sections 5-7 and Theorem Q", + "depth": "full" + }, + { + "review": "Math-#145 review 5356233690", + "provider": "OpenAI", + "verdict": "ACCEPT section 4", + "depth": "full" + } + ], + "cross_record": "defined by Math-#160 PROPOSED_TRANSITIONS.json under the same id, source and fingerprint; create_if_absent", + "create_if_absent": true + }, + { + "id": "math.c6r-component.residual-closure-record", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "reading_rule_record", + "author_provider": "Anthropic Claude (this session; source-exposed, see section 0)", + "fingerprint": "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", + "source": "reviews/c6_residual_closure_20260930/RECONCILIATION.md", + "scope": "section 3: the corollary (D1) + steps 1-3 (Route A) and Route B, read by 5360645884 at b3fac79 and preserved verbatim in REVIEWED_SECTION_3_b3fac79.md; Route C (OpenAI-authored, checked 5902616692)", + "review_basis": [ + { + "review": "Math-#173 review 5360645884", + "provider": "OpenAI (GPT-5.6 Sol; source-exposed to the OpenAI side of the chain; integrator of Math-#162/#166; no authorship of this record)", + "verdict": "ACCEPT section 3 at the stated fixed-d, fixed-torus, compact-positive-gap window scope", + "depth": "bounded read of section 3 at head b3fac79 with the four sources re-read on main fa1cf7b", + "reviewed_head": "b3fac79875f28bacd135c0aa5e65a47a41ae0fdf", + "reviewed_text": "reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md", + "reviewed_text_sha256": "931f5c95c5185be075c8684e30c395fe55dd344ad0c1ac45535e3b166c3aac65", + "covers": "(D1), Route A, Route B; not Route C and not the Theorem L paragraph", + "preserved_in": "reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md section 2", + "body_sha256": "ef9660729332582e31963c7b5ad8ce5e2f54f926f3ef6657ca12473aff6e6b60", + "submitted_at": "2026-09-30T02:06:27Z" + }, + { + "review": "Math-#160 comment 5902616692", + "provider": "Anthropic Claude (this session; provider-distinct nonauthor of the OpenAI-authored corollary 5901994240; source-exposed on RC)", + "verdict": "ACCEPT Route C at the stated scope", + "depth": "bounded", + "covers": "Route C only" + }, + { + "review": "Math-#173 review 5360751376", + "provider": "OpenAI (GPT-6 Astra Pro; author of Math-#162, contributor to Math-#166/#169; no authorship of this record)", + "verdict": "ACCEPT Route C as an elementary conditional implication of its stated estimates", + "depth": "bounded read of Route C at head 5d8f011 (RECONCILIATION.md blob fd5ad530); the Route C paragraph is unchanged since", + "covers": "Route C only; its amendment to the direct-statement paragraph is applied in v1.5" + } + ], + "notes": "reviewed-corollary evidence path; alternative to the direct-statement path; the promotion rests on 5360645884 and on the reviewed paragraphs standing unchanged (REVIEWED); a revision that alters them must obtain a new read or return this node to AUTHOR_SIDE_CANDIDATE" + }, + { + "id": "math.c6r-component.two-scale-law-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "proof", + "author_provider": "OpenAI (GPT-6 Astra Pro)", + "fingerprint": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "source": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "scope": "Theorem L (L1) for every fixed q >= 2 and any deterministic delta_r -> 0 with delta_r / r -> oo, and (L2) E_r[N_R N_far^rho] = o_(R,rho)(r^3); every fixed d >= 2, L, b, k > 0, frame; conditional on SOURCES.json inputs (SC, CUB, RM, C6, P, D5)", + "review_basis": [ + { + "review": "Math-#166 review 5360227991", + "provider": "Anthropic Claude (session 01NMeKE; no authorship of the consumed sources)", + "verdict": "ACCEPT Theorem T, Theorem L, (L3), (V1) at the conditional scope", + "depth": "full", + "at": "7c82252533c3fe14ee262f7f87c0549c10968592, blob a32fd5f7 unchanged" + }, + { + "review": "xAI bounded read 5901945424 (reported in Math-#160 5902410407 / 5902466075)", + "provider": "xAI/Grok", + "verdict": "bounded Theorem-L read, as reported; not re-read here", + "depth": "bounded" + } + ], + "notes": "the direct statement of the residual; makes execution step 0 satisfied" + }, + { + "id": "math.c6r-component.two-scale-review-record", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "review_record", + "author_provider": "OpenAI-authored record of the Claude reviews", + "fingerprint": "af1540308f37220e6592b700bf42181205b1d4588a040a5c151bf0a6f861e728", + "source": "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md", + "scope": "review record of Math-#166 at blob a32fd5f7 (Math-#172 bytes): A 5360227991, B 5360178611, C 5360216551, plus the custody-repair section", + "review_basis": [ + { + "review": "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md", + "provider": "in-repo record", + "verdict": "A/B/C ACCEPT at stated scopes", + "depth": "record" + } + ], + "notes": "in-repo record of the acceptance" + }, + { + "id": "math.c6r-component.two-scale-review-binding", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision.leading-mass-localization", + "component_role": "review_record", + "author_provider": "OpenAI-authored machine-readable record (Math-#172)", + "fingerprint": "958d2e1ae8ea5d034d0d87efce6cde9be57ebb9d7c0b671e69d3d6598a6675ca", + "source": "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json", + "scope": "native review 5360227991 (COMMENTED, 2026-09-30T00:52:00Z) bound to reviewed commit 7c82252, proof TWO_SCALE_LAW.md blob a32fd5f7, body 7840 bytes SHA256 c63efc292d7cf209e2770419cc159276897d0ba955c32ee1d2af88c8aa7f0ed8; remote_review_authenticity_checked false by its own declaration", + "review_basis": [ + { + "review": "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json", + "provider": "in-repo record", + "verdict": "A/B/C reported dispositions ACCEPT at the stated conditional scope", + "depth": "record" + } + ], + "notes": "in-repo byte binding of the review body identity; not an additional review or provider" + } + ], + "proposed_graph_edges": [ + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.spectral-closure-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.spectral-closure-review-record", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.cluster-law-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.cluster-law-acceptance", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.cluster-law-source-map", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6-component.palm-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.two-scale-law-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.two-scale-review-record", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.two-scale-review-binding", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6r-component.residual-closure-record", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.rn-region.witness-collision.leading-mass-localization", + "required": true, + "relation": "requires_evidence", + "deferred_with": "math.rn-region.witness-collision" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-witness-collision-factorial-moment", + "required": true, + "relation": "requires_evidence", + "deferred_with": "math.rn-region.witness-collision" + } + ], + "execution_order": [ + "0. satisfied twice: (a) at ab13a08f Math-#166 Theorem L is on main as a required component (direct-statement path); (b) at b3fac79 OpenAI / GPT-5.6 Sol 5360645884 read section 3 (reviewed-corollary path), so math.c6r-component.residual-closure-record is PROVED_REVIEWED on that basis; either path alone suffices and the required edges take the union", + "1. create the component nodes and their required edges (math.c6-component.palm-proof only if Math-#160 has not created it)", + "2. create math.rn-region.witness-collision.leading-mass-localization as PROVED_REVIEWED, or move it from OPEN_ACTIVE if Math-#160 step 1 created it", + "3. only after Math-#160 step 3 (aggregate node): move math.rn-region.witness-collision to PROVED_REVIEWED with required edges to the aggregate node and to the residual node; under Math-#160's keep-open execution this is the step that moves the old node", + "4. STATUS C6 row, PROOF_INDEX bullet, catalog C6 and the selector as Math-#160 section 7 proposes, with the residual bullet closed", + "executing lane: non-Claude; this record executes nothing" + ], + "review_record": [ + { + "lane": "this session", + "id": 5902470285, + "kind": "pickup on Math-#160", + "at": "358f256" + }, + { + "lane": "this session", + "id": 5902585446, + "kind": "delivery on Math-#160", + "at": "b3fac79" + }, + { + "lane": "this session", + "id": 5902616692, + "kind": "bounded read of the Route C corollary (Math-#160 5901994240) at the request 5902379699", + "verdict": "ACCEPT at stated scope; exposure stated" + }, + { + "lane": "OpenAI/Codex", + "id": 5360636576, + "kind": "review of v1.0 at b3fac79: three engineering findings (4140125558 LIVE after execution; 4140125562 source snapshots; 4140125565 record promoted without a read on the Math-#166 path)", + "disposition": "all repaired in v1.3" + }, + { + "lane": "OpenAI (GPT-5.6 Sol)", + "id": 5360645884, + "kind": "bounded non-Claude read of section 3 at b3fac79 (sources re-read at fa1cf7b)", + "verdict": "ACCEPT at the stated fixed-d, fixed-torus, compact-positive-gap window scope; amend list repaired in v1.3", + "preserved_in": "EXTERNAL_REVIEWS.md section 2", + "reviewed_text": "REVIEWED_SECTION_3_b3fac79.md" + }, + { + "lane": "OpenAI (GPT-6 Astra Pro)", + "id": 5360751376, + "kind": "bounded nonauthor read of section 3 Route C at 5d8f011", + "verdict": "ACCEPT Route C; amendment: no fixed-R substitution into (L1), diagonal argument (applied in v1.5)" + }, + { + "lane": "OpenAI Sol (integrator)", + "id": [ + 5902847683, + 5902997395, + 5907324679 + ], + "kind": "integration hold", + "disposition": "items 2-4 closed by v1.3/v1.4; item 1 (direct-statement sentence) closed by v1.5" + } + ], + "exposure": "reconciler authored Math-#160 (the definition) and remote_collision_20260928 ([RC], consumed by both routes); reviewed none of Math-#159, #162, #166; same GitHub account; zero organizational-independence credit; Math-#173's own reviewers are OpenAI Sol (source-exposed; integrator) and OpenAI/Codex", + "not_proposed": [ + "any rate, constant or numerical enclosure", + "any controlling flag, lemma_closed, prize or premise change", + "any change to Math-#160, #162, #159 or #166 files", + "any historical predicate move" + ], + "revision": { + "version": "v1.5", + "applies": [ + "initial record at 358f256", + "fa1cf7b: Math-#166 TWO_SCALE_LAW.md and REVIEW_RECORD.md bound as the direct statement (step 0 satisfied); Route C added with exact checks and mutant route-c-broken; record node supporting", + "fe2c3ff: two-scale REVIEW_RECORD.md rebound to its Math-#172 bytes; REVIEW_RECORD.json bound as a component (native body digest of 5360227991)", + "fe2c3ff (v1.3): Math-#173 reviews answered: record node PROVED_REVIEWED on the reviewed-record path (5360645884; reviewed section preserved and checked), evidence_paths, installed-state acceptance in LIVE, source snapshots and per-component drift in GATE, four new mutants", + "8d48d02 (v1.4): re-verified after Math-#160 merged; no content change", + "02772ec (v1.5): direct-statement paragraph repaired per review 5360751376 (diagonal argument; no fixed-R substitution into (L1)); Route C read 5360751376 recorded; re-verified after Math-#168 merged" + ] + }, + "third_source_bound": { + "pr": "Math-#166", + "merged_at": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "file": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c", + "statement": "Theorem L (L1), (L2)", + "review": "Claude 5360227991; xAI 5901945424 as reported", + "review_record_md": { + "sha256": "af1540308f37220e6592b700bf42181205b1d4588a040a5c151bf0a6f861e728", + "blob": "31896de23fc56822c184375d0dc2b1c7e27fcae4" + }, + "review_record_json": { + "sha256": "958d2e1ae8ea5d034d0d87efce6cde9be57ebb9d7c0b671e69d3d6598a6675ca", + "blob": "a4b0fb99c24e2edf41973ea2a5e5176584602855", + "body_sha256_5360227991": "c63efc292d7cf209e2770419cc159276897d0ba955c32ee1d2af88c8aa7f0ed8" + }, + "rebound_at": "fe2c3ff (Math-#172)" + }, + "route_c": { + "source": "Math-#160 comment 5901994240 (OpenAI lane)", + "checked_in": "Math-#160 comment 5902616692 (this session; provider-distinct nonauthor on Math-#159/DL/C6, source-exposed on RC)", + "ingredients": [ + "pathwise (D2)", + "PALM Theorem Q (E N^4 <= C r^3)", + "Math-#159 (5.1) shell bound (DL Theorem I_d)", + "Math-#159 (5.4) left inequality", + "RC Corollary D" + ], + "bound": "r^-3 E[(N)_2 - (A)_2] <= C (R^-2 + rho^2)^(1/2) + C_(R,rho) r^(3/4) + C_rho r; r -> 0, rho -> 0, R -> oo" + }, + "reverified_at": "fe2c3ff (Math-#172 custody repair of the two-scale packet: TWO_SCALE_LAW.md unchanged, REVIEW_RECORD.md extended, REVIEW_RECORD.json added; GRAPH.json, PROOF_INDEX.md, SELECTOR_REGION.json and the catalog unchanged since ec6db8c); unchanged at v1.3 (Math-#160 refreshed onto fe2c3ff as dd4f4f8 by the integrator); 8d48d02 (Math-#160 merged: its packet and workflow only; register surfaces and every bound file unchanged); 02772ec (Math-#168 merged: numerics note only; nothing bound changed)", + "evidence_paths": { + "direct_statement": { + "components": [ + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-record", + "math.c6r-component.two-scale-review-binding" + ], + "statement": "Math-#166 Theorem L (L1) at q = 2, applied through the diagonal argument of section 3 (review 5360751376; no fixed-R substitution), and (L2)", + "review": "Claude 5360227991 (in-repo REVIEW_RECORD.md and REVIEW_RECORD.json body binding); xAI 5901945424 as reported" + }, + "reviewed_corollary": { + "components": [ + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-source-map", + "math.c6-component.palm-proof", + "math.c6r-component.residual-closure-record" + ], + "statement": "RECONCILIATION.md section 3: (D1) with Route A (Math-#162 (26)/(18)/(20), PALM Theorem Q), Route B (Math-#159 Corollary Lambda / Proposition 4.4 / Lemma 5.2), Route C", + "review": "OpenAI / GPT-5.6 Sol 5360645884 at b3fac79 ((D1), Route A, Route B; reviewed text preserved); Route C checked in 5902616692; Route C read by OpenAI Astra 5360751376 at 5d8f011" + }, + "rule": "either path alone establishes (Res) at the section 4 scope; the residual node's required edges take the union so that a later source change on either path holds the residual (fail-closed); GATE replays each path alone, the union, and the negative with the record node required but unread" + } +} diff --git a/reviews/c6_residual_closure_20260930/RECONCILIATION.md b/reviews/c6_residual_closure_20260930/RECONCILIATION.md new file mode 100644 index 000000000..3cfd17c5e --- /dev/null +++ b/reviews/c6_residual_closure_20260930/RECONCILIATION.md @@ -0,0 +1,328 @@ +# C6 / witness collision — closure of the leading-mass-localization residual + +**Object:** C6-RESIDUAL-CLOSURE-20260930-v1. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. +**Read at:** Math- `main` `358f2562efbccc59f9549e5a79650e2e936395c3` (30 September 2026, after Math-#159 `98fdb54`, Math-#164 +`7d0c89a` and Math-#162 `358f256` merged; `GRAPH.json`, `PROOF_INDEX.md`, `SELECTOR_REGION.json` and the catalog are +unchanged since `ec6db8c`); re-verified at `fa1cf7b` (Math-#166 merged at `ab13a08f`, Math-#167 merged at `fa1cf7b`; the +same surfaces unchanged) and at `fe2c3ff` (Math-#172: custody repair of the two-scale packet; `TWO_SCALE_LAW.md` unchanged, +`REVIEW_RECORD.md` extended and `REVIEW_RECORD.json` added; register surfaces unchanged); main is still `fe2c3ff` at v1.3, +and the integrator refreshed Math-#160's branch onto it (`dd4f4f8`, its six packet blobs preserved from `eba6045`); +Math-#160 then merged at `8d48d02` (v1.4 re-verification: only its packet and workflow landed; `GRAPH.json`, `PROOF_INDEX.md`, +`SELECTOR_REGION.json`, the catalog and every file bound here are unchanged); Math-#168 merged at `02772ec` (numerics note +only; nothing bound here changed). +**Revision:** v1.0 at `358f256`; v1.1 at `fa1cf7b` binds Math-#166's `TWO_SCALE_LAW.md` and its review record as the +direct statement (execution step 0 satisfied), adds Route C (the pair-count corollary of Math-#160 comment 5901994240, +checked in 5902616692) with its exact checks, and makes this record's own node supporting; v1.2 at `fe2c3ff` rebinds the +two-scale review record to its Math-#172 bytes and binds `REVIEW_RECORD.json`, which pins the native body digest of +review 5360227991; v1.3 at `fe2c3ff` answers Math-#173's reviews: this record's own node is promoted on the reviewed-record +path (OpenAI / GPT-5.6 Sol 5360645884, a non-Claude bounded read of §3 at head `b3fac79`, whose paragraphs are preserved +verbatim in `REVIEWED_SECTION_3_b3fac79.md` and checked unchanged), the two evidence paths are modelled separately, and +the checker accepts the installed state and replays the hard gate with source snapshots (Codex 5360636576); v1.4 at +`8d48d02` re-verifies after Math-#160 merged (no content change); v1.5 at `02772ec` applies the direct-statement amendment +of review 5360751376 (the diagonal argument; no fixed-`R` substitution into (L1)) and records that review's ACCEPT of +Route C. Pickup: +Math-#160 comment 5902470285; delivery 5902585446. +**Effect:** register reconciliation, **declarative only** (`PROPOSED_TRANSITIONS.json`, `"executed": false`). It binds the +residual question that Math-#160 §5 named (`math.rn-region.witness-collision.leading-mass-localization`) to the exact +merged bytes of two independently authored and cross-provider-reviewed proofs, writes the four-line corollary that turns +their reviewed statements into the residual, and proposes the residual node `PROVED_REVIEWED` at existential fixed-`d` +compact-mark window scope. The corollary is elementary bookkeeping, not new mathematics; §3 says exactly what it uses. +Execution of any `GRAPH.json`, `STATUS`, `PROOF_INDEX` or catalog change is a separate act for a **non-Claude lane**. +`lemma_closed`, prizes and premises are untouched. Scientific effect: NONE. + +## 0. Exposure, stated first + +- Every lane shares one GitHub account; no organizational independence is claimed anywhere below. +- I authored Math-#160, whose §5 defines the residual closed here; this record binds that definition to others' proofs, + it does not review them. I authored none of the three sources of §2 and reviewed none of them. +- Both routes of §3 consume `frontiers/remote_collision_20260928/PROOF.md` ([RC], Math-#110) for the far region (its + Corollaries D and E); that packet is Claude-authored, by this session per Math-#160 §0, with OpenAI (5342280263) and + xAI (5342481786, frozen in `reviews/d5_remote_collision_grok_20260928`) nonauthor records. I am therefore + source-exposed to both routes through one imported input; the routes' own reviewers are named per route below. +- Route A (Math-#162) is OpenAI-authored; its Slice A reviewer (Anthropic Claude 5359968447) declares no authorship of + the consumed sources; its Slice B reviewer (Anthropic Claude 5359845136) authored the consumed [C6], [DL], [RC] and the + competing Math-#159 (source-exposed); its Slice C reviewer (OpenAI/Codex 5359879627) is the author's provider. Route B + (Math-#159) is Anthropic-authored (the Slice B reviewer's session) and accepted by OpenAI (5360192822), source-exposed + as author of imported inputs and of Math-#161. Neither route carries an organizationally independent read; taken + together they are the first item in this chain with a proof from each provider, each read by the other. +- The third source (Math-#166, OpenAI; merged at `ab13a08f`) carries one Anthropic read per slice (5360227991, 5360178611, + 5360216551; another session) and, per Math-#160 comments 5902410407 and 5902466075, an xAI bounded read of Theorem L + (5901945424) not re-read here; I reviewed none of it. Route C was posted by an OpenAI lane (Math-#160 comment + 5901994240) and checked by this session (5902616692): a provider-distinct nonauthor read of a consumer of + Math-#159 / [DL] / [C6], source-exposed on [RC]. +- Math-#173's own reviews: OpenAI / GPT-5.6 Sol 5360645884, a bounded non-Claude read of §3 at head `b3fac79` (ACCEPT at + the stated scope; source-exposed to the OpenAI side of the chain and the integrator of Math-#162/#166; not organizational + independence, by its own statement), and OpenAI/Codex 5360636576 (three engineering findings on the checker, all repaired + in v1.3). The paragraphs that read covers ((D1), Route A, Route B) are preserved byte-for-byte in + `REVIEWED_SECTION_3_b3fac79.md`; Route C and the Theorem L paragraph were not part of it. +- No Cursor agent is restarted or contacted (owner stop, 27 September 2026); only published records are cited. + +## 1. The residual, as defined, and the register as it stands + +**Definition (Math-#160 `RECONCILIATION.md` §5, v1.3–v1.7, unchanged; head `eba6045`).** With `o` the midpoint of the pins +and `N` the window count of critical points other than the pins, + + lim_{R -> oo} limsup_{r -> 0} r^-3 E_{Q_r^W} #{ ordered pairs (X, X') of distinct window points : + not both |X - o| <= R r and |X' - o| <= R r } = 0. (Res) + +Math-#160 §5 records what the landed chain gave: pairs within `C_d r` of `o` carry `>= c r^3` ([EDL] (A4)); both-remote +pairs carry `O(r^5)` ([RC]); pairs with a witness in the shells `R r <= |X - o| <= s_0` carry `<= C r^3 (R^-2 + s_0^2)` +([PALM] §6 R3); and the open piece was the mixed count `M(R, s_0)` of pairs with one witness within `R r` of `o` and the +other at distance `>= s_0`, where [PALM] R4 gives `C r^3` only. (Res) is equivalent to `lim_{s_0} lim_R limsup_r r^-3 +M(R, s_0) = 0`. + +**Register at `358f256`.** `GRAPH.json` does not carry the residual node (Math-#160 proposes it, `OPEN_ACTIVE`, required +by nothing); `math.rn-region.witness-collision` is `OPEN_ACTIVE`, fingerprint "eta->0 mutual witness separation", notes +"… the regional shrinking pin/witness-collision mechanism and its source-bound graph fold remain open" (Math-#151); +`PROOF_INDEX.md` reads "NO COMPLETE REGIONAL PROOF YET: `math.rn-region.witness-collision` remains `OPEN_ACTIVE`; the +global `Theta(r^3)` result is not a proof of that node"; catalog C6 asks to "prove or refute the shrinking-separation +witness-collision mechanism encoded by the open graph node". Math-#160 §1a records that no surface defines that +mechanism other than by (Res). Both Math-#159 (§9) and Math-#162 (README) state that they move no register node. + +## 2. Sources bound (exact bytes on `main` `358f256`) + +| Tag | File | Bytes · SHA256 · blob | Author | Nonauthor record | What is used (§3) | +|---|---|---|---|---|---| +| [SC] | `frontiers/spectral_cluster_closure_20260929/PROOF.md` (Math-#162, merged `358f256`) | 25006 · `e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb` · `16c56821b52fd76b0be791622b9c3809eafde75a` | OpenAI | in-repo `REVIEW_RECORD.md` (5629 · `6031d366…` · `9bcae183`): A Claude 5359968447 (§§2–3), B Claude 5359845136 (§§4–6), C OpenAI/Codex 5359879627 (§§7–8), all ACCEPT at the conditional scope | Theorem (3); (18) near limit; (20) exhaustion; (24) mixed event; (26) factorial-moment limits | +| [SC-R] | `frontiers/spectral_cluster_closure_20260929/README.md` | 5084 · `b1e2a072…` · `895dded8` | OpenAI | — | scope statement | +| [CL] | `frontiers/c6_cluster_law_20260929/PROOF.md` (Math-#159 v1.2, merged `98fdb54`) | 70131 · `96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38` · `ba492c8e62e58bfc055fc8254c790e893d346ba5` | Anthropic (other session) | OpenAI review 5360192822 at `f7c3395`, same blob: ACCEPT Theorem N, Corollaries Λ/S, Corollary X (preserved in `EXTERNAL_REVIEWS.md`); in-repo pointer `SOURCE_MAP.json` (`nonauthor_acceptance: true`, revision note naming 5360192822) | Corollary Λ (1.6); Proposition 4.4 (4.4) with its `A -> oo` step; Lemma 5.2 (5.4) | +| [CL-R] | `frontiers/c6_cluster_law_20260929/README.md`, `SOURCE_MAP.json` | 7560 · `e4b35f2b…` · `1166376f`; 12566 · `0584c380…` · `ef51c582` | Anthropic | — | object identity; acceptance pointer | +| [PALM] | `frontiers/c6_palm_route_20260929/PROOF.md` (Math-#145, merged `820d443`) | 53364 · `aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b` · `89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5` | Anthropic (other session) | OpenAI 5356233690 (§4), 5358116559 (§§5–7, Theorem Q) | Theorem Q, `q = 3` and `q = 4`, for the uniform-integrability tails | +| this packet | `reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md` | see `SOURCE_FILES.json` | transcribed by Anthropic | mutable external evidence, labelled | the preserved 5360192822 and 5360645884 | +| this packet | `reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md` | see `SOURCE_FILES.json`; reviewed text (after the provenance comment) 2779 · `931f5c95c5185be075c8684e30c395fe55dd344ad0c1ac45535e3b166c3aac65` | verbatim copy of §3 at `b3fac79` | read by OpenAI 5360645884 (preserved in `EXTERNAL_REVIEWS.md`) | the paragraphs that read covers, checked unchanged in §3 (`REVIEWED`) | +| [TSL] | `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md` (Math-#166, merged `ab13a08f`) | 17139 · `e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769` · `a32fd5f7d941bbe1fe943df045b1e0fbec8d691c` | OpenAI | in-repo `REVIEW_RECORD.md` (6732 · `af1540308f37220e6592b700bf42181205b1d4588a040a5c151bf0a6f861e728` · `31896de23fc56822c184375d0dc2b1c7e27fcae4`, extended by Math-#172 at `fe2c3ff`) and `REVIEW_RECORD.json` (4226 · `958d2e1ae8ea5d034d0d87efce6cde9be57ebb9d7c0b671e69d3d6598a6675ca` · `a4b0fb99c24e2edf41973ea2a5e5176584602855`: native review 5360227991, reviewed commit `7c82252`, proof `TWO_SCALE_LAW.md`, body SHA256 `c63efc29…`): Claude 5360227991 ACCEPT of Theorem T, Theorem L, (L3), (V1) at the conditional scope; xAI bounded read 5901945424 reported in Math-#160 5902466075 | Theorem L (L1), (L2): the direct statement | + +Identities are checked by `residual_check.py` (`IDENTITIES`, `NEGATIVES`); the quoted statements are checked as +substrings (`VERDICTS`). [SC] and [CL] are conditional on their own pinned inputs at those inputs' reviewed scopes +([SC] `SOURCES.json`: P, D5, C6, RM, RC, LM, CUB, RCL; [CL] `SOURCE_MAP.json`: LP, DL, C6, RM, RC, EDL, LM, RCL; [TSL] +`SOURCES.json`: SC, CUB, RM, C6, P, D5); this record inherits exactly those conditions and adds none. + +## 3. The corollary: (Res) from the reviewed statements + +Write `N_R` for the window count in the Euclidean ball of radius `R r` about `o` ([SC]'s `N_R`; [CL]'s `N^A` with +`A = R`), `N_out = N - N_R`, and `(n)_2 = n(n - 1)`. The quantity in (Res) is exactly + + M_R := E_{Q_r^W} #{ordered pairs of distinct window points not both within R r of o} = E[(N)_2] - E[(N_R)_2], (D1) + +since for every configuration `(N)_2 - (N_R)_2 = 2 N_R N_out + (N_out)_2 >= 2 N_R N_out >= 0` (exact identity, checked +in `DEDUCTION`). So (Res) says `lim_R limsup_r r^-3 (E[(N)_2] - E[(N_R)_2]) = 0`, and the mixed count of Math-#160 §5 is +`M(R, s_0) <= E[N_R N_out] <= M_R / 2`. + +**Route A ([SC], Math-#162).** +1. *Full second factorial moment.* [SC] (26): `E (N)_2 / r^3 -> 2 nu2 = 2 a_2`, a stated consequence of Theorem (3) + with `q = 2` (§8, Slice C). +2. *Near second factorial moment.* [SC] (18): `r^-3 Q_W(N_R = j) -> a_j^R` for `j = 1, 2` and `r^-3 Q_W(N_R >= 3) -> 0` + (§5, Slice B). Tail: `(N_R)_2 <= (N)_2`, and for integers `n > M >= 3`, `n(n-1) <= n(n-1)(n-2)/(M-1)` (exact, `DEDUCTION`), + so `E[(N_R)_2; N_R > M] <= E[(N)_3]/(M-1) <= C_3 r^3/(M-1)` by [PALM] Theorem Q with `q = 3`. Hence + `limsup_r | r^-3 E[(N_R)_2] - 2 a_2^R | <= C_3/(M-1)` for every `M`, i.e. `r^-3 E[(N_R)_2] -> 2 a_2^R` (the finite sum + over `n <= M` converges by (18), its `n >= 3` terms to zero). This is the same tail argument [SC] §8 uses for (3). +3. *Exhaustion.* [SC] (20): `a_2^R -> a_2` as `R -> oo` (dominated convergence with the integrable majorant (19), §6, + Slice B). + +Therefore `limsup_r r^-3 M_R = 2 a_2 - 2 a_2^R` (a full limit, by 1 and 2), which tends to `0` as `R -> oo` by 3. That is +(Res). ∎ + +**Route B ([CL], Math-#159).** Corollary Λ (1.6): `r^-3 E[(N)_2] -> Λ_2 = 2 nu(2) = 2 nu_near(2)`; Proposition 4.4: +`r^-3 Q{N^A = n} -> nu_near^A(n)` for `n = 1, 2`, `r^-3 Q{N^A >= 3} -> 0`, and `nu_near^A(n) -> nu_near(n)` as `A -> oo`; +the tail of step 2 is the one [CL] §6.3 uses ([PALM] Theorem Q). The same three lines give (Res). Directly, without the +factorial-moment limit: Lemma 5.2 (5.4), `E[N^A 1{N_far^rho >= 1}] <= C(A, rho) r^(9/2)`, together with +`E[N^A (N_far^rho - 1)_+] <= (E (N^A)^2)^(1/2) (E N_far^4)^(1/4) Q{N_far^rho >= 2}^(1/4) <= C r^(3/2) r^(3/4) r^(5/4) = C r^(7/2)` +([PALM] Theorem Q for the second and fourth moments through Stirling's identity `n^4 = (n)_4 + 6(n)_3 + 7(n)_2 + (n)_1`, +[RC] Corollary D for `Q{N_far^rho >= 2} <= E (N_far^rho)_2 / 2 = O(r^5)`), gives `E[N^A N_far^rho] = O(r^(7/2)) = o(r^3)` +at fixed `(A, rho)`, which is the mixed count of Math-#160 §5 with `s_0 = rho` (the far region `D_rho` is measured from +`o`; a point at distance `>= s_0` from both pins lies in `D_{s_0/2}` for `r < s_0`). Exponent ledgers checked exactly. + +**Route C (pair-count corollary; Math-#160 comment 5901994240, checked in 5902616692).** With `A`, `B`, `C` the window +counts in `|X| <= R r`, `R r < |X| < rho`, `|X| >= rho` (from `o`; `N = A + B + C`), pathwise + + 0 <= (N)_2 - (A)_2 <= N^2 [ 1{B > 0} + 1{A > 0, C > 0} + 1{C >= 2} ], (D2) + +exact: the right side vanishes only when `B = 0` and either `C = 0` or `A = 0, C <= 1`, and then the left side +`2A(B + C) + (B + C)(B + C - 1)` vanishes too (checked in `DEDUCTION`). Cauchy–Schwarz with `E N^4 <= C r^3` ([PALM] +Theorem Q through Stirling) and [CL] (5.1) (`E B <= C r^3 (R^-2 + rho^2)`, [DL] Theorem I_d, constant uniform in `R >= 4`, +`rho <= s_0`), the left inequality of [CL] (5.4) (`Q{A >= 1, C >= 1} <= C(A, rho) r^(9/2)`) and [RC] Corollary D +(`Q{C >= 2} <= E (C)_2 / 2 = O_rho(r^5)`) give + + r^-3 E[(N)_2 - (A)_2] <= C (R^-2 + rho^2)^(1/2) + C_(R,rho) r^(3/4) + C_rho r ; + +`r -> 0` first, then `rho -> 0`, then `R -> oo` gives (Res). This route uses no count-law limit: only the shell bound, +the cross-term bound, the far second factorial moment and Theorem Q (exponent ledgers `3/2 + 9/4 - 3 = 3/4` and +`3/2 + 5/2 - 3 = 1`, checked exactly). + +**Direct statement ([TSL], merged at `ab13a08f`).** Math-#166 Theorem L (L1): `r^-3 E_r[(N_r)_q - (N_in)_q] -> 0` for every +fixed `q >= 2` and every deterministic cutoff `delta_r` with `delta_r -> 0` and `delta_r / r -> oo` ([TSL] (T1)), `N_in` the +count within `delta_r` of `o`; and (L2): `E_r[N_R N_far^rho] = o_(R,rho)(r^3)`, the mixed count itself. (L1) cannot be +applied at a fixed `R` (`delta_r = R r` has `delta_r / r = R`, which does not tend to infinity); (Res) follows from (L1) by +the diagonal argument of Math-#173 review 5360751376 (OpenAI), reproduced here. Put `f_r(R) := r^-3 E[(N)_2 - (N_R)_2] +>= 0`; it is nonincreasing in `R`, because `N_R` and hence `(N_R)_2` are nondecreasing in `R` (`n -> (n)_2` is nondecreasing +on the nonnegative integers, checked in `DEDUCTION`), so `c := lim_(R -> oo) limsup_(r -> 0) f_r(R)` exists in `[0, oo]`. +Suppose `c > 0` and fix `0 < c' < c`. Since `limsup_(r -> 0) f_r(R) >= c > c'` for every `R`, choose `R_n -> oo` and `r_n` +decreasing to `0` with `r_n R_n <= 1/n` and `f_(r_n)(R_n) > c'`. Set `delta_r := r R_n` for `r in (r_(n+1), r_n]`: then +`delta_r <= 1/n -> 0` and `delta_r / r = R_n -> oo`, so (T1) holds, while `N_in = N_(R_n)` at `r = r_n` gives +`r_n^-3 E[(N)_2 - (N_in)_2] = f_(r_n)(R_n) > c'` along `r_n -> 0`, contradicting (L1) at `q = 2`. Hence `c = 0`, which is +(Res). (L2) is the `A > 0, C > 0` term of Route C, stated as a counted mixed pair. Theorem L is stronger than (Res) (moving +cutoff, all `q`) and is conditional on [SC]. It is bound in §2 and is a required component in §5, so execution step 0 is +satisfied by a reviewed merged source rather than by a read of this record. + +**What the corollary uses and nothing else.** Two reviewed limit statements about one law (the full and the near +second factorial moments), one reviewed exhaustion statement, and one reviewed moment bound for the tail. No +regression, no Kac–Rice step, no new estimate. The direct statement [TSL] and this section are two separate evidence +paths (§5). This section's (D1), Route A and Route B paragraphs were read by OpenAI / GPT-5.6 Sol (Math-#173 review +5360645884, at head `b3fac79`, ACCEPT at the stated scope; the reconciler is source-exposed, §0); those paragraphs are +preserved byte-for-byte in `REVIEWED_SECTION_3_b3fac79.md`, and `REVIEWED` checks that they stand unchanged above. Route C +is OpenAI-authored, Claude-checked (5902616692) and read by OpenAI Astra (Math-#173 review 5360751376 at `5d8f011`, ACCEPT as +an elementary conditional implication of its stated estimates; the paragraph is unchanged since); the Theorem L paragraph +is [TSL]'s own reviewed statement, applied through the diagonal argument above. A revision +that alters the reviewed paragraphs must obtain a new read or return this record's node to `AUTHOR_SIDE_CANDIDATE`. + +## 4. Scope, and what is not established + +**Scope of the proposed `PROVED_REVIEWED`.** Every fixed `d >= 2`, fixed torus `L`, fixed frame, compact birth and +positive-gap marks (`k_- > 0`), the height window `I_r = (b - k r^3, b)`, the original endpoint weight `W_r` and the full +normalizer `Z_r`; existential constants; conditional on the pinned inputs of [SC] and [CL] at their reviewed scopes +(§2). Under this scope the register's chain for the witness-collision region reads: torus-wide `c r^3 <= E N(N-1) <= C r^3` +([PALM] + [EDL] (A4), Math-#160 §4); the leading `r^3` mass of `E N(N-1)` is carried, in the iterated limit, by pairs both +within `R r` of the midpoint ((Res), this record); the coefficient is `2 a_2` ([SC] (26) / [CL] (1.6)); the count law is +`nu(1) delta_1 + nu(2) delta_2` ([SC] Theorem (3) / [CL] Theorem N). + +**Not established by this record.** No rate of convergence; no numerical value of `a_2`, `nu(1)`, `nu(2)`, `C_3` or of +the localization scale; no uniformity as `k -> 0`, in `L` or in `d`; no all-height statement (the window is +load-bearing in both sources); no statement about the elder selection of the extra points; no joint law of clusters +of distinct pin pairs; no configuration law beyond what [SC]/[CL] state (Math-#166 Theorem T supplies the configuration +form; not consumed); no `d_TV` statement (Math-#166 (V1); not consumed); the moving-cutoff form is [TSL] (L1) at its own +scope; no historical numerical certificate (RN annulus partition, +24-jet / OBL-H5-JETMOD) and no discharge of any historical predicate outside the analytic scope (Math-#160 §6). Neither +[SC] nor [CL] is re-reviewed here; their reviews stand at the scopes their records state. + +## 5. Proposed register transitions (for a non-Claude lane) + +| Node | Current (`358f256`) | Proposed | +|---|---|---| +| `math.rn-region.witness-collision.leading-mass-localization` | absent (Math-#160 proposes it `OPEN_ACTIVE`) | `PROVED_REVIEWED` at the §4 scope, with `fingerprint` unchanged from Math-#160's proposal, `scope`, `explicit_limits`, `review_disposition`, `review_basis`, and `required` edges to the component nodes below; created directly if Math-#160's step 1 has not run, moved if it has | +| `math.rn-region.witness-collision` | `OPEN_ACTIVE` (Math-#151 wording) | `PROVED_REVIEWED`, **conditional and deferred**: presupposes Math-#160's aggregate node `math.c6-witness-collision-factorial-moment` (count discharge) and this record's residual node (localization); executes only after both exist, with `required` edges to both. Under Math-#160's keep-open execution (§1a (iii) there), the residual's definition sits on the old node and this record moves the old node itself. If the executing lane holds that the node means something other than the count estimate and (Res), it must write that meaning down; no source defines one (Math-#160 §1a) | +| `SELECTOR_REGION.json`, `PROOF_INDEX.md`, catalog C6, `STATUS.md` C6 row | as at `358f256` | as Math-#160 §7 proposes, with the residual bullet closed: "leading-mass localization: resolved at existential scope by Math-#162 (26)/(18)/(20) and Math-#159 Corollary Λ / Proposition 4.4; direct statement Math-#166 Theorem L (merged at `ab13a08f`); open: rate, constants" | + +**Component nodes** (`kind: reading_rule_component`, `PROVED_REVIEWED`, `controlling: false`, `fingerprint` = SHA256 of the +source; one node per byte identity; none duplicates a source the live graph carries with a fingerprint): + +| Node | Source | Role | Provider | Review basis | +|---|---|---|---|---| +| `math.c6r-component.spectral-closure-proof` | [SC] | proof: Theorem (3), (18), (20), (26) | OpenAI | A 5359968447, B 5359845136, C 5359879627 (in-repo `REVIEW_RECORD.md`) | +| `math.c6r-component.spectral-closure-review-record` | `frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md` | review record | OpenAI-authored record of the Claude/Codex reviews | itself | +| `math.c6r-component.cluster-law-proof` | [CL] | proof: Corollary Λ, Proposition 4.4, Lemma 5.2 | Anthropic (other session) | OpenAI 5360192822 | +| `math.c6r-component.cluster-law-acceptance` | this packet's `EXTERNAL_REVIEWS.md` | external review record (5360192822, transcribed; mutable, labelled) | OpenAI review, transcribed by Anthropic | itself | +| `math.c6r-component.cluster-law-source-map` | `frontiers/c6_cluster_law_20260929/SOURCE_MAP.json` | in-repo acceptance pointer and consumed-input map | Anthropic | records 5360192822 | +| `math.c6-component.palm-proof` | [PALM] | proof: Theorem Q (`q = 3, 4`) | Anthropic (other session) | OpenAI 5358116559; **defined by Math-#160** (same id, source, fingerprint); created here only if absent | +| `math.c6r-component.two-scale-law-proof` | [TSL] | proof: Theorem L (L1), (L2), the direct statement | OpenAI | Claude 5360227991 (in-repo `REVIEW_RECORD.md`); xAI bounded read 5901945424 as reported | +| `math.c6r-component.two-scale-review-record` | `frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md` (Math-#172 bytes) | review record | OpenAI-authored record of the Claude reviews | itself | +| `math.c6r-component.two-scale-review-binding` | `frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json` | machine-readable review binding: native id 5360227991, reviewed commit `7c82252`, proof path and blob `a32fd5f7`, body SHA256 `c63efc29…` | OpenAI-authored record (Math-#172) | itself | +| `math.c6r-component.residual-closure-record` | this packet's `RECONCILIATION.md` | reading-rule record: the §3 corollary ((D1), Routes A, B) and Route C | Anthropic (this session; source-exposed) | **OpenAI / GPT-5.6 Sol 5360645884** (bounded non-Claude read of §3 at `b3fac79`; reviewed text preserved and checked, `REVIEWED`); Route C: OpenAI-authored, checked 5902616692. `PROVED_REVIEWED` on the reviewed-record path since v1.3 | + +Edges: from the residual node to each of the ten components, `required: true` (`requires_evidence`). From the old node +(deferred transition): `required` edges to the residual node and to Math-#160's aggregate node. + +**Two evidence paths** (`PROPOSED_TRANSITIONS.json` `evidence_paths`), each sufficient for (Res) on its own: the **direct +statement** ([TSL] Theorem L with its in-repo review record and JSON binding: three nodes) and the **reviewed corollary** +(§3 from [SC], [CL], [PALM] and this record, read by 5360645884: seven nodes). The residual's required edges take the +union, so that a later byte change on either path holds the residual (fail-closed); `GATE` replays each path alone (the +other path's edges supporting) as well as the union, and replays the negative in which the record node is required +without its read: the gate then holds the residual on exactly that node. + +**Execution order** (`PROPOSED_TRANSITIONS.json` `execution_order`): (0) satisfied twice: at `ab13a08f` +Math-#166's Theorem L is on main as a required component (direct path), and at `b3fac79` OpenAI 5360645884 read §3 +(reviewed-corollary path); (1) the component nodes and their edges; (2) the residual node +(created `PROVED_REVIEWED`, or moved from `OPEN_ACTIVE`); (3) only after Math-#160 step 3: the old node; (4) the +register rows. Nothing here edits `GRAPH.json`, `PROOF_INDEX.md`, `STATUS.md`, `SELECTOR_REGION.json` or the catalog +(`"executed": false`, checked). + +**Reverse-impact.** Every component node carries the SHA256 of its source; `GATE` calls `reverse_impact_between` with exact +source snapshots of both graph states (every node: `record_only`, `external_unresolved`, `blob` with bytes and SHA256, +or `tree`), asserts `source_snapshots_supplied`, and then, for each component in turn, changes only that component's +snapshot digest and checks that the residual node (and the old node, once it depends on the residual) is impacted. A +later byte change to [SC], [CL], [PALM], [TSL], a review record, the source map, the preserved reviews or this record +therefore reaches the residual through the graph, not through the fingerprint strings alone. + +## 6. Checks + +`residual_check.py` (stdlib only, run from the repository root): +- **IDENTITIES.** All thirteen inventoried files exist as regular files, no symlink on their paths, stated SHA256 and blob. +- **VERDICTS.** The quoted statements are present as substrings: [SC] Theorem (3), (18), (20), (24), (26) and the + Slice A/B/C lines of its review record with the core blob; [CL] Corollary Λ with `Λ_2 = 2 nu(2)`, Proposition 4.4's + `A -> oo` sentence, Lemma 5.2 with `r^(9/2)`, the object header, the source map's `nonauthor_acceptance` and its note + naming 5360192822; [PALM] object header and Theorem Q; [TSL] Theorem L with (L1) and (L2), its review record's + 5360227991 line at blob `a32fd5f7`, and the JSON binding's native id, proof path and body digest; the review ids, verdict lines, the `b3fac79` head and the + mutability label in `EXTERNAL_REVIEWS.md`; the heading and the Route A / Route B headers of the reviewed-section file. +- **LIVE.** `math.rn-region.witness-collision` is a live `OPEN_ACTIVE` region with its recorded fingerprint (or + `PROVED_REVIEWED` after execution); the residual node is absent, `OPEN_ACTIVE` with Math-#160's fingerprint, or + `PROVED_REVIEWED`; the selector lists the region. A live node on a component source is accepted only if it is the + proposed node on every proposed key (the palm node on its cross-record keys: kind, source, fingerprint, + classification, controlling); anything else on such a source, or anything else under a proposed id, is a mismatch. If + the residual is live `PROVED_REVIEWED`, every component, every edge and the reading rule must be present as proposed. + The same test runs on the executed graph built from the proposal (installed state, simulated) in every run, so that + execution cannot turn this check red (Codex 4140125558). +- **REVIEWED.** The record node's first review basis is Math-#173 review 5360645884 (provider OpenAI, ACCEPT, head + `b3fac79`); `REVIEWED_SECTION_3_b3fac79.md`, after its one-line provenance comment, has the SHA256 the proposal states, + begins at the §3 heading, contains Route A and Route B and not Route C, and is a verbatim substring of the current + `RECONCILIATION.md`; the review id and head are in `EXTERNAL_REVIEWS.md`. +- **DEDUCTION.** Exact finite content of §3: the pair identity (D1) for all `0 <= N_R, N_out <= 12`; the monotonicity + `(n+1)_2 >= (n)_2` on `n >= 0` used by the diagonal argument of the direct statement; the Route C + pathwise inequality (D2) for all `0 <= A, B, C <= 8` with its two exponent ledgers; the tail + inequality `n(n-1)(M-1) <= n(n-1)(n-2)` for `n > M >= 3`; the Stirling identity for `n^4`; the exponent ledgers + `3 + 3/2 = 9/2 > 3` and `3/2 + 3/4 + 5/4 = 7/2 > 3`; the truncation error of a finite distribution against + `E(N)_3/(M-1)`; and an exact rational instance of the iterated limit `2 a_2 - 2 a_2^R -> 0`. +- **TRANSITIONS.** `declarative` true, `executed` false; the residual node `PROVED_REVIEWED`, non-controlling, with + scope, explicit limits, review basis, Math-#160's fingerprint and a required edge to every component; every + component node fingerprinted to the inventory with role, provider and review basis, one per byte identity; the + every component `PROVED_REVIEWED` with at least one review basis and the residual's required edges covering all ten; + the two evidence paths partition the components, the record node on the corollary path and Theorem L on the direct + path; the old-node transition marked deferred with its two presuppositions; the palm node marked cross-record with + Math-#160's id. The builds are idempotent: installed components and edges are not re-created, and the installed state + rebuilds to itself. +- **GATE.** In every execution shape (interim; final; residual node pre-existing `OPEN_ACTIVE` as Math-#160 proposes; + each evidence path alone; the installed state, simulated; and the negative with the record node required but unread), + the proposed graph passes `validate_graph_fail_closed`, `closure_report` gives `gate_ok` with no illegal controlling + node, `reverse_impact_between` runs with source snapshots of both states and names the residual node and the new + components as changed and impacted (nothing, in the installed state); the residual is not held except in the negative, + where the gate holds it on exactly the record node; a single-component source drift reaches the residual for every + component; the deferred old-node transition is not applied (its presupposition is absent live) and is reported. +- **NEGATIVES.** Delete, one-byte change, symlink and symlinked parent on a temporary copy of the inventory are rejected. + +Fourteen mutants must fail: `allow-symlink`, `no-hash`, `stale-fingerprint`, `drop-required-edge`, `executed-flag`, +`controlling-true`, `tail-reversed`, `pair-identity-broken`, `route-c-broken`, `drop-review-needle`, and since v1.3 +`installed-drift` (an installed component with another fingerprint must fail `LIVE`), `snapshot-omitted` (the +reverse-impact replay without snapshots must fail `GATE`), `record-unreviewed` (the record node `PROVED_REVIEWED` with no +review basis must fail `TRANSITIONS` and `REVIEWED`), `reviewed-text-drift` (a changed reviewed text must fail `REVIEWED`). + +## 7. Relation to other lanes + +- **Math-#160 (this session):** defines (Res) and proposes the residual node `OPEN_ACTIVE`; this record is its successor + and composes with either execution of its §7. Math-#160 is not changed by this record; the integrator refreshed its + branch onto `fe2c3ff` (`dd4f4f8`) with the six packet blobs preserved and merged it at `8d48d02` (packet and workflow + only; its transitions remain declarative and unexecuted, so the deferred old-node transition here still presupposes + its step 3). +- **Math-#173 reviews (on v1.0 `b3fac79`):** OpenAI/Codex 5360636576, three findings: (i) `LIVE` rejected the installed + components (4140125558), repaired: the installed state is accepted when exact and simulated in every run; (ii) the + reverse-impact replay ran without source snapshots (4140125562), repaired: snapshots of both states and a + per-component drift test; (iii) the `final` build promoted this record without a read on the Math-#166 path + (4140125565), repaired: the two paths are modelled separately, the record node's promotion rests on 5360645884, and + the unread case is replayed as a negative. OpenAI / GPT-5.6 Sol 5360645884: ACCEPT of §3 at the stated scope with the + amend list above; the record node is promoted on that reviewed-record path, the #166 path stays a distinct alternative + with `TWO_SCALE_LAW.md` and its exact review binding as evidence, and the "pending" wording is gone. +- **Math-#173 review 5360751376 (OpenAI / GPT-6 Astra Pro, at `5d8f011`):** ACCEPT of Route C as an elementary conditional + implication of its stated shell, mixed-event, remote-second-moment and fourth-moment estimates (exposure: author of + Math-#162, contributor to Math-#166/#169). Its narrow amendment to the direct-statement paragraph (no fixed-`R` + substitution into (L1); the diagonal argument) is applied in v1.5, as the integrator's hold 5902847683 / 5902997395 / + 5907324679 asked; everything else in that hold was closed by v1.3 and v1.4. +- **Math-#162 (OpenAI; merged `358f256`, integrated by OpenAI Sol):** Route A. Not re-reviewed; its review record is + bound in-repo. +- **Math-#159 (Anthropic, other session; merged `98fdb54`, integrated by the OpenAI engineering lane):** Route B. Not + re-reviewed; its acceptance is preserved as external evidence. +- **Math-#166 (OpenAI; merged at `ab13a08f`, integrated by OpenAI Sol / C25):** the direct statement, bound in v1.1; not + re-reviewed. +- **Math-#167 (this session; merged at `fa1cf7b` by OpenAI Sol after review 5360524053):** disjoint (D2/D3/D4/D5-annulus/D6 + alignment); its execution is a separate non-Claude act. +- **Math-#172 (merged at `fe2c3ff`):** custody repair of the two-scale packet, binding its reviews to exact proof bytes; + the review record and its JSON binding are bound here at those bytes (v1.2); the proof bytes did not change. +- **Math-#151 (merged `ec6db8c`):** kept the old node open under the wording quoted in §1; this record supplies the + discharge of the only defined residual meaning. diff --git a/reviews/c6_residual_closure_20260930/RESULTS.json b/reviews/c6_residual_closure_20260930/RESULTS.json new file mode 100644 index 000000000..986a13200 --- /dev/null +++ b/reviews/c6_residual_closure_20260930/RESULTS.json @@ -0,0 +1,232 @@ +{ + "checks": { + "DEDUCTION": true, + "GATE": true, + "IDENTITIES": true, + "LIVE": true, + "NEGATIVES": true, + "REVIEWED": true, + "TRANSITIONS": true, + "VERDICTS": true + }, + "component_nodes": 10, + "gate": { + "final": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "final-corollary-only": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "final-direct-only": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "final-preexisting": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "final-record-unreviewed": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": true, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "installed-simulated": { + "changed_nodes": [], + "impacted": [], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "PROVED_REVIEWED", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + }, + "interim": { + "changed_nodes": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "impacted": [ + "math.c6-component.palm-proof", + "math.c6r-component.cluster-law-acceptance", + "math.c6r-component.cluster-law-proof", + "math.c6r-component.cluster-law-source-map", + "math.c6r-component.residual-closure-record", + "math.c6r-component.spectral-closure-proof", + "math.c6r-component.spectral-closure-review-record", + "math.c6r-component.two-scale-law-proof", + "math.c6r-component.two-scale-review-binding", + "math.c6r-component.two-scale-review-record", + "math.rn-region.witness-collision.leading-mass-localization" + ], + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)", + "residual_classification": "OPEN_ACTIVE", + "residual_held": false, + "source_drift_reaches_residual": 10, + "source_snapshots_supplied": true + } + }, + "inventory_files": 13, + "live_state": { + "edges_present": false, + "mismatched": [], + "present": [], + "residual": null + }, + "object": "C6-RESIDUAL-CLOSURE-20260930-v1", + "passed": true, + "scope": "identity, verdict-row, live-register (including the installed state), reviewed-text, exact finite bookkeeping of section 3, transition-shape, hard-gate replay with source snapshots in every execution shape, and filesystem-negative checks; the Gaussian limits are the sources' reviewed statements and are not re-proved; nothing is written" +} diff --git a/reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md b/reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md new file mode 100644 index 000000000..2fff7c277 --- /dev/null +++ b/reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md @@ -0,0 +1,36 @@ + + +## 3. The corollary: (Res) from the reviewed statements + +Write `N_R` for the window count in the Euclidean ball of radius `R r` about `o` ([SC]'s `N_R`; [CL]'s `N^A` with +`A = R`), `N_out = N - N_R`, and `(n)_2 = n(n - 1)`. The quantity in (Res) is exactly + + M_R := E_{Q_r^W} #{ordered pairs of distinct window points not both within R r of o} = E[(N)_2] - E[(N_R)_2], (D1) + +since for every configuration `(N)_2 - (N_R)_2 = 2 N_R N_out + (N_out)_2 >= 2 N_R N_out >= 0` (exact identity, checked +in `DEDUCTION`). So (Res) says `lim_R limsup_r r^-3 (E[(N)_2] - E[(N_R)_2]) = 0`, and the mixed count of Math-#160 §5 is +`M(R, s_0) <= E[N_R N_out] <= M_R / 2`. + +**Route A ([SC], Math-#162).** +1. *Full second factorial moment.* [SC] (26): `E (N)_2 / r^3 -> 2 nu2 = 2 a_2`, a stated consequence of Theorem (3) + with `q = 2` (§8, Slice C). +2. *Near second factorial moment.* [SC] (18): `r^-3 Q_W(N_R = j) -> a_j^R` for `j = 1, 2` and `r^-3 Q_W(N_R >= 3) -> 0` + (§5, Slice B). Tail: `(N_R)_2 <= (N)_2`, and for integers `n > M >= 3`, `n(n-1) <= n(n-1)(n-2)/(M-1)` (exact, `DEDUCTION`), + so `E[(N_R)_2; N_R > M] <= E[(N)_3]/(M-1) <= C_3 r^3/(M-1)` by [PALM] Theorem Q with `q = 3`. Hence + `limsup_r | r^-3 E[(N_R)_2] - 2 a_2^R | <= C_3/(M-1)` for every `M`, i.e. `r^-3 E[(N_R)_2] -> 2 a_2^R` (the finite sum + over `n <= M` converges by (18), its `n >= 3` terms to zero). This is the same tail argument [SC] §8 uses for (3). +3. *Exhaustion.* [SC] (20): `a_2^R -> a_2` as `R -> oo` (dominated convergence with the integrable majorant (19), §6, + Slice B). + +Therefore `limsup_r r^-3 M_R = 2 a_2 - 2 a_2^R` (a full limit, by 1 and 2), which tends to `0` as `R -> oo` by 3. That is +(Res). ∎ + +**Route B ([CL], Math-#159).** Corollary Λ (1.6): `r^-3 E[(N)_2] -> Λ_2 = 2 nu(2) = 2 nu_near(2)`; Proposition 4.4: +`r^-3 Q{N^A = n} -> nu_near^A(n)` for `n = 1, 2`, `r^-3 Q{N^A >= 3} -> 0`, and `nu_near^A(n) -> nu_near(n)` as `A -> oo`; +the tail of step 2 is the one [CL] §6.3 uses ([PALM] Theorem Q). The same three lines give (Res). Directly, without the +factorial-moment limit: Lemma 5.2 (5.4), `E[N^A 1{N_far^rho >= 1}] <= C(A, rho) r^(9/2)`, together with +`E[N^A (N_far^rho - 1)_+] <= (E (N^A)^2)^(1/2) (E N_far^4)^(1/4) Q{N_far^rho >= 2}^(1/4) <= C r^(3/2) r^(3/4) r^(5/4) = C r^(7/2)` +([PALM] Theorem Q for the second and fourth moments through Stirling's identity `n^4 = (n)_4 + 6(n)_3 + 7(n)_2 + (n)_1`, +[RC] Corollary D for `Q{N_far^rho >= 2} <= E (N_far^rho)_2 / 2 = O(r^5)`), gives `E[N^A N_far^rho] = O(r^(7/2)) = o(r^3)` +at fixed `(A, rho)`, which is the mixed count of Math-#160 §5 with `s_0 = rho` (the far region `D_rho` is measured from +`o`; a point at distance `>= s_0` from both pins lies in `D_{s_0/2}` for `r < s_0`). Exponent ledgers checked exactly. diff --git a/reviews/c6_residual_closure_20260930/SOURCE_FILES.json b/reviews/c6_residual_closure_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..692bb2b78 --- /dev/null +++ b/reviews/c6_residual_closure_20260930/SOURCE_FILES.json @@ -0,0 +1,47 @@ +{ + "schema": 1, + "object": "C6-RESIDUAL-CLOSURE-20260930-v1", + "scientific_effect": "NONE", + "disposition": "PROPOSAL_FOR_A_NON_CLAUDE_LANE", + "declarative": true, + "executed": false, + "base": "358f2562efbccc59f9549e5a79650e2e936395c3", + "files": [ + { + "path": "EXTERNAL_REVIEWS.md", + "bytes": 14108, + "sha256": "27b4d8a5f04981e7b01b20f767078df87b4d0c75aebc6f6ed665347a8b605a53", + "git_blob": "70d05080ed81c66b43261438cae3b9ec2840e637" + }, + { + "path": "PROPOSED_TRANSITIONS.json", + "bytes": 31461, + "sha256": "78a50b82fd9cbf04899aa5ddc62875850ce68b8b5a5babfb637b7db7e2a49c5c", + "git_blob": "3706e8ac4e906aa1b53cdf5634abfb324c35a9e3" + }, + { + "path": "RECONCILIATION.md", + "bytes": 34758, + "sha256": "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", + "git_blob": "3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c" + }, + { + "path": "RESULTS.json", + "bytes": 10376, + "sha256": "3c61cc49155cae7bba0fec30c0bf0a9fb085340f3a161d5898c527b87f9b5805", + "git_blob": "986a1320098b37238f117280b7925c683ca7c3c2" + }, + { + "path": "residual_check.py", + "bytes": 33664, + "sha256": "76ebda22aef26a70ab8ef5916e64bc7c5b3b34833fa702664fa00a7b4c1c4406", + "git_blob": "2667d4b35baa674c74c051244febb7f08118c572" + }, + { + "path": "REVIEWED_SECTION_3_b3fac79.md", + "bytes": 3157, + "sha256": "50daacbe7e244bc149dbf4568970d79f38a356daa1e88a850f2fc2d074db5b6c", + "git_blob": "2fff7c277b43fa8e967ce9a8d70cf01a69e5eee4" + } + ] +} diff --git a/reviews/c6_residual_closure_20260930/residual_check.py b/reviews/c6_residual_closure_20260930/residual_check.py new file mode 100644 index 000000000..2667d4b35 --- /dev/null +++ b/reviews/c6_residual_closure_20260930/residual_check.py @@ -0,0 +1,641 @@ +"""Checks for C6-RESIDUAL-CLOSURE-20260930-v1 (RECONCILIATION.md, PROPOSED_TRANSITIONS.json, EXTERNAL_REVIEWS.md, +REVIEWED_SECTION_3_b3fac79.md). + +Standard library only. Run from the repository root: python -B -S reviews/c6_residual_closure_20260930/residual_check.py +Checks: IDENTITIES, VERDICTS, LIVE, REVIEWED, DEDUCTION, TRANSITIONS, GATE, NEGATIVES. No mathematics is re-proved beyond the +exact finite bookkeeping of section 3 (pair identities, tails, ledgers); the proposed graph is replayed through the downstream +hard gate's own validators in every execution shape (interim, final, pre-existing residual node, each evidence path alone, the +unreviewed-record negative, and the installed state), with source snapshots. Nothing is written to the repository. +""" +import argparse +import copy +import hashlib +import importlib.util +import json +import os +import pathlib +import shutil +import sys +import tempfile +from fractions import Fraction as F + +MUTANTS = ("allow-symlink", "no-hash", "stale-fingerprint", "drop-required-edge", "executed-flag", "controlling-true", + "tail-reversed", "pair-identity-broken", "route-c-broken", "drop-review-needle", + "installed-drift", "snapshot-omitted", "record-unreviewed", "reviewed-text-drift") +MUT = None +HERE = "reviews/c6_residual_closure_20260930" +GRAPH = "frontiers/downstream_gate_20260925/GRAPH.json" +SELECTOR = "frontiers/downstream_gate_20260925/SELECTOR_REGION.json" +HARD_GATE = "frontiers/downstream_gate_20260925/hard_gate.py" +RES = "math.rn-region.witness-collision.leading-mass-localization" +WIT = "math.rn-region.witness-collision" +AGG = "math.c6-witness-collision-factorial-moment" +RECORD_NODE = "math.c6r-component.residual-closure-record" +PALM_NODE = "math.c6-component.palm-proof" +SC = "frontiers/spectral_cluster_closure_20260929/PROOF.md" +REVIEWED = HERE + "/REVIEWED_SECTION_3_b3fac79.md" +REVIEWED_HEAD = "b3fac79875f28bacd135c0aa5e65a47a41ae0fdf" +SOL_REVIEW = "Math-#173 review 5360645884" +FP_RES = "leading-order r^3 mass of E N(N-1): scale-r localization; open piece = mixed local/remote pairs M(R, s_0)" +FP_WIT = "eta->0 mutual witness separation" +INVENTORY = { + "frontiers/spectral_cluster_closure_20260929/PROOF.md": [ + "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "16c56821b52fd76b0be791622b9c3809eafde75a" + ], + "frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md": [ + "6031d36697789097185a8e65bfbdf80f447e229689eac5a706ee8b0558db64a5", + "9bcae183fecdf0ffe4d612a209438a4fb1fa0edd" + ], + "frontiers/spectral_cluster_closure_20260929/README.md": [ + "b1e2a0722945a6189d8338c2e5b1f1a74b3b5646b8dc10fe29168c165508342c", + "895dded8aa8e405ad8e9640d3ffebc6449a108d0" + ], + "frontiers/c6_cluster_law_20260929/PROOF.md": [ + "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "ba492c8e62e58bfc055fc8254c790e893d346ba5" + ], + "frontiers/c6_cluster_law_20260929/README.md": [ + "e4b35f2b0e919ba57ebfb121c403896a32782d58dd99d7ec6a3d6d9e5107c6ac", + "1166376f7b4d917b1bda3f9438adfd94d8831139" + ], + "frontiers/c6_cluster_law_20260929/SOURCE_MAP.json": [ + "0584c3808b633fceb3c308186b07d862eb4916053daf4e7bf1110592ec41f107", + "ef51c582d0b1c874aabb01fce3b35c10631f7fc6" + ], + "frontiers/c6_palm_route_20260929/PROOF.md": [ + "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5" + ], + "reviews/c6_residual_closure_20260930/EXTERNAL_REVIEWS.md": [ + "27b4d8a5f04981e7b01b20f767078df87b4d0c75aebc6f6ed665347a8b605a53", + "70d05080ed81c66b43261438cae3b9ec2840e637" + ], + "reviews/c6_residual_closure_20260930/RECONCILIATION.md": [ + "b647a186d0cfd1662fc69e9d31dfd447dbb1b2f6ef4364bbd59c4956f18aed25", + "3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c" + ], + "reviews/c6_residual_closure_20260930/REVIEWED_SECTION_3_b3fac79.md": [ + "50daacbe7e244bc149dbf4568970d79f38a356daa1e88a850f2fc2d074db5b6c", + "2fff7c277b43fa8e967ce9a8d70cf01a69e5eee4" + ], + "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md": [ + "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c" + ], + "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md": [ + "af1540308f37220e6592b700bf42181205b1d4588a040a5c151bf0a6f861e728", + "31896de23fc56822c184375d0dc2b1c7e27fcae4" + ], + "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json": [ + "958d2e1ae8ea5d034d0d87efce6cde9be57ebb9d7c0b671e69d3d6598a6675ca", + "a4b0fb99c24e2edf41973ea2a5e5176584602855" + ] +} +VERDICTS = { + "frontiers/spectral_cluster_closure_20260929/PROOF.md": [ + "sum_(n>=1) n^q |r^-3 Q_r^W(N_r=n)-nu1*1{n=1}-nu2*1{n=2}| -> 0", + "r^-3 Q_W(N_R=j) -> a_j^R=integral 1{n_R=j}dM", + "Dominated convergence proves a_j^R->a_j", + "Q_W(N_R>0,N_far^rho>0)=o_(R,rho)(r^3)", + "E (N)_2/r^3 -> 2nu2"], + "frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md": [ + "Git blob16c56821b52fd76b0be791622b9c3809eafde75a", + "A — Anthropic Claude5359968447", + "B — Anthropic Claude5359845136", + "C — OpenAI/Codex5359879627", + "ACCEPT Sections7-8, compact-origin-jet far conditioning, full mixed o(r^3), radius"], + "frontiers/spectral_cluster_closure_20260929/README.md": [ + "Core conditional proof has completed scoped nonauthor A/B/C acceptance."], + "frontiers/c6_cluster_law_20260929/PROOF.md": [ + "**Corollary Lambda (asymptotic factorial moments).**", + "Lambda_2 = 2 nu(2)", + "Moreover `nu_near^A(n) -> nu_near(n)` as `A -> oo`", + "**Lemma 5.2 (no near–far cross term).**", + "C(A, rho) r^(9/2)"], + "frontiers/c6_cluster_law_20260929/README.md": ["# C6 cluster law (CL-C6-CLUSTER-LAW-20260929-v1.2)"], + "frontiers/c6_cluster_law_20260929/SOURCE_MAP.json": ["nonauthor_acceptance", "review 5360192822"], + "frontiers/c6_palm_route_20260929/PROOF.md": [ + "**Object:** CL-C6-PALM-20260929-v1.", + "**Theorem Q (factorial moments, every fixed dimension).**"], + HERE + "/EXTERNAL_REVIEWS.md": [ + "pullrequestreview-5360192822", + "VERDICT: ACCEPT Theorem N, Corollaries Lambda/S", + "A pull-request review is a mutable external object", + "pullrequestreview-5360645884", + "MATHEMATICAL VERDICT ON §3: ACCEPT at the stated fixed-d, fixed-torus, compact-positive-gap window scope.", + REVIEWED_HEAD], + HERE + "/RECONCILIATION.md": ["**Object:** C6-RESIDUAL-CLOSURE-20260930-v1.", "(Res)", "(D1)", "(D2)", "5360645884"], + REVIEWED: ["## 3. The corollary: (Res) from the reviewed statements", "(D1)", + "**Route A ([SC], Math-#162).**", "**Route B ([CL], Math-#159).**"], + "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md": [ + "**Theorem L (factorial localization).** For every fixed integer q>=2,", + "r^-3 E_r[(N_r)_q-(N_in)_q] -> 0.", + "E_r[N_R N_far^rho]=o_(R,rho)(r^3),"], + "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md": [ + "Anthropic Claude review5360227991", + "bloba32fd5f7d941bbe1fe943df045b1e0fbec8d691c", + "ACCEPT of Theorem T, Theorem L, ordered-pair law L3"], + "frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.json": [ + "\"native_review_id\": 5360227991", + "\"proof_path\": \"TWO_SCALE_LAW.md\"", + "\"body_sha256\": \"c63efc292d7cf209e2770419cc159276897d0ba955c32ee1d2af88c8aa7f0ed8\"", + "\"git_blob\": \"a32fd5f7d941bbe1fe943df045b1e0fbec8d691c\""], +} +EXTERNAL_PREFIXES = ("https://", "http://", "external:") + + +def git_blob(data): + return hashlib.sha1(b"blob %d\0" % len(data) + data).hexdigest() + + +def no_symlink_on_path(root, path): + parts = pathlib.PurePosixPath(path).parts + return not any((root / pathlib.PurePosixPath(*parts[:i])).is_symlink() for i in range(1, len(parts) + 1)) + + +def identities_ok(root): + for path, (sha, blob) in INVENTORY.items(): + p = root / path + if not os.path.lexists(p): + return False + if MUT != "allow-symlink" and not no_symlink_on_path(root, path): + return False + if not p.is_file(): + return False + data = p.read_bytes() + if MUT == "no-hash": + continue + if hashlib.sha256(data).hexdigest() != sha or git_blob(data) != blob: + return False + return True + + +def check_verdicts(root): + needles = {k: list(v) for k, v in VERDICTS.items()} + if MUT == "drop-review-needle": + needles[SC].append("THIS NEEDLE IS NOT IN THE SOURCE") + for path, ns in needles.items(): + text = (root / path).read_text(encoding="utf-8") + if not all(n in text for n in ns): + return False + return True + + +def load_spec(root): + spec = json.loads((root / HERE / "PROPOSED_TRANSITIONS.json").read_text(encoding="utf-8")) + if MUT == "executed-flag": + spec["executed"] = True + if MUT == "controlling-true": + spec["transitions"][0]["proposed"]["controlling"] = True + if MUT == "stale-fingerprint": + for n in spec["proposed_graph_nodes"]: + if n["id"] == PALM_NODE: + n["fingerprint"] = "0" * 64 + if MUT == "drop-required-edge": + spec["proposed_graph_edges"] = [e for e in spec["proposed_graph_edges"] + if not (e["from"] == RES and e["to"] == "math.c6r-component.spectral-closure-proof")] + if MUT == "record-unreviewed": + for n in spec["proposed_graph_nodes"]: + if n["id"] == RECORD_NODE: + n["review_basis"] = [] # a PROVED_REVIEWED record with no nonauthor read must be rejected + return spec + + +def load_live(root): + graph = json.loads((root / GRAPH).read_text(encoding="utf-8")) + sel = json.loads((root / SELECTOR).read_text(encoding="utf-8")) + return graph, sel + + +def component_keys(comp): + """Keys a live node must reproduce to count as the proposed component. The palm node is cross-record (Math-#160 + defines its other fields); every node of this record must reproduce the proposal byte-for-byte on every proposed key.""" + if comp["id"] == PALM_NODE: + return ("kind", "source", "fingerprint", "classification", "controlling") + return tuple(k for k in comp if k not in ("id", "create_if_absent", "cross_record")) + + +def matches_proposal(live, comp): + return isinstance(live, dict) and all(live.get(k) == comp[k] for k in component_keys(comp)) + + +def installed_state(graph, spec): + """How much of the proposal the live graph already carries (Codex 4140125558). Fail-closed: a live node on a + component source that is not the proposed node on every proposed key is a mismatch, as is a proposed id carrying + anything else.""" + nodes = graph["nodes"] + by_source = {c["source"]: c for c in spec["proposed_graph_nodes"]} + present, mismatched = set(), set() + for nid, n in nodes.items(): + if not isinstance(n, dict) or n.get("source") not in by_source: + continue + comp = by_source[n["source"]] + (present if nid == comp["id"] and matches_proposal(n, comp) else mismatched).add(nid) + for c in spec["proposed_graph_nodes"]: + if c["id"] in nodes and c["id"] not in present: + mismatched.add(c["id"]) + res = nodes.get(RES) + live_edges = {(e["from"], e["to"], e["required"], e["relation"]) for e in graph["edges"]} + wanted = {(e["from"], e["to"], e["required"], e["relation"]) for e in spec["proposed_graph_edges"] + if not e.get("deferred_with")} + return {"present": sorted(present), "mismatched": sorted(mismatched), + "residual": res.get("classification") if isinstance(res, dict) else None, + "edges_present": wanted <= live_edges} + + +def live_ok(graph, sel, spec): + nodes = graph["nodes"] + wit = nodes.get(WIT) + ok = isinstance(wit, dict) and wit.get("kind") == "region" and wit.get("fingerprint") == FP_WIT + ok &= wit.get("classification") in ("OPEN_ACTIVE", "PROVED_REVIEWED") + res = nodes.get(RES) + if res is not None: + ok &= res.get("kind") == "region" and res.get("fingerprint") == FP_RES + ok &= res.get("classification") in ("OPEN_ACTIVE", "PROVED_REVIEWED") + ok &= "witness-collision" in sel.get("regions", []) + st = installed_state(graph, spec) + ok &= not st["mismatched"] + if st["residual"] == "PROVED_REVIEWED": + # a promoted residual must carry the whole proposal: every component by id and bytes, every edge, the reading rule + ok &= set(c["id"] for c in spec["proposed_graph_nodes"]) <= set(st["present"]) and st["edges_present"] + ok &= set(res.get("reading_rule", [])) >= set(spec["transitions"][0]["proposed"]["reading_rule"]) + return bool(ok), st + + +def check_live(root, spec): + graph, sel = load_live(root) + return live_ok(graph, sel, spec) + + +def reviewed_block(data): + """The reviewed text: the file after its one-line provenance comment.""" + if data.startswith(b"\n\n") + return rest if sep else b"" + return data + + +def check_reviewed(root, spec): + """The record node is PROVED_REVIEWED on the strength of a non-Claude read of section 3 (Math-#173 review 5360645884 + at head b3fac79): the paragraphs that review read are preserved verbatim, fingerprinted in the proposal, and present + unchanged in the current RECONCILIATION.md; the review's identity is preserved in EXTERNAL_REVIEWS.md.""" + rec = next((c for c in spec["proposed_graph_nodes"] if c["id"] == RECORD_NODE), None) + if rec is None or not rec.get("review_basis"): + return False + b0 = rec["review_basis"][0] + ok = rec["classification"] == "PROVED_REVIEWED" and b0.get("review") == SOL_REVIEW + ok &= str(b0.get("provider", "")).startswith("OpenAI") and "ACCEPT" in str(b0.get("verdict", "")) + ok &= b0.get("reviewed_head") == REVIEWED_HEAD and b0.get("reviewed_text") == REVIEWED + block = reviewed_block((root / REVIEWED).read_bytes()) + if MUT == "reviewed-text-drift": + block = block[:-1] + b"?" + ok &= len(block) > 1000 and hashlib.sha256(block).hexdigest() == b0.get("reviewed_text_sha256") + ok &= block.startswith(b"## 3. ") and b"**Route A ([SC], Math-#162).**" in block + ok &= b"**Route B ([CL], Math-#159).**" in block and b"Route C" not in block # the read does not cover Route C + ok &= block in (root / HERE / "RECONCILIATION.md").read_bytes() # verbatim in the current record + ext = (root / HERE / "EXTERNAL_REVIEWS.md").read_text(encoding="utf-8") + ok &= "pullrequestreview-5360645884" in ext and REVIEWED_HEAD in ext + return bool(ok) + + +def check_deduction(): + """Exact finite bookkeeping of RECONCILIATION.md section 3.""" + ok = True + f2 = lambda n: n * (n - 1) + f3 = lambda n: n * (n - 1) * (n - 2) + # (D1): (N)_2 - (N_R)_2 = 2 N_R N_out + (N_out)_2 >= 2 N_R N_out >= 0 + for nr in range(13): + for no in range(13): + rhs = 2 * nr * no + (no * no if MUT == "pair-identity-broken" else f2(no)) + ok &= f2(nr + no) - f2(nr) == rhs and rhs >= 2 * nr * no >= 0 + # monotonicity used by the diagonal argument of the direct statement: n -> (n)_2 is nondecreasing on n >= 0 + ok &= all(f2(n + 1) >= f2(n) for n in range(0, 41)) + # tail: for integers n > M >= 3, n(n-1) <= n(n-1)(n-2)/(M-1) + for M in range(3, 11): + for n in range(M + 1, 41): + lhs, rhs = f2(n) * (M - 1), f3(n) + ok &= (lhs >= rhs) if MUT == "tail-reversed" else (lhs <= rhs) + # (D2), Route C: 0 <= (N)_2 - (A)_2 <= N^2 [1{B>0} + 1{A>0, C>0} + 1{C>=2}] for all counts A, B, C + for a in range(9): + for b in range(9): + for c in range(9): + n = a + b + c + ind = (1 if b > 0 else 0) + (1 if (a > 0 and c > 0) else 0) + (0 if MUT == "route-c-broken" else (1 if c >= 2 else 0)) + ok &= 0 <= f2(n) - f2(a) <= n * n * ind + ok &= F(3, 2) + F(9, 4) - 3 == F(3, 4) and F(3, 2) + F(5, 2) - 3 == F(1) + # Stirling: n^4 = (n)_4 + 6 (n)_3 + 7 (n)_2 + (n)_1 + for n in range(0, 21): + ok &= n ** 4 == n * (n - 1) * (n - 2) * (n - 3) + 6 * f3(n) + 7 * f2(n) + n + # exponent ledgers of the two direct routes + ok &= F(3) + F(3, 2) == F(9, 2) and F(9, 2) > 3 + ok &= F(3, 2) + F(3, 4) + F(5, 4) == F(7, 2) and F(7, 2) > 3 + # truncation error of a finite distribution against E(N)_3 / (M - 1) + probs = [F(1, 2), F(1, 4), F(1, 8), F(1, 16), F(1, 32), F(1, 64), F(1, 128), F(1, 256), F(1, 256)] + assert sum(probs) == 1 + e3 = sum(p * f3(n) for n, p in enumerate(probs)) + for M in (3, 4, 5, 6): + tail = sum(p * f2(n) for n, p in enumerate(probs) if n > M) + ok &= tail <= e3 / (M - 1) + # iterated limit, exact rational instance: a_2 = 1, a_2^R = 1 - 1/R^2, limsup_r r^-3 M_R = 2 a_2 - 2 a_2^R = 2/R^2 -> 0 + vals = [2 - 2 * (1 - F(1, R * R)) for R in range(4, 40)] + ok &= all(v == F(2, R * R) for v, R in zip(vals, range(4, 40))) + ok &= all(vals[i] > vals[i + 1] for i in range(len(vals) - 1)) and vals[-1] < F(1, 500) + return bool(ok) + + +def evidence_paths(spec): + ep = spec["evidence_paths"] + return list(ep["direct_statement"]["components"]), list(ep["reviewed_corollary"]["components"]) + + +def build(spec, graph, stage): + """Apply the proposal to a copy of the live graph, idempotently: a component the live graph already carries + byte-for-byte is not re-created, an edge already present is not re-added. stage: 'interim' (before step 2: residual + node OPEN_ACTIVE with its evidence attached), 'final' (after step 2: residual PROVED_REVIEWED, both evidence paths + required), 'final-preexisting' (Math-#160 step 1 already created the residual node OPEN_ACTIVE), 'final-direct-only' + and 'final-corollary-only' (the other path's edges supporting: each path alone), 'final-record-unreviewed' (the + record node without its read but still required: the residual must be held). Returns (old, new).""" + old = copy.deepcopy(graph) + if stage == "final-preexisting": + old["nodes"][RES] = {"layer": "D5", "kind": "region", "classification": "OPEN_ACTIVE", "controlling": False, + "fingerprint": FP_RES, "notes": "Math-#160 execution_order step 1"} + new = copy.deepcopy(old) + present = set(installed_state(old, spec)["present"]) + live_by_source = {n.get("source"): nid for nid, n in old["nodes"].items() + if isinstance(n, dict) and n.get("fingerprint") and n.get("source")} + id_map = {} + for c in spec["proposed_graph_nodes"]: + nid = c["id"] + if nid in present: + id_map[nid] = nid # already installed as proposed + continue + if c.get("create_if_absent") and c["source"] in live_by_source: + id_map[nid] = live_by_source[c["source"]] # cross-record node already created by Math-#160 + continue + new["nodes"][nid] = {k: v for k, v in c.items() if k != "id"} + id_map[nid] = nid + t = spec["transitions"][0] + p = t["proposed"] + res = dict(old["nodes"].get(RES) or {}) + res.update({"layer": p["layer"], "kind": "region", + "classification": p["classification"] if stage.startswith("final") else "OPEN_ACTIVE", + "controlling": p["controlling"], "fingerprint": p["fingerprint"], "scope": p["scope"], + "explicit_limits": p["explicit_limits"], "review_disposition": p["review_disposition"], + "review_sources": [r["ref"] for r in p["review_sources"]], + "reading_rule": [id_map.get(x, x) for x in p["reading_rule"]], "notes": p["notes"]}) + new["nodes"][RES] = res + have = {(e["from"], e["to"], e["required"], e["relation"]) for e in new["edges"]} + added = [] + for e in spec["proposed_graph_edges"]: + if e.get("deferred_with"): + continue # the old-node transition presupposes Math-#160 step 3 + key = (e["from"], id_map.get(e["to"], e["to"]), e["required"], e["relation"]) + if key not in have: + added.append({"from": key[0], "to": key[1], "required": key[2], "relation": key[3]}) + have.add(key) + direct, corollary = evidence_paths(spec) + if stage == "final-direct-only": + for e in added: + if e["from"] == RES and e["to"] in corollary: + e["required"] = False + if stage == "final-corollary-only": + for e in added: + if e["from"] == RES and e["to"] in direct: + e["required"] = False + if stage == "final-record-unreviewed" and RECORD_NODE not in present: + new["nodes"][RECORD_NODE]["classification"] = "AUTHOR_SIDE_CANDIDATE" + new["nodes"][RECORD_NODE]["review_basis"] = [] + new["edges"].extend(added) + return old, new + + +def check_transitions(root, spec): + graph, sel = load_live(root) + ok = spec.get("declarative") is True and spec.get("executed") is False + t0, t1 = spec["transitions"] + p = t0["proposed"] + ok &= t0["node"] == RES and t0["kind"] == "region" and p["classification"] == "PROVED_REVIEWED" + ok &= p["controlling"] is False and p["fingerprint"] == FP_RES + ok &= bool(p.get("scope")) and bool(p.get("explicit_limits")) and len(p.get("review_sources", [])) >= 2 + ok &= bool(p.get("notes")) and p.get("review_disposition") == "ACCEPT_AT_EXISTENTIAL_SCOPE" + comps = spec["proposed_graph_nodes"] + cids = [c["id"] for c in comps] + ok &= len(cids) == len(set(cids)) and len({c["source"] for c in comps}) == len(comps) + for c in comps: + ok &= c["kind"] == "reading_rule_component" and c["controlling"] is False + ok &= c["source"] in INVENTORY and c["fingerprint"] == INVENTORY[c["source"]][0] + ok &= bool(c.get("component_role")) and bool(c.get("author_provider")) and len(c.get("review_basis", [])) >= 1 + ok &= c["component_of"] == RES and c["classification"] == "PROVED_REVIEWED" + if c["id"] == PALM_NODE: + ok &= c.get("create_if_absent") is True and "Math-#160" in str(c.get("cross_record")) + # two evidence paths, each sufficient, partitioning the components; the residual requires the union (fail-closed) + direct, corollary = evidence_paths(spec) + ok &= sorted(direct + corollary) == sorted(cids) and not set(direct) & set(corollary) + ok &= RECORD_NODE in corollary and "math.c6r-component.two-scale-law-proof" in direct and len(direct) >= 2 + ok &= sorted(p["reading_rule"]) == sorted(cids) and sorted(t0["required_premises_after"]) == sorted(cids) + req = {(e["from"], e["to"]) for e in spec["proposed_graph_edges"] if e["required"] is True and not e.get("deferred_with")} + ok &= all((RES, cid) in req for cid in cids) + seen = set() + for e in spec["proposed_graph_edges"]: + key = (e["from"], e["to"], e["relation"]) + ok &= key not in seen and type(e["required"]) is bool and bool(e["relation"]) + seen.add(key) + ok &= e["from"] in (RES, WIT) + # the deferred old-node transition + ok &= t1["node"] == WIT and t1.get("deferred") is True and t1["proposed"]["classification"] == "PROVED_REVIEWED" + ok &= sorted(t1["required_premises_after"]) == sorted([RES, AGG]) and t1["proposed"]["controlling"] is False + ok &= AGG not in graph["nodes"] or graph["nodes"][AGG].get("classification") == "PROVED_REVIEWED" + if not ok: + return False, None + st = installed_state(graph, spec) + installed = st["residual"] == "PROVED_REVIEWED" + stages = ("installed",) if installed else ("interim", "final", "final-preexisting", "final-direct-only", + "final-corollary-only", "final-record-unreviewed") + expected_required = {"final-direct-only": len(direct), "final-corollary-only": len(corollary)} + builds = {} + for stage in stages: + old, new = build(spec, graph, "final" if stage == "installed" else stage) + nodes, edges = new["nodes"], new["edges"] + ok &= all(old["nodes"][k] == nodes[k] for k in old["nodes"] if k != RES) # no live node other than the residual changes + ok &= edges[:len(old["edges"])] == old["edges"] + req_after = [e["to"] for e in edges if e["from"] == RES and e["required"] is True] + if stage == "interim": + ok &= nodes[RES]["classification"] == "OPEN_ACTIVE" + else: + ok &= nodes[RES]["classification"] == "PROVED_REVIEWED" + if stage == "installed": + ok &= new == old # the live graph carries the proposal exactly: a no-op + if stage == "final-record-unreviewed": + ok &= nodes[RECORD_NODE]["classification"] == "AUTHOR_SIDE_CANDIDATE" + else: + ok &= nodes[RECORD_NODE]["classification"] == "PROVED_REVIEWED" + ok &= all(nodes[d]["classification"] == "PROVED_REVIEWED" for d in req_after) + ok &= len(req_after) == expected_required.get(stage, len(cids)) + builds[stage] = (old, new) + if not installed: + # the installed state, simulated (Codex 4140125558): the executed graph passes LIVE and re-building it is a no-op + sim = build(spec, graph, "final")[1] + if MUT == "installed-drift": + sim["nodes"]["math.c6r-component.spectral-closure-proof"]["fingerprint"] = "0" * 64 + sim_ok, sim_st = live_ok(sim, sel, spec) + ok &= sim_ok and sim_st["residual"] == "PROVED_REVIEWED" and sim_st["present"] == sorted(cids) + ok &= sim_st["edges_present"] and not sim_st["mismatched"] + o2, n2 = build(spec, sim, "final") + ok &= o2 == n2 + builds["installed-simulated"] = (o2, n2) + return bool(ok), builds + + +def load_hard_gate(root): + hg_spec = importlib.util.spec_from_file_location("hard_gate", root / HARD_GATE) + hg = importlib.util.module_from_spec(hg_spec) + hg_spec.loader.exec_module(hg) + return hg + + +def snapshot(root, graph): + """Exact source snapshot covering every node of a graph, in the shape git_transition_audit.read_snapshot produces, + taken from the working tree: record_only, external_unresolved, blob (bytes and SHA256), tree (sorted file digests), + or missing.""" + out = {} + for nid, n in graph["nodes"].items(): + ref = n.get("source") if isinstance(n, dict) else None + if ref is None: + out[nid] = {"kind": "record_only"} + continue + if ref.startswith(EXTERNAL_PREFIXES): + out[nid] = {"kind": "external_unresolved", "reference": ref} + continue + p = root / ref.rstrip("/") + if p.is_file(): + data = p.read_bytes() + out[nid] = {"kind": "blob", "reference": ref, "bytes": len(data), "sha256": hashlib.sha256(data).hexdigest()} + elif p.is_dir(): + items = [[q.relative_to(p).as_posix(), hashlib.sha256(q.read_bytes()).hexdigest()] + for q in sorted(p.rglob("*")) if q.is_file() and "__pycache__" not in q.parts] + out[nid] = {"kind": "tree", "reference": ref, "files": len(items), + "sha256": hashlib.sha256(json.dumps(items, sort_keys=True).encode()).hexdigest()} + else: + out[nid] = {"kind": "missing", "reference": ref} + return out + + +def check_gate(root, spec, builds): + if builds is None: + return False, {} + hg = load_hard_gate(root) + out = {} + ok = True + comp_ids = [c["id"] for c in spec["proposed_graph_nodes"]] + for stage, (old, new) in builds.items(): + try: + hg.validate_graph_fail_closed(new) + rep_new = hg.closure_report(new) + rep_old = hg.closure_report(old) + old_src, new_src = snapshot(root, old), snapshot(root, new) + if MUT == "snapshot-omitted": + ri = hg.reverse_impact_between(old, new) + else: + ri = hg.reverse_impact_between(old, new, old_sources=old_src, new_sources=new_src) + except Exception: + return False, {} + ok &= ri.get("source_snapshots_supplied") is True and ri.get("promotion_permission") is False + ok &= rep_new.get("gate_ok") is True and not rep_new.get("illegal_controlling") and rep_old.get("gate_ok") is True + ok &= not any(n.get("controlling") for n in new["nodes"].values()) + changed, impacted = set(ri["changed_nodes"]), set(ri["impacted"]) + pre = set(installed_state(old, spec)["present"]) + if stage in ("installed", "installed-simulated"): + ok &= not changed and not impacted # nothing to apply: the proposal is live as declared + else: + ok &= RES in changed and RES in impacted + ok &= all((cid in changed) or (cid not in new["nodes"]) or (cid in pre) for cid in comp_ids) + ok &= WIT not in changed # the deferred transition is not applied + held = {h["node"]: h.get("unsatisfied_required", []) for h in rep_new.get("hold_proposals", [])} + if stage == "interim": + ok &= RES not in held or new["nodes"][RES]["classification"] == "OPEN_ACTIVE" + elif stage == "final-record-unreviewed": + ok &= RES in held and RECORD_NODE in held[RES] # the gate itself holds a residual resting on an unread record + else: + ok &= RES not in held # every required premise PROVED_REVIEWED + # a later source-byte change on any component reaches the residual (and the old node once it depends on it) + wit_depends = any(e["from"] == WIT and e["to"] == RES for e in new["edges"]) + tested = 0 + for cid in comp_ids: + if cid not in new["nodes"]: + continue + drift = copy.deepcopy(new_src) + drift[cid]["sha256"] = "0" * 64 + try: + ri2 = hg.reverse_impact_between(new, new, old_sources=new_src, new_sources=drift) + except Exception: + return False, {} + ok &= set(ri2["changed_nodes"]) == {cid} and RES in ri2["impacted"] + ok &= (not wit_depends) or WIT in ri2["impacted"] + tested += 1 + ok &= tested == sum(1 for cid in comp_ids if cid in new["nodes"]) >= len(comp_ids) - 1 + out[stage] = {"changed_nodes": sorted(changed), "impacted": sorted(impacted), + "residual_classification": new["nodes"][RES]["classification"], + "residual_held": RES in held, "source_snapshots_supplied": True, + "source_drift_reaches_residual": tested, + "old_node_transition": "deferred: presupposes Math-#160 step 3 (aggregate node absent live)"} + return bool(ok), out + + +def check_negatives(src): + rejected = [] + for fault in ("delete", "change", "symlink", "symlink-parent"): + with tempfile.TemporaryDirectory() as tmp: + root = pathlib.Path(tmp) + for path in INVENTORY: + dst = root / path + dst.parent.mkdir(parents=True, exist_ok=True) + shutil.copyfile(src / path, dst) + target = root / SC + if fault == "delete": + target.unlink() + elif fault == "change": + data = bytearray(target.read_bytes()) + data[0] ^= 1 + target.write_bytes(bytes(data)) + elif fault == "symlink": + real = target.with_name("real_" + target.name) + target.rename(real) + os.symlink(real.name, target) + else: + parent = target.parent + real = parent.with_name("real_" + parent.name) + parent.rename(real) + os.symlink(real.name, parent) + rejected.append(not identities_ok(root)) + return all(rejected) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument("--mutant", choices=MUTANTS) + MUT = ap.parse_args().mutant + root = pathlib.Path(".").resolve() + spec = load_spec(root) + ident = identities_ok(root) + live, live_state = check_live(root, spec) + trans_ok, builds = check_transitions(root, spec) + gate_ok, gate = check_gate(root, spec, builds) + checks = {"IDENTITIES": ident, "VERDICTS": ident and check_verdicts(root), "LIVE": live, + "REVIEWED": ident and check_reviewed(root, spec), "DEDUCTION": check_deduction(), + "TRANSITIONS": trans_ok, "GATE": gate_ok, "NEGATIVES": check_negatives(root)} + passed = all(checks.values()) and len(checks) == 8 + print(json.dumps({"object": "C6-RESIDUAL-CLOSURE-20260930-v1", "checks": checks, "passed": passed, + "inventory_files": len(INVENTORY), "component_nodes": len(spec["proposed_graph_nodes"]), + "live_state": live_state, "gate": gate, + "scope": "identity, verdict-row, live-register (including the installed state), reviewed-text, exact " + "finite bookkeeping of section 3, transition-shape, hard-gate replay with source snapshots in " + "every execution shape, and filesystem-negative checks; the Gaussian limits are the sources' " + "reviewed statements and are not re-proved; nothing is written"}, + indent=2, sort_keys=True)) + return 0 if passed else 1 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/reviews/c6_witness_collision_reconciliation_20260929/PROPOSED_TRANSITIONS.json b/reviews/c6_witness_collision_reconciliation_20260929/PROPOSED_TRANSITIONS.json new file mode 100644 index 000000000..8d70ee749 --- /dev/null +++ b/reviews/c6_witness_collision_reconciliation_20260929/PROPOSED_TRANSITIONS.json @@ -0,0 +1,948 @@ +{ + "schema": 1, + "object": "C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1", + "scientific_effect": "NONE", + "disposition": "PROPOSAL_FOR_A_NON_CLAUDE_LANE", + "base_commit": "8e61fc4c055b56e526b5b3488104945e8e0e29cc", + "reverified_at": "ec6db8c (Math- main after Math-#151 merged 2026-09-30T00:56:58Z; no inventoried path changed; PROOF_INDEX.md, GRAPH.json and reviews/candidates_pending_20260928/CANDIDATES.md changed as quoted in RECONCILIATION.md section 1a; SELECTOR_REGION.json unchanged)", + "revision": { + "version": "v1.7", + "applies": [ + "v1.1: OpenAI review 5359701805 items 1-3; OpenAI pickup 5900929519 residual wording", + "v1.2: merge of Math- main 5c484bf; candidate status after Math-#161 merged", + "v1.3: Codex review 5359693191 threads 4139312857 (catalog bound), 4139312869 (live LP node reused), 4139312863 (all six selector cells resolved, SELECTOR check), 4139312874 (residual defined at scale r)", + "v1.4: merge of Math- main aa28838 (Math-#158 merged); xAI 5901345590 recorded (ACCEPT as inventory, HOLD on execution); explicit execution_order", + "v1.5: merge of Math- main bb429d3; Math-#166 (TWO_SCALE_LAW.md a32fd5f7, math head 7c82252) recorded as the third residual candidate, matching the residual's definition, conditional on #162, A under review", + "v1.6: xAI bounded discharge read 5901915195 recorded (agrees sections 1 and 4; HOLD execution until Math-#151 on main; residual node first; not the PALM sections 4-6 second-provider read)", + "v1.7: re-verified at ec6db8c (Math-#151 merged); register surfaces re-quoted (section 1a); OBLIGATION check rebound to the current PROOF_INDEX/catalog text and the open node's reading rule; live D5 nodes bound to the D5 proposal's fingerprints; keep-open alternative recorded" + ], + "unchanged": "22-file inventory, status-row statement, historical predicates; one proposed node removed in favour of the live node" + }, + "review_record": [ + { + "lane": "OpenAI", + "id": 5900929519, + "kind": "pickup", + "at": "6a0e3b9" + }, + { + "lane": "OpenAI", + "id": 5359701805, + "kind": "review", + "verdict": "ACCEPT sections 1 and 4 (obligation reading and count-bound discharge); executable fold deferred to a separate engineering review after Math-#151", + "at": "6a0e3b9" + }, + { + "lane": "xAI", + "id": 5901345590, + "kind": "bounded read", + "at": "3e00f21", + "verdict": "ACCEPT as inventory (count-estimate obligation discharged at existential fixed-d compact-mark window scope); HOLD on graph execution until Math-#151 lands, residual node created first, xAI read of PALM sections 4-6 as a separate slice" + }, + { + "lane": "xAI", + "id": 5901915195, + "kind": "bounded discharge read", + "at": "c641df1", + "verdict": "count obligation discharged by Theorem Q + (A4) at existential fixed-d compact-mark window scope; monotonicity upper bound only; HOLD execution until Math-#151 lands; residual node before the old node moves; not the second-provider read of PALM sections 4-6" + } + ], + "presupposes": [ + "reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json executed by Math-#151, on main at ec6db8c: math.d5-pin-neighborhood-first-moment (PROVED_REVIEWED, aggregate) and its 13 component nodes are live; bound by TRANSITIONS" + ], + "status_rows": [ + { + "id": "C6", + "current": "STATUS C6 row cites the planar Fourier bound E N(N-1) <= C r^3 log(1/r) only; catalog C6 open; PROOF_INDEX line 45 NO COMPLETE PROOF YET", + "proposed": "ACCEPT_SCOPED_FACTORIAL_MOMENT", + "scope": "E_{Q_r^W}[(N)_q] <= C_q r^3 for every fixed integer q >= 2 and c r^3 <= E_{Q_r^W} N(N-1) <= C r^3; every fixed d >= 2; fixed torus; compact marks with k_- > 0; all frames and indices; window I_r; original W_r and full Z_r; existential constants. At d = 2 conditional on PALM's own steps only; for d >= 3 additionally on DL at its OpenAI acceptance. Regional monotonicity gives upper bounds only; the lower bound is global (EDL (A4) event).", + "graph_node": "math.c6-witness-collision-factorial-moment", + "required": [ + "frontiers/c6_palm_route_20260929/PROOF.md", + "frontiers/c6_palm_route_20260929/README.md", + "frontiers/c6_palm_route_20260929/SOURCE_MAP.json", + "frontiers/c6_palm_route_20260929/palm_exact_check.py", + "frontiers/c6_palm_route_20260929/RESULTS.json", + "frontiers/c6_factorial_moment_20260929/PROOF.md", + "frontiers/d5_dimension_lift_20260929/PROOF.md", + "frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md", + "frontiers/elder_dimension_lift_20260928/PROOF.md", + "reviews/elder_dimension_lift_claude_20260928/REVIEW.md", + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json" + ], + "supporting": [ + "frontiers/c6_fourier_cutoff_20260929/PROOF.md", + "reviews/c6_fourier_completion_20260929/READING_NOTE.md", + "frontiers/c6_count_cap_boundary_20260929/PROOF.md", + "frontiers/c6_sharpened_20260929/PROOF.md", + "frontiers/c6_rare_cluster_laws_20260929/PROOF.md", + "frontiers/remote_collision_20260928/PROOF.md", + "reviews/d5_remote_collision_grok_20260928/REVIEW.md", + "reviews/d5_offpin_second_moment_20260928/REVIEW.md", + "frontiers/remote_window_20260924/PROOF.md" + ] + } + ], + "graph": [ + { + "node": "math.rn-region.witness-collision", + "current": "OPEN_ACTIVE", + "proposed": "PROVED_REVIEWED", + "scope": "RECONCILIATION.md section 4: torus-wide factorial moments of the window count at every mutual separation and every distance from the pins; every fixed d >= 2; existential constants; the regional monotonicity of section 4 yields upper bounds only, for every Borel selector on ordered q-tuples (OpenAI 5359701805); the lower bound c r^3 is global, on the EDL (A4) event", + "required": [ + "frontiers/c6_palm_route_20260929/PROOF.md", + "frontiers/c6_palm_route_20260929/README.md", + "frontiers/c6_palm_route_20260929/SOURCE_MAP.json", + "frontiers/c6_palm_route_20260929/palm_exact_check.py", + "frontiers/c6_palm_route_20260929/RESULTS.json", + "frontiers/c6_factorial_moment_20260929/PROOF.md", + "frontiers/d5_dimension_lift_20260929/PROOF.md", + "frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md", + "frontiers/elder_dimension_lift_20260928/PROOF.md", + "reviews/elder_dimension_lift_claude_20260928/REVIEW.md", + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json" + ], + "required_nodes": [ + "math.c6-witness-collision-factorial-moment", + "math.d5-pin-neighborhood-first-moment", + "math.uniform-matrix-cap-lifetime" + ], + "supporting": [ + "frontiers/remote_collision_20260928/PROOF.md", + "reviews/d5_remote_collision_grok_20260928/REVIEW.md", + "reviews/d5_offpin_second_moment_20260928/REVIEW.md", + "frontiers/remote_window_20260924/PROOF.md" + ] + }, + { + "node": "math.rn-region.witness-collision.leading-mass-localization", + "current": "ABSENT", + "proposed": "OPEN_ACTIVE", + "scope": "RECONCILIATION.md section 5: leading-mass localization at scale r, i.e. lim_{R->oo} limsup_{r->0} r^-3 E #{ordered pairs of distinct window points not both within R r of the midpoint} = 0. Known: pairs within C_d r carry >= c r^3 (EDL (A4)), so local-local pairs between C_d r and R r are localized mass, not residual; both-remote pairs carry O(r^5) (P_eta); pairs with a witness in the shells R r <= |X - o| <= s_0 carry <= C r^3 (R^-2 + s_0^2) (PALM section 6 R3, marked shell bound, q = 2). Open piece: the mixed count M(R, s_0) of pairs with one witness within R r of the midpoint and the other at distance >= s_0, where PALM R4 gives C r^3 only; the residual is equivalent to lim_{s_0->0} lim_{R->oo} limsup_r r^-3 M(R, s_0) = 0, not supplied by the landed reviewed chain. Author-side candidates: Math-#161 (OpenAI, e7f8929, merged 5c484bf; planar; Anthropic analytic ACCEPT 5359733142 by the PALM/DL author session, source-exposed; Codex engineering ACCEPT 5359797442; sections 6-7 conditional on Math-#158, whose slices A/B/C/D carry verdicts and which merged at aa28838) and Math-#159 (Claude, 7188bfa, every fixed d, under OpenAI reassessment) and Math-#166 (OpenAI, TWO_SCALE_LAW.md a32fd5f7, math head 7c82252; states r^-3 E[(N)_q - (N_in)_q] -> 0 for q >= 2, conditional on #162; B/C accepted by session 01NMeKE, A under review); not premises, not inventoried, not consumed. Stays OPEN_ACTIVE for every fixed d; a planar sub-status is for an executing lane. Required by no theorem.", + "required": [], + "supporting": [ + "frontiers/elder_dimension_lift_20260928/PROOF.md", + "frontiers/remote_collision_20260928/PROOF.md" + ] + }, + { + "node": "regional.shrinking-regions.analytic-route", + "current": "ABSENT", + "proposed": "SUPERSEDED_NONBLOCKING", + "scope": "RECONCILIATION.md section 6: for the reviewed scopes of math.rn-region.pin-collision, math.rn-region.intermediate-r-to-rho and math.rn-region.witness-collision, the analytic route (planar D5 reconciliation, DL, PALM) bypasses hist.CH-LIFT, hist.Piece-2-annulus and hist.OBL-H5-JETMOD. The historical predicates keep their classifications in their original replay scopes. Scope retained: original W_r, full Z_r, compact marks with k_- > 0, fixed d and torus, window I_r where the source is window-only; no all-height bound, no numerical certificate.", + "required": [], + "supporting": [] + } + ], + "historical_predicates_unchanged": [ + "hist.CH-LIFT", + "hist.Piece-2-annulus", + "hist.OBL-H5-JETMOD", + "hist.ENV-RESCOV", + "hist.ALLCELL-FDZ-Q4" + ], + "selector_region_proposal": { + "regions": [ + "pin-collision", + "intermediate-r-to-rho", + "witness-collision" + ], + "cells": { + "CH-LIFT": "BYPASSED_BY_ANALYTIC_ROUTE", + "Piece-2-annulus": "BYPASSED_BY_ANALYTIC_ROUTE", + "OBL-H5-JETMOD": "BYPASSED_BY_ANALYTIC_ROUTE", + "H5-REMOTE-THRESHOLD": "BYPASSED_BY_ANALYTIC_ROUTE", + "ALLCELL-FDZ-Q4": "NOT_REQUIRED", + "ENV-RESCOV": "REDERIVED_QUALITATIVELY" + }, + "under_node": "regional.shrinking-regions.analytic-route", + "resulting_covered_region_ids": [ + "math.rn-region.fixed-remote", + "math.rn-region.pin-collision", + "math.rn-region.intermediate-r-to-rho", + "math.rn-region.witness-collision" + ], + "resulting_open_region_ids": [ + "math.rn-region.mesoscopic-scaled-annulus" + ], + "precedent": "fixed-remote row: ALLCELL-FDZ-Q4 NOT_REQUIRED, ENV-RESCOV REDERIVED_QUALITATIVELY; the analytic route uses the positive Fourier spectrum and the joint floors (PALM Lemma 6.1 / Lemma E_d), not the historical carriers. hist.ENV-RESCOV and hist.ALLCELL-FDZ-Q4 keep their classifications.", + "alternative": "node-only supersession as regional.fixed-annulus.high-jet-route (cells untouched, lists unchanged); the executing lane decides." + }, + "proposed_graph_nodes": [ + { + "id": "math.c6-witness-collision-factorial-moment", + "layer": "D5", + "kind": "candidate", + "classification": "PROVED_REVIEWED", + "controlling": false, + "scope": "RECONCILIATION.md section 4; every fixed d >= 2; fixed torus; compact marks with k_- > 0; window I_r; existential constants", + "source": "reviews/c6_witness_collision_reconciliation_20260929/RECONCILIATION.md" + }, + { + "id": "math.c6-component.palm-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "proof", + "author_provider": "Anthropic", + "fingerprint": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "source": "frontiers/c6_palm_route_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#145 5356233690", + "provider": "OpenAI", + "verdict": "section 4 sound", + "depth": "full", + "source_exposed": true + }, + { + "review": "Math-#145 5358116559", + "provider": "OpenAI", + "verdict": "ACCEPT sections 5-7, Theorem Q, Corollaries Theta and P", + "depth": "full", + "source_exposed": true + } + ] + }, + { + "id": "math.c6-component.palm-review-record", + "layer": "D5", + "kind": "reading_rule_component", + "classification": 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"component_role": "finite_companion", + "author_provider": "Anthropic", + "fingerprint": "3bd19ec659b2ebceaf2fadeff29c1913ab28998d5182503a6de012c018bb550b", + "source": "frontiers/c6_palm_route_20260929/palm_exact_check.py" + }, + { + "id": "math.c6-component.palm-results", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "finite_companion_result", + "author_provider": "Anthropic", + "fingerprint": "7812de453ff1efd77ffc2431874243713615ce4fcc0e55f2beae991e24f6f6bc", + "source": "frontiers/c6_palm_route_20260929/RESULTS.json" + }, + { + "id": "math.c6-component.c6l-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "proof", + "author_provider": "Anthropic", + "fingerprint": "b19927563e2e9ff8f38648b61cd3f2c09ee1a081494b98cc285283554746e6d3", + "source": "frontiers/c6_factorial_moment_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#140 5355120953", + "provider": "OpenAI", + "verdict": "ACCEPT sections 4-6", + "depth": "full" + }, + { + "review": "Math-#140 5355457002", + "provider": "OpenAI", + "verdict": "ACCEPT Lemma R and sections 2/7/8", + "depth": "full" + }, + { + "review": "Math-#140 5355682335", + "provider": "OpenAI", + "verdict": "rebind to v1.3", + "depth": "full" + }, + { + "review": "Math-#140 5894209739", + "provider": "xAI", + "verdict": "bounded read, no contradiction", + "depth": "partial" + } + ] + }, + { + "id": "math.c6-component.dimension-lift-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "proof", + "author_provider": "Anthropic", + "fingerprint": 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"component_role": "reading_rule", + "author_provider": "Anthropic", + "fingerprint": "02f91870b636635244a767ba92f2490dd480b1720d456889216a56e07705a68b", + "source": "reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json" + }, + { + "id": "math.c6-component.fourier-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "supporting_proof", + "author_provider": "OpenAI", + "fingerprint": "c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733", + "source": "frontiers/c6_fourier_cutoff_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#148 5356335148", + "provider": "Anthropic", + "verdict": "ACCEPT Theorem F, W as composition, (F4) planar", + "depth": "full" + } + ] + }, + { + "id": "math.c6-component.fourier-reading-note", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "review_record", + "author_provider": "OpenAI", + "fingerprint": "62f84b68b3bcd034f1427eacf3919c7fb418d311af64f0ab2807d17682cc00b3", + "source": "reviews/c6_fourier_completion_20260929/READING_NOTE.md" + }, + { + "id": "math.c6-component.count-cap-boundary-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "supporting_proof", + "author_provider": "OpenAI", + "fingerprint": "264a5e762f598145e9230aa0947acb2e0354efdb2e3adf65d331e7ee0ba52011", + "source": "frontiers/c6_count_cap_boundary_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#146 5355817385", + "provider": "Anthropic", + "verdict": "ACCEPT Theorems A, B, Proposition C (abstract)", + "depth": "full" + } + ] + }, + { + "id": "math.c6-component.sharpened-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "supporting_proof", + "author_provider": "Anthropic", + "fingerprint": "9f45274ab59d4f4417d1a51e9c97f5916240fdb0a4640ca6f0d3e9e0aea37dd7", + "source": "frontiers/c6_sharpened_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#142 5355652934", + "provider": "OpenAI", + "verdict": "ACCEPT (source-exposed)", + "depth": "full", + "source_exposed": true + } + ] + }, + { + "id": "math.c6-component.rare-cluster-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "downstream_consequence", + "author_provider": "OpenAI", + "fingerprint": "c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb", + "source": "frontiers/c6_rare_cluster_laws_20260929/PROOF.md", + "review_basis": [ + { + "review": "Math-#153 5358565704", + "provider": "Anthropic", + "verdict": "ACCEPT sections 1, 2, 5-7", + "depth": "full" + }, + { + "review": "Math-#153 5358623521", + "provider": "xAI", + "verdict": "ACCEPT sections 3-4 and addendum", + "depth": "full" + } + ] + }, + { + "id": "math.c6-component.remote-collision-proof", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "supporting_proof", + "author_provider": "Anthropic", + "fingerprint": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2", + "source": "frontiers/remote_collision_20260928/PROOF.md", + "review_basis": [ + { + "review": "reviews/d5_remote_collision_grok_20260928", + "provider": "xAI", + "verdict": "P_eta ACCEPT", + "depth": "full" + } + ] + }, + { + "id": "math.c6-component.remote-collision-review-grok", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.c6-witness-collision-factorial-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "f8289546ee82fb386a0d0b6b54fbb99f258a2dc2a959ae231ae80cb4baa61e1e", + "source": "reviews/d5_remote_collision_grok_20260928/REVIEW.md" + }, + { + "id": "math.rn-region.witness-collision.leading-mass-localization", + "layer": "D5", + "kind": "region", + "classification": "OPEN_ACTIVE", + "controlling": false, + "fingerprint": "leading-order r^3 mass of E N(N-1): scale-r localization; open piece = mixed local/remote pairs M(R, s_0)", + "notes": "Residual after the count-level discharge of math.rn-region.witness-collision. Definition: lim_{R->oo} limsup_{r->0} r^-3 E #{ordered pairs not both within R r of the midpoint} = 0. Known: pairs within C_d r carry >= c r^3 (EDL (A4)); both-remote pairs O(r^5) (P_eta); pairs with a shell witness <= C r^3 (R^-2 + s_0^2) (PALM section 6 R3, q = 2). Open: the mixed local/remote count M(R, s_0), o(r^3) in the iterated limit, not supplied by the landed reviewed chain; Math-#161 (e7f8929, merged 5c484bf, planar, Anthropic 5359733142 source-exposed + Codex engineering 5359797442, conditional on Math-#158) Math-#159 (7188bfa, under reassessment) and Math-#166 (a32fd5f7, states the residual exactly, conditional on #162, A under review); not premises, not consumed. Required by no theorem; recorded so that the register keeps a precise statement. Required by no theorem; recorded so that the register keeps a precise statement." + }, + { + "id": "regional.shrinking-regions.analytic-route", + "layer": "D0", + "kind": "regional_supersession", + "classification": "SUPERSEDED_NONBLOCKING", + "controlling": false, + "fingerprint": "regional-bypass-shrinking-regions-v1", + "scope": "pin-collision, intermediate-r-to-rho and witness-collision at their reviewed scopes (planar D5 reconciliation; DL in every fixed d; PALM Theorem Q); compact positive gaps; window I_r where the source is window-only; original W_r and full Z_r; fixed d and fixed torus; no all-height collision bound; no numerical certificate (RN annulus partition, 24-jet / OBL-H5-JETMOD)", + "notes": "For these scopes only, the reviewed analytic route bypasses CH-LIFT / Piece-2 / OBL-H5-JETMOD. Historical predicates remain OPEN/ABSENT in their original replay scopes." + } + ], + "proposed_graph_edges": [ + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-witness-collision-factorial-moment", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.d5-pin-neighborhood-first-moment", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.palm-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.palm-review-record", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.palm-source-map", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.palm-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.palm-results", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.c6l-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.dimension-lift-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.dimension-lift-crosswalk", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.elder-lift-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.elder-lift-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.uniform-matrix-cap-lifetime", + "required": true, + "relation": "requires_evidence_via_live_reconciled_node" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.planar-reading-rule", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.planar-reading-rule-transitions", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.remote-collision-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.c6-component.remote-collision-review-grok", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.d5-component.offpin-second-moment-review", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.rn-fixed-remote-window", + "required": false, + "relation": "supporting_evidence_via_live_node" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.palm-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.palm-review-record", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.palm-source-map", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.palm-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.palm-results", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.c6l-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.dimension-lift-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.dimension-lift-crosswalk", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.elder-lift-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.elder-lift-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.uniform-matrix-cap-lifetime", + "required": true, + "relation": "requires_evidence_via_live_reconciled_node" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.planar-reading-rule", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.planar-reading-rule-transitions", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.d5-pin-neighborhood-first-moment", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.fourier-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.fourier-reading-note", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.count-cap-boundary-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.sharpened-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.rare-cluster-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.remote-collision-proof", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.c6-component.remote-collision-review-grok", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.d5-component.offpin-second-moment-review", + "required": false, + "relation": "supporting_evidence" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "math.rn-fixed-remote-window", + "required": false, + "relation": "supporting_evidence_via_live_node" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6-component.elder-lift-proof", + "required": false, + "relation": "lower_bound_localized" + }, + { + "from": "math.rn-region.witness-collision.leading-mass-localization", + "to": "math.c6-component.remote-collision-proof", + "required": false, + "relation": "partial_cover_only" + }, + { + "from": "math.c6-witness-collision-factorial-moment", + "to": "regional.shrinking-regions.analytic-route", + "required": false, + "relation": "regionally_supersedes_historical_high_jet_route" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "regional.shrinking-regions.analytic-route", + "required": false, + "relation": "regionally_supersedes_historical_high_jet_route" + }, + { + "from": "regional.shrinking-regions.analytic-route", + "to": "hist.CH-LIFT", + "required": false, + "relation": "regional_bypass_only" + }, + { + "from": "regional.shrinking-regions.analytic-route", + "to": "hist.Piece-2-annulus", + "required": false, + "relation": "regional_bypass_only" + }, + { + "from": "regional.shrinking-regions.analytic-route", + "to": "hist.OBL-H5-JETMOD", + "required": false, + "relation": "regional_bypass_only" + } + ], + "note": "DECLARATIVE, NOT EXECUTED: no GRAPH, SELECTOR_REGION, PROOF_INDEX, STATUS or catalog edit is made here. Every required evidence file is a proposed graph node with its SHA256 as fingerprint, reached by a required edge from each consuming node. Supporting files are reached by non-required edges. The residual node is required by nothing. The supersession node changes no historical predicate. Execution is for a non-Claude lane. Math-#151's planar fold is on main (ec6db8c) and defines math.d5-pin-neighborhood-first-moment and math.d5-component.offpin-second-moment-review as live nodes (reused here rather than duplicated); the [LP] and remote_window bytes are referenced through the live nodes math.uniform-matrix-cap-lifetime and math.rn-fixed-remote-window (the latter now duplicated by the live math.d5-component.remote-window-proof; Math-#167 proposes the alignment).", + "declarative": true, + "executed": false, + "live_nodes_reused": [ + { + "id": "math.uniform-matrix-cap-lifetime", + "source": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "fingerprint": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "reading_rule_required": [ + "math.d1-component.congruence-erratum", + "math.d1-component.section9-replacement-v1_1", + "math.d1-component.marked-cylinder-cap", + "math.d1-component.reconciliation-record" + ], + "why": "the live graph already carries these bytes with four required reading-rule edges (D1 reconciliation, Math-#126); a second terminal node would not inherit a later hold on them (Codex 4139312869)" + }, + { + "id": "math.rn-fixed-remote-window", + "source": "frontiers/remote_window_20260924/PROOF.md", + "fingerprint": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "role": "supporting only", + "note": "live classification AUTHOR_SIDE_CANDIDATE although the source was reviewed at main#76; the D5 proposal creates math.d5-component.remote-window-proof for the same bytes. This record references the live node; collapsing the pair and repairing the classification is for the executing lane." + }, + { + "id": "math.d5-pin-neighborhood-first-moment", + "live_since": "ec6db8c (Math-#151)", + "classification": "PROVED_REVIEWED", + "kind": "candidate (aggregate, no fingerprint)", + "source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "role": "required evidence of the witness node and of the aggregate (planar reading rule)" + }, + { + "id": "math.d5-component.offpin-second-moment-review", + "live_since": "ec6db8c (Math-#151)", + "classification": "PROVED_REVIEWED", + "fingerprint": "43beb4ef7da5bf71e15164d5c4bc1862db387d83f906aebdf9831625b1ca6369", + "source": "reviews/d5_offpin_second_moment_20260928/REVIEW.md", + "role": "supporting only; since ec6db8c also the live reading_rule of math.rn-region.witness-collision" + }, + { + "id": "math.d5-component.remote-window-proof", + "live_since": "ec6db8c (Math-#151)", + "classification": "PROVED_REVIEWED", + "fingerprint": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "source": "frontiers/remote_window_20260924/PROOF.md", + "role": "not referenced; second live node for the math.rn-fixed-remote-window bytes (Math-#151 landed the D5 proposal as written); Math-#167 proposes the alignment of the older node" + } + ], + "register_read": { + "reviews/candidates_pending_20260928/CANDIDATES.md": { + "git_blob": "02455f0561d51990e1486e5d4fbb4229c8a34f5a", + "sha256": "cbaa71b55190ac234b8363614b7e57387f60c33b393ea102408eb88ec1e44513", + "at": "ec6db8c (rewritten by Math-#151; section 1a)", + "previous": { + "git_blob": "e3fc1a310c89ac9fd9249af26e28da509649ffd5", + "sha256": "3809e6147321df260085815b2d9e995094d63492534afa4ad1cd05b9a168ce4d", + "through": "bb429d3" + }, + "binding": "OBLIGATION check: the section 1 C6 heading with its open-task sentence, or the section 1a heading with the node named OPEN_ACTIVE, or this record referenced; workflow re-runs on change" + }, + "PROOF_INDEX.md": { + "git_blob": "030447a1130b52a09230e21c039409f87ef2ba32", + "sha256": "a1259b8bff4d73d97ee0835ea5c82568d4fac39fae352f5a3a2422c6bf0f8a0c", + "at": "ec6db8c (rewritten by Math-#151; section 1a)", + "previous": { + "git_blob": "dc7cea1568cd5562594bce6fae27d1422515253f", + "through": "bb429d3" + }, + "binding": "OBLIGATION check: the section 1 sentence, or the section 1a sentence, or this record referenced; workflow re-runs on change" + }, + "frontiers/downstream_gate_20260925/GRAPH.json": { + "git_blob": "220b56867f24c5ef6f55a3cce70490e9777cf3af", + "sha256": "8822e9618678321a342d69cd0b8ae6552de1b5d578c331de5072b2892ee9dd09", + "at": "ec6db8c (Math-#151 fold: 14 nodes and 30 edges added; witness node gains scope, review_source, review_disposition OPEN, reading_rule and notes; classification and fingerprint unchanged)", + "previous": { + "git_blob": "fa6ba930fac75e22079fb3b9f13e302a57d5f819", + "through": "bb429d3" + }, + "binding": "OBLIGATION, TRANSITIONS and SELECTOR checks; workflow re-runs on change" + }, + "frontiers/downstream_gate_20260925/SELECTOR_REGION.json": { + "git_blob": "ebffdb0969c7d3484f8fd1f4352ea290346f0318", + "sha256": "53fa0ec88e417882eff2a40e4b575c639d43b7e8787e7fb8917aa22ace79390d", + "at": "unchanged through ec6db8c", + "binding": "OBLIGATION and SELECTOR checks; workflow re-runs on change" + } + }, + "execution_order": [ + "1. create math.rn-region.witness-collision.leading-mass-localization (OPEN_ACTIVE, required by nothing)", + "2. create regional.shrinking-regions.analytic-route (SUPERSEDED_NONBLOCKING) with its three regional_bypass_only edges; selector cells per selector_region_proposal or the node-only alternative", + "3. create the component nodes and the required/supporting edges; reference the live nodes listed in live_nodes_reused", + "4. Math-#151's fold is on main (ec6db8c). Remaining precondition for this step: the xAI bounded read of PALM sections 4-6 (per xAI 5901915195) or an explicit integrator decision that OpenAI 5358116559 plus the section 1 reading suffices; then move math.rn-region.witness-collision to PROVED_REVIEWED and update STATUS/PROOF_INDEX/catalog rows. Register-equivalent alternative (section 1a): keep the node OPEN_ACTIVE, give it the residual's definition and fingerprint, record the discharge on the aggregate node, and do not create the residual node separately", + "executing lane: non-Claude; this record executes nothing" + ] +} diff --git a/reviews/c6_witness_collision_reconciliation_20260929/RECONCILIATION.md b/reviews/c6_witness_collision_reconciliation_20260929/RECONCILIATION.md new file mode 100644 index 000000000..e488c0afe --- /dev/null +++ b/reviews/c6_witness_collision_reconciliation_20260929/RECONCILIATION.md @@ -0,0 +1,407 @@ +# C6 / witness collision — source-bound reconciliation + +**Object:** C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. +**Read at:** Math- `main` `8e61fc4c055b56e526b5b3488104945e8e0e29cc` (29 September 2026); every inventoried byte +re-verified identical at the record's base `7a1cb09` (the merges of Math-#155 and Math-#156 in between touch no +inventoried path); re-verified again at `5c484bf` (Math-#161 merged), at `aa28838` (Math-#158 merged) and at +`bb429d3` (Math-#157 and #165 merged); no inventoried path changed at any of them. Re-verified at `ec6db8c` (Math-#151 +merged, 30 September 00:56Z): no inventoried path changed; three register surfaces did (§1a). +**Revision:** v1.1 applies the scope corrections and the residual wording of OpenAI nonauthor review 5359701805 +(at `6a0e3b9`) and pickup 5900929519; v1.2 merges Math- main `5c484bf` and updates the §5/§9 status of the +candidates after Math-#161 merged; v1.3 answers Codex review 5359693191 (the [LP] evidence reuses the live node +`math.uniform-matrix-cap-lifetime`; the selector proposal resolves all six cells per region and is checked; the +catalog is bound by the `OBLIGATION` check and the workflow paths; the residual is defined at scale `r`). §§1 and +3, the §4 statement and the 22-file inventory are unchanged; v1.3 removes one proposed node in favour of a live one. +v1.4 merges main `aa28838`, records xAI 5901345590 (ACCEPT as inventory; HOLD on execution until Math-#151 lands) and +makes the execution order explicit in §7. v1.5 merges main `bb429d3` and records Math-#166 as the third candidate +for the §5 residual, the one whose statement matches the residual's definition. v1.6 records xAI 5901915195. v1.7 re-verifies at `ec6db8c`, +quotes the register as Math-#151 rewrote it (§1a), rebinds the `OBLIGATION` check to the current text, lists the D5 nodes +now live, and records that Math-#151 took the keep-open alternative without defining the regional mechanism. +**Effect:** register reconciliation, **declarative only**. This record introduces **no new mathematical claim**. It binds +the merged C6 chain to exact bytes, quotes the open obligation as the register recorded it, states which reviewed +theorem discharges that obligation and at what scope, states precisely what is *not* discharged, and proposes the +register transitions in machine-readable form (`PROPOSED_TRANSITIONS.json`, `"executed": false`). Execution of any +STATUS, PROOF_INDEX, GRAPH or catalog change is a separate act for a **non-Claude lane**. `lemma_closed`, prizes and +premises are untouched. Scientific effect: NONE. + +## 0. Exposure, stated first + +Every lane shares one GitHub account; no organizational independence is claimed anywhere below. + +| Object | Author | Nonauthor verdict(s) | This reconciler's relation | +|---|---|---|---| +| [C6L] `frontiers/c6_factorial_moment_20260929/PROOF.md` (Math-#140) | Anthropic Claude, **this session** | OpenAI 5355120953 (§§4–6), 5355457002 (Lemma R, §§2/7/8), 5355682335 (rebind to v1.3); xAI bounded read 5894209739 | **author** | +| [PALM] `frontiers/c6_palm_route_20260929/PROOF.md` (Math-#145) | Anthropic Claude, session `015wNj8L…` | OpenAI 5356233690 (§4), 5357899713 (AMEND R3a), 5358116559 (ACCEPT §§5–7, Theorem Q, Corollaries Θ, P at `06bc0d6`); Codex integration review 5358643154 | none | +| [DL] `frontiers/d5_dimension_lift_20260929/PROOF.md` (Math-#141) | Anthropic Claude, session `015wNj8L…` | OpenAI 5357858391 (P_d, C_d), 5357882570 (I_d, G_d), source-exposed; xAI identities 5894274124, 5894406926; xAI planar P2 continuum 5894512272 | none | +| [EDL] `frontiers/elder_dimension_lift_20260928/PROOF.md` | OpenAI | Anthropic `reviews/elder_dimension_lift_claude_20260928/` ((A4) ACCEPT) | **reviewer** of (A4) | +| [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | OpenAI | D1 reconciliation (Math-#126); register row D1 | reviewer C1 and D1 reconciler | +| [P_η] `frontiers/remote_collision_20260928/PROOF.md` | Anthropic Claude, **this session** | xAI `reviews/d5_remote_collision_grok_20260928/`, `reviews/d5_offpin_second_moment_20260928/` | **author** | +| [FOU] `frontiers/c6_fourier_cutoff_20260929/PROOF.md` (Math-#148) | OpenAI | Anthropic review 5356335148 (session `01NMeKE…`), recorded in `reviews/c6_fourier_completion_20260929/READING_NOTE.md` | none | +| [BND] `frontiers/c6_count_cap_boundary_20260929/PROOF.md` (Math-#146) | OpenAI | Anthropic review 5355817385 (**this session**) | reviewer and integrator | +| [SHP] `frontiers/c6_sharpened_20260929/PROOF.md` (Math-#142) | Anthropic Claude, **this session** | OpenAI 5355652934 (source-exposed: the route was OpenAI's) | **author** | +| [CLU] `frontiers/c6_rare_cluster_laws_20260929/PROOF.md` (Math-#153) | OpenAI | Anthropic 5358565704 (session `01NMeKE…`), xAI 5358623521 | none | + +The three load-bearing proofs of the discharge, [C6L], [DL] and [PALM], are all Claude-authored (two sessions). Their +nonauthor verdicts are OpenAI's, and OpenAI states its own source exposure in each (it authored the planar first-moment +sources that [DL] lifts and that [C6L]/[PALM] consume). xAI's verdicts cover [DL]'s identities, the planar P2 continuum +and a bounded read of [C6L]. **An integrating lane that wants a second provider on the discharge itself should ask xAI +for a bounded read of [PALM] §§4–6 before executing §7.** That is the same caveat the D5 record carried for P2. + +**Verdicts on this record.** OpenAI nonauthor scope review 5359701805 (at `6a0e3b9`, `RECONCILIATION.md` blob +`66bb7cfd`): ACCEPT of §§1 and 4, the identification and the count-bound discharge of the original obligation, with +three scope corrections applied in v1.1 (§2 [P_η] row; §4 upper bounds only; §6 scope retention) and the §5 wording +"not supplied by the landed reviewed chain". That review explicitly does **not** accept the executable fold +(inventory, edges, selector transition, checker, hosted results); it defers that to a separate engineering review +after Math-#151 lands. No register transition occurs from it. + +xAI bounded read 5901345590 (at `3e00f21`): **ACCEPT as inventory** (§1 quotes a count estimate; §3 byte-binds the +chain; `MONOTONE` is correct; the line-45 sentence is discharged at existential fixed-`d` compact-mark window scope) +and **HOLD on execution** for three reasons this record already states: `executed: false` and the dependence on +Math-#151; the residual node must exist before the old node moves (§7 ordering); and the xAI read of [PALM] §§4–6 +asked for above is a separate slice, not that comment. The record may land declaratively; the +`OPEN_ACTIVE → PROVED_REVIEWED` step is a later non-Claude act. + +xAI bounded discharge read 5901915195 (at `c641df1`): agrees that §1 quotes a count estimate and that [PALM] +Theorem Q with [EDL] (A4) discharges it at existential, fixed-`d ≥ 2`, compact-mark, window-`I_r` scope, with the +regional monotonicity an upper bound only; holds execution until Math-#151 is on main; requires the residual node +to exist before the old node moves (the §7 order); and states that it is not the second-provider read of [PALM] +§§4–6, which the xAI lane (Benjamin) is taking separately. + +## 1. The obligation, as the register recorded it + +Four places define the open item. They are quoted at the bytes of `8e61fc4`. + +1. **`PROOF_INDEX.md`, line 45:** + > D5 shrinking multiple-witness collision: fixed-separation machinery is `frontiers/remote_window_20260924/PROOF.md` + > section 6, (15)–(16), at fixed `eta>0`. **NO COMPLETE PROOF YET: shrinking-separation factorial-moment/collision + > estimate.** +2. **`frontiers/downstream_gate_20260925/GRAPH.json`, node `math.rn-region.witness-collision`:** layer D5, kind + `region`, classification `OPEN_ACTIVE`, fingerprint **"eta->0 mutual witness separation"**, notes **"Fixed-eta + factorial moments do not cover shrinking separation"**. +3. **`frontiers/remote_window_20260924/PROOF.md`** (D4, reviewed at main#76), after (15)–(16): + > Witness tuples whose mutual separation tends to zero; (15) does not control that collision. +4. **`frontiers/remote_collision_20260928/PROOF.md`** ([P_η], xAI-reviewed), §1 and §8: + > […] records that shrinking separation is uncontrolled. This is the open "witness-collision" region of the D5 graph + > (`math.rn-region.witness-collision`, "eta->0 mutual witness separation"). […] + > Not established: **A torus-wide second factorial moment, or `math.rn-region.witness-collision` in its full + > torus-wide sense.** Pairs with either witness in the local, collar or intermediate regions are not estimated here. + +The D5 reconciliation (`reviews/d5_reconciliation_20260929/RECONCILIATION.md` §5, landed via Math-#138) restated it: +> **D5-open (catalog C6): shrinking multiple-witness collision.** Target. A torus-wide second factorial moment +> `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least one witness is within fixed `η` of a pin. +> […] **No upper bound.** The optimal order is open. + +And catalog C6 (`reviews/candidates_pending_20260928/CANDIDATES.md`, read at blob `e3fc1a31`, unchanged through +`5c484bf`; bound by the `OBLIGATION` check and the workflow paths) asked for "a valid full-window factorial upper +bound with the pin-neighborhood collisions included". + +**Reading.** In all five places the obligation is a **count estimate**: a factorial-moment (collision) bound for the +window count that covers (a) witness pairs at shrinking *mutual* separation, including coincidence limits, and (b) +witness pairs with a member at shrinking distance from a pin. The fixed-`η` results ([D4] (15)–(16); [P_η]) cover (a) +only when both witnesses stay in a fixed remote region; nothing covered (b). No source defines the node as a regional +decomposition of the leading-order mass, as a Palm or conditional statement, or as anything other than the count +estimate above. §5 returns to that point. + +### 1a. The register at `ec6db8c` (after Math-#151) + +Math-#151 merged at `ec6db8c` on 30 September (00:56Z) and rewrote three of the surfaces quoted above. The quotes in +§1 remain the obligation as the register recorded it when this record was opened and reviewed (OpenAI 5359701805, +xAI 5901915195 read that text); the `OBLIGATION` check now binds either text, so the workflow fails on any third +rewrite. + +1. **`PROOF_INDEX.md`** (blob `030447a1`), open-obligations section: + > D5 shrinking multiple-witness collision: fixed-separation machinery is `frontiers/remote_collision_20260928/PROOF.md`. + > The reviewed Fourier upper bound and the later merged sharp Palm route establish global count statements by routes + > that do not identify this regional shrinking-separation mechanism. **NO COMPLETE REGIONAL PROOF YET:** + > `math.rn-region.witness-collision` remains `OPEN_ACTIVE`; the global `Theta(r^3)` result is not a proof of that node. +2. **`GRAPH.json`** (blob `220b5686`), node `math.rn-region.witness-collision`: still `OPEN_ACTIVE`, fingerprint + unchanged ("eta->0 mutual witness separation"); new fields `scope: "catalog C6"`, + `review_source: reviews/d5_reconciliation_20260929/RECONCILIATION.md`, `review_disposition: "OPEN"`, + `reading_rule: [math.d5-component.offpin-second-moment-review]` with a `required` edge to that component; notes: + > Off-pin fixed-separation second-moment evidence is reviewed and sharp global order is reviewed and merged through + > Math-#145, but the regional shrinking pin/witness-collision mechanism and its source-bound graph fold remain open. +3. **Catalog C6** (`CANDIDATES.md`, blob `02455f05`), retitled "Torus-wide factorial moments: reviewed global order; + regional mechanism open": it records [PALM]'s `Θ(r³)` in every fixed `d ≥ 2` at its packet scope, then: + > They do not close the regional shrinking-witness mechanism: the planar graph node `math.rn-region.witness-collision` + > remains `OPEN_ACTIVE`. […] Open task: prove or refute the shrinking-separation witness-collision mechanism encoded by + > the open graph node, without inferring it from the global Palm bound. + +**Reading.** The count sentence of §1 ("shrinking-separation factorial-moment/collision estimate") is no longer in the +register: Math-#151 replaced it by a "regional shrinking pin/witness-collision mechanism" that no surface defines, i.e. +it took the alternative anticipated in §5 (keep the old node open) but under a new, undefined meaning and the old +fingerprint. Three consequences for the executing lane. (i) The count obligation, as the register recorded it through +`bb429d3` and as OpenAI 5359701805 and xAI 5901915195 read it, is discharged at the §4 scope; nothing in the new text +disputes that: "the global `Theta(r^3)` result is not a proof of that node" is a statement about the node's new +meaning, not about the count. (ii) The only definition on file for the new meaning is §5's residual, the scale-`r` +localization with the mixed count `M(R, s_0)` as its open piece; the new open task ("without inferring it from the +global Palm bound") is consistent with it and supplies no other. (iii) The proposal of §7 therefore stands, and its two +executions are register-equivalent: create the residual node and move the old node, or keep the old node open and give +it the residual's definition and fingerprint. Either way the reading rule now live on the old node +(`math.d5-component.offpin-second-moment-review`) is preserved; this record already lists that component as supporting +evidence of the old node and of the aggregate. + +## 2. What is now proved and reviewed + +All statements are for the exact pinned model of [LP] (period `T`, written `L` in [LP]/[PALM]): fixed `d ≥ 2`, fixed +torus, compact marks `b ∈ B`, `k ∈ K` with `k_− > 0`, all frames and indices, the original weight `W_r` and full +normalizer `Z_r`, the window `I_r = (b − k r³, b)`, and `N` = number of critical points of `f` in `X ∖ {M, S}` with +height in `I_r`. Constants are existential. + +| Tag | Statement | Scope | Verdict on file | +|---|---|---|---| +| [C6L] Theorem C6-L | `E_{Q_r^W} N(N−1) ≤ C r³ log²(1/r)`; `(N)_q ≤ C_q r³ log^{2(q−1)}(1/r)` | `d = 2`; D5 reading rule | OpenAI complete acceptance (5355120953 + 5355457002, rebound 5355682335) | +| [SHP] S₂ / S_d | `r³[L/log L]²` planar; `r³[L/log L]^{d(q−1)}` given G_d | fixed `d` | OpenAI 5355652934 (source-exposed) | +| [FOU] Theorem F, W | count-tail `C e^{−c x^{2/d}}`; `E N(N−1) ≤ C r³ log(1/r)` planar; `(N)_p ≤ C_p r³ [log(1/r)]^{d(p−1)/2}` given G_d | fixed `d` | Anthropic 5356335148 | +| [BND] Theorems A, B, Prop. C | cap tail + first moment alone cannot give `O(r³)`; the missing input is a size-biased cap moment | abstract | Anthropic 5355817385 | +| [DL] Theorems P_d, C_d, I_d, **G_d** | first moments in every fixed `d`; `E_{Q_r^W} N ≤ C r³` | fixed `d` | OpenAI 5357858391, 5357882570 (source-exposed); xAI identities | +| [PALM] **Theorem Q** | `E_{Q_r^W}[(N)_q] ≤ C_q r³` for every fixed integer `q ≥ 2` | every fixed `d ≥ 2` | OpenAI 5358116559 (source-exposed) | +| [PALM] **Corollary Θ** | `c r³ ≤ E_{Q_r^W} N(N−1) ≤ C r³` and `c r³ ≤ Q_r^W{N ≥ 2} ≤ C r³` | every fixed `d ≥ 2` | same, with [EDL] (A4) | +| [EDL] (A4) | `Q_r^W(N_r ≥ 2) ≥ c r³`, `E[N_r(N_r−1)] ≥ c r³`, from two extra index-`(d−1)` points within `C_d r` of the midpoint | every fixed `d` | Anthropic `reviews/elder_dimension_lift_claude_20260928/` | +| [P_η] Corollary D | `E N_η(N_η−1) ≤ C_η r⁵`, both witnesses at distance `≥ η` from the pins, all mutual separations | fixed `η`, every fixed `d` (Theorem C / Corollary D constants depend on `d, L, ρ, η_0, B, K`; the planar / `G_d` condition of [P_η] concerns its Corollary F upper half only) | xAI | +| [CLU] | conditional consequences of (H1), (H2), (Hq): Poisson obstruction at scale `r³`, cluster-law compactness | abstract given the above | Anthropic + xAI | + +[PALM] §1.4 records that at `d = 2` every regional input is the planar row it lifts, each with its own nonauthor +verdict, so the planar statement of Theorem Q depends on [PALM]'s own steps only; the every-`d` statement additionally +needs [DL]. [PALM] §2 pins [C6L] v1.3 and [DL] at the merged bytes, which are the bytes inventoried in §3 below. + +## 3. Components, exact bytes and verdict rows + +`reconciliation_check.py` binds every path below to its SHA256 and git blob, rejects symlinks anywhere on the path, +and checks the quoted verdict rows as exact substrings. The inventory is on `main` `8e61fc4`. + +| Role | Path | SHA256 (prefix) | Blob (prefix) | +|---|---|---|---| +| **Discharge (required)** | | | | +| [PALM] proof | `frontiers/c6_palm_route_20260929/PROOF.md` | `aa37f160…` | `89eb8adf…` | +| [PALM] review record | `frontiers/c6_palm_route_20260929/README.md` | `56c0b94f…` | `1ebf8ede…` | +| [PALM] source map | `frontiers/c6_palm_route_20260929/SOURCE_MAP.json` | `c7cc195f…` | `b528fe10…` | +| [PALM] checker, results | `palm_exact_check.py`, `RESULTS.json` | `3bd19ec6…`, `7812de45…` | `6b0c73ba…`, `cd8003da…` | +| [C6L] proof v1.3 | `frontiers/c6_factorial_moment_20260929/PROOF.md` | `b1992756…` | `f5bd013b…` | +| [DL] proof | `frontiers/d5_dimension_lift_20260929/PROOF.md` | `6fb94b24…` | `9d82c707…` | +| [DL] crosswalk | `frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md` | `bdb83c28…` | `45e4324f…` | +| [EDL] proof | `frontiers/elder_dimension_lift_20260928/PROOF.md` | `b529fe37…` | `7303bd79…` | +| [EDL] review | `reviews/elder_dimension_lift_claude_20260928/REVIEW.md` | `032bb4c7…` | `e2b9fcdc…` | +| [LP] parent | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `9350ad6e…` | `dfed3b8d…` | +| Planar reading rule | `reviews/d5_reconciliation_20260929/RECONCILIATION.md`, `PROPOSED_TRANSITIONS.json` | `14342b7e…`, `02f91870…` | `4062d76b…`, `d928969e…` | +| **Supporting (not required by the discharge)** | | | | +| [FOU] proof, reading note | `frontiers/c6_fourier_cutoff_20260929/PROOF.md`, `reviews/c6_fourier_completion_20260929/READING_NOTE.md` | `c1692379…`, `62f84b68…` | `1d9177a2…`, `682d7045…` | +| [BND] proof | `frontiers/c6_count_cap_boundary_20260929/PROOF.md` | `264a5e76…` | `3d4c28a4…` | +| [SHP] proof | `frontiers/c6_sharpened_20260929/PROOF.md` | `9f45274a…` | `70ba1972…` | +| [CLU] proof | `frontiers/c6_rare_cluster_laws_20260929/PROOF.md` | `c52a3cf1…` | `2ab625ce…` | +| [P_η] proof, xAI reviews | `frontiers/remote_collision_20260928/PROOF.md`, `reviews/d5_remote_collision_grok_20260928/REVIEW.md`, `reviews/d5_offpin_second_moment_20260928/REVIEW.md` | `b9b8b58f…`, `f8289546…`, `43beb4ef…` | `7b48a88e…`, `c18560f8…`, `62697c47…` | +| [D4] proof | `frontiers/remote_window_20260924/PROOF.md` | `a332bae9…` | `b383bfcc…` | + +Full hashes are in `reconciliation_check.py` (`INVENTORY`). The GitHub review ids are bound where an on-disk record +carries them: [PALM] `README.md` (review table), [C6L] header (5355120953, 5355457002), [DL] `README.md` and +`CONTINUUM_CROSSWALK.md` (5357858391, 5357882570, 5894512272), [FOU] reading note (5356335148), [EDL] review ((A4) +row). Reviews 5355817385, 5358565704, 5358623521 and 5355652934 exist on GitHub only; the record cites them and +`SOURCE_FILES.json` does not claim otherwise. + +## 4. Discharge + +**Claim reconciled.** The obligation of §1 — a shrinking-separation factorial-moment/collision estimate for the window +count, torus-wide, pin neighbourhoods included — is discharged by [PALM] Theorem Q, with the matching lower bound +[EDL] (A4), at the following scope: + +- **Statement.** `E_{Q_r^W}[(N)_q] ≤ C_q r³` for every fixed integer `q ≥ 2`, and `c r³ ≤ E_{Q_r^W} N(N−1) ≤ C r³`. + The optimal order asked for by catalog C6 is `r³`. +- **Scope.** Every fixed `d ≥ 2`; fixed torus; compact marks with `k_− > 0`; all frames and indices; the height window + `I_r`; the original `W_r` and full `Z_r`; existential constants. At `d = 2`, conditional on [PALM]'s own steps + (§§3–7) only; for `d ≥ 3`, additionally on [DL] at its OpenAI acceptance. +- **Why it is the recorded obligation and not a different quantity.** + 1. `N` counts *every* window critical point other than the pins. `(N)_2` counts every ordered pair of distinct + ones, at every mutual separation including the coincidence limit, and at every distance from the pins. [PALM] + proves this through the pathwise cap `N(N−1) ≤ N Ψ` and the marked Kac–Rice formula over the same tiling as + [DL] §7 (pin balls, collar, shells, remote region); the pin-neighbourhood pairs are the R1 regime of its §6. + 2. For any Borel region `A` of the torus, the count `N_A` of window critical points in `A` satisfies `N_A ≤ N` + pathwise, hence `(N_A)_q ≤ (N)_q` for every `q`. Every *regional* factorial-moment bound of the recorded type is + therefore implied by the torus-wide one. `reconciliation_check.py` records this reading exactly on integer count + vectors (`MONOTONE`). There is no sense of "factorial-moment/collision estimate" under which a torus-wide bound + leaves a regional bound of the same quantity open. This monotonicity supplies **upper bounds only**: the lower + bound `c r³` is global, realised on the [EDL] (A4) event by pairs within `C_d r` of the midpoint, and is not + claimed for an arbitrary subregion or shrinking selector. OpenAI review 5359701805 states the sharper pathwise + form, adopted here with credit: for any Borel selector `0 ≤ χ_r ≤ 1` on ordered `q`-tuples of distinct window + points (mutual-separation cutoffs tending to zero with `r`, positions relative to the pins and mixed + local/remote subsets included), `T_χ := Σ χ_r ≤ (N)_q` pathwise, hence `E T_χ ≤ C_q r³` uniformly in the + selector, with no separate regional constant. It gives no matching lower bound for a selected subregion, no + pointwise Kac–Rice density bound, no leading coefficient and no `o(r³)` for mixed pairs; those are §5's + residual. + 3. The lower bound is the one the D5 record already carried (`E N(N−1) ≥ 2 P(N ≥ 2) ≥ 2c r³`), now from [EDL] (A4) + in every fixed `d`; the upper bound matches it in order. The pair `(A4, Theorem Q)` is exactly "the eventual + disposition of C2" that catalog C6 said the answer must be compatible with. +- **Consistency across the chain.** [BND] shows that a cap tail plus the first moment cannot give `O(r³)`; [PALM] + uses more (the size-biased cap moment under the witness-conditioned regression, its Lemma M′ and Lemma 5.1), which is + precisely the input [BND] Proposition C names. [FOU]'s `r³ log(1/r)` and [SHP]'s `r³[L/log L]²` are upper bounds + superseded in order by Theorem Q and remain correct. [P_η]'s `O(r⁵)` for both-remote pairs is compatible with a + total of order `r³` whose lower bound (A4) is realised by pairs within `C_d r` of the midpoint. + +## 5. What is not discharged, and the one residual worth naming + +- **Numerical constants** (`C_q`, `c`, `r_*`): open, as in every source. Catalog C8. +- **All-height pairs:** [PALM] counts the window `I_r` only. The all-height statement is not part of the obligation + (which is the window count) and is not claimed. +- **Positional and cluster laws:** [CLU] gives conditional consequences (a Poisson obstruction at scale `r³`, + compactness of the conditional and size-biased laws) but, as it says, does not identify a unique limiting cluster + law. Not part of the obligation. +- **Uniformity** in `T` (`L`), in `d`, as `k ↓ 0` or as marks grow: not claimed anywhere. +- **Elder selection:** separate (D1; `frontiers/elder_lower_all_d_20260929/` for the every-`d` lower bound). +- **Historical numerical certificates** (RN annulus partition, 24-jet / OBL-H5-JETMOD): not discharged and not + claimed; see §6 for what the analytic route does to their *blocking role*. +- **The residual that the phrase "regional shrinking-witness mechanism" could legitimately mean.** Math-#151's catalog + and proof-index text (merged at `ec6db8c`; quoted in §1a) keep `math.rn-region.witness-collision` open on the ground that "the global `Θ(r³)` + result is not a proof of that node". §1 shows the node was never defined as anything but the count estimate. If the + integrating lanes nevertheless want the register to keep a *regional* question, it should be a new node with a + definition, not the old node under a new meaning. The one genuinely open regional statement is the + **localization of the leading-order mass at scale `r`**, defined (with `o` the midpoint) as + `lim_{R→∞} limsup_{r→0} r⁻³ E_{Q_r^W} #{ordered pairs (X, X′) of distinct window points : not both |X − o| ≤ R r + and |X′ − o| ≤ R r} = 0`. What the landed chain gives: (A4) shows pairs within `C_d r` of `o` carry `≥ c r³`, so + the scale-`r` mass is of order `r³`, and local–local pairs between `C_d r` and `R r` belong to it, not to the + residual; [P_η] shows pairs with both witnesses at distance `≥ s_0` from the pins carry `O(r⁵)`; [PALM] §6 R3, the + marked shell bound with `q = 2` whose dyadic sum is `C r³(A_0⁻² + s_0²)`, shows that every pair with a witness in + the shells `R r ≤ |X − o| ≤ s_0` carries at most `C r³(R⁻² + s_0²)`. What it does not give: an `o(r³)` bound for the + **mixed pairs**, one witness within `R r` of `o` and the other at distance `≥ s_0`, where [PALM] §6 R4 gives `C r³` + only. By the three facts above the residual is equivalent to `lim_{s_0→0} lim_{R→∞} limsup_{r→0} r⁻³ M(R, s_0) = 0` + for that mixed count `M(R, s_0)`, and this bound is **not supplied by the landed reviewed chain**. This is a + further theorem, not a gap in Theorem Q. Two author-side + candidates for it exist and are recorded here as **candidates only**: Math-#161 (OpenAI, head `e7f8929`, merged + to main at `5c484bf`; planar; its Lemma M gives joint probability `o(r³)` for the local and fixed-remote nonempty + events, and its §§6–7 one/two law is conditional on Math-#158 Theorems C/G/F) and Math-#159 (Anthropic Claude, + session `015wNj8L…`, v1.1 head `7188bfa`, every fixed `d`; its Lemma 5.2 claims a near–far cross term + `O(r^{9/2})`). Status at `5c484bf`: Math-#161 carries an Anthropic analytic ACCEPT (5359733142, by the session + that authored its consumed [PALM] and [DL], so source-exposed) and a Codex engineering ACCEPT (5359797442); the + four review slices of Math-#158 carry verdicts (A xAI 5359488967; B Codex 5359738045; C Anthropic, this session, + 5359814083; D Codex 5359601495), and Math-#158 merged at `aa28838`; Math-#159 is unreviewed. Neither candidate + is a premise of this record, neither is inventoried, and neither is consumed by any proposed node or edge. The + residual stays `OPEN_ACTIVE` for every fixed `d`: the Math-#161 chain is planar and source-exposed, so an executing + lane could at most record a planar sub-status from it, after its own fold; this record proposes none. A third + candidate, Math-#166 (OpenAI, `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md`, blob `a32fd5f7`, + math head `7c82252`), states the residual as defined here: `r⁻³ E[(N)_q − (N_in)_q] → 0` for every fixed `q ≥ 2` + and any `δ_r → 0` with `δ_r / r → ∞`, retaining counted mass (its near/remote product is `o(r³)`), conditional on + Math-#162; at `bb429d3` its Slices B and C carry Anthropic verdicts (5360178611, 5360216551, session `01NMeKE…`) and + Slice A, the two-scale law itself, is under review. It is unmerged, not a premise, not inventoried, not consumed. + When #162 and #166 are on main with their reviews, an additive successor record can bind the residual's closure + in every fixed `d`; this record does not. `PROPOSED_TRANSITIONS.json` proposes the residual as an explicitly + scoped `OPEN_ACTIVE` node, `math.rn-region.witness-collision.leading-mass-localization`, required by nothing, so that + the question is recorded precisely and blocks nothing. + +## 6. Historical predicates and the shrinking regions + +The graph already contains the pattern for this situation: `regional.fixed-annulus.high-jet-route` +(`SUPERSEDED_NONBLOCKING`, kind `regional_supersession`) records that, for the reviewed fixed-annulus scope, the +analytic route bypasses `hist.CH-LIFT`, `hist.Piece-2-annulus` and `hist.OBL-H5-JETMOD`, while those predicates stay +`OPEN`/`ABSENT` in their original replay scopes. `SELECTOR_REGION.json` still lists `CH-LIFT` and `Piece-2-annulus` as +`OPEN_ACTIVE` for `pin-collision` and `intermediate-r-to-rho`, and `Piece-2-annulus` as `OPEN_ACTIVE` for +`witness-collision`, with the notes "scaled charts reopen CH-LIFT" and "active mathematical blocker outside +fixed-remote". + +With the reviewed D5 chain (planar via `reviews/d5_reconciliation_20260929/`, every fixed `d` via [DL]) and Theorem Q, +the pin-collision, intermediate and witness-collision regions are covered by analytic proofs that use no shrinking +chart lift, no annulus piece and no 24-jet enclosure. This record therefore proposes a second supersession node, +`regional.shrinking-regions.analytic-route`, with the same three `regional_bypass_only` edges and the same discipline: +the historical predicates are **not** discharged, their classifications are **not** changed, and the bypass is scoped +to the reviewed regions and scopes only. The `SELECTOR_REGION.json` cells of those three regions are all resolved +under that node, so that `hard_gate.py::selector_region_report` shows no open cell for them: `CH-LIFT`, +`Piece-2-annulus`, `OBL-H5-JETMOD` and `H5-REMOTE-THRESHOLD` become `BYPASSED_BY_ANALYTIC_ROUTE`; `ALLCELL-FDZ-Q4` +becomes `NOT_REQUIRED` and `ENV-RESCOV` `REDERIVED_QUALITATIVELY`, the values the `fixed-remote` row already carries +for the same reason (the analytic route uses the positive Fourier spectrum and the joint floors of [PALM] Lemma 6.1 / +Lemma E_d, not the historical cell carrier or the historical covariance predicate). The three region ids then move +from `open_region_ids` to `covered_region_ids`; `mesoscopic-scaled-annulus` stays open. The `SELECTOR` check applies +the proposal to the live table and verifies exactly this. The file edit is for the executing lane; if it prefers the +node-only precedent of `regional.fixed-annulus.high-jet-route` (cells untouched), the lists stay unchanged and §7's +selector row does not apply. + +**Scope retained by the bypass.** The bypass inherits every restriction of §2: the original weight `W_r` and the full +normalizer `Z_r`, compact marks with `k_− > 0`, fixed `d` and fixed torus, and the height window `I_r` wherever the +source is window-only. It grants no all-height collision bound, no numerical certificate (RN annulus partition, +24-jet / OBL-H5-JETMOD) and no discharge of any historical predicate outside that analytic scope (OpenAI 5359701805, +item 3). + +## 7. Proposed register transitions (for a non-Claude lane) + +| Surface | Current (`8e61fc4`; `ec6db8c` where Math-#151 changed it) | Proposed | +|---|---|---| +| main `STATUS.md` C6 row | "C6 — Fourier count tail and planar factorial upper bound", `E N(N−1) ≤ C r³ log(1/r)` in `d = 2` | **"C6 — torus-wide factorial moments (witness collision): `Θ(r³)`"** with the §4 statement and scope; keep the Fourier row's count-tail sentence as a component; independence caveat of §0 in the notes column | +| `PROOF_INDEX.md` open-obligations bullet | at `8e61fc4`: "NO COMPLETE PROOF YET: shrinking-separation factorial-moment/collision estimate"; at `ec6db8c`: "NO COMPLETE REGIONAL PROOF YET: `math.rn-region.witness-collision` remains `OPEN_ACTIVE`; the global `Theta(r^3)` result is not a proof of that node" (§1a) | move to the reviewed section, citing [PALM] with reviews 5356233690/5358116559, [EDL] (A4), [DL], and this record; keep a separate open bullet for the §5 residual and for numerical constants | +| `reviews/candidates_pending_20260928/CANDIDATES.md` C6 | at `ec6db8c` (Math-#151): retitled "reviewed global order; regional mechanism open", open task "prove or refute the shrinking-separation witness-collision mechanism encoded by the open graph node" (§1a) | "Resolved at existential scope: optimal order `r³` in every fixed `d ≥ 2` (upper: Palm route; lower: (A4)). Open: numerical constants; leading-mass localization (§5); a unique limiting cluster law." | +| `GRAPH.json` `math.rn-region.witness-collision` | `OPEN_ACTIVE` (kept by Math-#151 at `ec6db8c`, with `reading_rule` `[math.d5-component.offpin-second-moment-review]` and `review_disposition` `OPEN`; §1a) | `PROVED_REVIEWED` at the §4 scope, realized by the aggregate node `math.c6-witness-collision-factorial-moment`, one component node per required file with `fingerprint` = SHA256 and `source` = path, and `required: true` edges from the region node and the aggregate to each; the [LP] evidence is the live reconciled node `math.uniform-matrix-cap-lifetime` (same SHA256, four required reading-rule edges), not a new node; plus a required edge to the planar reading-rule node `math.d5-pin-neighborhood-first-moment` (created by Math-#151) | +| `GRAPH.json` new node | — | `math.rn-region.witness-collision.leading-mass-localization`, `OPEN_ACTIVE`, required by nothing, defined as the scale-`r` localization of §5 with the mixed local/remote count as its open piece, with the §5 candidate note (Math-#161, Math-#159: not premises) | +| `GRAPH.json` new node | — | `regional.shrinking-regions.analytic-route`, `SUPERSEDED_NONBLOCKING`, non-required `regional_bypass_only` edges to `hist.CH-LIFT`, `hist.Piece-2-annulus`, `hist.OBL-H5-JETMOD` (§6) | +| `SELECTOR_REGION.json` | `Piece-2-annulus` / `CH-LIFT` `OPEN_ACTIVE`, `ENV-RESCOV` / `ALLCELL-FDZ-Q4` `OPEN_HISTORICAL` for the shrinking regions | all six cells resolved for `pin-collision`, `intermediate-r-to-rho`, `witness-collision` (§6: four `BYPASSED_BY_ANALYTIC_ROUTE`, `ALLCELL-FDZ-Q4` `NOT_REQUIRED`, `ENV-RESCOV` `REDERIVED_QUALITATIVELY`); the three region ids move to `covered_region_ids`; checked by `SELECTOR` | + +**Ordering.** These transitions presupposed Math-#151's execution of the planar D5 fold; it is on main at `ec6db8c` +(14 nodes, 30 edges), so the nodes this record reuses are live: `math.d5-pin-neighborhood-first-moment` +(`PROVED_REVIEWED`, aggregate, no fingerprint), `math.d5-component.offpin-second-moment-review` (`PROVED_REVIEWED`, +fingerprint `43beb4ef…`, now also the old node's live `reading_rule`) and `math.d5-component.remote-window-proof` +(`PROVED_REVIEWED`, fingerprint `a332bae9…`, a second node for the `math.rn-fixed-remote-window` bytes; Math-#167 +proposes the alignment of the older node). The executing lane, not this record, decides. Within the execution the +order is fixed (xAI 5901345590): first create the residual node +`math.rn-region.witness-collision.leading-mass-localization` (`OPEN_ACTIVE`) and the supersession node, then the +component nodes and edges, and only then move `math.rn-region.witness-collision` to `PROVED_REVIEWED`, so that the +register never loses the localization question under a renamed meaning; the xAI bounded read of [PALM] §§4–6 asked +for in §0 precedes that last step. The keep-open execution (§1a, point (iii)) performs the first three steps, records +the discharge on the aggregate node and then, instead of moving the old node, gives it the residual's definition and +fingerprint (the residual node is then not created separately); both executions leave the same question open. Nothing here edits `GRAPH.json`, `PROOF_INDEX.md`, `STATUS.md` or the catalog (`"executed": false`, checked). + +**Reverse-impact.** Every proposed component node carries the SHA256 of its source, so a later byte change to any of +them invalidates the fold through the existing reader; `coverage_source` metadata is not used. + +## 8. Checks + +`reconciliation_check.py` (stdlib only, run from the repository root): +- **IDENTITIES.** All 22 inventoried files exist as regular files, with no symlink anywhere on their paths, and with + the stated SHA256 and git blob. +- **VERDICTS.** The exact verdict rows and statements quoted in §§1–2 are present as substrings. +- **OBLIGATION.** The live graph still carries `math.rn-region.witness-collision` as an `OPEN_ACTIVE` region with the + fingerprint quoted in §1 (or as `PROVED_REVIEWED` after execution); while open, its `review_disposition`, if any, is + `OPEN` and every `reading_rule` component is live `PROVED_REVIEWED`; the selector still lists the region; + `PROOF_INDEX.md` carries the §1 sentence, or the §1a sentence ("NO COMPLETE REGIONAL PROOF YET" with the node named + `OPEN_ACTIVE`), or a reference to this record; the catalog carries the §1 C6 heading with its open-task sentence, or + the §1a heading with the node named `OPEN_ACTIVE`, or a reference to this record. Any other rewrite of those surfaces + fails the check, and the workflow re-runs on their change. +- **NEGATIVES.** Real filesystem faults on a temporary copy of the inventory: delete [PALM]'s proof, change one byte + of it (all quoted statements kept), replace it by a symlink to identical bytes, replace its directory by a symlink. + Each is rejected. +- **TRANSITIONS.** `PROPOSED_TRANSITIONS.json` names only inventoried files or known node ids; every required file is + a proposed node with matching fingerprint and source, reached by a required edge from each consuming node; the + supporting files are reached by non-required edges; the residual node is `OPEN_ACTIVE` with no required edges; the + supersession node has exactly the three `regional_bypass_only` edges and no historical predicate changes class; the + cross-record node is one the D5 proposal defines and, since `ec6db8c`, live with that proposal's fingerprint; a source already carried by a live fingerprinted node ([LP], + `math.uniform-matrix-cap-lifetime`) is referenced through that node, with equal SHA256 and its four required + reading-rule edges present, and no proposed node duplicates a live source; every proposed node passes the hard + gate's shape rules; `declarative` is true and `executed` is false. +- **SELECTOR.** The selector proposal names every selector of the live table for its three regions, assigns each a + value the hard gate classifies as covered or bypassed, and, applied to the live table, moves exactly those three + region ids from `open_region_ids` to `covered_region_ids`. +- **MONOTONE.** Exact enumeration: `(n_A)_q ≤ (n)_q` for every sub-count and `q ≤ 4`; `2·1{n≥2} ≤ n(n−1) ≤ nΨ`; + the `Θ(r³)` bracket needs both the upper and the lower row. +- **OPEN_RESIDUAL.** §5's open items are recorded (numerical constants; the residual node with its scale-`r` + definition; the mixed-pair bound + "not supplied by the landed reviewed chain", with the candidates named as candidates only) and the residual node + is proposed open. + +Ten mutants must fail: `allow-symlink`, `no-hash`, `drop-edge`, `stale-fingerprint`, `executed-flag`, +`close-residual`, `drop-lower-bound`, `regional-strict`, `clone-live-node`, `open-cell-left`. + +## 9. Relation to other lanes + +- **Math-#151 (Codex; merged at `ec6db8c`, 30 September 00:56Z, final head `609e559`):** executed the planar D5 fold + (14 nodes, 30 edges; the three planar first-moment regions to `PROVED_REVIEWED`) and kept witness-collision open under + the rewritten text quoted in §1a. This record does not conflict with its graph edits and now points at its live nodes; + it supersedes only the catalog and proof-index wording about C6, and only if the executing lane accepts §§1 and 4 + (OpenAI 5359701805 and xAI 5901915195 do). The D4 duplicate (`math.d5-component.remote-window-proof` beside + `math.rn-fixed-remote-window`, same bytes) landed uncollapsed; Math-#167 (this session) proposes the alignment of the + older node, and this record keeps referencing the older live node with supporting edges only. +- **Math-#155 (OpenAI, C7):** unrelated to the count; not consumed. +- **Math-#161 (OpenAI, `e7f8929`, merged at `5c484bf`) and Math-#159 (Anthropic Claude, other session, `7188bfa`):** + author-side candidates for the §5 residual; not premises, not inventoried, not consumed (§5, with their review + status at `5c484bf`). Their appearance changes none of this record's status rows, nodes or edges; a planar + sub-status of the residual would be a further transition by an executing lane, after its own fold. +- **Math-#166 (OpenAI, two-scale cluster geometry, math head `7c82252`, open):** the consumer whose `TWO_SCALE_LAW.md` + states the §5 residual exactly, conditional on Math-#162; Slices A, B and C carry the other Claude session's verdicts + (5360227991, 5360178611, 5360216551; one provider); unmerged; not consumed here; the residual closes by a successor + record after both land. +- **Math-#158 (OpenAI, planar cubic cluster law, `bfcc67dc`, merged at `aa28838`):** premise of Math-#161 §§6–7; this session + reviewed its Slice C (5359814083); not consumed here. +- **main#207 (closed 30 September 00:57Z, after Math-#151 merged):** the pickup for this record is comment 5900687550 + and the delivery 5900887059. The register execution stays with a non-Claude lane. diff --git a/reviews/c6_witness_collision_reconciliation_20260929/RESULTS.json b/reviews/c6_witness_collision_reconciliation_20260929/RESULTS.json new file mode 100644 index 000000000..f68f83614 --- /dev/null +++ b/reviews/c6_witness_collision_reconciliation_20260929/RESULTS.json @@ -0,0 +1,16 @@ +{ + "checks": { + "IDENTITIES": true, + "MONOTONE": true, + "NEGATIVES": true, + "OBLIGATION": true, + "OPEN_RESIDUAL": true, + "SELECTOR": true, + "TRANSITIONS": true, + "VERDICTS": true + }, + "inventory_files": 22, + "object": "C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1", + "passed": true, + "scope": "identity, verdict-row, obligation, filesystem-negative, transition-chain, monotonicity, open-item and selector-cell checks; no mathematics is re-proved" +} diff --git a/reviews/c6_witness_collision_reconciliation_20260929/SOURCE_FILES.json b/reviews/c6_witness_collision_reconciliation_20260929/SOURCE_FILES.json new file mode 100644 index 000000000..e94572a01 --- /dev/null +++ b/reviews/c6_witness_collision_reconciliation_20260929/SOURCE_FILES.json @@ -0,0 +1,36 @@ +{ + "schema": 1, + "object": "C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1", + "scientific_effect": "NONE", + "disposition": "PROPOSAL_FOR_A_NON_CLAUDE_LANE", + "declarative": true, + "executed": false, + "base": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "inventory_read_at": "8e61fc4c055b56e526b5b3488104945e8e0e29cc", + "files": [ + { + "path": "PROPOSED_TRANSITIONS.json", + "bytes": 42904, + "sha256": "5698a436db08b4c8942a2df77a1d1bddf52261fffed61c9a323d7e9a8d285f18", + "git_blob": "8d70ee7494327e4363548d4a74f0906b30acb818" + }, + { + "path": "RECONCILIATION.md", + "bytes": 41289, + "sha256": "96a0f0bf2700b64b933d08d3c81efbfa8388aa2094e5528a1a6bd627e69f902a", + "git_blob": "e488c0afed5c41c2872de0902f0e3d1c08e1b2ce" + }, + { + "path": "RESULTS.json", + "bytes": 482, + "sha256": "8fee5c3e02f2d10de095217d98e7b8a7a7bd7725821a2a9ec9680018cf728a33", + "git_blob": "f68f83614484a6e9f9ae3f96502c88a954dce6ee" + }, + { + "path": "reconciliation_check.py", + "bytes": 28037, + "sha256": "a19a01fec6957463229b2297195860105ac7be735f2dd297ffb2907ba2d5d2fa", + "git_blob": "a7de1d22c9177cb15133c425283ad990d928efb8" + } + ] +} diff --git a/reviews/c6_witness_collision_reconciliation_20260929/reconciliation_check.py b/reviews/c6_witness_collision_reconciliation_20260929/reconciliation_check.py new file mode 100644 index 000000000..a7de1d22c --- /dev/null +++ b/reviews/c6_witness_collision_reconciliation_20260929/reconciliation_check.py @@ -0,0 +1,452 @@ +"""Checks for C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1 (RECONCILIATION.md, PROPOSED_TRANSITIONS.json). +Stdlib only; run from the repository root. + + IDENTITIES every inventoried file exists as a regular file, with no symlink anywhere on its path (file or parent + directory), and with its stated SHA256 and git blob. + VERDICTS the exact statements and verdict rows quoted in RECONCILIATION.md sections 1-2 occur in the records. + OBLIGATION the live graph carries math.rn-region.witness-collision as a region node, either OPEN_ACTIVE with the + recorded fingerprint or already PROVED_REVIEWED; SELECTOR_REGION.json lists the region; PROOF_INDEX.md + carries the line-45 obligation sentence or a reference to this record. + NEGATIVES real filesystem faults on a temporary copy of the inventory applied to the PALM proof (delete, change one + byte with every quoted statement kept, symlink to identical bytes, symlinked parent directory) are each + REJECTED. + TRANSITIONS PROPOSED_TRANSITIONS.json names only inventoried files or known node ids; every required file is a + proposed node with matching fingerprint and source, reached by a required edge from each consuming node; + supporting files are reached by non-required edges; the residual node is OPEN_ACTIVE with no required + edges; the supersession node has exactly three regional_bypass_only edges and no historical predicate + appears among the transitions; the cross-record nodes are defined by the D5 proposal; every proposed + node passes the hard gate's shape rules; declarative is true and executed is false. + MONOTONE exact enumeration: (n_A)_q <= (n)_q for every sub-count and q <= 4; 2*1{n>=2} <= n(n-1) <= n*Psi for + Psi >= n; the Theta(r^3) bracket needs both the upper and the lower row. + OPEN_RESIDUAL the open items of section 5 are recorded and the residual node is proposed open. +Mutants (each must fail): allow-symlink, no-hash, drop-edge, stale-fingerprint, executed-flag, close-residual, +drop-lower-bound, regional-strict. +""" +import argparse +import hashlib +import itertools +import json +import os +import pathlib +import shutil +import sys +import tempfile + +MUTANTS = ("allow-symlink", "no-hash", "drop-edge", "stale-fingerprint", "executed-flag", "close-residual", + "drop-lower-bound", "regional-strict", "clone-live-node", "open-cell-left") +MUT = None +HERE = "reviews/c6_witness_collision_reconciliation_20260929" +PALM = "frontiers/c6_palm_route_20260929/PROOF.md" +EDL = "frontiers/elder_dimension_lift_20260928/PROOF.md" +EDL_REVIEW = "reviews/elder_dimension_lift_claude_20260928/REVIEW.md" +D5_TRANSITIONS = "reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json" +GRAPH = "frontiers/downstream_gate_20260925/GRAPH.json" +SELECTOR = "frontiers/downstream_gate_20260925/SELECTOR_REGION.json" +WITNESS = "math.rn-region.witness-collision" +AGGREGATE = "math.c6-witness-collision-factorial-moment" +RESIDUAL = "math.rn-region.witness-collision.leading-mass-localization" +SUPERSESSION = "regional.shrinking-regions.analytic-route" +HIST = ("hist.CH-LIFT", "hist.Piece-2-annulus", "hist.OBL-H5-JETMOD") +D5_AGGREGATE = "math.d5-pin-neighborhood-first-moment" +LIVE_LP = "math.uniform-matrix-cap-lifetime" +LP = "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md" +CATALOG = "reviews/candidates_pending_20260928/CANDIDATES.md" +OPEN_CELLS = ("OPEN_ACTIVE", "OPEN_HISTORICAL", "NOT_DISCHARGED", "PARTIAL_COVER_ONLY", "PARTIAL_PR7_REDUCTION", + "REOPENED", "PARTIAL_COVER_DECLARED_REGION") # hard_gate.py::selector_region_report +CLASSES = {"PROVED_REVIEWED", "SUPERSEDED_NONBLOCKING", "REFUTED", "BLOCKED_ABSENT", "OPEN_ACTIVE", "OPEN_HISTORICAL", + "AUTHOR_SIDE_CANDIDATE", "AUTHOR_SIDE_REDUCTION", "COVERED_BY_CANDIDATE", "ENGINEERING_CONTROL", "HOLD", + "FALSE", "REVALIDATION_REQUIRED"} + +# path: (sha256, git blob), read at main 8e61fc4 +INVENTORY = { + PALM: ("aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5"), + "frontiers/c6_palm_route_20260929/README.md": + ("56c0b94f9b1fcc1207b8e0bc4d1a7e89b6cba7b285fcbf550fd8dd1d9c30f3e1", "1ebf8ede049f23628db660c422612654b0a1f3fe"), + "frontiers/c6_palm_route_20260929/SOURCE_MAP.json": + ("c7cc195f00cc0c45577b7a27eb5e76f5cd9f1dbd1f11d855bcc9be427d699278", "b528fe10aed99f1ee0947bb58da8623b0fabe354"), + "frontiers/c6_palm_route_20260929/palm_exact_check.py": + ("3bd19ec659b2ebceaf2fadeff29c1913ab28998d5182503a6de012c018bb550b", "6b0c73baeeb2ac566160e23cc405daec07eb7585"), + "frontiers/c6_palm_route_20260929/RESULTS.json": + ("7812de453ff1efd77ffc2431874243713615ce4fcc0e55f2beae991e24f6f6bc", "cd8003da3bdbbe3a495e0bb81d6767c957740372"), + "frontiers/c6_factorial_moment_20260929/PROOF.md": + ("b19927563e2e9ff8f38648b61cd3f2c09ee1a081494b98cc285283554746e6d3", "f5bd013b7134be9efd246fc9bbb8d18ad5a728bc"), + "frontiers/c6_sharpened_20260929/PROOF.md": + ("9f45274ab59d4f4417d1a51e9c97f5916240fdb0a4640ca6f0d3e9e0aea37dd7", "70ba19726a9114bfceae04d021095e6c8eed5026"), + "frontiers/c6_fourier_cutoff_20260929/PROOF.md": + ("c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733", "1d9177a259654df0fb7c558fcb803685e09608d2"), + "reviews/c6_fourier_completion_20260929/READING_NOTE.md": + ("62f84b68b3bcd034f1427eacf3919c7fb418d311af64f0ab2807d17682cc00b3", "682d7045e71613efb92f6efb312b0bc210dfdabd"), + "frontiers/c6_count_cap_boundary_20260929/PROOF.md": + ("264a5e762f598145e9230aa0947acb2e0354efdb2e3adf65d331e7ee0ba52011", "3d4c28a4ca8784b6c71c13ca804a817a6cb52e6f"), + "frontiers/c6_rare_cluster_laws_20260929/PROOF.md": + ("c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb", "2ab625cedfc2e47575c4a7fa4853413418c532da"), + "frontiers/d5_dimension_lift_20260929/PROOF.md": + ("6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", "9d82c707fdb17d3072a8930f26dabedf59e456fc"), + "frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md": + ("bdb83c28ef77041ae8d5903b1e46ec96b8046660a07be439bcd60cac02f8c7dc", "45e4324ff262f6dba1880431a2cdfe25c231a9e2"), + EDL: ("b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", "7303bd791a68a1139251f0f6e403a9f7cc89b006"), + EDL_REVIEW: ("032bb4c79e4a985e5b53f5abb6bb06da4d643ec7345e4df9591e2046fb287e12", "e2b9fcdc8d24f6a4d532afb7d02627e4716158d1"), + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md": + ("9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", "dfed3b8d318a3ab1950957f393307733a4bef3f2"), + "frontiers/remote_collision_20260928/PROOF.md": + ("b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2", "7b48a88e2af54e759e89a8c8219573bb450a65ea"), + "reviews/d5_remote_collision_grok_20260928/REVIEW.md": + ("f8289546ee82fb386a0d0b6b54fbb99f258a2dc2a959ae231ae80cb4baa61e1e", "c18560f86cfb81e47515f65c16186e6a70577512"), + "reviews/d5_offpin_second_moment_20260928/REVIEW.md": + ("43beb4ef7da5bf71e15164d5c4bc1862db387d83f906aebdf9831625b1ca6369", "62697c471d4299590d807ab64b03b0ebb0d4dd43"), + "reviews/d5_reconciliation_20260929/RECONCILIATION.md": + ("14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432", "4062d76bdf3c60bc73f04295b88c245bcee395de"), + D5_TRANSITIONS: + ("02f91870b636635244a767ba92f2490dd480b1720d456889216a56e07705a68b", "d928969e0c880a3fc794d7935ed31172488c3f98"), + "frontiers/remote_window_20260924/PROOF.md": + ("a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", "b383bfcc88ec4ad497dff01fb6640e429ba24a84"), +} + +VERDICTS = { + PALM: ["**Theorem Q (factorial moments, every fixed dimension).**", + "E_(Q_r^W)[ (N)_q ] <= C_q r^3.", + "**Corollary Theta.** In every fixed `d >= 2`,", + "c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3,", + "c r^3 <= Q_r^W{ N >= 2 } <= C r^3.", + "The lower bounds are [EDL] (A4)."], + "frontiers/c6_palm_route_20260929/README.md": [ + "| OpenAI 5356233690 | `415044b` | §4 (parametric tail, regression facts, Proposition 4.5) sound, with the clarifications applied in v1.1 |", + "| OpenAI 5358116559 | `06bc0d6` | R3a repaired; ACCEPT §§5–7, Theorem Q, Corollaries Theta and P at the stated source scope, conditional only on the pinned inputs |"], + "frontiers/c6_factorial_moment_20260929/PROOF.md": [ + "That review ACCEPTs §§4–6 (Lemmas E, D and M)", + "- v1.3 makes the one indexing correction requested in OpenAI review 5355457002, which ACCEPTs Lemma R and the"], + "frontiers/d5_dimension_lift_20260929/README.md": [] if False else [], + "frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md": [ + "continuum record 5894512272 (Grok: (P10), (P11), (P12), (P18), region II ACCEPT for the planar source, no `d > 2`", + "refers to the OpenAI nonauthor review 5357858391", + "refers to the follow-up OpenAI nonauthor review 5357882570"], + EDL: ["Q_r^W(N_r>=2)>=c r^3,", "E_Qr^W[N_r(N_r-1)]>=c r^3."], + EDL_REVIEW: ["| §9 (363–415), **(A4)**: two extra index-(d−1) critical points, `Q^W(N_r ≥ 2) ≥ c r³`, factorial lower bound | **ACCEPT**; see §5 below. |"], + "reviews/c6_fourier_completion_20260929/READING_NOTE.md": [ + "review 5356335148", + "The review ACCEPTs Theorem F (F1)-(F2) for every fixed d>=2 at the stated original"], + "frontiers/c6_fourier_cutoff_20260929/PROOF.md": [ + "**Theorem W (window consequence).** Let G_d denote the dimension-matched", + "E_Qr^W N(N-1) <= C r^3 log(1/r),"], + "frontiers/c6_count_cap_boundary_20260929/PROOF.md": [ + "**Theorem B (limit of these hypotheses).**", "**Proposition C (the missing correlation).**"], + "frontiers/c6_sharpened_20260929/PROOF.md": ["**Theorem S_d (every fixed `d ≥ 2`, conditional).** Assume Theorem G_d of Math-#141:"], + "frontiers/c6_rare_cluster_laws_20260929/PROOF.md": ["(Hq) E (N)_q <= A_q e for every fixed integer q>=2."], + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md": ["00 mutual witness separation\").", + "- **A torus-wide second factorial moment, or `math.rn-region.witness-collision` in its full torus-wide sense.**"], + "frontiers/remote_window_20260924/PROOF.md": [ + "- Witness tuples whose mutual separation tends to zero; (15) does not control that collision."], + "reviews/d5_offpin_second_moment_20260928/REVIEW.md": [ + "| P_eta | `E N_eta(N_eta-1) <= C_eta r^5` for window points in `{|X|>=eta}` minus pins, fixed `eta>0` | ACCEPT |", + "| Torus-wide `E N(N-1)` | pair intensity with a witness near a pin | CANDIDATE pending review and test (catalog C6) |"], + "reviews/d5_reconciliation_20260929/RECONCILIATION.md": [ + "- **D5-open (catalog C6): shrinking multiple-witness collision.**", + "**No upper bound.** The optimal order is open."], +} +VERDICTS = {k: v for k, v in VERDICTS.items() if v} + +REQUIRED_SUPPORT = { + "frontiers/d5_dimension_lift_20260929/README.md": None, # not inventoried; review ids are bound through the crosswalk +} + + +def git_blob(data): + return hashlib.sha1(b"blob %d\0" % len(data) + data).hexdigest() + + +def no_symlink_on_path(root, path): + parts = pathlib.PurePosixPath(path).parts + return not any((root / pathlib.PurePosixPath(*parts[:i])).is_symlink() for i in range(1, len(parts) + 1)) + + +def check_identities(root): + for path, (sha, blob) in INVENTORY.items(): + p = root / path + if not os.path.lexists(p): + return False + if MUT != "allow-symlink" and not no_symlink_on_path(root, path): + return False + if not p.is_file(): + return False + data = p.read_bytes() + if MUT != "no-hash" and (hashlib.sha256(data).hexdigest() != sha or git_blob(data) != blob): + return False + return True + + +def check_verdicts(root): + for path, needles in VERDICTS.items(): + text = (root / path).read_text(encoding="utf-8") + if not all(n in text for n in needles): + return False + return True + + +def check_obligation(root): + graph = json.loads((root / GRAPH).read_text(encoding="utf-8")) + node = graph["nodes"].get(WITNESS) + if not node or node.get("kind") != "region": + return False + if node.get("classification") == "OPEN_ACTIVE": + ok = node.get("fingerprint") == "eta->0 mutual witness separation" + # since ec6db8c (Math-#151) the open node carries a review disposition and a reading rule + ok &= node.get("review_disposition") in (None, "OPEN") + ok &= all(graph["nodes"].get(x, {}).get("classification") == "PROVED_REVIEWED" for x in node.get("reading_rule", [])) + else: + ok = node.get("classification") == "PROVED_REVIEWED" + sel = json.loads((root / SELECTOR).read_text(encoding="utf-8")) + ok &= "witness-collision" in sel.get("regions", []) + index = (root / "PROOF_INDEX.md").read_text(encoding="utf-8") + ok &= ("NO COMPLETE PROOF YET: shrinking-separation factorial-moment/collision estimate." in index # through bb429d3 + or "**NO COMPLETE REGIONAL PROOF YET:** `math.rn-region.witness-collision` remains `OPEN_ACTIVE`" in index # ec6db8c + or HERE in index) + catalog = (root / CATALOG).read_text(encoding="utf-8") + ok &= (("## C6. Torus-wide second factorial moment" in catalog # through bb429d3 + and "a valid full-window factorial upper" in catalog) + or ("## C6. Torus-wide factorial moments: reviewed global order; regional mechanism open" in catalog # ec6db8c + and "`math.rn-region.witness-collision` remains `OPEN_ACTIVE`" in catalog) + or HERE in catalog) + return ok + + +def check_negatives(src): + """Copy the inventory to a temporary root, apply each real filesystem fault to the PALM proof, and require + rejection. The changed copy keeps every quoted statement, so only the byte binding can reject it.""" + rejected = [] + for fault in ("delete", "change", "symlink", "symlink-parent"): + with tempfile.TemporaryDirectory() as tmp: + root = pathlib.Path(tmp) + for path in INVENTORY: + dst = root / path + dst.parent.mkdir(parents=True, exist_ok=True) + shutil.copyfile(src / path, dst) + target = root / PALM + if fault == "delete": + target.unlink() + elif fault == "change": + target.write_bytes(target.read_bytes() + b"\n") + assert all(n in target.read_text(encoding="utf-8") for n in VERDICTS[PALM]) + elif fault == "symlink": + moved = root / "elsewhere" / "PROOF.md" + moved.parent.mkdir() + shutil.move(str(target), str(moved)) + os.symlink(moved, target) + else: + moved = root / "elsewhere_dir" + shutil.move(str(target.parent), str(moved)) + os.symlink(moved, target.parent, target_is_directory=True) + assert not target.is_symlink() and target.is_file() + rejected.append(not check_identities(root)) + return all(rejected) and len(rejected) == 4 + + +def check_transitions(root): + spec = json.loads((root / HERE / "PROPOSED_TRANSITIONS.json").read_text(encoding="utf-8")) + d5 = json.loads((root / D5_TRANSITIONS).read_text(encoding="utf-8")) + live = json.loads((root / GRAPH).read_text(encoding="utf-8"))["nodes"] + d5_nodes = {n["id"] for n in d5["proposed_graph_nodes"]} + nodes = {n["id"]: n for n in spec["proposed_graph_nodes"]} + edges = list(spec["proposed_graph_edges"]) + known = set(nodes) | d5_nodes | set(live) + entries = spec["status_rows"] + spec["graph"] + if MUT == "executed-flag": + spec["executed"] = True + if MUT == "close-residual": + for e in spec["graph"]: + if e["node"] == RESIDUAL: + e["proposed"] = "PROVED_REVIEWED" + nodes[RESIDUAL]["classification"] = "PROVED_REVIEWED" + if MUT == "drop-lower-bound": + for e in entries: + e["required"] = [p for p in e["required"] if p not in (EDL, EDL_REVIEW)] + if MUT == "drop-edge": + edges = [e for e in edges if not (e["from"] == WITNESS and nodes.get(e["to"], {}).get("source") == PALM)] + if MUT == "clone-live-node": + nodes["math.c6-component.lifetime-parent"] = dict(nodes[AGGREGATE], id="math.c6-component.lifetime-parent", + kind="reading_rule_component", source=LP, + fingerprint=INVENTORY[LP][0]) + if MUT == "stale-fingerprint": + for n in nodes.values(): + if n.get("source") == PALM: + n["fingerprint"] = "0" * 64 + ok = spec.get("declarative") is True and spec.get("executed") is False + # shape rules of the hard gate, applied to the proposed nodes and edges + for n in nodes.values(): + ok &= n.get("classification") in CLASSES and type(n.get("controlling")) is bool and n["controlling"] is False + seen = set() + for e in edges: + ok &= {"from", "to", "required", "relation"} <= set(e) and type(e["required"]) is bool + ok &= e["from"] in known and e["to"] in known + key = (e["from"], e["to"], e["relation"]) + ok &= key not in seen + seen.add(key) + # every named file is inventoried; every required or supporting file is a fingerprinted node reached by an edge + # of the right kind. Evidence nodes may be this record's or the D5 proposal's (cross-record reuse, same bytes). + # a source already carried by a live fingerprinted node is referenced through that node; then this record's + # nodes; then the D5 proposal's (cross-record reuse, same bytes) + live_fp = {nid: n for nid, n in live.items() if isinstance(n, dict) and n.get("fingerprint") and n.get("source")} + evidence = dict(live_fp) + for nid, n in nodes.items(): + evidence.setdefault(nid, n) + for n in d5["proposed_graph_nodes"]: + evidence.setdefault(n["id"], n) + by_source = {} + for r in spec["live_nodes_reused"]: # the live nodes this record names come first + n = live.get(r["id"]) + ok &= isinstance(n, dict) and (not r.get("fingerprint") or n.get("fingerprint") == r["fingerprint"]) + if isinstance(n, dict) and r.get("fingerprint") and n.get("source") in INVENTORY: + by_source.setdefault(n["source"], r["id"]) + for nid, n in evidence.items(): # then other live nodes, this record's, then D5's + if "fingerprint" in n and n.get("source") in INVENTORY: + by_source.setdefault(n["source"], nid) + # a source must not be proposed twice (one node per byte identity across the two records) + ok &= len({n.get("source") for n in nodes.values() if "fingerprint" in n and n.get("source")} + & {n.get("source") for n in d5["proposed_graph_nodes"] if "fingerprint" in n and n.get("source")}) == 0 + ok &= all(evidence[nid]["fingerprint"] == INVENTORY[src][0] for src, nid in by_source.items()) + # no proposed node duplicates a source the live graph already carries with a fingerprint (Codex 4139312869) + ok &= not ({n.get("source") for n in nodes.values() if "fingerprint" in n and n.get("source")} + & {n["source"] for n in live_fp.values()}) + # the reused live node is PROVED_REVIEWED with its four required reading-rule components, all PROVED_REVIEWED + lp = live.get(LIVE_LP, {}) + ok &= lp.get("classification") == "PROVED_REVIEWED" and lp.get("source") == LP and lp.get("fingerprint") == INVENTORY[LP][0] + rr = lp.get("reading_rule", []) + ok &= len(rr) == 4 and all(live.get(x, {}).get("classification") == "PROVED_REVIEWED" for x in rr) + ok &= by_source.get(LP) == LIVE_LP + ok &= by_source.get("frontiers/remote_window_20260924/PROOF.md") == "math.rn-fixed-remote-window" + have_req = {(e["from"], e["to"]) for e in edges if e["required"] is True} + have_sup = {(e["from"], e["to"]) for e in edges if e["required"] is False} + for e in entries: + consumer = e.get("node") or e.get("graph_node") + ok &= all(p in INVENTORY for p in e["required"] + e.get("supporting", [])) + ok &= all(p in by_source and (consumer, by_source[p]) in have_req for p in e["required"]) + ok &= all(p in by_source and (consumer, by_source[p]) in have_sup for p in e.get("supporting", [])) + for nid in e.get("required_nodes", []): + ok &= nid in known and (consumer, nid) in have_req + # the witness node is proposed PROVED_REVIEWED with the lower bound in its chain + wit = next(e for e in spec["graph"] if e["node"] == WITNESS) + ok &= wit["proposed"] == "PROVED_REVIEWED" and {EDL, EDL_REVIEW} <= set(wit["required"]) + row = next(e for e in spec["status_rows"] if e["id"] == "C6") + ok &= "c r^3 <= E_{Q_r^W} N(N-1) <= C r^3" in row["scope"] and {EDL, EDL_REVIEW} <= set(row["required"]) + # the residual node is open and blocks nothing + res = next(e for e in spec["graph"] if e["node"] == RESIDUAL) + ok &= res["proposed"] == "OPEN_ACTIVE" and nodes[RESIDUAL]["classification"] == "OPEN_ACTIVE" + ok &= res["required"] == [] and not any(e["from"] == RESIDUAL and e["required"] for e in edges) + ok &= not any(e["to"] == RESIDUAL and e["required"] for e in edges) + # the supersession node: exactly three regional_bypass_only edges, none required; no historical predicate moves + byp = [e for e in edges if e["from"] == SUPERSESSION] + ok &= nodes[SUPERSESSION]["classification"] == "SUPERSEDED_NONBLOCKING" + ok &= sorted(e["to"] for e in byp) == sorted(HIST) and all(e["relation"] == "regional_bypass_only" + and e["required"] is False for e in byp) + ok &= not any(e["node"].startswith("hist.") for e in spec["graph"]) + ok &= set(spec["historical_predicates_unchanged"]) >= set(HIST) + # cross-record nodes are the D5 proposal's + ok &= D5_AGGREGATE in d5_nodes and "math.d5-component.offpin-second-moment-review" in d5_nodes + ok &= (WITNESS, D5_AGGREGATE) in have_req and (AGGREGATE, D5_AGGREGATE) in have_req + ok &= (WITNESS, LIVE_LP) in have_req and (AGGREGATE, LIVE_LP) in have_req + # since ec6db8c (Math-#151) the reused D5 nodes are live: bound to the D5 proposal's classification and fingerprint + d5_by_id = {n["id"]: n for n in d5["proposed_graph_nodes"]} + for nid in (D5_AGGREGATE, "math.d5-component.offpin-second-moment-review"): + ln = live.get(nid, {}) + ok &= ln.get("classification") == "PROVED_REVIEWED" == d5_by_id[nid].get("classification") + ok &= (ln.get("fingerprint") or None) == (d5_by_id[nid].get("fingerprint") or None) + # the D4 bytes are carried by two live nodes (the D5 proposal landed as written); they must agree byte for byte + dup = live.get("math.d5-component.remote-window-proof") + if isinstance(dup, dict): + ok &= dup.get("fingerprint") == live.get("math.rn-fixed-remote-window", {}).get("fingerprint") \ + == INVENTORY["frontiers/remote_window_20260924/PROOF.md"][0] + return bool(ok) + + +def check_selector(root): + """Applied to the live selector table, the proposal leaves no open cell in its regions and moves exactly those + region ids from open to covered (Codex 4139312863).""" + spec = json.loads((root / HERE / "PROPOSED_TRANSITIONS.json").read_text(encoding="utf-8")) + live = json.loads((root / SELECTOR).read_text(encoding="utf-8")) + prop = spec["selector_region_proposal"] + cells = dict(prop["cells"]) + if MUT == "open-cell-left": + cells.pop("ENV-RESCOV", None) + regions = list(prop["regions"]) + ok = set(regions) <= set(live["regions"]) and len(regions) == 3 + ok &= set(cells) == set(live["selectors"]) + ok &= all(v not in OPEN_CELLS for v in cells.values()) + table = {s: dict(row) for s, row in live["selectors"].items()} + for s, v in cells.items(): + for reg in regions: + table[s][reg] = v + def is_open(reg): + return any(table[s].get(reg) in OPEN_CELLS for s in live["selectors"]) + ok &= not any(is_open(reg) for reg in regions) + ids = ["math.rn-region." + reg for reg in regions] + covered = list(live["covered_region_ids"]) + [i for i in ids if i not in live["covered_region_ids"]] + opened = [i for i in live["open_region_ids"] if i not in ids] + ok &= sorted(prop["resulting_covered_region_ids"]) == sorted(covered) + ok &= sorted(prop["resulting_open_region_ids"]) == sorted(opened) + ok &= all(("math.rn-region." + reg) in live["open_region_ids"] for reg in regions) + ok &= prop.get("under_node") == SUPERSESSION + return bool(ok) + + +def falling(n, q): + out = 1 + for i in range(q): + out *= max(n - i, 0) + return out + + +def check_monotone(): + """Regional counts are sub-counts: (n_A)_q <= (n)_q. Enumerate every integer vector of 3 regional counts with + total <= 6 and every sub-collection of regions.""" + ok = True + for counts in itertools.product(range(7), repeat=3): + n = sum(counts) + if n > 6: + continue + for mask in itertools.product((0, 1), repeat=3): + n_a = sum(c for c, m in zip(counts, mask) if m) + for q in (2, 3, 4): + lhs, rhs = falling(n_a, q), falling(n, q) + ok &= (lhs < rhs) if MUT == "regional-strict" else (lhs <= rhs) + for psi in (n, n + 2, n + 7): + ok &= 2 * (1 if n >= 2 else 0) <= n * (n - 1) <= n * psi + # the Theta(r^3) bracket: an upper row alone gives O, a lower row alone gives Omega; both give Theta + upper, lower = (3, 3) + ok &= upper == lower == 3 + return ok + + +def check_open_residual(root): + text = (root / HERE / "RECONCILIATION.md").read_text(encoding="utf-8") + ok = "**Numerical constants**" in text and "leading-mass-localization" in text + ok &= "localization of the leading-order mass at scale `r`" in text and "M(R, s_0)" in text + ok &= "not supplied by the landed reviewed chain" in text and "candidates only" in text + spec = json.loads((root / HERE / "PROPOSED_TRANSITIONS.json").read_text(encoding="utf-8")) + res = next(e for e in spec["graph"] if e["node"] == RESIDUAL) + ok &= res["proposed"] == "OPEN_ACTIVE" and MUT != "close-residual" + return ok + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument("--mutant", choices=MUTANTS) + MUT = ap.parse_args().mutant + root = pathlib.Path(".").resolve() + ident = check_identities(root) + checks = {"IDENTITIES": ident, "VERDICTS": ident and check_verdicts(root), "OBLIGATION": check_obligation(root), + "NEGATIVES": check_negatives(root), "TRANSITIONS": check_transitions(root), + "MONOTONE": check_monotone(), "OPEN_RESIDUAL": check_open_residual(root), + "SELECTOR": check_selector(root)} + passed = all(checks.values()) and len(checks) == 8 + print(json.dumps({"object": "C6-WITNESS-COLLISION-RECONCILIATION-20260929-v1", "checks": checks, "passed": passed, + "inventory_files": len(INVENTORY), + "scope": "identity, verdict-row, obligation, filesystem-negative, transition-chain, monotonicity, " + "open-item and selector-cell checks; no mathematics is re-proved"}, indent=2, sort_keys=True)) + return 0 if passed else 1 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/reviews/d2_remainder_full_depth_claude_20260930/RESULTS.json b/reviews/d2_remainder_full_depth_claude_20260930/RESULTS.json new file mode 100644 index 000000000..efa655a8c --- /dev/null +++ b/reviews/d2_remainder_full_depth_claude_20260930/RESULTS.json @@ -0,0 +1,126 @@ +{ + "checks": { + "C1_axial_and_transverse_rows": { + "info": { + "U3_second_order": "13/40 = f5/40", + "axial_rows": 4, + "transverse_rows": 2, + "vanishing_terms_asserted": { + "U0": [ + "r^1", + "r^3" + ], + "U1": [ + "r^-1", + "r^1", + "r^3" + ], + "U2": [ + "r^-1", + "r^1", + "r^3" + ], + "U3": [ + "r^-3", + "r^-2", + "r^-1", + "r^1", + "r^3" + ] + } + }, + "passed": true + }, + "C2_R7_affine_determinant": { + "info": { + "bound_checks": 390, + "identities": 150, + "inertia_crossings_both_endpoints_nonsingular": 12, + "singular_blocks_with_nonzero_adjugate": 1 + }, + "passed": true + }, + "C3_R6_filtered_determinant": { + "info": { + "checks": 10, + "inertia_crossings": 3 + }, + "passed": true + }, + "C4_reference_product": { + "info": { + "inertia_patterns": 3 + }, + "passed": true + }, + "C5_soft_integral_and_R15": { + "info": { + "grid_points": 1197, + "identity": true, + "points_where_hypothesis_fails": 1196 + }, + "passed": true + }, + "C6_exponent_ledger": { + "info": { + "a^(7/3)": [ + "7/3", + "0", + "-7" + ], + "delta = r/k at k = eta": [ + "1/9", + "0", + "0" + ], + "delta^3 k^(-2/3) at general k": [ + "1", + "-14/3", + "0" + ], + "ell eta^(-11/3)": [ + "7/18", + "0", + "0" + ], + "ell^(2/3) a^(-1/3)": [ + "1/3", + "0", + "1" + ], + "eta^(7/3)": [ + "7/18", + "0", + "0" + ], + "r k^(1/3)": [ + "1/3", + "0", + "0" + ], + "r^2 k^(-2/3)": [ + "2/3", + "-4/3", + "0" + ], + "riemann_cells": { + "int_a^1": 269, + "int_eta^1": 48 + } + }, + "passed": true + }, + "C7_loss_split_model": { + "info": { + "ell=1e-18": "3994923/3850000", + "ell=1e-36": "607474113/605194808", + "ell=1e-54": "77029/77000", + "ell=1e-72": "770029/770000" + }, + "passed": true + } + }, + "object": "CL-D2-REMAINDER-FULLDEPTH-20260930-v1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md b/reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md new file mode 100644 index 000000000..529c5264e --- /dev/null +++ b/reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md @@ -0,0 +1,208 @@ +# Nonauthor-lane full-depth review: D2 Theorem R (bounded unrestricted short-lifetime remainder) + +Scientific effect: **NONE**. This record changes no register, status, graph node, lemma flag, prize or source. +It supplies the second-provider read that the referee map lists as missing for the manuscript's remainder +statement (`ν(ℓ) = cℓ^{−1/3} + O(1)`, tagged `[R·1P]`: OpenAI author, xAI/Grok reviewer R1–R6 on main #67). +Integration is a separate act by a non-Claude lane. + +## Object and exposure + +| Field | Value | +|---|---| +| Object | LIFETIME-BOUNDED-REMAINDER-20260924-v1 (OpenAI/ChatGPT), Theorem R, review points R1–R6 as assigned on main #67 (5836435928, 5841269622) | +| Reviewed bytes | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf80bfbe896948d5d489b2d5842a81c481`, 17734 B, SHA256 `380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a` — the same immutable bytes the xAI review bound | +| Parent read with | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d…`) WITH the congruence erratum (blob `213594d6…`), the §9 Borel replacement v1.1 (blob `fe9b9ce4…`), W1 and the embedding radius `r < L/(4√2)`, exactly per `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` §1 and its consumption contract §7 (D2 row) | +| Cap theorem | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` (blob `0633aca3…`), §1 deterministic theorem only | +| Read at | Math- `main` `f430fde` (2026-09-30, after #169 merged); every bound blob byte-identical there | +| Reviewer | Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`) | +| Exposure | Same shared GitHub account as every lane; zero organizational independence. I authored none of D2, [P], the erratum, the Borel repair or the cap proof. I authored the D1-C/D/E full-depth review (`reviews/d1_cde_full_depth_claude_20260929`, PR #143) and the reconciliation's D3/§9 reads are by Claude sessions; those are reviews, not authorship, and D2's parent interfaces are consumed here at the reconciled scope, not revalidated. **Overlap disclosed:** I authored `frontiers/c7_total_bounded_20260929/PROOF.md` (Theorem K / Corollary T, PR #150, blob `28748b08…`), whose (K1) coincides with the first line of (R10) and whose (K2) is a `k`-explicit companion of (R12); D2 does not consume it, and I did not use it as evidence here — every R3/R4 step below is re-derived from D2's own text and the parent. I authored PR #180 and reviewed #170/#175/#179/#182 today; none of them is consumed by D2. | +| Own checks | `remainder_review_check.py`: exact `Fraction` controls C1–C7 (C7 evaluates the §7 split at `ℓ = 10^{−18m}`, where every fractional power involved is rational); byte-identical under `-O`; seven semantic mutants exit 1, unknown label exits 2. A clean-context referee agent read this record before landing; its findings (one false remark, one missing disclosure, script coverage claims) were applied, and the three Codex findings on Math-#186 (`r^{−3}` coefficient of `U3` unchecked; C6 compared a closed form to itself with the cutoff value unused; Theorem U cited without its review binding) were applied in v1.1. Finite controls only — not the Gaussian proof. | + +## Verdict + +**ACCEPT R1–R6 at Theorem R's stated existential `O(1)` scope**, read under the reconciliation's reading rule, with +no defect and three non-blocking remarks (§ Remarks). This makes the remainder theorem two-provider reviewed +(xAI R1–R6 + this record); it does not evaluate `C` or `ℓ_*`, does not prove convergence of the bounded +remainder, and does not extend the compact `O(ℓ^{2/3})` selection difference to unrestricted marks — exactly as +the note says in its §1. + +| Item | Note section | Verdict | +|---|---|---| +| **R1** second-order common regression coupling (R2)–(R5) | §2 | **ACCEPT** | +| **R2** inertia-filtered determinant lemmas (R6), (R7) | §3 | **ACCEPT** | +| **R3** target-uniform density perturbation (R5) and the refined intensity (R10)–(R11) | §§2, 4 | **ACCEPT** | +| **R4** unnormalized cap-loss bound with its `k`-dependence (R12), both eigenvalue branches, `m = 1` far branch, fourth-derivative exception | §5 | **ACCEPT** | +| **R5** lower cutoff `a = ℓ/r_0³`, `η = ℓ^{1/6}` split, (R16)–(R18) | §§6–7 | **ACCEPT** | +| **R6** derivation of (R1), (R19), (R20) with the far bound (14.1) | §§7–8 | **ACCEPT** | + +## R1 — the second-order coupling (§2) + +1. **Rows.** With `a = −r/2`, `c = r/2` and a Taylor expansion at the midpoint, the four axial rows of parent (3.1) + are `U0 = f + (r²/8)f_xx + O(r⁴)`, `U1 = f_x + (r²/24)f_xxx + O(r⁴)`, `U2 = f_xx + (r²/24)f_xxxx + O(r⁴)`, + `U3 = f_xxx + (r²/40)f_xxxxx + O(r⁴)`; the transverse rows are the `U0`/`U1` pattern applied to `f_{y_j}`. + I re-derived all four exactly on a degree-7 polynomial (checker C1: the `U3` second-order coefficient is + `f_5/40`, as the note states; a wrong `1/24` is rejected by mutant M1). Odd powers of `r` cancel by the centred + rules, so `U_r − U_0 = O(r²)` is the right order and the note's remark that `U3` is "not an `O(r)` error" is + correct. +2. **Uniform `L^p` control.** Each Fourier mode contributes a centred-rule error `≤ Cr²|n|^q`; with + `Σ√a_n(1+|n|)^q < ∞` (parent §2) Minkowski gives `‖U_r − U_0‖_{L^p} ≤ Cr²`, uniformly over frames, and the + same for `Cov(∂^αF(z), U_r − U_0)` uniformly in `z`. Hence (R2): `Σ_r − Σ_0 = O(r²)` (Cauchy–Schwarz on the + cross terms), `Σ_r^{−1} − Σ_0^{−1} = O(r²)` on the band where `Σ_r` is uniformly positive (parent §3 + compactness), and `‖C_r − C_0‖_{C^q} = O(r²)`. (All rows are even in `r`; checker C1 asserts that every + coefficient from the most negative power a row can carry — `r^{−3}` for `U3`, `r^{−1}` for `U1`, `U2` — through + `r³`, other than `r⁰` and `r²`, vanishes.) +3. **(R3) is a coupling of laws.** `F_r = F + C_rΣ_r^{−1}(v_r − U_r)` has the law of `F` given `U_r = v_r` + (Gaussian regression: `F − C_rΣ_r^{−1}U_r ⟂ U_r`); same for `F_0`. Then + `F_r − F_0 = (C_rΣ_r^{−1} − C_0Σ_0^{−1})(v_r − U_r) + C_0Σ_0^{−1}[(v_r − v_0) − (U_r − U_0)]`, and with + `|v_r − v_0| = |(−kr³/2, −kr², 0, …)| ≤ Cr²k`, `|v_r| ≤ C(|b| + k)`, `‖U_r‖_{L^p} ≤ C`, this is (R4): + `‖F_r − F_0‖_{L^p(C^q)} ≤ C_{p,q}r²P`, `P = 1 + |b| + k`. The moment bound on `1 + ‖F_r‖_{C^q}` is the mean + `C|v_r|` plus the residual. I agree it is a coupling of laws only; nothing is claimed about equality of + conditioned fields. +4. **(R5).** `π_r(v_r) ≤ Ce^{−c(b²+k²)}` is parent (13.2). For the difference the note interpolates + `Σ_θ = (1−θ)Σ_0 + θΣ_r` (positive definite, uniformly) and `v_θ`; `d/dθ log π_θ(v_θ)` has the trace term + `O(r²)`, the quadratic term `O(r²|v_θ|²) = O(r²P²)` and the target term `O(r²k|v_θ|) = O(r²kP)`; coercivity + `|v_θ|² ≥ (b − θkr³/2)² + 144k² ≥ c(b² + k²)` (using `b² ≤ 2(b − θkr³/2)² + k²/2` for `r ≤ 1`, the parent's + (13.1) argument) keeps the Gaussian factor; integrating the derivative gives + `|π_r(v_r) − π_0(v_0)| ≤ Cr²P²e^{−c(b²+k²)}`. Correct, including for arbitrarily large `b, k`. + +## R2 — the two determinant lemmas (§3) + +1. **Lemma R3.1 (R6).** Same inertia: `||det X| − |det Y|| ≤ |det X − det Y| ≤ n max(‖X‖,‖Y‖)^{n−1}‖X − Y‖` by + telescoping columns and Hadamard. Different inertia: eigenvalues move continuously along the segment, so a + singular `Z` lies on it; at most one endpoint has `F_j ≠ 0`, and its `|det|` equals `|det X − det Z|`, which the + same bound controls with `‖Z‖ ≤ max(‖X‖,‖Y‖)`, `‖X − Z‖ ≤ ‖X − Y‖`. A singular endpoint has `F_j = 0` by + convention. The lemma is right and the note is right that the inertia indicator itself is not Lipschitz — + the bound is on the filtered determinant. Checker C3 exercises three rational pairs, each changing inertia + along the segment (inertia computed exactly by congruence diagonalisation, multiplicities counted). +2. **Lemma R3.2 (R7).** `det K_t = α det A − t βᵀadj(A)β` is a polynomial identity (block expansion), valid for + singular `A` — checked exactly on `3×3` and `4×4` blocks including a corank-one `A` with nonzero adjugate + (checker C2, 150 identities; mutant M2 flips the sign). Same inertia at `t = 0, r`: difference of absolute values of an affine function of + `t`, at most `r|βᵀadj(A)β|`. Different inertia: a singular `K_{t₀}`, `t₀ ∈ [0, r]`, exists by continuity of the eigenvalues (it may be + the non-contributing endpoint itself, e.g. `α = 0`, `A = I`, `β = e_1`, where `K_0` is singular and every `K_t`, + `t > 0`, has index 1 and `|det| = t`), and the contributing endpoint's `|det|` is `|t − t₀|·|slope| ≤ + r|βᵀadj(A)β|`. Checked on 390 endpoint/index + combinations, 12 of which cross inertia between nonsingular endpoints. The note's point — that a naive use of (R6) on the `√r` entries would + lose a square root, while (R7) does not, "also as `k` tends to zero" — is exactly the mechanism that makes + (R10) linear in `r`. No inverse of `A` and no eigenvalue margin is used. + +## R3 — the refined intensity and its contact error (§4) + +1. **Congruence.** `K_i = D_r^{−1}H_iD_r^{−1}` with `D_r = diag(√r, I)` gives `[[α_i, √rβ_iᵀ],[√rβ_i, A_i]]` and + `|det H_i|/r = |det K_i|`: this is the *corrected* congruence of erratum E1 (`diag(r^{−1/2}, I)` applied to + `H_i`), so the note reads the parent as the erratum requires. Sylvester's law keeps the inertia, so + `W_r/r² = F_d(K_M)F_{d−1}(K_S)`. +2. **(R8)–(R9).** `|α_M + 6k| + |α_S − 6k| ≤ CrM_4` is parent (5.2); `‖β_i‖ ≤ M_3/2` is (5.1); + `‖A_i − A_0‖ ≤ (r/2)‖F_r‖_{C³} + ‖F_r − F_0‖_{C²}` and the note's `T` (which includes `r^{−2}‖F_r − F_0‖_{C²}`, + of `L^p` size `≤ CP` by (R4)) gives `‖A_i − A_0‖ ≤ CrT` for `r ≤ 1` (with `C = 3/2` for the displayed + choice; the note's `T` absorbs it); `T ≥ 1 + k` with all moments `≤ C_pP^p`. + No independence of `T` and `A_0` is needed, as stated. +3. **Endpoint errors.** First (R7) removes the off-diagonal block: `≤ r|β_iᵀadj(A_i)β_i| ≤ CrT^{m+1} = CrT^d`. + Then (R6) between `diag(α_i, A_i)` and `diag(±6k, A_0)`: `≤ d(CT)^{d−1}(CrT + rT) = CrT^d`. The displayed + size bounds `F_j(K_i) ≤ C(k + r)T^d`, `F_d(diag(−6k, A_0)) ≤ CkT^{d−1}`, `F_{d−1}(diag(6k, A_0)) ≤ CkT^{d−1}` + follow from the block identity and `T ≥ 1 + k`. +4. **Reference product.** Index `d` for `diag(−6k, A_0)` and index `d − 1` for `diag(6k, A_0)` both force + `A_0 < 0` (checker C4 on three inertia patterns), so `W_0 = (6k)²det(A_0)²1{A_0 < 0}` and `E W_0 = z_0` is + exactly parent (5.4). +5. **(R10)–(R11).** Pathwise `|W_r/r² − W_0| ≤ Cr(k + r)T^{2d}`; expectations give + `|Z_r/r² − z_0| ≤ Cr(k + r)P^N` and `Z_r/r² ≤ C(k + r)²P^N`. With (R5): `|A_r − A_0| ≤ 12|π_r − π_0|Z_r/r² + + 12π_0|Z_r/r² − z_0| ≤ r(k + r)H(b,k)` after absorbing `r(k + r) ≤ k + 1` into the polynomial, and + `A_r ≤ (k + r)²H`, `A_0 ≤ k²H`. No division by `k` or `Z` anywhere — this is what allows the `k ↓ 0` region + in §§6–7. + +## R4 — the unnormalized cap-loss bound (§5) + +1. **(R13).** Appending the independent entries of `A_M` to `U_r` keeps the joint covariance uniformly positive at + contact and for `r ≤ r_0` (parent §3, last paragraph); the joint Gaussian density at `(v_r, A)` is therefore + `≤ Ce^{−c(b²+k²+‖A‖²)}` by coercivity of `(v_r, A)`, which is the product of `π_r(v_r)` and the conditional + density. Correct and uniform in eigenvector orientation. +2. **Depth failure, `m ≥ 2`.** With `M_3 ≤ K(T + Λ)` (mean `≤ C(P + ‖A‖)`, residual `J` independent of the + appended vector, `‖A‖_F ≤ √mΛ`), failure of `λ_min(−A_M) > (4/(3k))rM_3²` gives `λ_1 ≤ (4K²/3)(r/k)(T + Λ)² = + Dδ(T + Λ)²`, i.e. the depth width is `δ = r/k` times a polynomial, with the largest eigenvalue *inside* the + width. The envelope (R14) is parent (6.2) divided by `r²`, both soft factors retained. The soft integral + `∫_0^{DδU²} λ(λ + ErU)dλ = (D³/3)δ³U⁶ + (ED²/2)rδ²U⁵` (checker C5) and `rδ² = kδ³` make both terms + `δ³ × polynomial(P, J, Λ)`; the Vandermonde `≤ Λ^{m(m−1)/2}`, the Gaussian in `λ_2, …, λ_m` from (R13) and the + finite `J` moments give `π_rE_Q[(W/r²)1{depth}] ≤ δ³H`. Every corank stratum is included; no lower cutoff on + `λ_2`. +3. **`m = 1` (R15).** `λ ≤ Dδ(T + λ)²` implies `λ ≤ 4DδT²` or `λ ≥ 1/(4Dδ)` — from `(T + λ)² ≤ 2T² + 2λ²` and + `2Dδλ² ≤ λ/2` when `λ ≤ 1/(4Dδ)` (checker C5 tests it on a rational grid whose parameters keep the two + branches disjoint, so the hypothesis fails on most of the grid). Near branch: `h ≤ CT²` for `δ ≤ 1`, direct integration `δ³H`. Far branch: fourth-moment Markov, + `(4Dδ)⁴π_rE_Q[(W/r²)λ⁴] ≤ δ⁴H ≤ δ³H`. Correct; the scalar far branch is retained, as in the parent. +4. **Fourth-derivative exception.** `rM_4 > 3k/10 ⇔ M_4 > 3/(10δ)`; Markov with the actual weight gives + `≤ (10δ/3)⁴π_rE_Q[(W/r²)M_4⁴] ≤ δ⁴H`. Correct. +5. **Containment.** `A_r(1 − p_r) = 12π_rE_Q[(W/r²)1_F] ≤ 12π_rE_Q[(W/r²)1{G_r^c}] = B_r ≤ A_r` uses the cap + implication `F ⊂ G_r^c` on typed support (parent §8 with the cap theorem, reconciled), i.e. the consumption + contract's "§8 comparison `1 − p_r ≤ Q^W(G_r^c)`" in unnormalized form. (R12) `B_r ≤ δ³H` for `δ ≤ 1` is + proved; the note is right to say it is not a globally uniform *normalized* estimate. + +## R5 — cutoffs and the `η` split (§§6–7) + +1. **(R16).** `ℓ^{1/3}ν_cand^{near} = ∫_{k ≥ a}A_r/(3k^{2/3})` with `a = ℓ/r_0³` is parent (13.5); the contact + coefficient is `∫_{k > 0}A_0/(3k^{2/3}) = c` (13.6). Missing contact mass: `∫_0^a k²H/(3k^{2/3}) = O(a^{7/3})`. + The two monomials of `r(k + r)H/(3k^{2/3})` are exactly `rk^{1/3} = ℓ^{1/3}` and `r²k^{−2/3} = ℓ^{2/3}k^{−4/3}` + (checker C6, which binds every fractional power to its base by an exact cube identity, verifies the + antiderivatives by exponent differentiation and brackets the integrals by exact Riemann sums on a perfect-cube + geometric partition); `∫_a^1 k^{−4/3}dk = 3a^{−1/3} − 3`, so the second contributes at most `Cℓ^{2/3}a^{−1/3} = Cr_0ℓ^{1/3}` + (the constant from the integral of `H`) plus an `O(ℓ^{2/3})` tail. Hence `|ℓ^{1/3}ν_cand^{near} − c| ≤ Cℓ^{1/3}`. The note correctly observes that dropping the + lower cutoff would make the `r²k^{−2/3}` integral diverge; the cutoff is exactly the near/far split. +2. **(R17)–(R18).** On `a ≤ k ≤ η = ℓ^{1/6}`: `B_r ≤ A_r ≤ (2k² + 2r²)H` gives `Cη^{7/3} = Cℓ^{7/18}` and + `Cℓ^{2/3}a^{−1/3} = Cr_0ℓ^{1/3}`. On `k ≥ η`: `δ = ℓ^{1/3}k^{−4/3} ≤ ℓ^{1/9} ≤ 1`, so (R12) applies; + `B_r/(3k^{2/3}) ≤ Cℓk^{−14/3}H` and `∫_η^∞ k^{−14/3} = (3/11)η^{−11/3}`, giving `Cℓ^{1 − 11/18} = Cℓ^{7/18}`. + Since `7/18 > 1/3`, the scaled loss is `≤ Cℓ^{1/3}`. The choice `η = ℓ^{1/6}` balances the two error terms + exactly (`η^{7/3} = ℓη^{−11/3} ⇔ η⁶ = ℓ`), which I note as a sanity check on the exponent bookkeeping; checker + C7 evaluates the same split as a model integral with `H = 1`, `r_0 = 1/2`, exactly at `ℓ = 10^{−18m}`, + `m = 1, …, 4`, and finds the ratio to `ℓ^{1/3}` equal to `1.0376…`, `1.0038…`, `1.00038…`, `1.000038…` — + decreasing to the limit `2r_0 = 1` set by the cutoff term. + +## R6 — Theorem R and its corollaries (§§7–8) + +1. **Far pairs.** Parent (14.1) gives `0 ≤ ν_eld^{far} ≤ ν_cand^{far} ≤ C` for `0 < ℓ ≤ 1` on torus separations + `≥ r_0`, from the nonsingular full-pin covariance on a compact set; the note consumes it as stated in the + consumption contract. +2. **(R1).** `|ν_cand − cℓ^{−1/3}| ≤ ℓ^{−1/3}·Cℓ^{1/3} + C`, `0 ≤ ν_cand − ν_eld ≤ ℓ^{−1/3}·Cℓ^{1/3} + C`, and their + difference — all `≤ C`. The relative error `O(ℓ^{1/3})` follows. +3. **(R19)–(R20).** Integrating the density versions: `∫_0^t ℓ^{−1/3} = (3/2)t^{2/3}` and, for `q > −2/3`, + `∫_0^t ℓ^{q−1/3} = t^{q+2/3}/(q + 2/3)` with the `O(1)` remainder contributing `O(t^{q+1})` for `q > −1`; + the nonselected measure alone has weight `≤ C_qt^{q+1}` for `q > −1`, meaningful on `(−1, −2/3]` because it + bounds a single nonnegative measure. Correct, and the note's caveat that this is a sufficient inverse + integrability range for nonselected candidates, not an exact threshold, is right (the exact fixed-`r` + threshold is a different object; see Math- #182). + +## Remarks (non-blocking) + +1. **Embedding radius.** §2 chooses `r_0 ≤ 1` "inside the torus injectivity scale"; for the cap implication + `A_r(1 − p_r) ≤ B_r` (the step before (R12)) the reconciled requirement is the cap cylinder's embedding, + `r < L/(4√2)` (RECONCILIATION §5), which matters only when `L < 4√2` (for `L = 24` it is implied by `r_0 ≤ 1`). + Since every constant here is existential, taking `r_0 < min(1, L/(4√2))` costs nothing; a successor should + say so. +2. **Where the `O(1)` comes from.** At fixed `(b, k, u)` the contact error is in fact `O(r²)`, not `O(r)`: every + observation row is even in `r` (checker C1), the covariance and coupling are even, the only odd target + coordinate is `b − kr³/2` (an `O(r³)` shift that enters no Hessian and changes `Z_r` by `O(r⁵)`), and the + unsigned product `det H(−ru/2)·det H(ru/2)` is symmetric under the pin swap; the typed indicator's boundary + layer carries a vanishing determinant factor and is `o(r)`. So `A_r = A_0 + O(r²)`, and (R10)–(R11)'s + `r(k + r)` is valid but not sharp in `r` at fixed `k` (the pathwise (R7)/(R6) bounds are first-order; the + cancellation is one of expectations). This does not improve Theorem R: the near remainder is dominated by + the cutoff region `k ≍ a`, i.e. pairs at distance `≍ r_0`, which contribute a genuine `O(1)` (the term + `Cr_0ℓ^{1/3}` in (R16)), and the far density (14.1) is `O(1)`; for its rejected part a positive lower bound + is the content of Theorem U (`frontiers/unrestricted_selection_difference_20260929/PROOF.md`, blob + `5a55b179`, an Anthropic candidate whose header still reads "nonauthor analytic review required"; it carries + the nonauthor OpenAI record `reviews/c7_nonvanishing_openai_20260929/REVIEW.md`, blob `64794e5e`, verdict + ACCEPT at that blob). This remark is context only: no step of R1–R6 uses Theorem U, and this record does not + re-review it. The note's "we do not prove + convergence of the bounded remainder, [nor] identify a second coefficient" is the right boundary; a second + coefficient would need the second-order expansion of the typed determinant expectation (beyond the + first-order pathwise bounds) together with the zero-gap far pair density, and is not claimed anywhere. +3. **Consumption map.** Beyond the reconciliation §7 D2 row (§8 comparison, §9 as E2, §10, (14.1), (15.2) as + `c`), the note re-uses — without revalidating — parent §2 (Fourier summability and distinct-site rank), §3 + (rows (3.1)–(3.3) with the factor 12; uniform positivity of `Cov(U_r)` and of `Cov(U_r ⊕ A_M)`, which feeds + (R13)), §4 (the independent residual, in its (13.3) growth form), (5.1)–(5.4), (6.1)–(6.2), (7.2), + (13.1)–(13.6) and the cap theorem §1. All of these are reconciled interfaces and none depends on the + target, so nothing outside the reconciled scope is imported; the reconciliation's D2 row could list them. + +## What this record does not do + +No numerical `C`, `ℓ_*`, `r_0`, `z_*` or `c_{d,L}`; no convergence of the remainder; no second coefficient; no +unrestricted `O(ℓ^{2/3})` difference (the positive lower bound for the far rejected density on `(0, ℓ_0]` is +Theorem U, cited above with its OpenAI review; the limit of that density is the separate candidate Theorem Z, +`frontiers/c7_zero_gap_limit_20260929/PROOF.md`, blob `5b6328ea`, whose header reads "non-OpenAI review +required" — neither is reviewed or relied on here); no `d ≥ 3` lower bound; no register, STATUS, GRAPH, +PROOF_INDEX or catalog change; no merge. diff --git a/reviews/d2_remainder_full_depth_claude_20260930/SOURCE_FILES.json b/reviews/d2_remainder_full_depth_claude_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..d1aebbfdc --- /dev/null +++ b/reviews/d2_remainder_full_depth_claude_20260930/SOURCE_FILES.json @@ -0,0 +1,103 @@ +{ + "author": "Anthropic Claude", + "cited_only": [ + { + "blob": "5a55b179a974a90c65d257f3f93765cc6fb30bb8", + "bytes": 13028, + "path": "frontiers/unrestricted_selection_difference_20260929/PROOF.md", + "role": "Theorem U (Anthropic candidate; 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support other lanes); OWNER_STOP left in place, no self-merge, scientific effect NONE", + "files": [ + { + "bytes": 20595, + "path": "REVIEW.md", + "sha256": "8679afd19ad313ade3602bba48b744183fdbefc9596a7793cfe7f921988a5dfd" + }, + { + "bytes": 24294, + "path": "remainder_review_check.py", + "sha256": "fec1ec7cad865d2f5d1399fc93f1a4def554d862d9fb32ec6dae34ef64d293f5" + }, + { + "bytes": 2043, + "path": "RESULTS.json", + "sha256": "310e1c26b6b0fcd3d64367e1c2c7a0f5b6892e0f9f113c31a8db9520999dd761" + } + ], + "mathematical_acceptance_by_this_record": "scoped ACCEPT R1-R6, nonauthor lane (second provider after xAI/Grok), existential O(1) scope only", + "mutants": [ + "M1", + "M2", + "M3", + "M4", + "M5", + "M6", + "M7" + ], + "object": "CL-D2-REMAINDER-FULLDEPTH-20260930-v1.1", + "read_at_head": "f430fdecbb1d8802d8af40419b701f12af87c039", + "referee_pass": "clean-context Anthropic Claude subagent (same provider, same account; not a nonauthor review): no blocking finding; one false remark (contact error is O(r^2), not O(r)), one missing disclosure (#150 overlap), and script coverage claims corrected before landing", + "reviewed_sources": [ + { + "blob": "247b3ecf80bfbe896948d5d489b2d5842a81c481", + "bytes": 17734, + "path": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md", + "role": "reviewed object (D2, Theorem R)", + "sha256": "380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a" + }, + { + "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "bytes": 40261, + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "role": "parent (P)", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7" + }, + { + "blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "bytes": 1782, + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "role": "reading rule E1", + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028" + }, + { + "blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "bytes": 9062, + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "role": "reading rule E2 (section 9 replacement)", + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f" + }, + { + "blob": "75da2597971510f843f8d90c743950cb8c177342", + "bytes": 23312, + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "role": "reading rule W1, embedding radius, consumption contract", + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da" + }, + { + "blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "bytes": 15160, + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "role": "cap theorem (deterministic part)", + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc" + } + ], + "revision": "v1.1 (2026-09-30): applied the three Codex findings on Math-#186 \u2014 C1 forbids the r^-3 coefficient of U3 (all powers from -offset through r^3 other than r^0, r^2); C6 binds a^(-1/3), eta^(-11/3) to the cutoff values by exact cube identities, verifies the antiderivatives by exponent differentiation and brackets the integrals by exact Riemann sums (mutant M7 corrupts a^(-1/3)); C7 binds every fractional power to its base; REVIEW.md cites Theorem U with its nonauthor OpenAI review record and marks it non-load-bearing", + "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU" +} diff --git a/reviews/d2_remainder_full_depth_claude_20260930/remainder_review_check.py b/reviews/d2_remainder_full_depth_claude_20260930/remainder_review_check.py new file mode 100644 index 000000000..0185e3d4b --- /dev/null +++ b/reviews/d2_remainder_full_depth_claude_20260930/remainder_review_check.py @@ -0,0 +1,491 @@ +#!/usr/bin/env python3 +"""Exact finite controls accompanying the D2 (Theorem R) full-depth review. Standard library only. + +They check, exactly in rational arithmetic, the displayed algebra that the review re-derives; they do not prove +the Gaussian estimates of the note and are not acceptance. Scientific effect: NONE. + + C1 the four axial rows (3.1) of the parent applied to a degree-7 polynomial: U0 = f0 + (r^2/8) f2 + O(r^4), + U1 = f1 + (r^2/24) f3 + O(r^4), U2 = f2 + (r^2/24) f4 + O(r^4), U3 = f3 + (r^2/40) f5 + O(r^4) + (note S2: "U_r - U_0 = O(r^2)", "(r^2/40) f_xxxxx"); every coefficient from the most negative power the + row can carry (r^-3 for U3, r^-1 for U1, U2) through r^3, other than r^0 and r^2, is asserted to vanish + (no unscaled residual, no odd term); the two transverse rows are the U0/U1 pattern applied to f_y and are + checked on a second polynomial; + C2 Lemma R3.2's affine determinant identity det K_t = alpha det A - t beta^T adj(A) beta on rational + 3x3 and 4x4 blocks, INCLUDING a singular A of corank one with NONZERO adjugate, and the bound + |F_j(K_r) - F_j(K_0)| <= r |beta^T adj(A) beta| on paths whose inertia changes (a singular crossing in + [0, r]); inertia is computed exactly by congruence diagonalisation (eigenvalue multiplicities counted); + C3 Lemma R3.1 on rational symmetric pairs whose inertia changes along the segment: the filtered + determinant difference is bounded by n max(||X||, ||Y||)^(n-1) ||X - Y|| (operator norms replaced by the + larger Frobenius norms, which only weakens the bound); + C4 the reference product: W_0 = (6k)^2 det(A_0)^2 1{A_0 < 0} is the only inertia pattern giving index d at + diag(-6k, A_0) and index d-1 at diag(6k, A_0) (three inertia patterns for A_0); + C5 the soft-eigenvalue integral of note S5: int_0^(D delta U^2) lambda (lambda + E r U) d lambda + = (D^3/3) delta^3 U^6 + (E D^2/2) r delta^2 U^5 (exact, at rational parameters); the scalar split (R15): + lambda <= D delta (T + lambda)^2 implies lambda <= 4 D delta T^2 or lambda >= 1/(4 D delta), on a rational + grid whose parameters satisfy 4 D delta T^2 < 1/(4 D delta), so that the two branches do not overlap and + the hypothesis fails on part of the grid (the check is informative); + C6 the exponent ledger of notes S6-S7 by exact monomial substitution r = (ell/k)^(1/3), a = ell r_0^-3, + eta = ell^(1/6): r k^(1/3) = ell^(1/3); r^2 k^(-2/3) = ell^(2/3) k^(-4/3); a^(7/3) = ell^(7/3) r_0^-7; + eta^(7/3) = ell^(7/18); delta = r/k at k = eta is ell^(1/9); ell eta^(-11/3) = ell^(7/18); 7/18 > 1/3; + and the antiderivatives int_a^1 k^(-4/3) dk = 3 a^(-1/3) - 3 and int_eta^oo k^(-14/3) dk = (3/11) eta^(-11/3) + evaluated exactly at ell = 10^-18 (all fractional powers rational there, each bound to its base by an exact + cube identity), verified by exponent differentiation and bracketed by exact lower/upper Riemann sums on a + perfect-cube geometric partition (tail on [1, oo) by geometric series); + C7 the split of note S7 as a model integral with H = 1, r_0 = 1/2, in EXACT rationals at ell = 10^(-18 m), + m = 1..4 (where ell^(1/6), ell^(1/3), a^(-1/3), a^(7/3) are rational): piece 1 = int_a^eta (2k^2 + 2r^2) + /(3 k^(2/3)) dk = (2/7)(eta^(7/3) - a^(7/3)) + 2 ell^(2/3)(a^(-1/3) - eta^(-1/3)); piece 2 = + int_eta^oo ell k^(-14/3)/3 dk = ell eta^(-11/3)/11; their sum divided by ell^(1/3) stays below 3/2 (every + fractional power bound to its base by an exact integer-power identity); + M semantic mutants (--mutant M1|M2|M3|M4|M5|M6|M7) exit 1; an unknown label exits 2 (M7 corrupts the + hard-coded a^(-1/3) of C6, which the cube identity must catch). +""" +import argparse +import json +import sys +from fractions import Fraction as F + + +# ---------------------------------------------------------------------------------------------------------- +# polynomial helpers (coefficient lists in r, exact) +def poly_mul(p, q): + out = [F(0)]*(len(p)+len(q)-1) + for i, a in enumerate(p): + for j, b in enumerate(q): + out[i+j] += a*b + return out + + +def poly_add(p, q): + n = max(len(p), len(q)) + return [(p[i] if i < len(p) else F(0)) + (q[i] if i < len(q) else F(0)) for i in range(n)] + + +def poly_scale(p, c): + return [c*a for a in p] + + +def taylor_eval(fcoef, point_poly): + """Sum_j f_j point^j / j! where point is a polynomial in r and f_j are the derivatives at 0.""" + total = [F(0)] + power = [F(1)] + fact = F(1) + for j, fj in enumerate(fcoef): + if j > 0: + power = poly_mul(power, point_poly) + fact *= j + total = poly_add(total, poly_scale(power, fj/fact)) + return total + + +def coeff(p, i): + return p[i] if 0 <= i < len(p) else F(0) + + +def rows_from(fc): + """The four axial rows of parent (3.1) as Laurent polynomials in r: returns dict name -> (offset, coeffs) + where the row equals sum_i coeffs[i] r^(i-offset).""" + fxc = [fc[j+1] for j in range(len(fc)-1)] + a = [F(0), F(-1, 2)] + c = [F(0), F(1, 2)] + fa, fcv = taylor_eval(fc, a), taylor_eval(fc, c) + fxa, fxcv = taylor_eval(fxc, a), taylor_eval(fxc, c) + U0 = (0, poly_scale(poly_add(fa, fcv), F(1, 2))) + d1 = poly_add(fcv, poly_scale(fa, F(-1))) # f(c) - f(a), then divide by r + U1 = (1, d1) + d2 = poly_add(fxcv, poly_scale(fxa, F(-1))) + U2 = (1, d2) + two_U1 = poly_scale(d1, F(-2)) # -2 (f(c)-f(a)) as polynomial; /r later + # inner = f_x(a) + f_x(c) - 2 (f(c)-f(a))/r : represent with a common offset 1: multiply first part by r + s = poly_add(fxa, fxcv) + inner_times_r = poly_add(poly_mul(s, [F(0), F(1)]), two_U1) # r * inner + U3 = (3, poly_scale(inner_times_r, F(6))) # (6/r^2) inner = 6 * (r inner)/r^3 + return {'U0': U0, 'U1': U1, 'U2': U2, 'U3': U3} + + +def check_rows(mutant): + fc = [F(v) for v in (2, 3, 5, 7, 11, 13, 17, 19)] # f_j = f^(j)(0), distinct primes + rows = rows_from(fc) + expect = {'U0': (fc[0], fc[2]/8), 'U1': (fc[1], fc[3]/24), 'U2': (fc[2], fc[4]/24), 'U3': (fc[3], fc[5]/40)} + if mutant == 'M1': + expect['U3'] = (fc[3], fc[5]/24) + vanishing = {} + for name, (off, p) in rows.items(): + # coefficient of r^e is coeff(p, e + off); every exponent from the most negative one the representation + # can carry (r^-off) through r^3, other than r^0 and r^2, must vanish: for U3 this includes r^-3 + # (a leftover constant in r*inner would make U3 diverge and break the second-order coupling) + forbidden = [e for e in range(-off, 4) if e not in (0, 2)] + for e in forbidden: + if coeff(p, e + off) != 0: + return False, 'row %s has a nonzero r^%d term' % (name, e) + vanishing[name] = ['r^%d' % e for e in forbidden] + c0, c2 = expect[name] + if coeff(p, off) != c0 or coeff(p, off + 2) != c2: + return False, 'row %s expansion mismatch' % name + # transverse rows: V0 = (g(a)+g(c))/2, V1 = (g(c)-g(a))/r with g = f_y: the U0/U1 algebra on another polynomial + gc = [F(v) for v in (23, 29, 31, 37, 41, 43, 47, 53)] + rows_g = rows_from(gc) + off, p = rows_g['U0'] + if coeff(p, off) != gc[0] or coeff(p, off + 2) != gc[2]/8 or coeff(p, off + 1) != 0: + return False, 'transverse row V0 mismatch' + off, p = rows_g['U1'] + if coeff(p, off) != gc[1] or coeff(p, off + 2) != gc[3]/24 or coeff(p, off + 1) != 0: + return False, 'transverse row V1 mismatch' + return True, {'axial_rows': 4, 'transverse_rows': 2, 'U3_second_order': str(fc[5]/40) + ' = f5/40', + 'vanishing_terms_asserted': vanishing} + + +# ---------------------------------------------------------------------------------------------------------- +# exact rational linear algebra +def det(M): + n = len(M) + if n == 1: + return M[0][0] + if n == 2: + return M[0][0]*M[1][1]-M[0][1]*M[1][0] + total = F(0) + for j in range(n): + minor = [row[:j]+row[j+1:] for row in M[1:]] + total += (-1)**j*M[0][j]*det(minor) + return total + + +def adjugate(A): + n = len(A) + adj = [[F(0)]*n for _ in range(n)] + for i in range(n): + for j in range(n): + minor = [row[:j]+row[j+1:] for k, row in enumerate(A) if k != i] + adj[j][i] = (-1)**(i+j)*det(minor) if n > 1 else F(1) + return adj + + +def inertia(M): + """Number of negative eigenvalues of a symmetric rational matrix, exactly, by congruence diagonalisation + (Sylvester's law of inertia). Repeated eigenvalues are counted with multiplicity.""" + n = len(M) + A = [row[:] for row in M] + neg = 0 + for p in range(n): + # find a nonzero diagonal pivot at or after p; else create one by a congruence if an off-diagonal + # entry exists in the trailing block; else the trailing block is zero + piv = None + for i in range(p, n): + if A[i][i] != 0: + piv = i; break + if piv is None: + found = None + for i in range(p, n): + for j in range(i+1, n): + if A[i][j] != 0: + found = (i, j); break + if found: break + if found is None: + break + i, j = found + # congruence e_i -> e_i + e_j : A <- E^T A E with E = I + e_j e_i^T (row/col i += row/col j) + for k in range(n): + A[i][k] += A[j][k] + for k in range(n): + A[k][i] += A[k][j] + piv = i if A[i][i] != 0 else None + if piv is None: + raise RuntimeError('congruence failed to create a pivot') + if piv != p: + A[p], A[piv] = A[piv], A[p] + for row in A: + row[p], row[piv] = row[piv], row[p] + d = A[p][p] + if d < 0: + neg += 1 + for i in range(p+1, n): + f = A[i][p]/d + if f != 0: + for k in range(p, n): + A[i][k] -= f*A[p][k] + for k in range(p, n): + A[k][i] -= f*A[k][p] + return neg + + +def filtered_det(M, j): + d = det(M) + if d == 0: + return F(0) + return abs(d) if inertia(M) == j else F(0) + + +def frob2(M): + return sum(x*x for row in M for x in row) # squared Frobenius norm (exact) + + +def embed(alpha, beta, A, s): + n = len(A) + K = [[F(0)]*(n+1) for _ in range(n+1)] + K[0][0] = alpha + for i in range(n): + K[0][i+1] = s*beta[i]; K[i+1][0] = s*beta[i] + for j in range(n): + K[i+1][j+1] = A[i][j] + return K + + +def check_R7(mutant): + """det K_t identity and Lemma R3.2's bound on inertia-changing paths (t = s^2 with rational s).""" + A1 = [[F(-2), F(1, 2), F(0)], [F(1, 2), F(-1), F(1, 3)], [F(0), F(1, 3), F(-3, 2)]] # negative definite + A2 = [[F(-1), F(1, 2), F(0)], [F(1, 2), F(-2), F(0)], [F(0), F(0), F(0)]] # corank one, adj != 0 + A3 = [[F(-1), F(0), F(0), F(0)], [F(0), F(-1, 2), F(1, 4), F(0)], [F(0), F(1, 4), F(-1, 3), F(0)], [F(0), F(0), F(0), F(-2)]] + betas = [[F(1), F(-1), F(2)], [F(3, 2), F(0), F(1, 5)], [F(2), F(1), F(1), F(-1)]] + alphas = [F(-6), F(6), F(1, 100), F(-1, 100), F(0)] + n_id = 0; n_bound = 0; crossings = 0; singular_adj_nonzero = 0 + for A in (A1, A2, A3): + n = len(A) + adjA = adjugate(A); dA = det(A) + if dA == 0 and any(x != 0 for row in adjA for x in row): + singular_adj_nonzero += 1 + for beta in betas: + if len(beta) != n: + continue + q = sum(beta[i]*sum(adjA[i][j]*beta[j] for j in range(n)) for i in range(n)) # beta^T adj(A) beta + for alpha in alphas: + for s in (F(0), F(1, 3), F(1, 2), F(1), F(3, 2), F(2)): + t = s*s + lhs = det(embed(alpha, beta, A, s)); rhs = alpha*dA - t*q + if mutant == 'M2': + rhs = alpha*dA + t*q + if lhs != rhs: + return False, 'det K_t identity fails (alpha=%s, t=%s)' % (alpha, t) + n_id += 1 + for s in (F(1, 2), F(1), F(2)): + r = s*s + K0 = embed(alpha, beta, A, F(0)); Kr = embed(alpha, beta, A, s) + if det(K0) != 0 and det(Kr) != 0 and inertia(K0) != inertia(Kr): + crossings += 1 + for j in range(n+2): + diff = abs(filtered_det(Kr, j) - filtered_det(K0, j)) + if diff > r*abs(q): + return False, 'Lemma R3.2 bound fails at alpha=%s r=%s j=%s' % (alpha, r, j) + n_bound += 1 + if crossings == 0 or singular_adj_nonzero == 0: + return False, 'test set lacks inertia crossings or a corank-one block with nonzero adjugate' + return True, {'identities': n_id, 'bound_checks': n_bound, 'inertia_crossings_both_endpoints_nonsingular': crossings, + 'singular_blocks_with_nonzero_adjugate': singular_adj_nonzero} + + +def check_R6(mutant): + """Lemma R3.1 on rational symmetric pairs with an inertia change along the segment (Frobenius norms).""" + pairs = [ + ([[F(-1), F(0)], [F(0), F(-1)]], [[F(1), F(0)], [F(0), F(-1)]]), + ([[F(-2), F(1)], [F(1), F(-2)]], [[F(2), F(1)], [F(1), F(1, 2)]]), + ([[F(-1), F(1, 2), F(0)], [F(1, 2), F(-1), F(0)], [F(0), F(0), F(-3)]], + [[F(1), F(1, 2), F(1, 3)], [F(1, 2), F(-1), F(0)], [F(1, 3), F(0), F(2)]]), + ] + checks = 0; crossings = 0 + if inertia([[F(-1), F(0)], [F(0), F(-1)]]) != 2 or inertia([[F(1), F(0)], [F(0), F(-1)]]) != 1: + return False, 'inertia routine miscounts a repeated eigenvalue' + for X, Y in pairs: + n = len(X) + if inertia(X) != inertia(Y): + crossings += 1 + nx, ny = frob2(X), frob2(Y) + dXY = frob2([[X[i][j]-Y[i][j] for j in range(n)] for i in range(n)]) + bound2 = F(n*n)*max(nx, ny)**(n-1)*dXY # squared bound, exact + for j in range(n+1): + diff = abs(filtered_det(X, j) - filtered_det(Y, j)) + if mutant == 'M3': + diff = diff*10 + if diff*diff > bound2: + return False, 'Lemma R3.1 bound fails on pair with j=%s' % j + checks += 1 + if crossings != len(pairs): + return False, 'expected every pair to change inertia' + return True, {'checks': checks, 'inertia_crossings': crossings} + + +def check_reference_product(mutant): + k = F(1, 3) + mats = { + 'negdef': [[F(-1), F(1, 4)], [F(1, 4), F(-2)]], + 'indef': [[F(-1), F(0)], [F(0), F(1)]], + 'posdef': [[F(1), F(0)], [F(0), F(2)]], + } + for name, A0 in mats.items(): + m = len(A0); d = m+1 + DM = embed(-6*k, [F(0)]*m, A0, F(0)); DS = embed(6*k, [F(0)]*m, A0, F(0)) + prod = filtered_det(DM, d)*filtered_det(DS, d-1) + expected = (6*k)**2*det(A0)**2 if name == 'negdef' else F(0) + if mutant == 'M4' and name == 'indef': + expected = (6*k)**2*det(A0)**2 + if prod != expected: + return False, 'reference product mismatch on %s' % name + return True, {'inertia_patterns': 3} + + +def check_soft_integral(mutant): + D, E, U, dl, k = F(4, 3), F(3, 2), F(7, 2), F(1, 5), F(2, 3) + r = k*dl + X = D*dl*U*U + lhs = X**3/3 + E*r*U*X*X/2 # closed form of int_0^X lambda(lambda + E r U) + rhs = (D**3/3)*dl**3*U**6 + (E*D*D/2)*r*dl*dl*U**5 + if lhs != rhs: + return False, 'soft integral identity fails' + # (R15) with non-overlapping branches: D delta small so that 4 D delta T^2 < 1/(4 D delta) + D2, dl2 = F(4, 3), F(1, 100) + hyp_false = 0; grid = 0 + for T in (F(1), F(3, 2), F(2)): + lo, hi = 4*D2*dl2*T*T, 1/(4*D2*dl2) + if not lo < hi: + return False, 'branch overlap; grid uninformative' + for lam in [F(n, 20) for n in range(1, 400)]: + hyp = lam <= D2*dl2*(T+lam)**2 + if mutant == 'M5': + hyp = True + if hyp and not (lam <= lo or lam >= hi): + return False, '(R15) branch logic fails at T=%s lam=%s' % (T, lam) + hyp_false += (0 if hyp else 1); grid += 1 + if hyp_false == 0: + return False, 'the hypothesis never fails on the grid; check uninformative' + return True, {'identity': True, 'grid_points': grid, 'points_where_hypothesis_fails': hyp_false} + + +class Mono: + """Monomial ell^a k^b r0^c with exact rational exponents.""" + def __init__(self, a=F(0), b=F(0), c=F(0)): + self.a, self.b, self.c = F(a), F(b), F(c) + def __mul__(self, o): + return Mono(self.a+o.a, self.b+o.b, self.c+o.c) + def __pow__(self, e): + e = F(e); return Mono(self.a*e, self.b*e, self.c*e) + def key(self): + return (self.a, self.b, self.c) + + +def check_exponents(mutant): + ell, k, r0 = Mono(1, 0, 0), Mono(0, 1, 0), Mono(0, 0, 1) + r = (ell * k**(-1))**F(1, 3) + if mutant == 'M6': + r = (ell * k**(-1))**F(1, 2) + a = ell * r0**(-3) + eta = ell**F(1, 6) + claims = { + 'r k^(1/3)': ((r * k**F(1, 3)).key(), (F(1, 3), F(0), F(0))), + 'r^2 k^(-2/3)': (((r**2) * k**F(-2, 3)).key(), (F(2, 3), F(-4, 3), F(0))), + 'a^(7/3)': ((a**F(7, 3)).key(), (F(7, 3), F(0), F(-7))), + 'ell^(2/3) a^(-1/3)': (((ell**F(2, 3)) * a**F(-1, 3)).key(), (F(1, 3), F(0), F(1))), + 'eta^(7/3)': ((eta**F(7, 3)).key(), (F(7, 18), F(0), F(0))), + 'delta = r/k at k = eta': (((ell**F(1, 3)) * eta**F(-4, 3)).key(), (F(1, 9), F(0), F(0))), + 'ell eta^(-11/3)': ((ell * eta**F(-11, 3)).key(), (F(7, 18), F(0), F(0))), + 'delta^3 k^(-2/3) at general k': ((((ell**F(1, 3)) * k**F(-4, 3))**3 * k**F(-2, 3)).key(), (F(1), F(-14, 3), F(0))), + } + for name, (got, want) in claims.items(): + if got != want: + return False, 'monomial claim fails: %s got %s' % (name, got) + if not (F(7, 18) > F(1, 3) and F(1, 9) > 0): + return False, 'exponent comparison fails' + # antiderivatives evaluated exactly at ell = 10^-18, r_0 = 1/2 (a = 8e-18, a^(-1/3) = 10^6/2, eta = 10^-3). + # The fractional powers are written down and then BOUND to the cutoff values by exact cube identities, so a + # wrong value cannot pass; the antiderivatives are verified by exponent differentiation and by exact + # Riemann brackets, not by re-evaluating their own closed form. + L = F(1, 10**18); a_v = L*8; eta_v = F(1, 1000) + a_third = F(10**6, 2) # a^(-1/3) + if mutant == 'M7': + a_third = F(10**6) # a wrong a^(-1/3) must be caught by the cube identity + if (1/a_third)**3 != a_v: + return False, 'a^(-1/3) does not match the cutoff a = ell r_0^-3' + if eta_v**6 != L: + return False, 'eta^6 != ell' + # d/dk [-3 k^(-1/3)] = k^(-4/3) and d/dk [-(3/11) k^(-11/3)] = k^(-14/3): coefficient and exponent, exactly + for (cst, p, want_c, want_p) in ((F(-3), F(-1, 3), F(1), F(-4, 3)), (F(-3, 11), F(-11, 3), F(1), F(-14, 3))): + if cst*p != want_c or p - 1 != want_p: + return False, 'antiderivative differentiation fails for exponent %s' % p + int_a_1 = 3*a_third - 3 # F(1) - F(a) with F(k) = -3 k^(-1/3) + # exact Riemann bracket of int_a^1 k^(-4/3) dk on the geometric partition k_i = a q^(3i), q = 21/20, closed + # by [k_N, 1]: the integrand is decreasing, k_i^(-1/3) = a^(-1/3) q^(-i) is rational, so the lower and + # upper sums are exact rationals and must bracket the closed form (their ratio is at most q^4 < 1.22) + q = F(21, 20) + ks, invs = [a_v], [a_third] # k_i and k_i^(-1/3) + while ks[-1]*q**3 < 1: + ks.append(ks[-1]*q**3); invs.append(invs[-1]/q) + ks.append(F(1)); invs.append(F(1)) + lower = sum((ks[i+1]-ks[i])*invs[i+1]**4 for i in range(len(ks)-1)) + upper = sum((ks[i+1]-ks[i])*invs[i]**4 for i in range(len(ks)-1)) + if not (lower <= int_a_1 <= upper): + return False, 'Riemann bracket fails for int_a^1 k^(-4/3)' + if not (upper < q**4*lower): + return False, 'Riemann bracket is looser than expected' + cells_a = len(ks) - 1 + eta_m113 = F(10**11) # eta^(-11/3) at eta = 10^-3 + if eta_m113**3 != eta_v**(-11): + return False, 'eta^(-11/3) does not match eta' + # int_eta^oo k^(-14/3) dk = (3/11) eta^(-11/3): bracket the piece on [eta, 1] by the same partition and the + # tail on [1, oo) by the geometric partition q^(3i) summed as exact geometric series (no antiderivative used): + # (q^3 - 1) q^(-14) / (1 - q^(-11)) <= int_1^oo k^(-14/3) dk <= (q^3 - 1) / (1 - q^(-11)) + int_eta_inf = F(3, 11)*eta_m113 + ks, invs = [eta_v], [F(10)] # k_i and k_i^(-1/3) with eta^(-1/3) = 10 + if invs[0]**(-3) != eta_v: + return False, 'eta^(-1/3) does not match eta' + while ks[-1]*q**3 < 1: + ks.append(ks[-1]*q**3); invs.append(invs[-1]/q) + ks.append(F(1)); invs.append(F(1)) + tail_lower = (q**3 - 1)*q**(-14)/(1 - q**(-11)) + tail_upper = (q**3 - 1)/(1 - q**(-11)) + lower = sum((ks[i+1]-ks[i])*invs[i+1]**14 for i in range(len(ks)-1)) + tail_lower + upper = sum((ks[i+1]-ks[i])*invs[i]**14 for i in range(len(ks)-1)) + tail_upper + if not (lower <= int_eta_inf <= upper): + return False, 'Riemann bracket fails for int_eta^oo k^(-14/3)' + info = {name: [str(x) for x in got] for name, (got, want) in claims.items()} + info['riemann_cells'] = {'int_a^1': cells_a, 'int_eta^1': len(ks) - 1} + return True, info + + +def check_loss_split(mutant): + """Model check of S7's split with H = 1, r_0 = 1/2, exactly, at ell = 10^(-18 m).""" + ratios = {} + for m in (1, 2, 3, 4): + L = F(1, 10**(18*m)) + eta = F(1, 10**(3*m)) # ell^(1/6) + l13 = F(1, 10**(6*m)); l23 = F(1, 10**(12*m)) + a = 8*L # ell / r0^3, r0 = 1/2 + a_m13 = F(10**(6*m), 2) # a^(-1/3) + a_73 = F(128, 10**(42*m)) # a^(7/3) = 2^7 10^(-42 m) + eta_73 = F(1, 10**(7*m)); eta_m13 = F(10**m); eta_m113 = F(10**(11*m)) + # every fractional power above is bound to its base by an exact integer-power identity + if (eta**6 != L or l13**3 != L or l23**3 != L*L or a_m13**(-3) != a or a_73**3 != a**7 + or eta_73**3 != eta**7 or eta_m13**(-3) != eta or eta_m113**3 != eta**(-11)): + return False, 'a fractional power does not match its base at m=%d' % m + if not a < eta < 1: + return False, 'cutoff order a < eta < 1 fails' + piece1 = F(2, 7)*(eta_73 - a_73) + 2*l23*(a_m13 - eta_m13) + if mutant == 'M6': + piece1 = piece1 + l13 + piece2 = L*eta_m113/11 + ratio = (piece1 + piece2)/l13 + if ratio > F(3, 2): + return False, 'loss split ratio exceeds 3/2 at m=%d: %s' % (m, ratio) + ratios['ell=1e-%d' % (18*m)] = str(ratio.limit_denominator(10**9)) + return True, ratios + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument('--mutant', default=None) + args = ap.parse_args() + mutant = args.mutant + if mutant is not None and mutant not in ('M1', 'M2', 'M3', 'M4', 'M5', 'M6', 'M7'): + sys.stderr.write('unknown mutant label\n'); sys.exit(2) + results = {} + ok_all = True + for name, fn in (('C1_axial_and_transverse_rows', check_rows), ('C2_R7_affine_determinant', check_R7), + ('C3_R6_filtered_determinant', check_R6), ('C4_reference_product', check_reference_product), + ('C5_soft_integral_and_R15', check_soft_integral), ('C6_exponent_ledger', check_exponents), + ('C7_loss_split_model', check_loss_split)): + ok, info = fn(mutant) + results[name] = {'passed': bool(ok), 'info': info} + ok_all = ok_all and ok + out = {'object': 'CL-D2-REMAINDER-FULLDEPTH-20260930-v1', 'scientific_effect': 'NONE', 'passed': ok_all, + 'checks': results} + print(json.dumps(out, indent=1, sort_keys=True)) + sys.exit(0 if ok_all else 1) + + +if __name__ == '__main__': + main() diff --git a/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json b/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json new file mode 100644 index 000000000..0270312de --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/AUTHOR_COMMENT_5903080058.json @@ -0,0 +1,7 @@ +{ + "id": 5903080058, + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5903080058", + "created_at": "2026-09-30T02:47:55Z", + "updated_at": "2026-09-30T02:47:55Z", + "body": "Author-side corollary of unchanged §7, offered to the Slice-B reader (not a new independent verdict): at the exact aligned-planar scope, (1+12A²)/(1+A²)>=1 and D_aG0<=0. Since B<0, equation(13) therefore gives the model-independent leading-coefficient lower bound\n\n`C2/C0 > 11 U2/(2I) = 4587/856`.\n\nStrictness follows from positive Gaussian mass at a!=0. Consequently the leading coefficient of the radial TV/CDF error in(16) exceeds\n\n`(4587/856)*(2/13)*(11/13)^(11/2)`\n\n(approximately0.328937168357..., decimal diagnostic only). This is a bound on the asymptotic coefficient for each fixed model, NOT a uniform-in-L,k remainder or finite-threshold inequality.\n\nLikewise (19) rewrites as `Bsign=(11 Uabs/(2I)) E_rho[|A|/sqrt(1+A²)]` with rho proportional to H gamma^11G0, so `0t]\n = 11/(11-p)\n - [2p c / ((13-p)(11-p))] t^-2\n + O_p(t^-4).\n\nThe coefficient follows exactly from\n13/(13-p)-11/(11-p) = -2p/[(13-p)(11-p)].\nThe restriction p<11 is sharp for the Pareto limiting moment. This is AFTER the original r->0 limit and does not imply finite-r moment convergence.\n\nDifferentiating the moment identity at p=0 (justified by the same q^-12 envelope on a fixed neighborhood of p=0) gives the log-radius correction\n\nE[log(R/t) | R>t]\n = 1/11 - (2c/143) t^-2 + O(t^-4).\n\nThere is also an explicit fixed-upper-tail quantile correction. If Q_alpha(t) is defined by\nP(R/t > Q_alpha(t) | R>t)=alpha, 04587/856>0, so every fixed upper-tail quantile approaches its Pareto limit from below at second order, and E log(R/t) approaches 1/11 from below. This is an asymptotic coefficient/sign statement for each fixed model; it is NOT a model-uniform finite-threshold inequality because the O(t^-4) remainder is not uniform in L,k,b.\n\nThese are direct consequences of the frozen proof65f24b1f and can be checked inside Slice B without enlarging the source interface." +} diff --git a/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json b/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json new file mode 100644 index 000000000..5b405d890 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/COROLLARY_CUSTODY.json @@ -0,0 +1,91 @@ +{ + "artifacts": [ + { + "bytes": 11110, + "path": "MOMENT_CONSEQUENCE.md", + "sha256": "29bba56b56fd3223a2ea71d012f9abe04eb30240bc5bbfe8f5b131e0bba6f4dd" + }, + { + "bytes": 8259, + "path": "moment_consequence_checks.py", + "sha256": "0c35140179e0345eb03764bf29c3c81453ea278ce1a6319ab33edf6a548e03e6" + }, + { + "bytes": 1219, + "path": "moment-checks-normal.json", + "sha256": 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"observed_control_replay": { + "all_order_controls": "normalized finite Pareto mixtures", + "exact_rational_powers": [ + "-100", + "-7", + "-1", + "-1/2", + "0", + "1/2", + "1", + "2", + "10", + "109/10", + "10999/1000" + ], + "log_correction": "-2/143", + "log_leading": "1/11", + "moment_rows": [ + { + "coefficient_of_c_t_minus_2": "200/12543", + "leading": "11/111", + "p": "-100" + }, + { + "coefficient_of_c_t_minus_2": "7/180", + "leading": "11/18", + "p": "-7" + }, + { + "coefficient_of_c_t_minus_2": "1/84", + "leading": "11/12", + "p": "-1" + }, + { + "coefficient_of_c_t_minus_2": "4/621", + "leading": "22/23", + "p": "-1/2" + }, + { + "coefficient_of_c_t_minus_2": "0", + "leading": "1", + "p": "0" + }, + { + "coefficient_of_c_t_minus_2": "-4/525", + "leading": "22/21", + "p": "1/2" + }, + { + "coefficient_of_c_t_minus_2": "-1/60", + "leading": "11/10", + "p": "1" + }, + { + "coefficient_of_c_t_minus_2": "-4/99", + "leading": "11/9", + "p": "2" + }, + { + "coefficient_of_c_t_minus_2": "-20/3", + "leading": "11", + "p": "10" + }, + { + "coefficient_of_c_t_minus_2": "-2180/21", + "leading": "110", + "p": "109/10" + }, + { + "coefficient_of_c_t_minus_2": "-21998000/2001", + "leading": "11000", + "p": "10999/1000" + } + ], + "quantile_controls": "exact rational brackets for Pareto mixtures", + "scientific_effect": "NONE", + "scope": "PR176 equation17 conditional moment consequence only", + "tests": 11 + }, + "observed_control_log": "test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests) ... ok\ntest_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests) ... ok\ntest_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests) ... ok\ntest_direct_log_tail_integral (__main__.MomentConsequenceTests) ... ok\ntest_hand_checked_special_cases (__main__.MomentConsequenceTests) ... ok\ntest_offered_rational_coefficient_factors (__main__.MomentConsequenceTests) ... ok\ntest_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests) ... ok\ntest_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests) ... ok\ntest_threshold_and_above_diverge (__main__.MomentConsequenceTests) ... ok\ntest_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests) ... ok\ntest_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests) ... ok\n\n----------------------------------------------------------------------\nRan 11 tests in 0.004s\n\nOK\n" + }, + "checks": [ + "C28 source P size+SHA256+blob", + "C28 source ERR size+SHA256+blob", + "C28 source BOREL size+SHA256+blob", + "C28 source REC size+SHA256+blob", + "C28 source MARK_D size+SHA256+blob", + "C28 source MARK_2 size+SHA256+blob", + "C28 source CAP size+SHA256+blob", + "C27 artifact MOMENT_CONSEQUENCE.md size+SHA256", + "C27 artifact moment_consequence_checks.py size+SHA256", + "C27 artifact moment-checks-normal.json size+SHA256", + "C27 artifact moment-checks-normal.log size+SHA256", + "C27 artifact moment-checks-optimized.json size+SHA256", + "C27 artifact moment-checks-optimized.log size+SHA256", + "C27 artifact AUTHOR_COMMENT_5907459466.json size+SHA256", + "C27 artifact AUTHOR_COMMENT_5903080058.json size+SHA256", + "C27 artifact SOURCES.json size+SHA256", + "C27 source PR176 size+SHA256+blob", + "C27 source R size+SHA256+blob", + "C27 source SC size+SHA256+blob", + "C27 source P size+SHA256+blob" + ] +} diff --git a/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md b/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md new file mode 100644 index 000000000..fb3bef3a9 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/MOMENT_CONSEQUENCE.md @@ -0,0 +1,228 @@ +# Conditional moments, quantiles, and coefficient bounds for the radial ratio + +Read-only nonauthor consequence review of Math- PR #176 equation (17), exact head +`71772a8906cf6663d16be2cd5f63e5d4ebdcee2b`, proof blob +`65f24b1f9a7b80522a2ee059f5f6877c9f970f0d`, 15,755 bytes, SHA-256 +`a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710`. + +This is an elementary conditional implication, not a second Slice B verdict. +Claude's Slice A and the competing actual Slice B reviewer retain their scopes. +The reviewer here authored none of this packet or its mathematical sources; +all are OpenAI-source-exposed, and organizational-independence credit is zero. +R17 claim: `13dc84b1-bc12-42df-8c7f-96f23cb1dd7d`, WE317; root coordinates +the actual public disposition. No proof branch or scientific status was changed. + +Exposure chronology: the second-order power/log derivations and their first +seven exact arithmetic controls were completed before this reviewer read +author comment [5907459466](https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5907459466), +posted at 2026-09-30T08:38:26Z. That comment already states these power/log and +fixed-alpha quantile formulas. They are credited to the author's offered +corollaries here; no novelty or source-unexposed credit is claimed. The quantile +analysis below was made after that exposure. The all-order extension was +suggested by root, then reconstructed here using tail integration. Author +comment [5903080058](https://github.com/d6g8k5htny-coder/Math-/pull/176#issuecomment-5903080058) +was read before analysis of its coefficient inequalities. + +## Consumed hypothesis and scope + +Write Q_t=R/t under the source's point-intensity sampling conditional on R>t, +after the original r limit has already formed its microscopic point measure. +Retain the source's full normalizer; no cluster-maximum selection or extra +factor for multiplicity is introduced. Let c=C_2/C_0, with C_0>0. Consume (17) +in its quantified form: for each fixed source model there are T,K=T and q>=1, + + S_t(q)=P(Q_t>q) + =q^-11[1+c t^-2(q^-2-1)]+E_t(q), + |E_t(q)| <= K t^-4 q^-11. (M1) + +This pointwise remainder, uniform over the unbounded q range, is essential. +Unweighted L1 or total-variation convergence alone would not justify integrating +the unbounded test function Q_t^p or log Q_t. The original source's positive +tail asymptotic also supplies F(x)~C_0 x^-11 as x tends to infinity. + +## Every fixed real power below eleven + +For every fixed real p<11, + + E Q_t^p = 11/(11-p) + - [2p c/((13-p)(11-p))] t^-2 + O_p(t^-4). (M2) + +For p=0, this means the exact identity E Q_t^0=1 and a zero correction. +For p!=0, the deterministic identity for x>=1 is + + x^p = 1 + p integral_1^x q^(p-1)dq. + +For 00, the leading +correction is negative for positive p and positive for negative p. These are +asymptotic statements at each fixed model, not uniform finite-threshold bounds. + +## The divergence boundary + +For any finite t>0 for which this conditional law is defined, and every p>=11, +`E Q_t^p=infinity`. Indeed, + + S_t(q)=F(tq)/F(t) ~ [C_0 t^-11/F(t)] q^-11, + +and its positive leading coefficient yields a lower bound by a positive +multiple of q^-11 for all sufficiently large q. The positive-power tail +integral diverges at p=11 logarithmically and at every p>11 as a power. +No limit interchange at p=11 and no claim about the original finite-r field +is involved. The finite capped eleventh moment in #176 is a different object. + +## Logarithmic moment + +Since log x=integral_1^x q^-1dq for x>=1, direct Tonelli integration gives + + E log Q_t = 1/11 - (2c/143)t^-2 + O(t^-4). (M4) + +The first integral is `1/11`; the correction is +`c(1/13-1/11)=-2c/143`. The absolute remainder is at most `K t^-4/11`. +This derivation does NOT differentiate (M2) with respect to p or assume any +unstated differentiable uniformity in the p-indexed remainder. + +## Conditional all-order extension + +Consume in addition the finite-order remainder in equation (5), obtained from +the smooth even scalar C in Theorem 1. Let alpha_j=11+2j and D(t)=C(1/t)>0. +For every fixed nonnegative integer N and every t>=T, q>=1, + + t^11 F(tq) = sum_(j=0)^N C_(2j) t^(-2j) q^(-alpha_j) + + R_(N,t)(q), + |R_(N,t)(q)| <= K_N t^(-2N-2) q^(-13-2N). (M5) + +The scalar bound at s=tq provides this uniform remainder directly; no derivative +of a rough O term is taken. For every fixed real p<11, tail integration gives + + E Q_t^p = [sum_(j=0)^N alpha_j/(alpha_j-p) + C_(2j)t^(-2j) + + O_(p,N)(t^(-2N-2))] / C(1/t). (M6) + +To see the normalization explicitly, multiply the exact tail identity for +E Q_t^p by D(t). Its right side is D(t) plus p times the integral of (M5) +against q^(p-1). Expand D(t) at the same finite order and use +`1+p/(alpha_j-p)=alpha_j/(alpha_j-p)`. The remainder in this numerator is +bounded by `K_N[1+|p|/(13+2N-p)]t^(-2N-2)`. The p=0 moment is exactly one. +Keeping the denominator exact avoids a hidden normalization change. Replacing +it by `sum_(j=0)^N C_(2j)t^(-2j)+O_N(t^(-2N-2))` is equivalent. + +For every fixed integer m>=1, direct nonnegative tail integration similarly gives + + E (log Q_t)^m = [m! sum_(j=0)^N C_(2j)t^(-2j)/(11+2j)^m + + O_(m,N)(t^(-2N-2))] / C(1/t). (M7) + +Indeed, use `m integral_1^infinity (log q)^(m-1)S_t(q)dq/q` and substitute +y=log q. Each integral is m!/alpha_j^m. The numerator remainder is at most +`m! K_N/(13+2N)^m t^(-2N-2)`. These are finite-order expansions for fixed +indices, with no claim of convergence of an infinite Taylor series. + +## Fixed upper-tail quantiles + +Let 01, +and its q derivative at (0,q0) is -11 q0^-12, which is nonzero. The implicit +function theorem therefore gives a smooth even local solution q_alpha(delta). +For sufficiently large t, the separately supplied differentiated expansion +`-F'(s)=11C_0 s^-12+O(s^-14)` makes F strictly decreasing for every s>=t. +Together with S_t(1)=1 and S_t(q)->0, this identifies the local solution as +the unique upper-tail quantile of the conditional law. + +Put q=q0(1+a delta^2). Substitution in (17) gives + + S_delta(q)=alpha[1+(-11a+c(q0^-2-1))delta^2]+O_alpha(delta^4). + +Consequently + + Q_alpha(t)=alpha^(-1/11)[1+(c/11)(alpha^(2/11)-1)t^-2 + + O_alpha(t^-4)]. (M8) + +The author's quantile formula is correct under these source inputs. Strict +density at sufficiently large radii supplies uniqueness; an unquantified +pointwise distributional limit alone would not supply the error term. At +the aligned planar scope c>0 makes the displayed correction strictly negative. +The eventual approach from below is for each fixed alpha and fixed model; +no uniform claim as alpha tends to an endpoint, or over L,k,b, is needed. + +## Scope of the signed-law coefficient bound + +Let rho have probability density proportional to H gamma^11 G0 in lambda. +The positive finite normalizer is supplied by C_0 and the source Gaussian +majorant. Equations (13) and (19), with the constants in (12), yield exactly + + B_sign = (11 U_abs/(2I)) E_rho[|A|/sqrt(1+A^2)]. + +At every finite A, the integrand lies in [0,1). Positivity of G0 on the +nondegenerate Gaussian support gives positive rho mass at a!=0, so its mean +is strictly positive. Its mean is strictly below one because 1 minus the +integrand is a strictly positive measurable function under a probability law. +Thus, throughout the fixed dimension/frame scope of sections 1--5, + + 0 < B_sign < 11 U_abs/(2I) = 4587821/876544. (M9) + +This bound does not use a sign of D_a G0 and does not require the aligned +planar factorization of section 7. The author's comment 5903080058 states the +bound after an aligned-planar discussion using the word "Likewise"; (M9) +makes its valid broader scope explicit. By contrast, the offered strict bound +`c>11U_2/(2I)=4587/856` does use section 7's D_a G0<=0, with B<0. In that +case the first geometric integrand ratio `(1+12A^2)/(1+A^2)` is >1 whenever +a!=0, which proves strictness already from the positive Gaussian mass. +Neither coefficient inequality supplies a parameter-uniform remainder. + +## Source and verification boundary + +`SOURCES.json` is the exact source map from #176. The cached `R.md`, `SC.md`, +and `P.md` were each checked against its declared size, SHA-256 and Git blob. +R and P reused matching local source bytes; SC was fetched at its explicit +historical commit/path. No author program was used for the calculations. + +`moment_consequence_checks.py` independently compares tail- and density-based +rational integrals, fixed hand-derived cases, the divergence threshold and log +coefficient. A genuine mixture of Pareto(11), Pareto(13), Pareto(15) validates +the quantitative remainder bound and supplies the nonuniform-in-p counterexample. +Wrong normalization, sign, omitted p, and missing factor-two controls are rejected. +Additional exact mixture checks compare the all-order normalized numerator +against direct component moments. Log-power integrals, quantile coefficients, +and the two rational coefficient factors are checked separately. +These finite checks support arithmetic only; they do not replace (M1), its +Gaussian assumptions, or the pending source-bound A/B analytic reviews. + +Preliminary reading of #176 §§4–7 found no blocking error before coordination +redirected this reviewer; that preliminary observation is not a completed B +verdict. The results here are conditional corollary validation (M2)--(M9) and +the positive-tail divergence implication, not acceptance of the consumed +Gaussian theorem or the full PR. Scientific effect: NONE. diff --git a/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md b/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..fcd55a978 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/NONAUTHOR_REVIEW.md @@ -0,0 +1,40 @@ +# Nonauthor review of the radial moment consequences + +Reviewer: OpenAI / Codex subagent `/root/workflow_bottleneck_audit`. +Date: 30 September 2026. + +**VERDICT: ACCEPT (M2)–(M9), including the positive-tail divergence statement, as conditional consequences of the exact specified PR176 tail, smoothness and coefficient inputs. No blocking amendment found. This is not another Slice B verdict.** + +I did not draft or amend this consequence manuscript or its control script. I read the frozen consequence, the exact PR176 proof, the source/custody map, and the exact controls after the derivation was delivered. Same provider and shared-account/source exposure imply zero organizational-independence credit. The author's prior power/log/quantile formulas are expressly credited by the manuscript; I make no novelty claim for them. + +## Exact object + +- `MOMENT_CONSEQUENCE.md`: 11,110 bytes; SHA256 `29bba56b56fd3223a2ea71d012f9abe04eb30240bc5bbfe8f5b131e0bba6f4dd`; Git blob `fb3bef3a9a46cc0135a88bae6c26e1bade4e7384`. +- `COROLLARY_CUSTODY.json`: SHA256 `b532794b61e6ffd5539ce9f7dc5509b44d04d2c238a14f33e214c8e387532a64`. +- Consumed PR176 proof: head `71772a8906cf6663d16be2cd5f63e5d4ebdcee2b`, blob `65f24b1f9a7b80522a2ee059f5f6877c9f970f0d`, SHA256 `a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710`. + +All nine custody-listed artifacts and all four declared source bodies matched their recorded sizes and hashes; source Git blob identities also matched. The observation file is `IDENTITY_AND_CONTROL_OBSERVATIONS.json`. + +## Analytic reconstruction + +1. The consumed remainder really has the strength used here: PR176(17) is uniform for q≥1 with an O(t^-4 q^-11) tail envelope. The consequence does not derive unbounded moments from unweighted total variation. It keeps the already-formed microscopic point-intensity law, conditioning at R>t, and the original source normalization. + +2. For a positive power p<11, nonnegative tail integration yields 1+p∫q^(p-1)S_t(q)dq. For p<0, absolute integrability follows already from S_t≤1 and ∫_1^∞q^(p-1)dq<∞. Thus the same signed identity holds throughout p<11, including the separate exact p=0 case. Inserting the uniform remainder gives a bound |p|K/(11-p)t^-4. The displayed correction is p c[1/(13-p)-1/(11-p)] = -2pc/[(13-p)(11-p)]. All special cases and sign statements follow. + +3. The range is sharp for this actual limiting law, not merely for a formal approximation. F(tq)/F(t) has a positive constant times q^-11 tail for every finite t>0 with F(t)>0. Therefore every p≥11 moment diverges, logarithmically at eleven and as a power above it. This is distinct from the capped eleventh moment and asserts nothing about finite-r field moments. + +4. The logarithmic correction can be obtained directly from ∫S_t(q)dq/q. This gives 1/11 and c(1/13-1/11) = -2c/143, with absolute error at most Kt^-4/11. No differentiation in p of a remainder is required. + +5. For the finite-order Mellin expansion, applying the source scalar expansion at s=tq gives a uniform remainder O(t^(-2N-2)q^(-13-2N)) after multiplying by t^11. Its weighted tail integral converges for every fixed p<11. The numerator is D(t)+p times that integral, so expanding D(t) at the same order yields (11+2j)/(11+2j-p), exactly the proposed coefficient. Retaining the exact denominator C(1/t) preserves normalization; its positive limit C0 keeps division stable. The stated remainder bound and p=0 convention are consistent. There is no claim that the infinite Taylor series converges. + +6. For log powers, the change of variable y=log q reduces each term to an exponential Gamma integral. Multiplication by m gives m!/(11+2j)^m. The remainder integral is bounded by m!K_N/(13+2N)^m at the stated t order. Every fixed integer m≥1 is covered; there is no uniform-in-m claim. + +7. The quantile expansion uses more than the CDF error: the exact scalar representation is smooth and even in delta, and its q derivative at delta=0 is -11q0^-12≠0. The implicit function theorem gives the local even expansion. The separately supplied differentiated tail makes F strictly decreasing for all sufficiently large arguments, identifying the local root with the unique conditional upper-tail quantile. Substitution gives coefficient c(alpha^(2/11)-1)/11. The restriction to fixed alpha in (0,1) and fixed source model is essential and present. + +8. The signed coefficient bound follows directly from the source C0 and B_sign identities. After normalization by the positive measure with density proportional to H gamma^11 G0, its remaining ratio is E[|A|/sqrt(1+A²)]. It is strictly between zero and one because the source density gives positive mass at nonzero a and A is finite. This proves 0 < B_sign < 11U_abs/(2I) at the stated general fixed-dimension/frame scope. It needs no sign for D_a G0. The stricter lower bound on c uses the aligned planar sign from source §7, as the consequence correctly states: its second contribution is nonnegative and its geometric ratio (1+12A²)/(1+A²) exceeds one on positive mass. + +## Verification and limits + +I inspected and replayed `moment_consequence_checks.py` once with `python3 -B -S`; all eleven test methods passed. The finite Pareto mixtures independently exercise tail-versus-density normalization, powers of both signs, threshold divergence, log coefficients, finite-order numerator normalization, quantile brackets and the rational coefficient factors. They support arithmetic and the claimed failure of a uniform-in-p remainder; they do not certify the source Gaussian theorem. + +No defect was found in these conditional implications. Their premise remains the exact source smooth expansion and retained Gaussian coefficient identities, whose Slice A/B reviews are separate. Constants need not be uniform as p approaches eleven, alpha approaches zero or one, or physical parameters leave fixed compact scopes. The original r limit precedes the t limit. No whole-cluster maximum law, simultaneous t(r) theorem, source acceptance or scientific-register change follows from this review. diff --git a/reviews/radial_moment_corollaries_20260930/P.md b/reviews/radial_moment_corollaries_20260930/P.md new file mode 100644 index 000000000..dfed3b8d3 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/P.md @@ -0,0 +1,494 @@ +# Uniform matrix-cap selection and unrestricted short-lifetime asymptotics + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1. +**Author:** OpenAI / ChatGPT, 24 September 2026. +**Disposition:** author-side mathematical candidate; no nonauthor acceptance, formal verification, numerical finite-band certificate, or program-level closure is asserted. + +This is an additive continuation of MARKED-CYLINDER-CAP-20260924-v1, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`, reproduced unchanged in `sources/`. It addresses the matrix-probability, angular-uniformity, compact-mark, and lifetime-composition interfaces of campaign #61. It does **not** extend the predecessor's numerical constants to new parameters. Instead it derives new, non-numerically-evaluated constants by a uniform Gaussian argument. + +## 1. Model, convention, and the three theorems + +Fix a dimension d >= 2 and a side length L > 0. On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance + + K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2) + / sum_{n in Z^d} exp(-|Ln|^2/2). + +In particular d=3, L=24 is an actual three-dimensional SIDE24 application. No replacement by a nonperiodic covariance is made. Fix compact intervals B=[b_-,b_+] and K=[k_-,k_+] with 00, use the embedded local cylinder centered at the origin with pins + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=s=b-k r^3, + grad f(M)=grad f(S)=0. + +Q_{r,b,k,R} is the continuous Gaussian regression law at these 2(d+1) observations, not a conditioning on a positive-probability event. A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. Set + + W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}, + Z_r = E_Q W_r, dQ^W = (W_r/Z_r)dQ. + +The index counts negative eigenvalues. The FULL normalizer Z_r has no extra pin-density, coordinate Jacobian, or adjacency condition. Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. + +**Theorem A (uniform selection candidate).** There exist r_*>0 and C=z_* r^2 and the full transverse-matrix boundary layer, including simultaneous small transverse eigenvalues. It does not presume isotropy under arbitrary rotations of the torus. + +**Theorem B (compact-mark density candidate).** Suppose B and K have positive lengths. Count ordered maximum/saddle pairs with 00, + 0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3), (1.2) + +as ell decreases to zero. An exact finite-dimensional Gaussian integral for c_{B,K} appears in Section 11. + +Theorem B is a COMPACT MARK WINDOW result, especially its quantitative difference estimate. Theorem C below separately removes the mark and distance restrictions for the LEADING asymptotic. There is no stated numerical value of C, r_*, z_*, or c_{B,K}, and no claim that r_*=1/20. The new asymptotic normalizer proof does not replace or re-certify the historical H3 finite-band floor. + +**Theorem C (unrestricted finite-lifetime leading density candidate).** For the same fixed torus and dimension, let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. There is a positive finite constant c_{d,L} such that + + nu_cand^all(ell) ~ c_{d,L} ell^(-1/3), + nu_eld^all(ell) ~ c_{d,L} ell^(-1/3). (1.3) + +All birth heights, all positive scaled-gap marks and all spatial separations on the torus are included in (1.3). The essential global-maximum class is excluded from the finite-lifetime count. The constant is an explicitly identified finite Gaussian/angular integral, with the birth and gap integrals analytically eliminated in Section 15. No elementary closed form or numerical enclosure is claimed. Crucially, the compact-window O(ell^(2/3)) DIFFERENCE estimate in (1.2) is not asserted for the unrestricted densities. Theorem C has a separate domination and off-diagonal proof, not a change of label on Theorem B. + +## 2. Positive Fourier spectrum and finite-jet rank + +Poisson summation gives + + K_L(z)=sum_{n in Z^d} a_n exp(2 pi i n.z/L), + a_n=exp(-2 pi^2 |n|^2/L^2) / sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +Thus every Fourier mode has positive variance, and sum_n sqrt(a_n)(1+|n|)^q is finite for every q. A real sine/cosine expansion therefore converges in every C^q norm in every finite L^p norm. In particular the field has a smooth version and all finite moments of its global derivative suprema exist. + +Any finite list of distinct derivative evaluation functionals at distinct sites is linearly independent. Indeed a vanishing-variance linear combination must annihilate every Fourier mode because every a_n is positive. As a distribution it is a finite sum of derivatives of point masses with every Fourier coefficient zero, so it is zero. Smooth test functions supported at one site and with arbitrarily prescribed finite jets show that every coefficient is zero. At one site in a rotated frame the same assertion also follows from a polynomial P(R^T n) vanishing on all n in Z^d: the invertible rotation reduces this to a polynomial vanishing on the full integer lattice, hence the polynomial is zero. + +Only lists without repeated derivative functionals are called distinct. A Hessian is parametrized by its independent symmetric entries, not by d^2 allegedly independent entries. + +These arguments justify positive covariance for the original pins at each pair of distinct points and for every residual jet list modulo the pins used below. They also explain why finitely many observed modes or a numerically positive sampled covariance would not suffice for this proof. + +## 3. An exact nonsingular contact frame for all orientations + +Write x for the axial coordinate and y_1,...,y_m for the transverse coordinates, m=d-1. On the line y=0 set a=-r/2, c=r/2. Order the original observations as + + O_r=(f(a), f_x(a), f(c), f_x(c), + f_y1(a),f_y1(c), ..., f_ym(a),f_ym(c)). + +This is a permutation and orthogonal re-expression of the specified physical pins. Define U_r=T_r O_r by four axial rows + + U0 = (f(a)+f(c))/2, + U1 = (f(c)-f(a))/r, + U2 = (f_x(c)-f_x(a))/r, + U3 = (6/r^2)[f_x(a)+f_x(c)-2(f(c)-f(a))/r], (3.1) + +and, for every transverse direction j, two rows + + Vj0=(f_yj(a)+f_yj(c))/2, + Vj1=(f_yj(c)-f_yj(a))/r. + +The transformation is invertible for r>0 and has exact absolute determinant + + |det T_r| = (12/r^4)(1/r)^m = 12 r^(-(d+3)). (3.2) + +The prescribed target becomes exactly + + v_r=(b-k r^3/2, -k r^2, 0, 12k, 0,...,0). (3.3) + +This is not an approximate transformation, and the factor 12 in (3.2) is retained in the lifetime ledger. + +Taylor formulas with integral remainders give mean-square convergence, uniformly over frames, of U_r to the contact jet + + U0*=(f, f_x, f_xx, f_xxx, + f_y1,f_xy1,...,f_ym,f_xym) at 0. (3.4) + +The potentially dangerous U3 cancellation is an averaged third derivative, not a subtraction of uncontrolled random values. Uniform convergence of its covariances, and of cross-covariances with any fixed finite derivative order, follows either from those integral formulas or from the rapidly summable Fourier series. The target v_r converges uniformly on B x K to v_0=(b,0,0,12k,0,...,0). + +The functionals in (3.4) are distinct. Hence their covariance Sigma_0(R) is positive definite by Section 2. Its entries are continuous in R. The orthogonal group is compact, so its least eigenvalue has a positive uniform minimum. Consequently Sigma_r(R) remains uniformly positive definite for all sufficiently small r; Sigma_r and Sigma_r^{-1} converge uniformly. + +This compactness argument establishes existence, not an evaluated interval enclosure. It never equates Sigma_r at different orientations. Full torus rotational invariance is neither assumed nor true in general. + +Let A_r=D_y^2 f(M), represented by its m(m+1)/2 independent entries. Append these entries to U_r. At r=0 the extra functionals are f_yi_yj, i<=j; none duplicates a functional in (3.4). The enlarged covariance is therefore uniformly positive definite at contact and for small r. Its Schur complement shows that the Q-law of A_r has covariance uniformly bounded above and uniformly bounded away from zero. Its mean is uniformly bounded. In symmetric-matrix Lebesgue coordinates it has a density satisfying + + density_Q(A) <= C_0 exp(-c_0 ||A||_F^2), (3.5) + +for fixed positive C_0,c_0. This remains valid after any change of eigenvector orientation. Orthogonal invariance of the density is not required. + +## 4. Uniform conditional derivative moments and the independent residual + +Because Sigma_r^{-1} and v_r are uniformly bounded, the regression energy v_r^T Sigma_r^{-1}v_r is uniformly bounded. For each unconditioned standardized Fourier coefficient xi, covariance Cauchy-Schwarz bounds its conditional mean by the square root of this energy, and its conditional centered variance by one. Minkowski's inequality and the summable Fourier coefficients then give, for every finite p and derivative order q, + + sup_{r,b,k,R} || ||f||_{C^q(X)} ||_{L^p(Q)} < infinity. (4.1) + +The same proof applies to a centered residual after further conditioning on A_r. Write, in one full-field Gaussian coupling, + + f(z)=mu_r(z)+B_r(z).(A_r-E_Q A_r)+g_r(z), (4.2) + +where g_r is jointly independent of the entire symmetric matrix A_r. The regression coefficient B_r and its derivatives through order four are uniformly bounded, by the uniform inverse covariance from Section 3 and bounded derivative cross-covariances. The residual has uniformly bounded C^4 moments of every finite order. + +Independence in (4.2) is not asserted separately for Hessian entries, endpoint variables, or sampled derivative maxima. All residual evaluations are orthogonal to A_r; Gaussian independence on a countable dense collection and continuity give independence of the random smooth field. The residual's internal correlations are retained. + +Let J_r=1+||g_r||_{C^4(X)}, with a fixed coordinate norm. The partial-block operator norms of the cap theorem differ from this coordinate norm by finite constants depending only on d. Hence, for M3,M4 on the local cylinder, + + M3 <= K_0(J_r+||A_r||_F), + M4 <= K_0(J_r+||A_r||_F), + J_r >= 1, J_r independent of A_r, + sup ||J_r||_p < infinity for every finite p. (4.3) + +No inverse Hessian moment is used. Gaussian pin covariance and random field Hessians are different objects. + +## 5. Endpoint identities and the actual full normalizer + +For either pin i in {M,S}, put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, + A_i=D_y^2 f(i), + H_i=[[r alpha_i, r beta_i^T], [r beta_i,A_i]]. + +The two gradient pins imply the vector and scalar average bounds + + |alpha_i| <= M3/2, ||beta_i|| <= M3/2, + ||A_S-A_M||_op <= r M3. (5.1) + +For example integral_a^c f_xx=0; subtract this average at an endpoint and integrate the M3-Lipschitz bound. For beta use the average of grad_y f_x and its vector derivative. This does not invoke a simultaneous vector Rolle zero. + +The exact height identity gives a weighted mean of f_xxx equal to 12k on [a,c]. Variation of f_xxx over that interval is at most rM4. The same average calculation yields + + |alpha_M+6k| <= r M4/2, + |alpha_S-6k| <= r M4/2. (5.2) + +The block determinant identity, valid even when alpha_i or A_i is zero, is + + det H_i/r = alpha_i det A_i + -r beta_i^T adj(A_i) beta_i. (5.3) + +By (4.1),(5.1), det H_i/r and W_r/r^2 have uniformly bounded moments of every finite order. + +Let A_0 be D_y^2 f(0) under regression on U0*=v_0. Its law is a full-support nondegenerate Gaussian on symmetric matrices. Finite-dimensional Gaussian regression gives uniform convergence of the means and covariances of A_M to those of A_0. Equations (5.1),(5.2) imply that the congruence-scaled endpoint Hessians converge, in probability uniformly over parameters, to + + diag(-6k,A_0) at M, diag(6k,A_0) at S. + +Here the congruence uses diag(sqrt(r),I), so its off-diagonal entries are sqrt(r) beta_i. The limiting matrices are nonsingular almost surely. This follows from the full matrix density and k>=k_->0. Thus the type indicator converges to 1{A_0<0}. Uniform integrability from (5.3) gives + + Z_r/r^2 -> z_0(b,k,R) + := (6k)^2 E[det(A_0)^2 1{A_0<0}], (5.4) + +uniformly on the compact parameter set. + +For completeness, uniformity can be checked by a subsequence argument: any allegedly bad sequence of parameters has a convergent subsequence; continuous Gaussian means/covariances, (5.2), and uniform integrability give the stated limit on that subsequence. The discontinuity set of the type indicator has Gaussian probability zero. Equation (3.5) supplies a common integrable majorant. + +The right side of (5.4) is continuous and strictly positive: the Gaussian law has positive density on an open negative-definite matrix ball. Compactness therefore yields + + 00, with eigenvalues + + 00. + +Determinant monotonicity on positive-definite matrices therefore gives + + |det H_M| <= r (h/2) det B. (6.1) + +This is a canceled-pivot bound: no estimate of 1/alpha_M is taken. + +From ||A_S+B||<=rh, the ordered singular values of A_S are at most lambda_j+rh. The adjugate norm is at most the product of the largest m-1 such bounds. With the empty product equal to one when m=1, (5.3) gives + + |det H_S| <= r(h/2) [lambda_1+(3/2)rh] + product_{j=2}^m(lambda_j+rh). + +Hence the actual typed weight satisfies + + W_r <= (r^2 h^2/4) lambda_1[lambda_1+(3/2)rh] + product_{j=2}^m lambda_j(lambda_j+rh). (6.2) + +The second soft factor includes the saddle mixed-square term. Omitting it, or replacing it by an unrelated order-one factor, changes the radius power. No Gaussian-entry independence is needed for (6.2). + +## 7. Full matrix boundary-layer integration, including corank intersections + +By the predecessor's deterministic marked-cylinder theorem, pairing and the one-branch conclusion hold on + + G_r={lambda_min(-A_M)>[4/(3k)]rM3^2, + rM4<=3k/10}. + +On typed support A_M<0 automatically. It suffices to control G_r^c directly, not infer a converse from a local path-event bound. + +First suppose m>=2. In (4.3) absorb sqrt(m) into K and set U=J_r+Lambda>=1. Then h<=KU. Failure of the depth condition is contained in + + 00 and use h<=K(J_r+lambda). Depth failure implies + + lambda<=Dr(J_r+lambda)^2 + <=2DrJ_r^2+2Dr lambda^2. + +It is contained in the union + + 01/(4Dr). (7.5) + +The large-lambda branch is retained. On the near branch, h<=K(1+4D)J_r^2 for r<=1, J_r>=1. The bounded scalar density, (6.2), and direct integration in lambda give a numerator <=C r^5 E J_r^10. On the far branch, direct fourth-moment Markov with the actual weight gives + + E_Q[(W_r/r^2)1_far] + <=(4Dr)^4 E_Q[(W_r/r^2)lambda^4] <=C r^4. (7.6) + +Thus the scalar numerator is <=C r^5+C' r^6. This separate argument is why the all-dimension statement does not silently drop the scalar far branch. + +Finally, for every dimension, the fourth-derivative exception is bounded directly by + + E_Q[(W_r/r^2)1{rM4>3k_-/10}] + <=[10r/(3k_-)]^4 E_Q[(W_r/r^2)M4^4] <=C r^4. (7.7) + +This is a pointwise Markov bound followed by a joint moment estimate. It is not Cauchy-Schwarz with a missing square root. All joint moments in (7.6),(7.7) are finite by Sections 4-5. + +Divide the numerators by the FULL lower bound (5.5). The result is + + Q^W(G_r^c)<=C_3 r^3+C_4 r^4<=C r^3. (7.8) + +## 8. Pinned genericity and global event measurability + +The following argument applies at every fixed positive r and parameter, not on an asserted common probability-one set for uncountably many laws. + +Away from the pins, consider (grad f(x),Hf(x)v), with ||v||=1. Its range has dimension 2d; its parameter manifold has dimension 2d-1. Modulo the pin observations, these functionals have full covariance rank: gradient and Hessian orders are distinct, and the map H -> Hv from symmetric matrices onto R^d is surjective. For two distinct unpinned points, the map + + (grad f(x),grad f(y),f(x)-f(y)) + +has range dimension 2d+1 and parameter dimension 2d. For a tie to a pinned height use (grad f(x),f(x)-b), or the analogous s map, with dimensions d+1 and d. + +Exhaust the relevant domains by compact sets separated from pins and from coincident points; use finitely many sphere charts when v occurs. Covariance positivity from Section 2 and compactness give bounded Gaussian densities on each such set. An elementary overdetermined-zero lemma applies: on a mesh of size epsilon, a random C^1 map from dimension q to dimension p>q, with derivative norm bounded by N, could have a zero only if one of O(epsilon^-q) mesh values has norm O(N epsilon). The density bound makes the probability O(N^p epsilon^(p-q)), tending to zero. First fix N, then let N increase; the random derivative norm is almost surely finite. This proves absence of all these zeros. + +Pinned Hessians have a full density on their symmetric entries for fixed distinct pins, so their determinants are nonzero almost surely. The pinned heights differ because k,r>0. Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. The transfer to Q^W follows by absolute continuity and 0f(M)} min_t f(gamma(t)), + +with supremum of an empty set equal to -infinity. The event d_f(M)>h can be represented by a countable union of polygonal-path tests, with rational interior vertices and endpoints, all staying strictly above h and with endpoint value above f(M). Strict clearances allow approximation by such paths. For variable M, use countably many coordinate charts and an initial segment depending continuously on M. These are Borel tests in (M,f). Thus equality d_f(M)=f(S) is Borel. On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. + +On G_r the unchanged deterministic cap theorem controls every boundary exit and supplies a connection through S to a point higher than b. Therefore the GLOBAL ordinary elder partner is S, not merely a local adjacency candidate. Equations (7.8) and the full-probability locus prove Theorem A. The cap theorem also gives exactly one actual ascending branch to M. It does not identify that branch with the transverse ridge. + +## 9. A source-bound weighted Kac-Rice interface + +For clarity, this section specifies why a global elder mark is admissible; a formula for local Hessian index alone would not suffice. + +Apply Kac-Rice first on compact spatial pair domains separated from the diagonal, to the Gaussian vector field (grad f(x),grad f(y)). It is C^1 and has positive definite point covariance for x!=y by Section 2. Its derivative is block diagonal with Jacobian |det H_x det H_y|. Conditional laws of the whole smooth field depend continuously on the imposed gradient values by finite-dimensional Gaussian regression. Derivative moments are locally bounded. + +The expected-integral extension in Armentano-Azais-Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1, gives the identity for bounded nonnegative continuous weights in the location and an additional Gaussian field mark. It is enough first to use cylinder weights depending on finitely many fixed evaluations of f and its derivatives through order two. Their conditional distributions depend continuously on the imposed gradients by Gaussian regression, so the paper's continuity requirements hold. The field is a random element of the separable C^2 space on the torus. On each compact pair domain the two measures on location x C^2 are finite, by the unweighted formula. Cylinder evaluations on a countable dense set, together with location coordinates, generate this Borel sigma-algebra. Equality on their bounded continuous test-function algebra gives equality of the Borel measures by the monotone-class theorem. This extends the identity to bounded Borel global marks, rather than pretending the elder indicator is continuous or lower semicontinuous. No independence between the mark and gradient field is assumed. + +Use the Borel elder mark from Section 8 and the type indicators. Disintegrate the two heights using their nondegenerate joint Gaussian density conditional on the gradients. This gives the full-pin density times E_Q[W] for candidates, and the same quantity times p_r for selected pairs. The continuous Gaussian regression kernel defines the canonical density version; the underlying measure identity is unaffected by height-null-set choices. + +Exhaust the off-diagonal spatial domain toward r=0. The r dr bound proved in the next section makes the expectations locally integrable, and nonnegative monotone convergence completes the near-diagonal formula. No finite-grid approximation of the continuum field is substituted. + +## 10. Exact radial ledger: the dimension cancels + +Let pi_r(R;v_r) denote the density of U_r at (3.3). Since U_r=T_r O_r, + + density_{O_r}(specified pins) + =12 r^(-(d+3)) pi_r(R;v_r). (10.1) + +Use midpoint z, directed separation h=r u, birth b, and gap mark k. The change from the ordered pair (M,S) to (z,h) has absolute determinant one. Polar separation contributes r^(d-1) dr d sigma(u), where sigma is ordinary, NOT probability-normalized, surface area on S^(d-1). The height change (f(M),f(S))=(b,b-k r^3), at fixed r, contributes r^3 db dk. There is no factor 1/2: maximum/saddle pairs are ordered by their different roles. + +Consequently the candidate intensity per unit midpoint volume is + + r A_r(b,k,u) dr db dk d sigma(u), + A_r=12 pi_r(R;v_r) (Z_r/r^2). (10.2) + +The powers are + + spatial (d-1) + height-mark 3 + pin -(d+3) + + determinant-normalizer 2 = 1. + +By Sections 3 and 5, + + A_r -> A_0=12 pi_0(R;v_0) z_0(b,k,R)>0 (10.3) + +uniformly and with a uniform finite upper bound. Although frames were used in the proof, (10.2) is invariant under a rotation of the transverse frame: the original zero-gradient law and determinant weight are unchanged by that orthogonal coordinate change. Thus it is a function of u alone. Uniformity was proved on the compact orthogonal group and descends to the sphere; no nonexistent global continuous transverse frame on the sphere is assumed. + +The selected intensity is r A_r p_r with the same measures. This supplies the domination needed below, rather than assuming a fixed-axis statement can be integrated. + +## 11. Lifetime pushforward and the exact compact coefficient + +For ell=k_->0. This proves (1.2). It is an O(ell^(2/3)) difference, not a claim that either density separately has a second-order expansion with that remainder. Convergence of A_r to A_0 was not assigned a numerical rate. + +## 12. Short bars, moments, and the exact remaining boundary + +Integrating the density versions gives the following expected per-unit-volume results for the same spatial/mark-restricted populations: + + E N_eld(0,t] ~ (3/2)c_{B,K} t^(2/3), + 0<=E[N_cand(0,t]-N_eld(0,t]]<=C t^(5/3). (12.1) + +For any real q>-2/3, + + E sum_{selected, ell<=t} ell^q + ~ c_{B,K} t^(q+2/3)/(q+2/3). (12.2) + +The same leading expression holds for candidates. For q>-5/3 the nonselected difference alone satisfies + + E sum_{candidate but not selected, ell<=t} ell^q + <= C_q t^(q+5/3). (12.3) + +For q<=-2/3 the two individual expectations in (12.2) are infinite for every sufficiently small positive t, because their densities have a positive ell^(-1/3) leading term. Equation (12.3) concerns the nonselected counting measure directly, not subtraction of two infinities. In particular the compact-window inverse-lifetime threshold is p<2/3 for integrability of ell^(-p), with logarithmic divergence at p=2/3. This is unrelated to the inverse-Hessian/Schur thresholds in issue #56. + +The corresponding unrestricted cumulative and individual-moment corollaries follow from Theorem C with c_{d,L} in place of c_{B,K}, by the same integrable-power argument. The compact-window nonselected difference rates (12.1),(12.3) are not automatically unrestricted rates. + +## 13. Removing all birth and gap cutoffs by a direct intensity majorant + +The covariance Sigma_r of U_r depends on radius and frame, NOT on the imposed target. Section 3 therefore supplies a fixed r_0>0, independent of b and k, on which its eigenvalues are bounded above and away from zero. Reduce r_0 below the torus injectivity scale and below one. Do NOT require a globally uniform lower bound for Z_r or a globally uniform constant in (1.1). + +For every b in R, k>0 and 0= c(b^2+k^2), (13.1) + +with fixed c,C>0. For the second inequality it suffices to use its two coordinates b-k r^3/2 and 12k: if r<=1, then b^2<=2(b-k r^3/2)^2+k^2/2. Thus the invertible covariance bound yields + + pi_r(R;v_r) <= C exp[-c(b^2+k^2)]. (13.2) + +Conditional Fourier-coefficient means are bounded by C|v_r| and their centered variances by one. The same summability/Minkowski proof as Section 4 now gives the TARGET-GROWTH bound + + E_Q (1+||f||_{C^3})^p <= C_p(1+|b|+k)^p (13.3) + +for every fixed finite p, uniformly over r and frames on this r_0 band. This is not the compact-target bound (4.1) applied outside its scope; it is its explicit polynomial-growth version. + +By (5.1),(5.3), for r<=1, + + |det H_i|/r <= C (1+||f||_{C^3})^d, + 0<=W_r/r^2<=C(1+||f||_{C^3})^(2d). + +These inequalities remain true when k is arbitrarily small; no division by k or by Z_r has occurred. Combining with (13.2),(13.3) gives the full unnormalized-intensity majorant + + 0<=A_r(b,k,u)=12 pi_r(R;v_r) Z_r/r^2 + <= C(1+|b|+k)^(2d) exp[-c(b^2+k^2)] =: H(b,k). (13.4) + +For every fixed positive r and every b,k>0, Z_r is positive and finite: append BOTH endpoint Hessians to the distinct-site pins; the full residual Gaussian density is positive on a product of an open maximum cone and an open index-(d-1) cone. Thus p_r is defined, even though no globally uniform lower floor has been used. For every fixed b,k>0, Theorem A on a compact neighborhood implies p_r->1 as r->0. + +At a fixed lifetime ell, the near-pair restriction r<=r_0 is exactly k>=ell/r_0^3. After the exact change (11.1), + + ell^(1/3) nu_cand^near(ell) + = integral_{R x (0,infinity) x S^(d-1)} + 1{k>=ell/r_0^3} A_{(ell/k)^(1/3)}(b,k,u) + /(3 k^(2/3)) db dk d sigma, (13.5) + +with the integrand defined to be zero where the indicator is zero. The selected near density has the additional factor p_r. Both integrands are dominated by H(b,k)/(3 k^(2/3)). This majorant is integrable: the k exponent -2/3 is strictly greater than -1 at zero, and Gaussian tails absorb the polynomial at infinity and in b. This explicitly handles the shrinking-k boundary where a compact-window selection constant can blow up. + +For each fixed b,k>0,u, the indicator tends to one, A_r->A_0 and p_r->1. Dominated convergence therefore gives the same limit for the two near densities: + + c_{d,L}=4 integral_{R x (0,infinity) x S^(d-1)} + k^(-2/3) pi_0(R;v_0) z_0(b,k,R) + db dk d sigma(u), (13.6) + 00, + D_u=E[(det A_u)^2 1{A_u<0} | V_u=0]. (15.1) + +Here A_u conditional on V_u=0 is an actual centered nondegenerate Gaussian symmetric matrix, not automatically GOE. Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and D_u>0. All are continuous in the frame and invariant under transverse-frame changes. + +The contact density and cone moment factor as + + pi_0(R;v_0)=p_(f,V_u)(b,0) p_G(0) phi_tau_u(12k), + z_0=(6k)^2 E[(det A_u)^2 1{A_u<0} | f=b,V_u=0], + +where phi_tau is the centered scalar Gaussian density of standard deviation tau. Integrating b by ordinary Gaussian disintegration gives p_Vu(0) D_u. Thus (13.6) becomes + + c_{d,L}=144 integral_{S^(d-1)} p_G(0) p_Vu(0) D_u + [integral_0^infinity k^(4/3) + phi_tau_u(12k) dk] d sigma. + +The scalar integral can be evaluated exactly. With t=12k and the elementary gamma integral, + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + = Gamma(7/6) tau^(4/3) / [24^(1/3) sqrt(pi)]. + +Consequently the full leading coefficient is + + c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] + * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) + D_u d sigma(u). (15.2) + +This eliminates both unbounded mark integrals. The remaining angular and negative-definite-cone expectation are finite, strictly positive and specified by derivatives of the EXACT periodized covariance. Formula (15.2) is an explicit finite-dimensional Gaussian representation, not a reported elementary closed form or numerical enclosure. The factor24 is a numerical Jacobian/gamma factor, NOT the side length L; it is the same factor for every L. Changing the field amplitude by a>0 scales the coefficient by a^(-2/3), consistent with nu_(af)(ell)=a^(-1)nu_f(ell/a); this is an additional normalization check. + +## 16. Supplied interfaces, retained limits, and review + +**Interfaces supplied by this candidate:** the actual transverse matrix boundary argument (including d=3); a uniform compact-mark asymptotic full normalizer; all orientations without false isotropy; the global Borel marked Kac-Rice interface; exact pin/radial/lifetime factors; compact-window leading density and quantitative selection difference; a separate integrable unnormalized Gaussian majorant removing ALL birth/gap cutoffs; an off-diagonal bound completing ALL spatial separations; the unrestricted finite-bar leading density and its cumulative/moment consequences; and analytic elimination of birth and gap in its coefficient. + +**Not supplied:** a numerical r_* or C on a prescribed band; a globally uniform O(r^3) probability constant over unbounded b,k; the compact O(ell^(2/3)) difference as an unrestricted remainder; an elementary closed form or numerical enclosure of (15.2); a rigorously evaluated lower coefficient for pairing failure; the original numerical H3/LPW replay; a shrinking-annulus RN certificate; a24-jet displacement enclosure; formal proof verification; nonauthor mathematical acceptance; or unrestricted P15 prize closure. No historical scientific status is changed by this author-side proof. + +The exact-arithmetic program checks pin transforms/targets, block determinants, finite typed-matrix envelopes, spectral Jacobians, radius/lifetime exponents, explicit small-eigenvalue integral models, global target coercivity and coefficient scaling. Semantic mutations challenge the load-bearing factors. These are finite algebra/implementation controls, NOT Gaussian simulation or proof of continuum estimates. + +Priority nonauthor review: (i) desingularized full residual covariance; (ii) uniform all-field moments and independent-matrix regression; (iii) both soft determinant factors; (iv) eigenvalue integration retaining the random largest-eigenvalue width and all corank strata; (v) scalar far branch; (vi) full-normalizer type convergence; (vii) continuous-to-Borel marked Kac-Rice extension; (viii) every factor in the pin/lifetime ledger; (ix) target-growth bound (13.3) and integrability at k=0 in (13.5); (x) distinct-site compact covariance in the far domain; and (xi) parity, b-disintegration and the gamma factor in (15.2). + +### Primary-source attribution + +Armentano, Azais and Leon, arXiv:2304.07424v3, Sections 2 and 7, supply established Gaussian Kac-Rice/expected-integral background. The precise full-field/Borel and pair application is explained in Section 9, not attributed wholesale to that paper. Curry, *The Fiber of the Persistence Map for Functions on the Interval*, arXiv:1706.06059v2, Section 3, supplies elder-rule background; the all-dimensional separating cap is the explicitly identified in-project predecessor. Beliaev, Cammarota and Wigman, *No repulsion between critical points for planar Gaussian random fields*, arXiv:1911.03455, is neighboring two-point/index-asymptotic work, not a source of our elder-selection conclusion. No priority or novelty claim follows from this bounded reconnaissance. diff --git a/reviews/radial_moment_corollaries_20260930/PROOF.md b/reviews/radial_moment_corollaries_20260930/PROOF.md new file mode 100644 index 000000000..65f24b1f9 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/PROOF.md @@ -0,0 +1,334 @@ +# Smooth inverse-radius expansion and sharp rates for microscopic extreme marks + +Object: OA-RADIAL-RATE-20260930-v1. Author: OpenAI / GPT-6 Astra Pro. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect NONE. No parent proof, register, Boolean, prize, or earlier +review is changed. This conversation authored SC/P and contributed to the +related cluster line; no independent revalidation of those premises is implied. + +## 1. Exact object and scope + +Fix d>=2, L>0, b real, k>0 and an orthonormal frame, under the original model +and full endpoint normalizer. This paper concerns the ALREADY-FORMED limiting +microscopic POINT intensity of [R]. First the original field radius r tends to +zero; only then does the microscopic distance threshold t tend to infinity. +No finite-r convergence rate is proved. Whole-cluster maximum selection and +its moving tie boundaries are not consumed or treated here. + +Exact source identities are in SOURCES.json. [R] is PR166 RADIAL_TAIL.md at +80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4: +R7-R13 supply the root-resolved integral, Gaussian domination and positive leading +coefficient. [SC] is PR162 PROOF.md at382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf, +blob16c56821b52fd76b0be791622b9c3809eafde75a: the nondegenerate raw Gaussian contact +jet, orthogonal spectral convention and finite nonempty microscopic measure. +[P] is imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md at +7a1cb09a9d58d179393e6d29146252f714b9c9bf, blobdfed3b8d318a3ab1950957f393307733a4bef3f2: +only the exact periodized covariance and contact conditioning, used in Section7. +Neither PR169's whole-cluster candidate, its companion-ratio addendum, nor any +numerical coefficient note is a premise. Source reviews are not reviews here. + +Use [R]'s variables lambda=(a,h,O,xi), with ordered positive hard curvatures h, +normalized Haar O, and remaining cubic entries xi. Put + + A=a/(12k), gamma=sqrt(1+A^2), + beta0=a^2/(12k), c0=a^3/(144k^2), + A0=O diag(0,-h) O^T, + H(h)=product h_j^3 product_(i0 on S. + +For y=1/Z define + + g_lambda,u,v(y)=G_lambda(beta0+k b0 y^2, + c0+(a b0/4)y^2+2k(v-b0u)y^3). (2) + +[R, R11] says exactly that the tail F(t)=integral sum_i 1{|z_i|>t}dM is + + F(t)=K0 integral_(S,lambda) Q H + integral_R |Z|^-12 g(1/Z) + 1{(u-AZ)^2+Z^2>t^2}dZ du dv d lambda. (3) + +The multiplicity of roots is already correct in this formula. No extra factor +for a double cluster, pin density, or endpoint normalizer is introduced. + +## 2. A fixed-domain smooth even representation + +**Theorem 1.** There is a smooth even function C(delta), defined for +|delta|<2/3 and with every fixed derivative bounded on |delta|<=1/3, such that + + F(t)=t^-11 C(1/t), t>=3. (4) + +Smoothness also holds in L1 for the underlying UNSIGNED shape/parameter measures. +The claim is C-infinity, not analyticity or convergence of an infinite Taylor +series. Consequently for every fixed N>=0 there are finite coefficients C_(2j) +with + + F(t)=sum_(j=0..N) C_(2j) t^(-11-2j)+O_N(t^(-13-2N)), + -F'(t)=sum_(j=0..N)(11+2j)C_(2j)t^(-12-2j) + +O_N(t^(-14-2N)). (5) + +C_(0)>0 is [R]'s C_star. Constants are at the fixed source parameters; no growing +dimension, volume, vanishing-gap, or parameter-uniform claim is made. + +**Proof.** The reciprocal substitution in (3) gives |Z|^-12 dZ=|y|^10 dy on +both signs. For t>|u| the physical radius condition is + + (t^2-u^2)y^2+2uAy-gamma^2<0. + +Set delta=1/t and y=delta w. Its two endpoints in w are + + w_plus/minus(delta)=m(delta) +/- d(delta), + m(delta)=-uA delta/(1-u^2 delta^2), + d(delta)=sqrt(gamma^2-u^2 delta^2)/(1-u^2 delta^2). (6) + +Thus (4) holds with + + J_delta(u,v,lambda)=d(delta) integral_(-1)^1 + [m(delta)+d(delta)x]^10 + g(delta[m(delta)+d(delta)x]) dx, + C(delta)=K0 integral Q H J_delta du dv d lambda. (7) + +There is no moving integration boundary left. On |delta|<=delta0<2/3 the +denominator is uniformly positive and the square-root radicand is bounded +below by a positive constant, since |u|<=3/2 and gamma>=1. Each fixed derivative +of m,d is bounded by a polynomial in1+|a| with constants depending on delta0,k. + +For every fixed j the nondegenerate Gaussian contact density satisfies + + sup_(beta,c) |partial_(beta,c)^j G_lambda(beta,c)| + <= C_j exp[-c_j(|h|^2+a^2+|xi|^2)] (8) + +uniformly in O. Indeed each derivative is a polynomial times the same Gaussian; +absorb that polynomial into a weaker Gaussian exponent and then discard beta,c +from its quadratic lower bound. A bounded mean changes the constants only. +Uniform tensor norm equivalence under O supplies the displayed retained variables. +No inverse hard eigenvalue is used. This is an elementary derivative extension +of [R]'s raw Gaussian bound, not an extra stochastic assumption. + +The chain rule in (7), for any fixed number of delta derivatives, now gives an +integrable majorant of the form + + C Q H (1+|a|)^M exp[-c(|h|^2+a^2+|xi|^2)], (9) + +on the bounded shape domain and x in[-1,1]. The actual shifted beta,c values +may be large: (8) is a supremum over them, not an invalid compactness assumption. +This proves differentiation under all integrals and smoothness in L1 as well +as for the total mass. The exact identities m(-delta)=-m(delta), +d(-delta)=d(delta), followed by x->-x, show J_(-delta)=J_delta. This cancellation +holds even when g(y) is not even; its cubic y^3 term is retained. + +Taylor's theorem with the integrated derivative bounds proves (5), including +its differentiated remainder. It is NOT obtained by formally differentiating +an unspecified big-O estimate. C(0) is the positive source coefficient. QED. + +## 3. Explicit second coefficient + +Write C_0=C_(0), C_2=C_(2), to distinguish coefficients from their tail powers11,13. +Set the differential operator acting on G_lambda at fixed a,h,O,xi + + D_a=k partial_beta+(a/4)partial_c. + +The pointwise density has expansion g(y)=G0+g2 y^2+g3 y^3+O(y^4), with + + g2=b0 D_a G0. (10) + +There is no linear y term. Expanding the exact endpoints, or differentiating +(7), gives + + j0=(2/11)gamma^11 G0, + j2=u^2 gamma^9(1+12A^2)G0+(2/13)b0 gamma^13 D_a G0, + J_delta=j0+j2 delta^2+O_L1(delta^4). (11) + +For the geometric part, write w_+=gamma-uA delta+ +[u^2 gamma-u^2/(2gamma)]delta^2+O(delta^3), with the corresponding reflected +negative endpoint. Then (w_+^11-w_-^11)/11 equals +2gamma^11/11+u^2 gamma^9(1+12A^2)delta^2+O(delta^4). +The g2 term contributes2g2 gamma^13 delta^2/13. The odd g3 term first contributes +at order delta^4 after the two physical sides are summed; it is not dropped +from the all-order argument. + +The necessary exact shape integrals are + + I=integral_S Q=246528/35, + U2=integral_S u^2 Q=240192/35, + B=integral_S b0 Q=-428544/7=3I-12U2, + Uabs=integral_S |u|Q=5898627/880. (12) + +They follow by integrating the polynomial Q in v and then on the strips cut +at u=-1/2 and u=0. The companion computes them in exact Fraction arithmetic. +In particular + + C_0=K0*(2I/11) integral H gamma^11 G0 d lambda, + C_2=K0 integral H [U2 gamma^9(1+12A^2)G0 + +(2B/13)gamma^13 D_a G0]d lambda. (13) + +Both integrals are finite by (8). C_0>0. No general sign or nonzero assertion +for C_2 is made in arbitrary dimension/frame; Section7 supplies an exact +noncancellation result at a narrower source scope. No numerical evaluation of +either Gaussian integral is claimed. + +## 4. Rates for the radial ratio and unsigned marks + +Let P_t be point-INTENSITY sampling conditioned on physical radius R>t. Define +q=R/t>=1 and retain Y=(u,v,lambda), but FORGET sign Z. These are the root-resolved +latent coordinates of [R], not a configuration-uniform selection. Define the +finite measures with densities + + dmu0=K0 Q H j0 dY, dmu2=K0 Q H j2 dY, + c=C_2/C_0. (14) + +mu0 has mass C_0 and mu2 mass C_2. The joint law of (q,Y) has the L1 expansion + + dP_t(q,Y)=11q^-12 dq dmu0(Y)/C_0 + +t^-2 [13q^-14 dq dmu2(Y)/C_0 + -11c q^-12 dq dmu0(Y)/C_0]+O_L1(t^-4). (15) + +In particular its total-variation distance to the stated independent limiting +product is O(t^-2). Projection to Y, or to the downward height +eta=u+1/2+v/6, retains this rate. This supplies an O(t^-2) TV rate for the +intensity-selected height law H_eta/I in [R, H1-H2]. No spatial-configuration +TV claim is made: the map to actual positions depends on t. + +Proof: for fixed Y, the radius tail density is t^-11 K0 QH J_(1/t). +Differentiate the smooth expression already justified in Section2; at tq its +radial density times t equals + +t^-11 K0 QH q^-12[11J_(1/(tq))+(1/(tq))J'_(1/(tq))]. + +Divide by F(t)=t^-11 C(1/t) and use the L1 Taylor estimates. The remainder is +bounded after integration by a constant times t^-4 q^-12, integrable on q>=1. +This proves (15) without differentiating a merely pointwise asymptotic. QED. + +Let Pareto_11 have density11q^-12 on q>=1. For the scalar radial ratio, (15) gives + + p_t(q)=11q^-12[1+c t^-2((13/11)q^-2-1)]+O_L1(t^-4). + +**Sharp scalar coefficient:** + + dTV(Law_(P_t)(R/t),Pareto_11) + =kappa |c| t^-2+O(t^-4), + kappa=(2/13)(11/13)^(11/2). (16) + +The same leading coefficient holds for the supremum distance of their CDFs. +The correction changes sign only at q0=sqrt(13/11). Its positive integral is +q0^-11-q0^-13=kappa; its total integral is zero. Half its L1 norm is therefore +kappa, proving (16). The uniform CDF remainder follows from the L1 remainder; +the maximum of q^-11(1-q^-2) is also kappa. If C_2=0 the stated error is O(t^-4), +not an unjustified claim of a nonzero second-order term. + +As another quantitative consequence, uniformly over q>=1, + + F(tq)/F(t)=q^-11[1+c t^-2(q^-2-1)]+O(t^-4 q^-11). (17) + +This is a source-specific second-order regular-variation statement. Generic +second-order tail theory is not claimed novel. + +## 5. Retaining the sign has a different, strictly slower sharp rate + +The cancellation in Section2 used both signs. Keep epsilon=sign Z and the +same Y. The two half integrals are exactly + + J_delta,epsilon=integral_0^(d+epsilon m) w^10 g(epsilon delta w)dw, + J_delta,epsilon=j0/2+epsilon h1 delta+O_L1(delta^2), + h1=-uA gamma^10 G0. (18) + +The linear term comes from the physical cutoff shift m. g'(0)=0, so no density +term cancels it. Applying the differentiated density argument to each half yields + + dTV(Law_(P_t)(epsilon,R/t,Y), + UniformSign x Pareto_11 x mu0/C_0) + =B_sign t^-1+O(t^-2), + B_sign=(K0/C_0) Uabs integral H |A|gamma^10 G0 d lambda>0. (19) + +Indeed the first signed correction has density +12 epsilon h1 q^-13 K0 QH/C_0; its half-L1 norm, summed over the two signs and +integrated over q, is the displayed B_sign because integral_1^infinity12q^-13dq=1. +Finiteness follows from (8); strict positivity follows from the full positive +Gaussian density on open sets with a!=0 and positive hard spectrum. The gauge +convention for O is exactly that in [R]; (19) is a result for those retained +latent variables, not a statement that a sign alone has a nonzero marginal bias. +Marginalizing can cancel a correction, and specifically forgetting epsilon +cancels this one. Thus signed and unsigned rates must not be conflated. + +## 6. A finite part for the critical eleventh moment + +Let M11(t)=integral sum_i min(R_i,t)^11 dM. The prior theorem gave its leading +logarithm. There is now a finite constant L11 such that + + M11(t)=11 C_0 log t+L11-(11/2)C_2 t^-2+O(t^-4). (20) + +One exact definition is + + L11=11 integral_0^1 x^10 F(x)dx + +11 integral_1^infinity[x^10 F(x)-C_0/x]dx. + +The first integral is finite by finite total point mass. The second converges +by (5) at N=1. Tonelli gives M11(t)=11 integral_0^t x^10 F(x)dx; subtracting +its limiting finite part and integrating the controlled remainder proves (20). +No numerical value for L11 is asserted. This is again after the r limit only. + +## 7. Exact positive second correction in the aligned planar periodic model + +**Theorem 2.** In d=2, with the axial coordinate aligned with one of the lattice +axes of [P]'s EXACT periodized kernel, C_2>0 for every fixed L,k>0 and b real. +Consequently the radial TV and CDF errors in (16) are genuinely Theta(t^-2), +with the positive coefficient in (16); not merely an upper bound. + +Proof: [P]'s kernel factors as K_L(x,z)=k_L(x)k_L(z). Let m2,m4,m6 be the even +spectral moments of the one-dimensional variance-one factor; m2>0 and the full +positive lattice spectrum gives m4-m2^2>0 and m6-m4^2/m2>0. At contact condition +on (f,fx,fxx,fxxx,fz,fxz)=(b,0,0,12k,0,0). + +Even-total-order jets and odd-total-order jets are independent. For the odd +soft cubic jets (a,beta,c)=(fxxz,fxzz,fzzz), separability and reflection in each +axis give a centered DIAGONAL conditional Gaussian with variances + + va=vbeta=m2(m4-m2^2), vc=m6-m4^2/m2. (21) + +For transparency: Cov(beta,(fx,fxxx))=(-m2^2,m2 m4) is -m2 times the first row +of Cov(fx,fxxx), so its conditional mean is -m2 fx=0 even though fxxx=12k. +a and c correlate only with fz in that odd conditioning block, with covariances +-m2^2 and -m4; their raw covariance m2 m4 is removed by that regression. Their +conditional covariance with beta vanishes by coordinate reflection. The diagonal +variances follow by subtracting these rank-one regressions. The even soft +curvature has its own strictly positive conditional density at zero and merely +multiplies this cubic Gaussian by a positive b-dependent scalar. + +Therefore on the cusp + + D_a G0=-[a^2/(12vbeta)+a^4/(576k^2 vc)]G0<=0. (22) + +In (13), U2>0, B<0 and G0>0. The geometric term is strictly positive and the +second term is nonnegative. Thus C_2>0, without replacing finite L by a continuum +covariance or numerically approximating a lattice sum. The same common even +b-dependent factor cancels from c=C_2/C_0 and B_sign at this planar aligned +scope. We do not extend this diagonal regression or noncancellation assertion +to arbitrary rotations or d>=3 without a separate argument. QED. + +## 8. Verification and review boundary + +A: reciprocal physical cutoff, fixed-domain map, C-infinity L1 Gaussian derivative +domination, exact parity and differentiated all-order asymptotic expansion. +B: j0/j2 coefficients, all four exact shape integrals, signed/unsigned TV and CDF +constants, finite-part moment, and aligned planar regression/noncancellation. + +The stdlib companion checks shape integrals, formal endpoint equations, exact +Taylor coefficients, cancellation and signed bias, independent scalar controls +and deliberately false variants. These are finite checks, not a formal proof +of any Gaussian input or differentiation under an infinite-dimensional integral. +No Lean formalization, global status change, coefficient enclosure or self-merge. + +No rate for r->0, no uniform simultaneous t(r)->infinity statement, no claim +about the remote singleton population, no convergence rate for whole-cluster +maximum selectors across ties, and no spatial independence/elder pairing is +added by this packet. Sign-forgetting is part of (15), not an optional omission. diff --git a/reviews/radial_moment_corollaries_20260930/R.md b/reviews/radial_moment_corollaries_20260930/R.md new file mode 100644 index 000000000..0f14417ef --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/R.md @@ -0,0 +1,277 @@ +# An eleven-power microscopic radius tail and its universal extreme-height law + +Object: OA-TWO-SCALE-CLUSTER-GEOMETRY-20260929-R1. +Author: OpenAI / GPT-6 Astra Pro, foreground recovery, 29 September 2026 (Central). +Disposition: AUTHOR-SIDE CONDITIONAL PROOF; NONAUTHOR REVIEW REQUIRED. +Scientific effect NONE. No scientific-status transition or numerical Gaussian +coefficient is asserted. This is an analytic calculation for the explicit +limiting MICROSCOPIC measure M of TWO_SCALE_LAW.md (M1), not for the entire +finite-r global field or its remote singleton component. + +## 1. Which radius and which sampling law? + +Use the exact source conventions and fixed d,L,b,k,frame of TWO_SCALE_LAW.md. +For each additional root (X_i,Z_i) of the unsheared planar cubic, its physical +rescaled position is + + v_i=X_i e_axis+Z_i e_soft(O), R_i=|v_i|. + +Define the finite point-intensity measure by integrating the sum over these roots +against M. Its tail mass is + + F(t)=integral sum_i 1{R_i>t} dM. (R1) + +This is point-intensity sampling: a cluster with two qualifying points contributes +two, not one. Every assertion below is a statement about the limiting microscopic +measure. In an application to the Gaussian field, r->0 is taken before t->infinity. +No finite-r uniform spatial moment or rate is inferred. + +**Theorem R.** There is an explicit finite positive C_star such that + + F(t) ~ C_star t^-11. (R2) + +For p>=0 the microscopic intensity moment integral sum_i R_i^p dM is finite +exactly when p<11. The same threshold holds for the maximum radius of a nonempty +cluster under its normalized limiting law. At the boundary, the capped intensity +moment satisfies + + integral sum_i min(R_i,t)^11 dM ~ 11 C_star log t. (R3) + +A full formula for C_star is given in (R13). No numerical evaluation of its +Gaussian integral is claimed. + +## 2. Resolve one stationary point by an exact change of variables + +Retain the canonical shear invariants of CUB: + + u=X+aZ/(12k), B=beta-a^2/(12k), + D=(c-a beta/(4k)+a^3/(72k^2))/2, x=-s>0. + +At fixed a the inverse raw-jet map is + + beta=a^2/(12k)+B, + c=a^3/(144k^2)+aB/(4k)+2D, + |d(beta,c)/d(B,D)|=2. (R4) + +The factor 2 is essential. This is a change in the JET coordinates, separate +from the determinant-one spatial shear. The remaining third-jet entries, denoted +xi, are unchanged. + +Let + + b0(u)=3-12u^2, v=xZ^2/k. + +For an extra root, the canonical conic and line give the inverse parametrization + + x=kv/Z^2, + B=k b0(u)/Z^2, + D=k(v-b0(u)u)/Z^3, Z!=0. (R5) + +Conversely, every parameter in the region defined below with Z!=0 gives a strict +window root at (X,Z)=(u-aZ/(12k),Z). The normalized DOWNWARD height is + + eta=-P/k=u+1/2+v/6. (R6) + +Typed endpoints require v>|b0(u)|/2. The open height window requires 0-6u-3 is automatic on R7: +for -1/2<=u<1/2 its right side is nonpositive; for -3/2=0. The upper height inequality and v>|b0|/2 are +compatible exactly on the displayed u interval. Boundaries are excluded. + +An explicit determinant computation gives the ORIENTED Jacobian + + det d(x,B,D)/d(v,u,Z) + =6k^3(12u^2+4uv-3)/Z^8 + =-6k^3[b0(u)-4uv]/Z^8. (R8) + +On R7 both b0(u)-4uv and 4v^2-b0(u)^2 are strictly positive. The first is also +the saddle determinant sign from CUB; it can be checked directly: for u<-1/2, +use v>|b0|/2, and for u>=0 use v<3-6u, giving a lower bound +12(u-1/2)^2, with the remaining interval -1/2<=u<0 immediate. The second follows +from v>|b0|/2. Thus the Jacobian is nonzero in the open region. + +For each fixed (x,B,D) in the typed domain there are at most two extra window +roots. The usual change of variables with multiplicity is legitimate here: +cover R7 times (R minus {0}) by countably many neighborhoods where R5 is a +diffeomorphism, make them disjoint measurably, apply the ordinary change of +variables to each, and sum the resulting nonnegative integrals. The multiplicity +is exactly the number of counted roots, rather than an omitted or extra factor +of two. Degenerate/boundary jets are algebraically null and not in the strict +window. This justifies applying R5 to a point-intensity sum. + +Define + + Q(u,v)=[4v^2-b0(u)^2][b0(u)-4uv]. (R9) + +The endpoint weight is w=9k^4[4v^2-b0(u)^2]/Z^4. Combining this with the absolute +Jacobian R8 and the raw-jet Jacobian 2 of R4 gives exactly + + w ds d beta dc =108 k^7 Q(u,v)|Z|^-12 du dv dZ, (R10) + +in the point-counted integral, with a and all other spectral/tensor variables +retained. This identity incorporates the root multiplicity through the map, +not by an additional constant. The two signs of Z will be integrated next. + +## 3. Tail scaling, domination and the complete coefficient + +Let h,O,xi be the remaining variables of M, and let tau(a,beta,c,xi) denote the +full rotated cubic tensor in those coordinates. Its raw contact Gaussian density +remains evaluated at + + h0(A0,Rot_O(tau)), A0=O diag(0,-h)O^t. + +It is NOT replaced by an isotropic density. Combining M1 and R10 gives the exact +representation of F(t) as the integral over R7 and Z!=0 of + + (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12 + h0(A0,Rot_O tau(a, a^2/(12k)+k b0/Z^2, + a^3/(144k^2)+a b0/(4 Z^2)+2k(v-b0 u)/Z^3, xi)) + *1{ (u-aZ/(12k))^2+Z^2 > t^2 }, (R11) + +with measure du dv dZ da dxi dh dO, ordered positive h, and the empty-product +conventions for d=2. The coefficient a b0/(4Z^2) in c includes the k cancellation +in aB/(4k). Formula R11 is often where confusing the spatial and jet Jacobians, +or integrating out a before testing the actual radius, gives a wrong exponent +or coefficient. + +Put Z=t z and gamma(a)=sqrt(1+(a/(12k))^2). Then |Z|^-12 dZ=t^-11 |z|^-12 dz. +At each fixed nonzero z the raw jet tends to the cusp slice + + beta_cusp=a^2/(12k), c_cusp=a^3/(144k^2), + +and the radius indicator tends almost everywhere to 1{gamma(a)|z|>1}. +For t>=3 and |u|<=3/2, the indicator in R11 implies + + |z|>1/[2gamma(a)] + +by the triangle inequality t<|u|+gamma|Z|. Orthogonal norm equivalence and SC's +Gaussian density bound imply, uniformly in t,z,u,v,O, + + h0(A0,Rot_O tau(a,beta,c,xi)) + <= C exp[-c(|h|^2+a^2+|xi|^2)]. + +Dropping beta and c from this upper bound is legitimate; it never introduces +an inverse hard eigenvalue. Integration in z over |z|>1/(2gamma) costs at most +a constant times gamma^11. Q is bounded on the bounded region R7, H(h) is a +nonnegative polynomial, and Gaussian integration in h,a,xi dominates gamma^11. +Dominated convergence therefore applies to t^11 F(t). + +Let + + J_cusp=integral H(h) gamma(a)^11 + h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi)) + da dxi dh dO. (R12) + +This is finite by the same majorant and positive by positivity of the Gaussian +density and H on open sets. Integrating both signs of z gives +integral_(gamma|z|>1)|z|^-12 dz=2 gamma^11/11. Finally, + + I_shape=integral_Dshape Q(u,v) du dv=246528/35, + + C_star=(216/11) k^7 (c_m/z0) I_shape J_cusp + =(53250048/385) k^7 (c_m/z0) J_cusp. (R13) + +For an exact check of I_shape, integrate v first using + + (4b0/3)v^3 -4u v^4 -b0^3 v +2u b0^2 v^2. + +The strips -3/2=2 under the source interface. The exponent 11 +is dimension-independent here because the hard directions affect only the +finite integral J_cusp; the soft jet weights are s,B of degree -2, D of degree -3, +and endpoint weight of degree -4, for total 2+2+3+4=11. This explanatory homogeneity +is not substituted for the dominated-convergence proof. + +For p>0, Tonelli gives the moment identity integral R^p=p integral_0^infinity +t^(p-1)F(t)dt. Finite total point mass handles t<=1 and R2 handles infinity. +This proves the threshold in Theorem R and the boundary formula R3. If R_max +is the maximum radius of a nonempty cluster, its unnormalized tail lies between +F(t)/2 and F(t), since there are at most two points. Its normalized law therefore +has the same exact moment threshold, although no exact maximum-tail coefficient +is claimed. QED. + +## 4. A universal law for an intensity-selected far local point + +Normalize the point-intensity measure restricted to R_i>t. For that selected +point retain its canonical shape variables (u,v), downward height eta, radial +ratio q=R_i/t, sign epsilon=sign Z, and the remaining variables (a,h,O,xi). + +**Theorem H (extreme local marks).** As t->infinity, the joint law converges +weakly to the product of: + +1. shape density Q(u,v)/I_shape on R7; +2. a Pareto radial ratio q>=1 with density 11q^-12; +3. a symmetric sign epsilon in {-1,1}; +4. the parameter measure with density H(h) gamma(a)^11 times the cusp Gaussian + in R12, divided by J_cusp. + +In particular eta=u+1/2+v/6 is independent of the limiting radial ratio and the +remaining parameter measure. Its probability density on 01. The limiting Gaussian depends only on (a,h,O,xi), while +Q depends only on (u,v). Division by R13 gives exactly the four factors above. +No claim of spatial independence in the original field is involved. + +To obtain H1, substitute v=6e-6u-3, with dv=6de. Conditions R7 become + + -1/2-sqrt(e) < u < 1/2-sqrt(1-e), 0=0 a reproducible rational formula before dividing by I_shape is + + integral e^j H_eta(e) de + =(110592/7)[1/(j+9/2)-B(j+1,9/2)+1/(j+1)] + +13824[1/(j+5)-2/(j+4)+5/(j+3)-4/(j+2)], (H4) + +with B(j+1,9/2)=j!/product_(l=0..j)(9/2+l). No special-function numeric library +is needed to check these moments. Seven such moments are checked independently +against exact double polynomial integration on R7. QED. + +## 5. Evidence boundary and useful falsifiers + +The finite companion independently checks the oriented root Jacobian, the inverse +raw-jet map and its factor 2, stationary heights and gradients at 648 rational +parameter choices, the two exact shape integrals, H4 versus direct integration, +and the k^7/radius-power ledgers. Eight explicit false variants must fail. +These controls do not prove the Gaussian bound, change-of-variables multiplicity, +weak convergence or domination; those are the written analytic obligations above. + +The result would fail if the contact slice lost full positive Gaussian density, +if the physical radius were incorrectly replaced by |Z| before retaining a, +or if the invariant density were assumed isotropic rather than kept under Haar. +A finite-r spatial-moment convergence claim, a global tail including remote +singletons, an ordinary configuration-selected version of H1, or a numerical +Gaussian value of C_star would require new work and is not supplied here. diff --git a/reviews/radial_moment_corollaries_20260930/README.md b/reviews/radial_moment_corollaries_20260930/README.md new file mode 100644 index 000000000..3e161ed95 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/README.md @@ -0,0 +1,20 @@ +# Moments of the radial tail + +For the microscopic point-intensity law, let `Q_t=R/t` conditional on `R>t`. The tail exponent **11** is the sharp moment boundary: every fixed real power below 11 is finite, and every power at or above 11 diverges in the actual conditional microscopic law. + +If `c=C₂/C₀`, the source's uniform tail expansion gives + +```text +E Q_t^p = 11/(11-p) − 2pc/((13-p)(11-p)) t^-2 + O_p(t^-4), p<11 +E log Q_t = 1/11 − 2c/143 t^-2 + O(t^-4). +``` + +[The derivation](MOMENT_CONSEQUENCE.md) also supplies every fixed finite-order power/log-moment expansion, checks the author's fixed quantile formula, and shows that `0=2, m=d-1, torus side L>0, birth b, gap mark k>0, and an orthonormal +axial/transverse frame. Use exactly [P]'s variance-one periodized Gaussian +covariance, original endpoint observations + + M=(-r/2,0), S=(r/2,0), + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0, + +and endpoint regression Q_r. Let W_r=F_d(H_M)F_(d-1)(H_S), Z_r=E_Q W_r and +Q_r^W=(W_r/Z_r)Q_r. N_r counts ALL additional critical points with heights in +(b-k r^3,b), excluding the two pins. Constants below depend on these fixed +parameters, d, and L; where stated they also depend on R or rho. No uniformity +in varying dimension, volume, or unbounded/vanishing marks is claimed. + +Exact source commits, paths, sizes and hashes are in SOURCES.json. The inputs are: + +[P] parent Sections2-5: positive Fourier spectrum, nonsingular contact frame U_r, +its target v_r, smooth finite-jet regression, every fixed C^q moment, and +Z_r/r^2 -> z0>0. The endpoint-gradient identity gives W_r/r^2 <= C K^(2d), where +K>=1 is a fixed multiple of 1+||f||C4 bounding operator derivatives through four. + +[D5] the fixed-d global first moment and height-window shell estimate + + E_W N_r <= C r^3, + E_W N({Rr<|x|=4. (1) + +[RM,RC] for fixed rho>0, on D_rho={|x|>=rho}, with Lambda the SUM of the +index-specific normalized contact kernels, + + E_W N(D_rho)=k r^3 integral_Drho Lambda+O_rho(r^4), + E_W (N(D_rho))_2=O_rho(r^5). (2) + +[C6] E_W (N_r)_q<=C_q r^3 for every fixed q>=2. With (1), Stirling's identity +also gives E_W N_r^q=O_q(r^3). We use this only for moment uniform integrability +and consequences, not for the new near-event domination or mixed-event estimate. + +[CUB] the exact deterministic classifier in #158, Theorem C, at its xAI-reviewed +algebraic scope. Its Gaussian sector theorem is NOT consumed. [DEEP] is credited +for the planar exclusion construction; Section3 proves the vector version here. +[RCL] the previously reviewed generic Poisson/compound-Poisson consequences. +The defective global proof in #159 is NOT a premise. This note gives an alternative +spectral-coordinate proof, not a retrospective acceptance of its shear argument. + +Theorem (conditional global cluster law). There are explicitly specified finite +positive numbers a1,a2,beta_far such that, putting + + nu1=a1+beta_far, nu2=a2, + +for every fixed nonnegative integer q, + + sum_(n>=1) n^q |r^-3 Q_r^W(N_r=n)-nu1*1{n=1}-nu2*1{n=2}| -> 0. (3) + +The coefficients are (17) and (22) below. Thus the full limiting nonempty count +measure is unique and supported on {1,2}. This is not a joint law of clusters +from distinct pin pairs, and not a spatial independence statement. Unscaled +Law(N_r) tends to delta_0. The zero mass must be CENTERED before multiplying +by r^-3; r^-3 Q_r^W(N_r=0) diverges. + +## 2. Physical jets, residual coupling, and exact orthogonal coordinates + +Let A=D_y^2 f(0) be the m-by-m midpoint transverse Hessian and T the list of +independent third derivatives at 0 EXCEPT f_xxx. Set J=(A,T). At contact, +(U_0,J) is the complete nonrepeated jet list through degree three: U_0 supplies +f, all first derivatives, f_xx, f_xyi, and f_xxx. Positive Fourier spectrum +therefore gives a positive-definite covariance. For sufficiently small r, the +joint covariance and its inverse are uniformly bounded. The conditional density +h_r(A,T) under Q_r converges locally uniformly to h_0 and obeys + + h_r(A,T)<=C exp[-c(||A||F^2+|T|^2)]. (4) + +The bounded conditional means have been absorbed by reducing c. + +Take an independent copy F of the unconditioned smooth field. For each raw target +j, construct the conditional field by regression on V_r=(U_r,J): + + f_r^j=F+Cov(F,V_r)Cov(V_r)^-1((v_r,j)-V_r(F)). (5) + +All coefficient functions are bounded in C4 and converge there. Averaged contact +functionals in U_r are bounded by a constant times ||F||C3; this uses the integral +forms of the pin divided differences, not separate uncontrolled r^-p terms. +Consequently, with S=C(1+||F||C4), independent of the target j, + + K(f_r^j)<=C(S+|j|), E S^p=1, and let K bound the operator norms of +all derivatives through four. Let lambda=lambda_min(-A). If + + rK<=k/(4R), lambda>(8RK+56RK^2/k)r, (11) + +there are only the two pinned critical points in the cylinder. + +Proof. Throughout the cylinder f_yy<=-lambda/2, since its variation from A is at +most 2RKr. Put g(x)=f_y(x,0). The pin gradients give g(-r/2)=g(r/2)=0. Applying +the scalar two-node interpolation estimate to every unit projection of g gives +||g(x)||<=KR^2r^2, also outside the pin interval. No vector-valued Rolle zero is +assumed. Likewise + + g'(x)=r^-1 integral_(-r/2)^(r/2)[g'(x)-g'(t)]dt + +implies ||g'(x)||<=K(R+1/2)r. For rho=4R^2Kr^2/lambda=11k>0. +The strictly convex function psi' already has the two pinned zeros and cannot +have a third. All critical points in the cylinder lie on its graph. This proves +(11) in every fixed dimension. + +On the typed support H_M<0, A has largest eigenvalue less than rK/2 by the +Hessian Lipschitz bound, so sk/(4R), + + |s|<=C_R K^2<=a_R U^2+a_R r^2 |s|^2, a_R>=1. (12) + +Here K>=1, k is fixed, and (10) was used. The elementary absorption dichotomy is + + x<=a U^2+a r^2 x^2 implies + x<=2a U^2 OR x>=1/(2a r^2), x>=0. (13) + +If x>2aU^2, subtracting aU^2 from (12) proves the second alternative. +On the second alternative |A|op>=r|s|>=c_R/r, hence K>=c_R/r. The first excluded +event also implies K>=c_R/r. All exceptional mass can therefore be bounded in +ORIGINAL Q_r coordinates, before any spectral rescaling. For every p,M>=0, + + E_W[K^p;exceptional] <= C E_Q[K^(p+2d);K>=c_R/r] + <= C_(p,M,R) r^M. (14) + +The last bound uses one higher fixed Q_r moment. The normalizer is the full +Z_r>=z_*r^2 and W_r<=Cr^2K^(2d). There is no need to posit a conditional Gaussian +tail that is uniform in an unbounded scaled target. The exceptional branch has +not been discarded; it is negligible to every fixed power of r. + +## 4. One global majorant, including multiple soft eigenvalues + +At each pin the axial Hessian column has norm <=rK/2 by the vector gradient +average. The TRANSVERSE part of the midpoint column in the soft eigenvector direction is +r s times that eigenvector; its endpoint transverse perturbation is bounded by +rK/2. Its axial entry at a pin is already bounded by the short axial column. +Thus the full soft column is bounded by r(|s|+K), up to a fixed harmless constant. +The axial and soft columns correspond to orthogonal coordinate directions. Hadamard therefore gives + + W_r/r^4 <= C K^(2d-2)(|s|+K)^2. (15) + +This does not divide by any hard eigenvalue. On the regular near-occurrence +branch |s|<=2a_R U^2, (10) gives K<=C_R U^2 (r<=1), so (15)<=C_R U^(4d). +Writing v=m(m-1)/2, (9) gives |J_r|<=C_R U^(2v). Thus the density of the rescaled +NONEMPTY near measure, in spectral variables and the coupling variable F, is +bounded by + + G_R=C_R exp[-c(|h|^2+|tau|^2)] + U^(4d+2v) 1{|s|<=2a_R U^2}. (16) + +Indeed disintegration gives r^-3 times r from (9), times r^4 from the weight, +divided by Z_r: the prefactor is c_m r^2/Z_r, which is bounded. Integrating s in +(16) contributes only another U^2. Gaussian h,tau integration and every fixed +moment of S make G_R integrable. Multiplication by any fixed polynomial in +K, |s|, |h|, |tau| remains integrable on the regular branch. Formula (14) controls +the weighted exceptional branch as well: for a scaled-jet power, use |s|<=K/r +and choose the arbitrarily large power M in (14) to absorb that extra r^-p. + +Consequences: the r^-3 near-occurrence measures are tight in all the spectral +jet variables; their tails outside increasing compact sets tend to zero uniformly +as r->0. Their occurrence-weighted K moments are O(r^3). If m>=2, the limsup of +scaled near-occurrence mass with |h2|<=eta tends to zero as eta->0, directly by +integrating G_R over that strip. The same holds near repeated hard spectra. No +inverse determinant or 1/eta bound appears. This proves the needed multiple-soft +exclusion IN MEASURE, without a pointwise assertion that two soft directions +cannot occur. + +Using C6 only now, Cauchy-Schwarz also yields +E_W[N_R K^p]<=sqrt(E_W N_R^2)*sqrt(E_W[K^(2p);N_R>0])=O_(R,p)(r^3). +Thus the LOCAL derivative-weighted bound is established, not assumed. The global +analogue is not needed for the rest of this proof. + +## 5. Pointwise normal form, typed-boundary control, and the near limit + +Fix spectral variables s,h,O,tau with 0 P_s,tau(X,Z) in C2, + r^-2 grad_w f(rX,rZ,r^2V) -> -diag(h)V+Q_tau(X,Z) in C1, + +where + + P_s,tau=2kX^3-3kX/2-k/2+sZ^2/2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3, + a=tau_xxz, beta=tau_xzz, c=tau_zzz. + +Q_tau contains the hard-index cubic terms. For fixed positive h the hard +Hessian stays negative definite for small r. On the full physical Rr ball, +grad_w f=-diag(h)w+O(Kr^2)+O(Kr|w|). Every critical point therefore has +|w|<=C_(h,R,F,tau) r^2. The hard implicit function is unique and smooth, and +substitution into f/r^3 changes the planar potential by O(r) in C2 on fixed +sets. Its inverse-hard-eigenvalue constants are used ONLY for pointwise +convergence, never in (16). + +At the endpoints the soft Hessian blocks divided by r converge to + + M_M=[[-6k,-a/2],[-a/2,s-beta/2]], + M_S=[[ 6k, a/2],[ a/2,s+beta/2]]. + +A Schur complement across the fixed hard block, or the determinant expansion, +gives W_r/r^4 -> product h_j^2 * w(s,a,beta), where + + w=[-6k(s-beta/2)-a^2/4]_+ [a^2/4-6k(s+beta/2)]_+. + +It is positive exactly on s<-|B|/2, B=beta-a^2/(12k), and equals +9k^2(4s^2-B^2) there. Outside this open typed domain its limiting value is zero; +on the typed boundary the determinant limit is also zero. Consequently no count +convergence is needed at a point with zero limiting weight. This is the explicit +handling of the typed boundary, not an assumed compact exclusion from it. + +Within the typed domain use [CUB]'s exact classifier n(s,a,beta,c) in {0,1,2}. +Let n_R count those additional cubic critical points in X^2+Z^2=1, +Z!=0 at any critical root, and the two gradient equations solve for (beta,s) +as smooth functions of (X,Z,a,c). A circle coordinate plus a,c is three parameters, +so its image in four-dimensional (s,a,beta,c) space is null. Cover by countably +many compact charts with |Z| bounded away from zero. Height boundary relations +are the nonzero polynomial threshold of [CUB]. The pins are nondegenerate and +remain EXACT, so they cannot split into new counted roots. + +Choose disjoint neighborhoods of relevant zeros. C2 convergence and the inverse +function theorem preserve precisely one root in each; on the rest of the compact +set in a slightly enlarged height window the gradient is bounded away from zero. +No condition on lower-height degeneracies is necessary. Hard slaving then proves +N_R -> n_R pointwise for these fixed variables. The hard coordinates vanish on +the r spatial scale. The extra points have full index d-1. + +For precision, extend the coefficient kernel by zero off ordered positive h and +the typed domain. With T=Rot_O(tau), define + + dM(s,h,O,tau) = (c_m/z0) h_0(O diag(0,-h)O^t,T) + * product_(j=2..m) h_j^3 + * product_(2<=i=1. To see this, on the regular branch sum the absolute conditional +errors over the count index j: it is bounded by the weight error plus twice +the weighted probability of a count mismatch. Pointwise convergence gives zero +and (16) dominates. Cases of zero limiting weight give zero directly. On a fixed typed jet with n_R>0 and finite S, K is bounded as r->0, so the +exceptional branch is eventually absent and the regular-branch indicator is +eventually one. No part of the proposed limit is lost by that split. The +exceptional mass is o(r^3) by (14). This proves + + r^-3 Q_W(N_R=j) -> a_j^R=integral 1{n_R=j}dM, j=1,2, + r^-3 Q_W(N_R>=3) -> 0. (18) + +There is NO finite uncentered n=0 formula. For fixed R, the right side is a +finite measure only on nonempty counts. + +## 6. Integrability at arbitrary scaled radius + +For the planar classifier put D=(c-a beta/(4k)+a^3/(72k^2))/2 and +T_*=-(s-B)^2(B+2s)/(12k). It says: s>B gives n=2 below D^2=T_* and n=1 at or above; +s<=B gives n=1 strictly above that threshold and n=0 otherwise. On n>=1, x=-s>0 +satisfies + + x<=2|B|+(48kD^2)^(1/3). + +Indeed s>B forces x<|B|. Otherwise D^2>T_*, and if x>=2|B| then +T_*>=(x/2)^2*x/(12k)=x^3/(48k). Since w<=36k^2 x^2, + + integral_s w 1{n>=1} <=384k^2 |B|^3+2304k^3 D^2. (19) + +This is a polynomial bound in the cubic coordinates at fixed k. The remaining +factor in (17) is a nonnegative polynomial in h times a uniformly Gaussian-decaying +function of h,tau, by ORTHOGONAL norm preservation. Therefore + + a_j=integral 1{n=j}dM, j=1,2 (20) + +are finite. They are strictly positive: take the open two-point cubic sector +(s,a,beta,c)=(-3k/2,0,-2k,0), or the one-point sector (-k,0,0,2k), choose distinct +positive hard h in a compact box, and arbitrary remaining cubic entries in a +small box. Gaussian densities are positive, and the spectral/weight factors are +positive on an open set. For m=1 the empty hard products are 1. + +For every fixed jet, n_R=n eventually as R->infinity. Exact-count indicators +{n_R=1} need NOT be monotone, but are bounded by 1{n>=1}; (19) supplies an +integrable majorant. Dominated convergence proves a_j^R->a_j. No inverse shear +parameter is suppressed by integrating out variables that a spatial observable +still uses. + +## 7. Remote singleton coefficient and vanishing mixed event + +By (1) and [RM]'s fixed-remote mean limit, for 00; equivalently +choose R_r=rho'/r, which is permitted because the shell constant is uniform in +R_r>=4. The kernel is nonnegative and the annulus is fixed, so its mean limit +is supplied by [RM]. Thus + + beta_far=k integral_(X\{0}) Lambda(x)dx (22) + +exists, is finite, and differs from the fixed-rho integral by O(rho^2). +It is positive by positivity on any fixed remote region of positive volume. +Corollary D of [RC] and the mean limit give +r^-3 Q_W(N_far^rho=1)->k integral_Drho Lambda and +Q_W(N_far^rho>=2)=O_rho(r^5). + +Fix R and rho. To control the mixed event, first restrict the spectral jets +(s,h,tau) to a fixed compact box H, retaining all O. Under the conditional law +(5) with these targets, the covariance of (f(x),grad f(x)), |x|>=rho, has a +uniform positive lower bound. At r=0 the conditioning is the COMPLETE degree-three +jet at the origin and x is a distinct site; positive Fourier spectrum gives +full joint rank. Compactness in x, O, and the target-independent covariance, +plus continuity through r=0, proves the bound. Conditional Hessian moments are +bounded on H and on the height window. The usual unweighted Gaussian Kac-Rice +formula, integrated over the window of length k r^3, gives + + E[N_far^rho | U_r,J]<=C_(H,rho) r^3. (23) + +On H, (15) and (6) give a uniform second moment of W_r/r^4. Conditional +Cauchy-Schwarz therefore gives + + E[(W_r/r^4)1{N_far^rho>0} | U_r,J]<=C_(H,rho) r^(3/2). + +Disintegrating over H with (9), the full normalizer, and bounded density and +Vandermonde gives Q_W(spectral jets in H,N_far^rho>0)=O_(H,rho)(r^(9/2)). +In particular this bounds the compact-jet near/far intersection. +Outside H, its probability is bounded by Q_W(N_R>0,jets outside H). By (16) +and (14), its scaled limsup tends to zero as H exhausts the spectral variables. +Thus + + Q_W(N_R>0,N_far^rho>0)=o_(R,rho)(r^3). (24) + +The full mixed term has no asserted r^(9/2) rate: that rate was proved only on +a fixed compact jet box. This proof avoids conditioning on a near WITNESS whose +observation frame degenerates. It uses fixed-origin physical jets instead. + +## 8. The full count limit and all polynomial weights + +Write N=N_R+N_mid+N_far. For fixed R,rho, the bad events N_mid>0, +{N_R>0,N_far>0}, N_far>=2, and N_R>=3 have probabilities bounded by (1), (24), +(2), and (18), respectively. On their complement N equals either the near +count in {0,1,2} or a remote singleton, but not both. Therefore, with an absolute +constant C independent of R,rho, + + limsup |r^-3 Q(N=2)-a_2^R| <= C(R^-2+rho^2), + limsup |r^-3 Q(N=1)-a_1^R-k integral_Drho Lambda| <= C(R^-2+rho^2), + limsup r^-3 Q(N>=3) <= C(R^-2+rho^2). (25) + +First r->0 at fixed R,rho, then R->infinity and rho->0, using (20)-(22). +This proves coordinate convergence to nu1=a1+beta_far and nu2=a2, and vanishing +scaled tail mass. For a fixed integer q>=0, C6 supplies E N^(q+1)<=C r^3; hence + + r^-3 E[N^q;N>M]<=C/M. + +Finite-coordinate convergence plus this uniform tail proves (3). No interchange +of a nonuniform radius limit and r limit has been assumed. No finite unscaled +polynomial moment theorem alone would imply (3); the new domination and regional +sandwich are the essential ingredients. + +In particular + + E N/r^3 -> nu1+2nu2, + E (N)_2/r^3 -> 2nu2, + E (N)_q/r^3 -> 0 for every fixed q>=3. (26) + +The positive-conditioned count converges in TV to +(nu1 delta1+nu2 delta2)/(nu1+nu2), and its size-biased version to +(nu1 delta1+2nu2 delta2)/(nu1+2nu2). The two laws are different. Their formulas +use the normalized full tilt, not a restricted normalizer. + +For independent replicas with floor(t/r^3) counts, the identified compound limit +is P1+2P2, with independent P1~Pois(t nu1), P2~Pois(t nu2), as follows either by +the exact generating function or [RCL]'s proved independent-replica bound. +There is no independence claim for regions of a single field. + +The sharp [RCL] Poisson comparison also now has coefficients: +optimal ordinary-Poisson TV error/r^3 -> nu2 and mean-matched error/r^3 ->2nu2, +so their ratio tends to 2 (not just liminf>=2). The canonical approximant using +the ACTUAL conditional mark law still has Theta(r^6) error. Replacing its mark +law by the identified LIMIT does not inherit that rate: without a convergence +rate for the coefficients, this replacement is only o(r^3) in one-count TV. + +## 9. What is solved here, conditional on review, and what is not + +The proof supplies a replacement for scaled jet tails, typed-boundary control, +nearly singular hard blocks, the near/far mixed event and the radius sandwich. +It works for every FIXED d>=2 via orthogonal spectral coordinates. It does not +require the stronger global derivative-weighted insertion estimate proposed in +#157; Section4 proves the local version as a consequence and the global law does +not need the global version. Thus that former sufficient route is bypassed, not +silently assumed true. + +The local coefficients in d=2 reduce exactly to #158's alpha1,alpha2, and its +contact birth factorization remains a local statement. In higher dimensions the +Haar/stable integration may couple birth-dependent even jets to orientation; +no higher-dimensional or global birth cancellation is asserted. + +No convergence rate, numerical coefficient enclosure, unbounded-mark theorem, +growing-volume limit, inter-pin-pair cluster independence, elder selection of the +extra points, or full Lean formalization is claimed. The input Gaussian and +regional sources remain conditional premises at their exact stated scopes. +Theorem (3) is a new author-side candidate, not accepted merely because finite +controls or existing formal gates pass. + +Review division: A, vector exclusion, absorption, and orthogonal spectral measure; +B, integrable majorant, typed/hard-boundary handling and pointwise normal form; +C, compact-jet remote conditioning, mixed event, radius sandwich and consequences. +The prior #159 diagnosis and the distinction between that proof and this replacement +are retained in AUDIT_AND_REPAIR.md. 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[{"coefficient_of_c_t_minus_2": "200/12543", "leading": "11/111", "p": "-100"}, {"coefficient_of_c_t_minus_2": "7/180", "leading": "11/18", "p": "-7"}, {"coefficient_of_c_t_minus_2": "1/84", "leading": "11/12", "p": "-1"}, {"coefficient_of_c_t_minus_2": "4/621", "leading": "22/23", "p": "-1/2"}, {"coefficient_of_c_t_minus_2": "0", "leading": "1", "p": "0"}, {"coefficient_of_c_t_minus_2": "-4/525", "leading": "22/21", "p": "1/2"}, {"coefficient_of_c_t_minus_2": "-1/60", "leading": "11/10", "p": "1"}, {"coefficient_of_c_t_minus_2": "-4/99", "leading": "11/9", "p": "2"}, {"coefficient_of_c_t_minus_2": "-20/3", "leading": "11", "p": "10"}, {"coefficient_of_c_t_minus_2": "-2180/21", "leading": "110", "p": "109/10"}, {"coefficient_of_c_t_minus_2": "-21998000/2001", "leading": "11000", "p": "10999/1000"}], "quantile_controls": "exact rational brackets for Pareto mixtures", "scientific_effect": "NONE", "scope": "PR176 equation17 conditional moment consequence only", "tests": 11} diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log new file mode 100644 index 000000000..e75147417 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.log @@ -0,0 +1,16 @@ +test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests.test_actual_probability_mixture_has_claimed_moment_remainder_bound) ... ok +test_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests.test_all_order_finite_tail_matches_direct_pareto_mixture) ... ok +test_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests.test_all_order_log_powers_and_normalization) ... ok +test_direct_log_tail_integral (__main__.MomentConsequenceTests.test_direct_log_tail_integral) ... ok +test_hand_checked_special_cases (__main__.MomentConsequenceTests.test_hand_checked_special_cases) ... ok +test_offered_rational_coefficient_factors (__main__.MomentConsequenceTests.test_offered_rational_coefficient_factors) ... ok +test_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests.test_quantile_expansion_brackets_an_actual_conditional_law_exactly) ... ok +test_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests.test_tail_and_density_derivations_agree_for_negative_zero_positive_p) ... ok +test_threshold_and_above_diverge (__main__.MomentConsequenceTests.test_threshold_and_above_diverge) ... ok +test_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests.test_uniform_in_p_remainder_overclaim_has_exact_counterexample) ... ok +test_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests.test_wrong_normalizer_sign_and_missing_p_are_detected) ... ok + +---------------------------------------------------------------------- +Ran 11 tests in 0.003s + +OK diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json new file mode 100644 index 000000000..d311bc520 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.json @@ -0,0 +1 @@ +{"all_order_controls": "normalized finite Pareto mixtures", "exact_rational_powers": ["-100", "-7", "-1", "-1/2", "0", "1/2", "1", "2", "10", "109/10", "10999/1000"], "log_correction": "-2/143", "log_leading": "1/11", "moment_rows": [{"coefficient_of_c_t_minus_2": "200/12543", "leading": "11/111", "p": "-100"}, {"coefficient_of_c_t_minus_2": "7/180", "leading": "11/18", "p": "-7"}, {"coefficient_of_c_t_minus_2": "1/84", "leading": "11/12", "p": "-1"}, {"coefficient_of_c_t_minus_2": "4/621", "leading": "22/23", "p": "-1/2"}, {"coefficient_of_c_t_minus_2": "0", "leading": "1", "p": "0"}, {"coefficient_of_c_t_minus_2": "-4/525", "leading": "22/21", "p": "1/2"}, {"coefficient_of_c_t_minus_2": "-1/60", "leading": "11/10", "p": "1"}, {"coefficient_of_c_t_minus_2": "-4/99", "leading": "11/9", "p": "2"}, {"coefficient_of_c_t_minus_2": "-20/3", "leading": "11", "p": "10"}, {"coefficient_of_c_t_minus_2": "-2180/21", "leading": "110", "p": "109/10"}, {"coefficient_of_c_t_minus_2": "-21998000/2001", "leading": "11000", "p": "10999/1000"}], "quantile_controls": "exact rational brackets for Pareto mixtures", "scientific_effect": "NONE", "scope": "PR176 equation17 conditional moment consequence only", "tests": 11} diff --git a/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log new file mode 100644 index 000000000..e75147417 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-optimized.log @@ -0,0 +1,16 @@ +test_actual_probability_mixture_has_claimed_moment_remainder_bound (__main__.MomentConsequenceTests.test_actual_probability_mixture_has_claimed_moment_remainder_bound) ... ok +test_all_order_finite_tail_matches_direct_pareto_mixture (__main__.MomentConsequenceTests.test_all_order_finite_tail_matches_direct_pareto_mixture) ... ok +test_all_order_log_powers_and_normalization (__main__.MomentConsequenceTests.test_all_order_log_powers_and_normalization) ... ok +test_direct_log_tail_integral (__main__.MomentConsequenceTests.test_direct_log_tail_integral) ... ok +test_hand_checked_special_cases (__main__.MomentConsequenceTests.test_hand_checked_special_cases) ... ok +test_offered_rational_coefficient_factors (__main__.MomentConsequenceTests.test_offered_rational_coefficient_factors) ... ok +test_quantile_expansion_brackets_an_actual_conditional_law_exactly (__main__.MomentConsequenceTests.test_quantile_expansion_brackets_an_actual_conditional_law_exactly) ... ok +test_tail_and_density_derivations_agree_for_negative_zero_positive_p (__main__.MomentConsequenceTests.test_tail_and_density_derivations_agree_for_negative_zero_positive_p) ... ok +test_threshold_and_above_diverge (__main__.MomentConsequenceTests.test_threshold_and_above_diverge) ... ok +test_uniform_in_p_remainder_overclaim_has_exact_counterexample (__main__.MomentConsequenceTests.test_uniform_in_p_remainder_overclaim_has_exact_counterexample) ... ok +test_wrong_normalizer_sign_and_missing_p_are_detected (__main__.MomentConsequenceTests.test_wrong_normalizer_sign_and_missing_p_are_detected) ... ok + +---------------------------------------------------------------------- +Ran 11 tests in 0.003s + +OK diff --git a/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py b/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py new file mode 100644 index 000000000..952759716 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment_consequence_checks.py @@ -0,0 +1,171 @@ +"""Independent exact controls for the consequence of PR176 equation (17). + +No author code is imported. These finite arithmetic/probability controls do not +prove the source Gaussian theorem, its remainder, or any original finite-r rate. +""" +from fractions import Fraction as F +import json +from math import factorial +import unittest + + +POWERS = [F(-100), F(-7), F(-1), F(-1, 2), F(0), F(1, 2), + F(1), F(2), F(10), F(109, 10), F(10999, 1000)] + + +def integral(exponent): + """Exact integral of q^exponent over [1,infinity), or divergent.""" + if exponent >= -1: + raise ValueError('divergent power integral') + return -1 / (exponent + 1) + + +def tail_moment(p): + return (1 + p * integral(p - 12), + p * (integral(p - 14) - integral(p - 12))) + + +def density_moment(p): + return (11 * integral(p - 12), + 13 * integral(p - 14) - 11 * integral(p - 12)) + + +def proposed_moment(p): + return F(11) / (11 - p), -2 * p / ((13 - p) * (11 - p)) + + +class MomentConsequenceTests(unittest.TestCase): + def test_tail_and_density_derivations_agree_for_negative_zero_positive_p(self): + for p in POWERS: + with self.subTest(p=p): + self.assertEqual(tail_moment(p), proposed_moment(p)) + self.assertEqual(density_moment(p), proposed_moment(p)) + + def test_hand_checked_special_cases(self): + for p, wanted in [(F(-1), (F(11, 12), F(1, 84))), + (F(0), (F(1), F(0))), + (F(1), (F(11, 10), F(-1, 60))), + (F(2), (F(11, 9), F(-4, 99)))]: + self.assertEqual(tail_moment(p), wanted) + + def test_threshold_and_above_diverge(self): + for p in (F(11), F(111, 10), F(12), F(100)): + with self.assertRaises(ValueError): + integral(p - 12) + + def test_direct_log_tail_integral(self): + self.assertEqual(integral(F(-12)), F(1, 11)) + self.assertEqual(integral(F(-14)) - integral(F(-12)), F(-2, 143)) + + def test_actual_probability_mixture_has_claimed_moment_remainder_bound(self): + # A genuine mixture of Pareto(11), Pareto(13), Pareto(15): its tail + # obeys (17) with c=1 and remainder (1/2)t^-4 q^-11(q^-4-1). + for t in (4, 8, 16): + delta, bound = F(1, t * t), F(1, 2) + weights = [1 - delta - bound * delta**2, delta, bound * delta**2] + self.assertEqual(sum(weights), 1) + self.assertTrue(all(w > 0 for w in weights)) + for p in POWERS: + exact = sum(w * F(alpha) / (alpha - p) + for w, alpha in zip(weights, (11, 13, 15))) + leading, correction = proposed_moment(p) + error = exact - leading - delta * correction + self.assertLessEqual(abs(error), abs(p) * bound * delta**2 / (11 - p)) + exact_log = sum(w / alpha for w, alpha in zip(weights, (11, 13, 15))) + self.assertLessEqual(abs(exact_log - F(1, 11) + F(2, 143) * delta), + bound * delta**2 / 11) + + def test_uniform_in_p_remainder_overclaim_has_exact_counterexample(self): + p, delta, bound = F(10999, 1000), F(1, 16), F(1, 2) + actual_error = abs(bound * delta**2 * (F(15) / (15 - p) - F(11) / (11 - p))) + self.assertGreater(actual_error, bound * delta**2) + + def test_wrong_normalizer_sign_and_missing_p_are_detected(self): + cases = [ + ('missing denominator normalization', F(0), F(1)), + ('reversed correction sign', F(1), F(1, 60)), + ('omitted p factor', F(0), F(-2, 143)), + ('omitted factor two', F(1), F(-1, 120)), + ] + for name, p, false_coefficient in cases: + with self.subTest(mutant=name): + self.assertNotEqual(tail_moment(p)[1], false_coefficient) + self.assertNotEqual(integral(F(-14)) - integral(F(-12)), F(-1, 143)) + + def test_all_order_finite_tail_matches_direct_pareto_mixture(self): + # F(s)=sum C_j s^(-11-2j), s>=1, divided by F(1), is an actual + # survival function. Conditioning at t gives component weights + # proportional to C_j t^(-2j); its conditional moments are exact. + coefficients = (F(2), F(3), F(5), F(7)) + for t in (3, 7, 13): + pieces = [c / t**(2*j) for j, c in enumerate(coefficients)] + normalizer = sum(pieces) + weights = [part / normalizer for part in pieces] + for p in POWERS: + direct = sum(w * F(11+2*j) / (11+2*j-p) + for j, w in enumerate(weights)) + from_tail = 1 + p * sum(part * integral(p-12-2*j) + for j, part in enumerate(pieces)) / normalizer + numerator = sum((11+2*j) * part / (11+2*j-p) + for j, part in enumerate(pieces)) + self.assertEqual(direct, from_tail) + self.assertEqual(direct, numerator / normalizer) + self.assertNotEqual(direct, numerator) # missing denominator + + def test_all_order_log_powers_and_normalization(self): + for m in (1, 2, 3, 4): + for alpha in (11, 13, 15, 17): + # Integration by parts in y=log(q): + # integral y^n exp(-alpha*y)dy = n/alpha times the n-1 integral. + gamma_integral = F(1, alpha) + for n in range(1, m): + gamma_integral *= F(n, alpha) + tail_log_moment = m * gamma_integral + exponential_log_moment = F(factorial(m), alpha**m) + self.assertEqual(tail_log_moment, exponential_log_moment) + if m > 1: + self.assertNotEqual(tail_log_moment, F(1, alpha**m)) + + def test_quantile_expansion_brackets_an_actual_conditional_law_exactly(self): + # The exact conditional tail q^-11(1+c t^-2 q^-2)/(1+c t^-2) + # comes from F(s)=s^-11(1+c s^-2), with an irrelevant global normalizer. + # Bracket the true quantile using only rational comparisons, avoiding + # floating point root finding or differentiation of a remainder. + for q0 in (F(5, 4), F(3, 2), F(2)): + alpha = q0**-11 + for c in (F(1), F(2)): + correction = c * (q0**-2 - 1) / 11 + for t in (8, 16, 32): + delta = F(1, t*t) + def survival(q): + return q**-11 * (1+c*delta*q**-2) / (1+c*delta) + qlo = q0*(1+correction*delta-delta**2) + qhi = q0*(1+correction*delta+delta**2) + self.assertGreater(qlo, 1) + self.assertGreater(survival(qlo), alpha) + self.assertLess(survival(qhi), alpha) + # Reversing the coefficient produces an O(delta) error + # that lies outside this O(delta^2) bracket. + wrong = q0*(1-correction*delta) + self.assertGreater(wrong, qhi) + + def test_offered_rational_coefficient_factors(self): + integral_shape, u2, uabs = F(246528, 35), F(240192, 35), F(5898627, 880) + self.assertEqual(11*u2/(2*integral_shape), F(4587, 856)) + self.assertEqual(11*uabs/(2*integral_shape), F(4587821, 876544)) + + +if __name__ == '__main__': + suite = unittest.defaultTestLoader.loadTestsFromTestCase(MomentConsequenceTests) + result = unittest.TextTestRunner(verbosity=2).run(suite) + if not result.wasSuccessful(): + raise SystemExit(1) + print(json.dumps({'scope': 'PR176 equation17 conditional moment consequence only', + 'exact_rational_powers': [str(p) for p in POWERS], + 'moment_rows': [{'p': str(p), 'leading': str(tail_moment(p)[0]), + 'coefficient_of_c_t_minus_2': str(tail_moment(p)[1])} + for p in POWERS], + 'log_leading': '1/11', 'log_correction': '-2/143', + 'quantile_controls': 'exact rational brackets for Pareto mixtures', + 'all_order_controls': 'normalized finite Pareto mixtures', + 'tests': result.testsRun, 'scientific_effect': 'NONE'}, sort_keys=True)) diff --git a/reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md b/reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md new file mode 100644 index 000000000..2be80e35b --- /dev/null +++ b/reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md @@ -0,0 +1,265 @@ +# External review comments preserved for REGISTER-ALIGNMENT-20260930-v1 + +**Object:** REGISTER-ALIGNMENT-EXTERNAL-REVIEWS-20260930-v1 (component of the alignment record; source of the proposed +graph node `math.align-component.external-issue-reviews`). +**What this is.** The four nonauthor review comments and three owner reconciliation comments on GitHub issues of +`d6g8k5htny-coder/main` that the D2, D3, D4 and D6 rows of `RECONCILIATION.md` §2 cite, read back on 30 September 2026 +through the session's GitHub connector and transcribed here with their identity fields (comment id, URL, posting +account, provider and session named in the body, `created_at`, `updated_at`). The Cursor "Open in Web / Open in Cursor" +HTML footer appended to the four `cursor[bot]` comments is omitted; everything else is reproduced as returned. +**Status of this evidence.** An issue comment is a mutable external object: this file preserves what was read, it does +not make the comment immutable, and an offline check cannot re-authenticate it (the packet checker is stdlib-only and +has no network). What the checker binds is this file's bytes (the SHA256 fingerprint of the node) and the presence of +the quoted verdict lines in it (`VERDICTS`). OpenAI nonauthor review 5360327058 on Math-#167 reports having directly +authenticated comments 5841270276, 5841782206, 5841269490 and 5841783172 against GitHub. The canonical claim manifest +`claims/LANDING_CLAIMS.json` cites the same issues as the review objects of `lifetime-remainder` (main#67), +`side24-coefficient` (main#65), `rn-fixed-remote-window` (main#76) and `p15-full-price` (main#74). No Cursor agent was +contacted or restarted (owner stop of 27 September 2026); only published comments are cited. Same GitHub account +throughout; zero organizational-independence credit. + +## Index + +| # | Object | Issue | Comment | Posted by | Provider / session named | `created_at` | `updated_at` | Verdict line | +|---|---|---|---|---|---|---|---|---| +| 1 | D2 `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`, SHA256 `380b7d0a…`) | main#67 | 5841270276 | `cursor[bot]` | xAI, Grok 4.7 (`grok-4.7-high-fast`), Cursor cloud session `bc-81faa729-b944-48d7-991e-0411ea4e3f90` | 2026-09-25T23:59:05Z | 2026-09-26T00:07:25Z | R1–R3 ACCEPT, R4 ACCEPT for (R12), R5–R6 IMPORTED-OPEN | +| 2 | D2, same blob | main#67 | 5841782206 | `cursor[bot]` | xAI, Grok 4.7, Cursor cloud session `bc-dfbce246-8a04-471f-8d9b-8909a2e4eaa0` | 2026-09-26T01:04:21Z | 2026-09-26T01:07:56Z | "R5 and R6 are **ACCEPT**. Theorem R is accepted at its stated O(1) remainder scope." | +| 3 | D3 `coefficients/side24_v1/PROOF.md` (SHA256 `c06daccc…`, commit `e329fba1`) | main#65 | 5841269490 | `cursor[bot]` | Cursor cloud agent, Grok 4.7 (`grok-4.7-high-fast`), run `bc-be5dc8d9-8a69-4ca9-983b-af79d710c1c1` | 2026-09-25T23:58:59Z | 2026-09-26T00:09:32Z | "All six coefficient interfaces check out. The disposition is **COEFFICIENT-CALC-REVIEWED / PARENT-IMPORTED-OPEN**." | +| 4 | D3 reconciliation | main#65 | 5841779222 | `d6g8k5htny-coder` (owner) | — | 2026-09-26T01:03:54Z | 2026-09-26T01:03:54Z | "Closing #65 as the coefficient-review work package." | +| 5 | D4 `frontiers/remote_window_20260924/PROOF.md` (SHA256 `a332bae9…`, commit `191ea7d5`) | main#76 | 5841783172 | `cursor[bot]` | Cursor cloud session `bc-593a8246-91d3-4d99-90bc-7448dff875dc`; identified as the xAI/Grok review by the owner's reconciliation 5841861362 | 2026-09-26T01:04:29Z | 2026-09-26T01:12:36Z | "This accepts only the fixed-ρ and fixed-η interfaces below. It does not accept a full RN or 24-jet theorem." Items 1–7 ACCEPT | +| 6 | D4 reconciliation | main#76 | 5841861362 | `d6g8k5htny-coder` (owner) | — | 2026-09-26T01:15:52Z | 2026-09-26T01:15:52Z | "returns ACCEPT on all seven fixed-rho/fixed-eta interfaces … Closing #76 as the fixed-remote REVIEW work package only." | +| 7 | D6 reconciliation (the analytic review itself is the in-repo file `reviews/p15_full_price_nonauthor_20260926/REVIEW.md`) | main#74 | 5842112010 | `d6g8k5htny-coder` (owner) | — | 2026-09-26T01:48:31Z | 2026-09-26T01:48:31Z | "D6 ANALYTIC REVIEW COMPLETE … Closing #74 as the D6 review work package" | + +## 1. main#67 comment 5841270276 (D2, xAI review, R1–R6) + +URL: https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841270276 · posted by `cursor[bot]` · created 2026-09-25T23:59:05Z · updated 2026-09-26T00:07:25Z + +`````text +R1, R2, R3, and the cap-loss bound in R4 check. R5 and R6 stay IMPORTED-OPEN: their density and count conclusions still require open #63 interfaces. No counterexample turned up in the new estimates. Nothing here accepts issue #63 or promotes a status. + +**Reviewer.** xAI, Grok 4.7 (`grok-4.7-high-fast`), Cursor cloud session `bc-81faa729-b944-48d7-991e-0411ea4e3f90`. This is not an OpenAI session. Finite tests were not run and are not used as evidence. + +**Sources read.** Successor `LIFETIME-BOUNDED-REMAINDER-20260924-v1`: Math- `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf80bfbe896948d5d489b2d5842a81c481`, 17734 bytes, SHA256 `380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a`. That blob is unchanged from commit `8a13ee977530` through Math- `232a182fee316d2bb428b7351b5c5caae3b8cc2e`, and it is the file at `9b5fb7fa0ce3271afb4168dbada4893a53eaf307`. Parent #63: Drive `1foDgiDi4XIKOfbrZEE8dWKV_BkZU8LIb`, 40261 bytes, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. Cap statement read for the event shape only: Drive `1BnPods7Lf-ECdD34noQihZcEfcqpy7R5`, 15160 bytes. + +## Dependency crosswalk + +| New step | Uses from #63 / cap | Disposition of that import | +|---|---|---| +| R1 coupling | §2 Fourier summability; §3 uniform positive-definiteness of the contact covariance on a fixed band | Read and used. This is not an acceptance of Theorems A–C. | +| R2 determinant lemmas | None beyond symmetric-matrix algebra | No parent probability. | +| R3 densities | §3 joint nonsingularity of `(U_r, A_M)`, target-independent | Same limited import as R1. | +| R4 bound `(R12)` | Parent (6.2), re-derived here; cap event `λ > (4/(3k)) r M_3^2`, `r M_4 ≤ 3k/10` | Integral bound stands on its own. | +| `A_r(1-p_r) ≤ B_r` | Cap global-elder conclusion; parent §8 Morse/Borel | **IMPORTED-OPEN** | +| R5–R6 densities `ν` | §9 Borel marked Kac–Rice; §10 radial ledger; §14 far `O(1)`; `c` named as (15.2) via §15 | **IMPORTED-OPEN** | + +## Verdicts + +**R1 — ACCEPT.** Centered rules meet the contact jet at order `r^2`. The fourth axial row is `f_xxx + (r^2/40) f^{(5)} + O(r^4)`. With `|v_r − v_0| ≤ C k r^2` and `Σ_r − Σ_0 = O(r^2)`, the common regression coupling satisfies `‖F_r − F_0‖_{L^p(C^q)} ≤ C_{p,q} r^2 (1+|b|+k)`. + +**R2 — ACCEPT.** For `K_t = [[α, √t βᵀ], [√t β, A]]`, `det K_t = α det A − t βᵀ adj(A) β` holds on singular `A`. If the indices differ, the affine determinant vanishes in between, so `|F_j(K_r) − F_j(K_0)| ≤ r |βᵀ adj(A) β|`. A direct Lipschitz bound on the `√r` entries would lose a square root; this cancellation keeps the later intensity error `O(r(k+r))`. Endpoint comparison with `diag(±6k, A_0)` then gives `(R10)` and `(R11)`. + +**R3 — ACCEPT.** Along the linear path in covariance and target, `Σ` stays uniformly positive and `|v(θ)|^2` stays coercive in `b^2+k^2`. Differentiating the Gaussian density produces a trace `O(r^2)`, a quadratic `O(r^2 P^2)`, and a target term `O(r^2 k P)`. Integrating `|dπ/dθ|` gives `(R5)` without a small-log restriction. The joint bound `(R13)` is the same uniformly nondegenerate Gaussian density at `(v_r, A)`. + +**R4 — ACCEPT for `(R12)`.** Both soft factors in parent (6.2) were re-derived, including the saddle factor `λ_1 + (3/2) r h`. Depth failure has width `O((r/k) U^2)`. The `λ_1` integral is `(D^3/3) δ^3 U^6 + (E D^2/2) r δ^2 U^5`, and `r δ^2 = k δ^3`. For `m=1`, failure sits in `{λ ≤ 4 D δ T^2} ∪ {λ > 1/(4 D δ)}`; the far piece and the `M_4` piece are fourth-moment Markov bounds, and `δ ≤ 1` turns `δ^4` into `δ^3`. The comparison `A_r(1−p_r) ≤ B_r` stays **IMPORTED-OPEN**. + +**R5 — IMPORTED-OPEN.** Conditional on `(R11)` and the parent intensity `r A_r dr db dk dσ`, the cutoff arithmetic is right. The missing mass `k < a = ℓ/r_0^3` is `O(a^{7/3})`. The two error monomials are `k^{−2/3} r k = ℓ^{1/3}` and `k^{−2/3} r^2 = ℓ^{2/3} k^{−4/3}`; the second, cut at `a`, is `O(ℓ^{1/3})`. Replacing `a` by `0` makes `∫ k^{−4/3} dk` diverge. The split `η = ℓ^{1/6}` makes `δ ≤ ℓ^{1/9}` on `k ≥ η`, and both loss pieces are `O(ℓ^{7/18})` inside the scaled density. Passage from these Gaussian integrals to `ν_cand` and `ν_eld` still requires open §8, §9, and §10. + +**R6 — IMPORTED-OPEN.** Given the `O(1)` density bound on `0 < ℓ ≤ ℓ_*`, `∫_0^t ν = (3/2) c t^{2/3} + O(t)` and the nonselected count is `O(t)`. For `q > −2/3` one also has `q > −1`, so `∫_0^t ℓ^q O(1) dℓ = O(t^{q+1})` and `(R20)` is the correct integration of that bound. The far `O(1)` that puts the unrestricted densities on the whole torus is parent §14 and remains open. The individual threshold `p < 2/3` and the nonselected range `p < 1` match these exponents. +````` + +## 2. main#67 comment 5841782206 (D2, xAI delta review, R5–R6 ACCEPT) + +URL: https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206 · posted by `cursor[bot]` · created 2026-09-26T01:04:21Z · updated 2026-09-26T01:07:56Z + +`````text +R5 and R6 are **ACCEPT**. Theorem R is accepted at its stated O(1) remainder scope. The parent imports left open in comment 5841270276 are the five interfaces accepted in #63 comment 5841570965. No successor-specific gap turned up. I did not replay R1–R4 and I did not edit the proof. + +**Reviewer.** xAI, Grok 4.7 (`grok-4.7-high-fast`), Cursor cloud session `bc-dfbce246-8a04-471f-8d9b-8909a2e4eaa0`. This delta uses the arithmetic already recorded in comment 5841270276 (session `bc-81faa729`) and the parent acceptance in comment 5841570965 (session `bc-ce4bf0bc`). Same provider family as those two reviews. This is not an OpenAI session. Finite tests were not run. + +**Immutable D2 source, re-read and hashed.** Math- `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, blob `247b3ecf80bfbe896948d5d489b2d5842a81c481`, 17734 bytes, SHA256 `380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a`. Parent bytes checked against that citation: Math- blob `dfed3b8d318a3ab1950957f393307733a4bef3f2`, 40261 bytes, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. + +## Import mapping + +| Prior open import | Where #63 comment 5841570965 accepts it | What R5/R6 use it for | +|---|---|---| +| `1-p_r ≤ Q^W(G_r^c)`, hence `A_r(1-p_r) ≤ B_r` | D1-A §8, lines 215–217 and 276–296. The cap event is the same pair `λ_min(-A_M) > (4/(3k)) r M_3^2` and `r M_4 ≤ 3k/10`. D1-A concludes `1-p_r ≤ Q^W(G_r^c)` for the embedded small-`r` cylinder, with `0 < Z_r < ∞`. | With D1-C’s `A_r = 12 π_r (Z_r/r^2)` and `B_r = 12 π_r E_Q[(W/r^2) 1_{G_r^c}]`, this is the comparison in successor §5. The quantitative bound on `B_r` remains the already checked `(R12)`. | +| §9 Borel marked Kac–Rice | D1-B, lines 298–308. Intensity equals the full-pin density times `E_Q[W]`; the selected intensity is that quantity times `p_r`. | Passage from the Gaussian integrands to `ν_cand` and `ν_eld`. | +| §10 radial ledger | D1-C, lines 88–94 and 310–335. `r A_r dr db dk dσ` with ordinary sphere area and no role-order `1/2`, and `A_r = 12 π_r (Z_r/r^2)`. | Successor §6. The lifetime Jacobian `r (dr/dℓ) = (1/3) k^{-2/3} ℓ^{-1/3}` turns that measure into `ℓ^{1/3} ν = ∫ A/(3 k^{2/3})`. | +| §14 far `O(1)` | D1-D, lines 386–445, equation (14.1): `0 ≤ ν_eld^far(ℓ) ≤ ν_cand^far(ℓ) ≤ C` for `0 < ℓ ≤ 1`. | Successor §7, added to `(R16)`–`(R18)` to reach `(R1)` on `0 < ℓ ≤ ℓ_*`. | +| `c` as (15.2) | D1-E, lines 447–480. | Successor §1 and §6. `A_0 = 12 π_0 z_0`, so `∫ A_0/(3 k^{2/3}) = 4 ∫ k^{-2/3} π_0 z_0`, which is parent (13.6). D1-E reduces that integral to (15.2). | + +The cutoff arithmetic recorded in comment 5841270276 still stands: missing mass `k < ℓ/r_0^3` is `O(a^{7/3})`; the monomials `k^{-2/3} r k` and `k^{-2/3} r^2` are both `O(ℓ^{1/3})` once the lower cutoff is kept; the split `η = ℓ^{1/6}` makes both pairing-loss pieces `O(ℓ^{7/18})` inside the scaled density. Given the resulting `O(1)` density bound, `∫_0^t ν = (3/2) c t^{2/3} + O(t)`, the nonselected count is `O(t)`, and `(R20)` is the same integration for `q > -2/3`. + +R5 and R6 take the elder comparison and the integral definition of `c`. They do not take the parent’s uniform `C r^3` selection bound (1.1), and they do not take the dominated-convergence route to `p_r → 1`. Successor §1 and §6 already replace that route by `(R11)` and `(R12)`. The §§2–§7 chain is outside this delta. + +## Scope of the acceptance + +Theorem R is the existential statement: some `ℓ_* > 0` and some finite `C` give + +`|ν_cand(ℓ) - c ℓ^{-1/3}| ≤ C`, `0 ≤ ν_cand(ℓ) - ν_eld(ℓ) ≤ C`, and `|ν_eld(ℓ) - c ℓ^{-1/3}| ≤ C` + +for `0 < ℓ ≤ ℓ_*`, with `c` equal to parent (15.2). That includes `(R16)`, `(R18)`, `(R19)`, and `(R20)`. + +A numerical value of `C` or `ℓ_*`, convergence of the bounded remainder, a second coefficient, an RN expected-critical-count estimate, and a 24-jet certificate are outside this verdict. Scientific status of those separate items is unchanged. +````` + +## 3. main#65 comment 5841269490 (D3, xAI coefficient review, C1–C6) + +URL: https://github.com/d6g8k5htny-coder/main/issues/65#issuecomment-5841269490 · posted by `cursor[bot]` · created 2026-09-25T23:58:59Z · updated 2026-09-26T00:09:32Z + +`````text +All six coefficient interfaces check out. The disposition is **COEFFICIENT-CALC-REVIEWED / PARENT-IMPORTED-OPEN**. This is a same-session nonauthor check of the stated expression, and parent issue #63 is still open. + +Provider/model/session: Cursor cloud agent, Grok 4.7 (`grok-4.7-high-fast`), run `bc-be5dc8d9-8a69-4ca9-983b-af79d710c1c1`, https://cursor.com/agents/bc-be5dc8d9-8a69-4ca9-983b-af79d710c1c1. Source exposure: public `Math-` commit `e329fba1e927a12dbb4f0d1556f85f39284d17a9`, directory `coefficients/side24_v1/`; parent Drive proof `UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (40,261 bytes, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`) only to transcribe (15.2) and recheck its scalar integral; NIST DLMF 5.11; issue #63’s displayed formula. No Vault99 reading, no source edits, no child agents. Local interpreter: CPython 3.12.3. + +## Interfaces + +| Interface | Result | +|---|---| +| C1 negative-cone moment \(D_1=4/3\) | **VERIFIED** | +| C2 \(D_2=29/6-\sqrt{6}\), negative-definite cone retained | **VERIFIED** | +| C3 angular / all-direction reduction | **VERIFIED** | +| C4 SIDE24 image / periodization bound | **VERIFIED** | +| C5 covariance / Schur transfer | **VERIFIED** | +| C6 outward Gamma / Stirling arithmetic | **VERIFIED** | + +**C1.** For the reference \(K_\infty(z)=\exp(-|z|^2/2)\), \(\mathrm{Cov}(H_{ij},H_{kl})=\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\) and \(\mathrm{Cov}(V)=\mathrm{diag}(3,1,\ldots,1)\). Conditioning the transverse entry on \(V=0\) gives variance \(8/3\). The \(1\times1\) negative cone is \(A<0\), and \(A^2\) is even, so \(D_1=\tfrac12\cdot\tfrac83=\tfrac43\). + +**C2.** For \(m=2\), \(A=\begin{pmatrix}s+x&y\\y&s-x\end{pmatrix}\) with independent \(s,x,y\), \(\mathrm{Var}(s)=5/3\), \(\mathrm{Var}(x)=\mathrm{Var}(y)=1\). Both eigenvalues are negative exactly on \(s<-R\), \(R=\sqrt{x^2+y^2}\). With \(R^2\) of density \(e^{-z/2}/2\), +\[ +\int_0^a(a-z)^2 e^{-z/2}\,dz/2=a^2-4a+8-8e^{-a/2}. +\] +Symmetry then gives +\[ +D_2=\tfrac12\bigl(\tfrac{25}3-\tfrac{20}3+8-8\sqrt{3/8}\bigr)=\tfrac{29}6-\sqrt6, +\] +using \(E[s^4]=25/3\) and \(E[e^{-s^2/2}]=\sqrt{3/8}\). The identity \(4\sqrt{3/8}=\sqrt6\) is the square check \(16\cdot(3/8)=6\). Half the untruncated determinant-square moment is exactly \(29/6\); the cone cutoff is the \(-\sqrt6\) term. + +**C3.** Parent (15.2) is +\[ +c_{d,L}=\frac{\Gamma(7/6)}{24^{1/3}\sqrt\pi}\int_{S^{d-1}}p_G(0)\,p_{V_u}(0)\,\tau_u^{4/3}D_u\,d\sigma(u). +\] +The scalar identity was recomputed from the parent’s intermediate factors: \(4\cdot(6\kappa)^2=144\) and \(k^{-2/3}\cdot k^2=k^{4/3}\), then +\[ +144\int_0^\infty k^{4/3}\varphi_\tau(12k)\,dk=\Gamma(7/6)\,\tau^{4/3}/(24^{1/3}\sqrt\pi). +\] +For the isotropic reference, \(\tau^2=6\), \(p_G(0)p_V(0)=(2\pi)^{-d}/\sqrt3\), and \(|S^1|=2\pi\), \(|S^2|=4\pi\). Those areas equal \(2^{d-1}\pi\) for \(d=2,3\), and \(6^{2/3}/24^{1/3}=(3/2)^{1/3}\), which is formula (1). The same multiplicative bound holds in every orthonormal frame, so the sphere integral keeps it. + +**C4.** Coordinate partials of \(\varphi(x)=\exp(-|x|^2/2)\) factor as Hermite products. For every multi-index of order at most 6 and every \(|y|\ge24\), +\[ +\prod_i|He_{\alpha_i}(y_i)|\le76\,|y|^6. +\] +The check covered all 84 multi-indices in \(d\le3\); the worst ratio to \(76\cdot24^6\) is about \(0.0135\), on the pure sixth derivative. At the origin the same partials are at most 15. The lattice estimate +\[ +\sum_{n\neq0}|n|^6 e^{-288|n|^2}\le1458\,e^{-288} +\] +holds, and the degree-20 Taylor sum for \(e^{288/125}\) is greater than 10, so \(e^{-288}<10^{-125}\). Therefore +\[ +E=1458(76\cdot24^6+15)\cdot10^{-125}=21175738586478\cdot10^{-125}, +\] +and \(60E<10^{-108}\). + +**C5.** Entrywise differences are at most \(2E\) after the \(\sqrt2\) duplication on off-diagonal Hessian coordinates. In dimension at most 10 the spectral norm is at most \(20E\). The reference odd block \(\begin{pmatrix}1&-3\\-3&15\end{pmatrix}\) shifted by \(I/3\) has leading minor \(2/3\) and determinant \(7/9\), and \((23/3)^2-58=7/9>0\), so its smaller eigenvalue exceeds \(1/3\). Hessian eigenvalues are \(2\) and \(d+2\). Thus \(C_{\mathrm{ref}}\ge I/3\) and \((1-\varepsilon)C_{\mathrm{ref}}\le C_{24}\le(1+\varepsilon)C_{\mathrm{ref}}\) at \(\varepsilon=10^{-108}\). Schur complements are infima of the quadratic form, so the sandwich passes to \(\tau^2\) and to \(A\mid V=0\). The Gaussian density comparison and the degree-\(2m\) cone scaling then give integrand powers \(a=m+2/3+n/2\), \(b=d+n/2\). For \(d=3\), \(a+2b=79/6<14\) and \(2a+b=77/6<13\), and \(32\varepsilon<10^{-106}\). + +**C6.** Bernoulli numbers \(B_2,\ldots,B_{22}\) match the \(B_1=-1/2\) recurrence, including \(B_{22}=854513/138>0\). DLMF 5.11(ii) puts the positive-real remainder after the \(B_{20}\) term between 0 and the first omitted \(B_{22}\) term; the code encloses that way, then removes 32 logarithms from \(z=199/6\) by DLMF 5.5(i). An independent Stirling shift by 48, with a separate Machin/atan log/exp implementation, landed inside the package interval for \(\Gamma(7/6)\), width about \(1.4\cdot10^{-31}\), midpoint \(0.9277193336300392\). The \(10^{-106}\) allowance is applied to an already valid enclosure of \(c_{d,\mathrm{ref}}\). Author tests: `python3 -m unittest test_coefficient.py` in that directory, **30 runs, 0 failures**. The 20-digit outward endpoints are the published ones: +\[ +0.073406919306034271030`, because the conditional law of the transverse Hessian has full support on an open negative-definite ball. Remote conditioning changes the law inside the numerator expectation and leaves this denominator in place. The height split uses Lebesgue `dt` on the window. The factor `12` that appears in the unconditional pin intensity is absent from this conditional count. + +4. **Contact kernel and `O(k r^4 |E|)` remainder — ACCEPT.** `Λ_j` in (13) is continuous on the stated compact sets, finite, and bounded above and below by positive constants. Positivity uses the joint conditional support of `(A_0,H_x)` and keeps their dependence inside the expectation. The density error is `O(r^2)` and the endpoint-weight error is `O(r)`, so the Kac–Rice integrand differs from `Λ_j` by `O(r)`. Integration over a window of length `k r^3` gives the `O(k r^4 |E|)` count remainder, which is (2). The matching lower bound is for the expectation on a fixed positive-volume set. Markov is applied only from that expectation to the probability upper bound. + +5. **Fixed-separation factorial moments — ACCEPT.** For each fixed `m≥2` and `η>0`, one endpoint weight and one normalizer are used. There are `m+2` determinant factors. The ordered mean is `(k r^3)^m C_m+O(r^{3m+1})`, with a joint contact kernel. Each unordered `η`-separated `m`-set contributes `m!` ordered tuples, so its probability is at most `E[T]/m! = O(r^{3m})`. A separated pair is `O(r^6)`. + +6. **Selector and height-window mapping — ACCEPT for the implication in §7 only.** A `0/1` selector is dominated by `N_{r,j}(E)` when its type, location, and value conditions imply (1): index `j`, location in a Borel subset of the fixed `D_ρ`, and height strictly between the pinned heights. Boundary heights have expected count zero. No historical RN witness predicate, inner-wedge cell, whitened-jet cell, or all-cell partition is shown to imply (1). Those objects stay outside this acceptance. Ambient Lebesgue measure `dx` is the coordinate system of (14); a polar or scaled chart has to carry its own Jacobian. + +7. **Uncovered complement — ACCEPT as stated in §7.** The regions that remain open are exactly the ones listed in the crosswalk below. `ρ` and `η` stay fixed. + +## Region crosswalk + +| Region | This note | +|---|---| +| Fixed `D_ρ`, height in `(b-k r^3,b)`, index `j` | Theorem A and numerator (14) | +| Fixed `η>0`, ordered `m`-tuples of index `j` in that window | (15); probability ≤ `E[T]/m!` | +| Mesoscopic neighborhood `x=ry` | Open. Distance to the pins tends to zero | +| Pin collision, witness within `O(r)` of `M` or `S` | Open. Same distance restriction | +| Intermediate annulus, a multiple of `r` out to a shrinking cutoff, including `r≪\|x‖≪ρ` | Open. Compactness constants are not claimed there | +| Witness collision, mutual separation tending to zero | Open. (15) stops at fixed `η` | +| Remote critical points with no shrinking height window | Open. The factor `r^3` is the window length | +| Legacy all-cell, 24-jet, `ENV-RESCOV` through `CH-LIFT`, fixed-`r` inner wedge | Open. §7 requires a separate proof that the legacy selector implies (1). `STATUS_RN_UNIF` still records those carriers as absent | +| Probability lower bound, Poisson law, numerical `C` or `r_*`, elder selection | Not conclusions of this argument | + +## Checks and one citation correction + +I recomputed the pin Jacobian, the target `v_r`, the axial coefficient `r^2/40` on `s^5`, and the Hermite values `α_M=-6k`, `α_S=6k`, `f_xxx=12k`. The local exact suite, 28 tests, passed under Python 3. Those tests check finite algebra. They do not evaluate `Λ_j`. + +The operative Kac–Rice input is Theorem 6.1, Remark 8, and §7.1 of arXiv:2304.07424v3: the weight may depend on an extra field, the critical-point Jacobian is `|det Hess|` once, and the height split is the joint density in `dt`. In that HTML, Theorem 7.1 is the i.i.d. sum model, and the critical-point discussion is §7.1 rather than §8.1. The formula in (12) matches Theorem 6.1 and §7.1. A later bibliographic edit can retarget the sentence in §8; that mismatch is not a false step in the count. +````` + +## 6. main#76 comment 5841861362 (D4, owner reconciliation) + +URL: https://github.com/d6g8k5htny-coder/main/issues/76#issuecomment-5841861362 · posted by `d6g8k5htny-coder` (owner) · created 2026-09-26T01:15:52Z · updated 2026-09-26T01:15:52Z + +`````text +D4 REVIEW DELIVERY RECONCILED. The complete proof is in Math at `frontiers/remote_window_20260924/PROOF.md`, and the fresh xAI/Grok review on this issue returns ACCEPT on all seven fixed-rho/fixed-eta interfaces: conditioned covariance, all three determinant factors and O(k r^5) numerator, original endpoint normalizer, contact-kernel remainder, fixed-separation factorial moments, selector/window mapping, and the explicit uncovered complement. Closing #76 as the fixed-remote REVIEW work package only. This does not shrink rho or eta and does not close mesoscopic, pin, intermediate, or shrinking witness-collision regions. +````` + +## 7. main#74 comment 5842112010 (D6, owner reconciliation of the in-repo review) + +URL: https://github.com/d6g8k5htny-coder/main/issues/74#issuecomment-5842112010 · posted by `d6g8k5htny-coder` (owner) · created 2026-09-26T01:48:31Z · updated 2026-09-26T01:48:31Z + +`````text +D6 ANALYTIC REVIEW COMPLETE. The full proof is in Math at `frontiers/full_price_20260924/PROOF.md`, and merged Math PR63 records a cross-provider nonauthor ACCEPT on all six requested slices: hazard/separate concavity, odd-majority worst case, global independence/hazard direction, realized-family cover, sharpness, and demand-one boundary. Exact accepted scope is the realized disjoint capacity/clutter family with d_i>=2, every independent probability, prices c_v<=phi(p_v), same palette K>=K_H(d), and rho*=1/(3-log(3e-2)). The p=1 strictness sentence is corrected to interior 1/2=rho and between-pin height" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "dist>=rho and between-pin height", + "source": null, + "scope": "the fixed-remote region at the D4 scope: dist >= rho from the pins, between-pin height window", + "explicit_limits": "as the covering node", + "review_disposition": "ACCEPT", + "review_sources": [ + { + "ref": "main#76 comment 5841783172", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "covering theorem accepted at this exact region scope; region crosswalk table" + } + ], + "coverage_source": "math.rn-fixed-remote-window", + "notes": "PROVED_REVIEWED region: covered by math.rn-fixed-remote-window (required covered_by edge; coverage_source retained). Register alignment v1.3." + }, + "required_premises_after": [ + "math.rn-fixed-remote-window" + ], + "notes": "Covered by a PROVED_REVIEWED node after the previous transition; coverage_source retained and a required covered_by edge added so that a change to the covering proof reaches the region (Codex 4139807558; OpenAI 5360327058 item 1). The current hard gate reports PROVED_REVIEWED regions in its own proved_reviewed_regions bucket since ec6db8c (Math-#151), so the earlier region-reporting complaint (Codex 4139807545) is superseded." + }, + { + "node": "math.rn-fixed-annulus-window", + "kind": "candidate", + "current": { + "classification": "AUTHOR_SIDE_CANDIDATE", + "fingerprint": "081abc13c5e66342c13df2af2bd6b114f320486f" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b", + "source": "frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md", + "scope": "d = 2; fixed L; fixed scaled annulus 1 < A0 <= |y| <= B0 < infinity; compact positive gaps k_- > 0; all frames; between-pin height window (live node scope)", + "explicit_limits": "no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighborhood, intermediate-scale, collision, or elder-selection closure; excludes PR28 and later blobs", + "review_disposition": "ACCEPT", + "review_sources": [ + { + "ref": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "ACCEPT at the exact scoped theorem (covariance floor, cutoff r^(1/24), rare target density, conditioned C^3 sixth moment, ...)", + "binds": "commit 2804dc1, blob 081abc13, SHA256 1fd9fe71" + } + ], + "notes": "PROVED_REVIEWED at the fixed d = 2 scaled-annulus window scope; see explicit_limits. Evidence: pr22 review record; consumed S6-S21 stitching input with its review. Single nonauthor provider. Register alignment v1.3." + }, + "required_premises_after": [ + "math.rn-count-interface", + "math.align-component.pr22-annulus-review", + "math.align-component.two-scale-addendum" + ], + "notes": "Fingerprint kind changes from git blob 081abc13 (current) to SHA256, matching the other component nodes; same bytes. Single nonauthor provider. Its consumed S6-S21 stitching input (TWO_SCALE_ADDENDUM.md) and that input's review are component nodes since v1.3 (Codex 4139807567; OpenAI 5360327058 item 2)." + }, + { + "node": "math.rn-region.fixed-annulus-window", + "kind": "region", + "current": { + "classification": "COVERED_BY_CANDIDATE", + "fingerprint": "fixed-d2-annulus-between-pin-window-reviewed" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "fixed-d2-annulus-between-pin-window-reviewed", + "source": null, + "scope": "the fixed d = 2 scaled annulus at the covering node's exact declared scope", + "explicit_limits": "as the covering node", + "review_disposition": "ACCEPT", + "review_sources": [ + { + "ref": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "covering theorem accepted at this exact scope" + } + ], + "coverage_source": "math.rn-fixed-annulus-window", + "notes": "PROVED_REVIEWED region: covered by math.rn-fixed-annulus-window (required covered_by edge; coverage_source retained). Register alignment v1.3." + }, + "required_premises_after": [ + "math.rn-fixed-annulus-window" + ], + "notes": "Covered by a PROVED_REVIEWED node after the previous transition; coverage_source retained and a required covered_by edge added (as for the fixed-remote region)." + }, + { + "node": "math.p15-full-price", + "kind": "candidate", + "current": { + "classification": "AUTHOR_SIDE_CANDIDATE", + "fingerprint": "full-price-20260924" + }, + "proposed": { + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9", + "source": "frontiers/full_price_20260924/PROOF.md", + "scope": "realized disjoint capacity/clutter family with d_i >= 2, all independent probabilities, c_v <= phi(p_v), same palette K >= K_H(d), sharp rho* = 1/(3 - log(3e-2)) (STATUS D6 row)", + "explicit_limits": "demand one remains an obstruction; no arbitrary-downset or unrestricted prize-problem extension", + "review_disposition": "ACCEPT", + "review_sources": [ + { + "ref": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor (Math- issue 57, merged Math-#63)", + "verdict": "ACCEPT on all six slices", + "binds": "blob 582180e4, SHA256 87521901" + }, + { + "ref": "main#74 comment 5842112010", + "provider": "owner", + "verdict": "D6 analytic review complete; #74 closed as the D6 review work package" + } + ], + "notes": "PROVED_REVIEWED on the realized family with d_i >= 2; see explicit_limits. Evidence: p15 review record, consumed setwise interfaces of P15_REALIZED_COVERS.md, main#74 reconciliation (external record node). Single nonauthor provider. Register alignment v1.3." + }, + "required_premises_after": [ + "math.align-component.external-issue-reviews", + "math.align-component.p15-review", + "math.align-component.p15-realized-covers-setwise" + ], + "notes": "Fingerprint changes from the label full-price-20260924 to SHA256; the live note 'review open' is stale against the merged review record. Single nonauthor provider. The consumed setwise interfaces (P1), (P2), Theorem P2 of P15_REALIZED_COVERS.md are a component node since v1.3, at the D6 review's scope only." + } + ], + "execution_order": [ + "0. create the component nodes and edges of proposed_graph_nodes / proposed_graph_edges (review records, consumed sources, covered_by, the interface's two live premises); they are evidence records and change no classification of an existing node", + "1. math.rn-count-interface -> PROVED_REVIEWED (premise of 4 and 6)", + "2. math.lifetime-remainder and math.side24-coefficient -> PROVED_REVIEWED (premise D1 is live PROVED_REVIEWED)", + "3. Math-#151 is on main (ec6db8c): the D4 bytes are carried by two live nodes (math.rn-fixed-remote-window AUTHOR_SIDE_CANDIDATE; math.d5-component.remote-window-proof PROVED_REVIEWED; same fingerprint, same main#76 review); execute math.rn-fixed-remote-window -> PROVED_REVIEWED, then math.rn-region.fixed-remote; collapsing the component node onto the live node is an optional one-edge follow-up", + "4. math.rn-fixed-annulus-window -> PROVED_REVIEWED (fingerprint to SHA256), then math.rn-region.fixed-annulus-window", + "5. math.p15-full-price -> PROVED_REVIEWED (fingerprint to SHA256)", + "6. regenerate the hard gate's manifest/results per frontiers/downstream_gate_20260925/README.md; run git_transition_audit base->head; expect promotion_permission false and no controlling node", + "executing lane: non-Claude; this record executes nothing; a lane that wants a second provider on the single-provider rows (D2, D4, D5-annulus, D6) asks for it before step 2-5" + ], + "not_proposed": [ + "math.rn-region.mesoscopic-scaled-annulus, pin-collision, intermediate-r-to-rho: Math-#151 (D5 fold)", + "math.rn-region.witness-collision and its residual: Math-#160", + "math.rn-mesoscopic-reduction (legacy PR7 route), eng.* controls, hist.* predicates: unchanged", + "any controlling flag, STATUS, PROOF_INDEX, SELECTOR_REGION or catalog edit" + ], + "independence_caveat": "All cited nonauthor reviews except D3's second read and the interface review are xAI/Grok via Cursor cloud sessions on the shared account; no organizational independence. The D1 precedent for PROVED_REVIEWED used xAI plus Anthropic reads. No Cursor agent is restarted or contacted (owner stop 2026-09-27); only published comments are cited.", + "declarative": true, + "executed": false, + "review_record": [ + { + "lane": "xAI", + "id": 5901912476, + "kind": "inventory read", + "at": "670e4ae", + "verdict": "ACCEPT as declarative inventory; HOLD execution until Math-#151 Cut A is on current tip; execute #151 first, then this alignment", + "hold_precondition_math_151": "satisfied: Math-#151 merged at ec6db8c (2026-09-30T00:56:58Z)" + }, + { + "lane": "xAI/Grok (Lucas)", + "id": 5360240245, + "kind": "review", + "at": "670e4ae", + "verdict": "ACCEPT at classification-lag scope; HOLD execution until #151 on main and single-provider rows recorded as such; interface flip conditional on this packet's Anthropic note, not an xAI re-read", + "hold_precondition_math_151": "satisfied: Math-#151 merged at ec6db8c (2026-09-30T00:56:58Z)" + }, + { + "lane": "OpenAI/ChatGPT", + "id": 5360320845, + "kind": "source-custody and RN-interface review", + "at": "f2250bf", + "verdict": "historical lag and logical-interface inequalities sound at stated scopes; AMEND REQUIRED before the proposal is represented as a complete executable evidence-bound fold: (1) review bytes not carried by the graph, production read_snapshot reads node.source only; (2) N4/N5 checks are floating, relabel; (3) the interface is not a quantitative rate; N6 may be recorded via the canonical-kernel interface", + "answered_by": "v1.3: review records and consumed sources as fingerprinted component nodes with required edges; external comments preserved in EXTERNAL_REVIEWS.md and labelled mutable; PROPAGATION two-commit cases; INTERFACE exact/numerical labels; interface scope narrowed with its premises as edges" + }, + { + "lane": "OpenAI/Codex (delegated)", + "id": 5360327058, + "kind": "engineering/interface review", + "at": "f2250bf", + "verdict": "ACCEPT the bounded historical inventory; AMEND the executable proposal; HOLD execution: (1) covering-proof loss does not reach the promoted regions; (2) TWO_SCALE_ADDENDUM, P15_REALIZED_COVERS, the interface's consumed C6-Palm/cap items are not tracked; (3) apply() drops explicit_limits and keeps stale notes; region-reporting complaint superseded on main; D2/D3/D4 comments directly authenticated", + "answered_by": "v1.3: covered_by edges; consumed-source component nodes; interface premises as edges to the live D1 node and cap component (Lemma 5.1 cited, not consumed); apply() carries explicit_limits, replacement notes and review_basis; PROPAGATION regressions; the superseded complaint recorded, not replayed" + }, + { + "lane": "OpenAI/Codex (bot review 5360253242)", + "id": "4139807545/4139807551/4139807558/4139807562/4139807567", + "kind": "review threads", + "at": "670e4ae", + "verdict": "region reporting (superseded on main); stale notes on apply; region-to-covering-proof dependency; authenticate external verdicts; track consumed source dependencies", + "answered_by": "v1.3 (all five); 4139807545 by Math-#151's hard_gate.py proved_reviewed_regions bucket, recorded in the region transition notes" + } + ], + "revision": { + "version": "v1.3", + "applies": [ + "xAI 5901912476 and 5360240245 recorded; #151-first ordering of the D4 flips made explicit", + "re-verified at ec6db8c (Math-#151 merged); live D4 duplicate bound (D4_DUPLICATE check, duplicate-drift mutant); HOLD precondition on Math-#151 recorded as met", + "OpenAI 5360320845 and 5360327058 and Codex threads 4139807545-67: eight review-record / consumed-source component nodes and sixteen required edges (edges_unchanged false); covered_by edges from regions; interface premises as edges to the live D1 node and cap component and its scope narrowed; apply() carries explicit_limits, replacement notes, review_basis; PROPAGATION check with nine two-commit cases and mutants region-unlinked / review-metadata-only; INTERFACE exact versus numerical labels; RN_COUNT_INTERFACE_REVIEW.md evidence labels; EXTERNAL_REVIEWS.md preserved and labelled mutable" + ], + "unchanged": "the eight classifications and every scope exclusion; the ten original inventory identities" + }, + "reverified_at": "ec6db8c (Math- main after Math-#151 merged 2026-09-30T00:56:58Z; the ten inventoried sources and the eight live nodes unchanged; math.d5-component.remote-window-proof live as PROVED_REVIEWED with fingerprint a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 beside math.rn-fixed-remote-window at AUTHOR_SIDE_CANDIDATE)", + "live_duplicate": { + "node": "math.d5-component.remote-window-proof", + "since": "ec6db8c (Math-#151)", + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "source": "frontiers/remote_window_20260924/PROOF.md", + "component_of": "math.d5-pin-neighborhood-first-moment", + "review_basis": "STATUS D4 (main#76, external), nonauthor, ACCEPT, full", + "relation_to_this_record": "same bytes and same review as the math.rn-fixed-remote-window transition; the flip aligns the older node with it; collapsing the pair is a one-edge follow-up for the executing lane, not proposed here", + "check": "D4_DUPLICATE; mutant duplicate-drift" + }, + "proposed_graph_nodes": [ + { + "id": "math.align-component.rn-interface-review", + "layer": "D4", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-count-interface", + "component_role": "review_record", + "author_provider": "Anthropic (nonauthor; session 017Mi3hx)", + "fingerprint": "52d9038a3f27335d3c219babec785984e00b77f28d7cd7b390477c0bc6a749d1", + "source": "reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md", + "scope": "ACCEPT at the logical-interface scope: (N1)-(N5) and the section 3 counterexamples self-contained; (N6)-(N7) and section 5 as formula-level implications conditional on the parent regression law / section 9 convention and the cap support source", + "review_basis": [ + { + "review": "reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md", + "provider": "Anthropic", + "verdict": "ACCEPT at the logical-interface scope", + "depth": "full at that scope; exact rational checks plus labelled floating-point corroborations in alignment_check.py" + } + ], + "notes": "This packet's own review record; a nonauthor read by a second provider was not available at v1.3 (xAI 5360240245 did not re-read the note)." + }, + { + "id": "math.align-component.external-issue-reviews", + "layer": "D2", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": null, + "components_of": [ + "math.lifetime-remainder", + "math.side24-coefficient", + "math.rn-fixed-remote-window", + "math.p15-full-price" + ], + "component_role": "review_record_external", + "author_provider": "xAI/Grok 4.7 via Cursor (posted by cursor[bot]) and owner reconciliations; transcribed by Anthropic from the GitHub API read-back of 2026-09-30", + "fingerprint": "fa6677f16f688352f3fc9b1703d39324fe57a3b2bf61960591f524f858e99c9d", + "source": "reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md", + "scope": "seven main-repository issue comments: main#67 5841270276 and 5841782206 (D2), main#65 5841269490 and 5841779222 (D3), main#76 5841783172 and 5841861362 (D4), main#74 5842112010 (D6), with comment id, URL, posting account, created_at, updated_at and body", + "review_basis": [ + { + "review": "main#67 comment 5841270276", + "provider": "xAI/Grok 4.7 via Cursor (session bc-81faa729)", + "verdict": "R1-R3 ACCEPT, R4 ACCEPT for (R12), R5-R6 IMPORTED-OPEN", + "created_at": "2026-09-25T23:59:05Z", + "updated_at": "2026-09-26T00:07:25Z" + }, + { + "review": "main#67 comment 5841782206", + "provider": "xAI/Grok 4.7 via Cursor (session bc-dfbce246)", + "verdict": "R5 and R6 ACCEPT; Theorem R accepted at its stated O(1) remainder scope", + "created_at": "2026-09-26T01:04:21Z", + "updated_at": "2026-09-26T01:07:56Z" + }, + { + "review": "main#65 comment 5841269490", + "provider": "xAI/Grok 4.7 via Cursor (run bc-be5dc8d9)", + "verdict": "C1-C6 VERIFIED; COEFFICIENT-CALC-REVIEWED / PARENT-IMPORTED-OPEN", + "created_at": "2026-09-25T23:58:59Z", + "updated_at": "2026-09-26T00:09:32Z" + }, + { + "review": "main#65 comment 5841779222", + "provider": "owner", + "verdict": "reconciliation; #65 closed as the coefficient-review work package", + "created_at": "2026-09-26T01:03:54Z" + }, + { + "review": "main#76 comment 5841783172", + "provider": "xAI/Grok via Cursor (session bc-593a8246; provider named by the owner's reconciliation)", + "verdict": "items 1-7 ACCEPT at fixed-rho/fixed-eta scope; not a full RN or 24-jet theorem", + "created_at": "2026-09-26T01:04:29Z", + "updated_at": "2026-09-26T01:12:36Z" + }, + { + "review": "main#76 comment 5841861362", + "provider": "owner", + "verdict": "reconciliation; ACCEPT on all seven interfaces; #76 closed as the fixed-remote REVIEW work package only", + "created_at": "2026-09-26T01:15:52Z" + }, + { + "review": "main#74 comment 5842112010", + "provider": "owner", + "verdict": "D6 ANALYTIC REVIEW COMPLETE; cross-provider nonauthor ACCEPT on all six slices (the review itself is reviews/p15_full_price_nonauthor_20260926/REVIEW.md); #74 closed", + "created_at": "2026-09-26T01:48:31Z" + } + ], + "authentication": "mutable external evidence, read back through the session's GitHub connector and transcribed; not byte-authenticated offline (stdlib checker, no network); OpenAI 5360327058 reports direct authentication of 5841270276, 5841782206, 5841269490 and 5841783172; claims/LANDING_CLAIMS.json cites the same issues as review objects", + "notes": "One node for one byte object (this file); loss-only: any edit to the record revalidates D2, D3, D4 and D6." + }, + { + "id": "math.align-component.side24-review-claude", + "layer": "D3", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.side24-coefficient", + "component_role": "review_record", + "author_provider": "Anthropic (session 01NMeKE)", + "fingerprint": "258c414c46be93142ee552cfa4c32c03dd89b6bf7b0b470334f152e1e0a57eef", + "source": "reviews/side24_v1_coefficient_claude_20260929/REVIEW.md", + "scope": "second-provider read of coefficients/side24_v1/PROOF.md: ACCEPT at the arithmetic-enclosure scope", + "review_basis": [ + { + "review": "reviews/side24_v1_coefficient_claude_20260929/REVIEW.md", + "provider": "Anthropic", + "verdict": "ACCEPT at the arithmetic-enclosure scope", + "depth": "full at that scope" + } + ], + "notes": "Makes D3 the two-provider row (with main#65 5841269490)." + }, + { + "id": "math.align-component.pr22-annulus-review", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-fixed-annulus-window", + "component_role": "review_record", + "author_provider": "xAI/Grok 4.7 via Cursor", + "fingerprint": "29a0c6d00ce739baf1d29a3163977d8e707e2079e216fba4fee3669cebec127d", + "source": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "scope": "ACCEPT of FIXED_ANNULUS_CANDIDATE.md (blob 081abc13, SHA256 1fd9fe71) at the exact scoped theorem: covariance floor, cutoff r^{1/24}, rare target, conditioned C^3 sixth moment, inner two-scale stitching; PR28 and later blobs excluded", + "review_basis": [ + { + "review": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "ACCEPT at the exact scoped theorem", + "depth": "full" + } + ], + "notes": "Single nonauthor provider." + }, + { + "id": "math.align-component.two-scale-addendum", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-fixed-annulus-window", + "component_role": "consumed_proof", + "author_provider": "OpenAI", + "fingerprint": "079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d", + "source": "frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md", + "scope": "S6-S21 and (S3)-(S5): the inner two-scale stitching input (S4) consumed by FIXED_ANNULUS_CANDIDATE.md for |t| < h_r on A <= |u| <= B (claims/LANDING_CLAIMS.json dependency two-scale-s6-s21)", + "review_basis": [ + { + "review": "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "ACCEPT S6-S21 and (S3)-(S5) at blob 89cae3a9", + "depth": "full" + }, + { + "review": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "inner two-scale stitching ACCEPT; the addendum read as the imported inner bound, not re-derived", + "depth": "gluing step" + } + ], + "notes": "The addendum's own review record is the next node; the addendum requires it." + }, + { + "id": "math.align-component.two-scale-review", + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.align-component.two-scale-addendum", + "component_role": "review_record", + "author_provider": "xAI/Grok 4.7 via Cursor (session bc-ff620630)", + "fingerprint": "04cf4c986f29ac7ac33b3cc87feac23f3cf494e0c1b3bf4476e4e6666de46574", + "source": "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md", + "scope": "ACCEPT of TWO_SCALE_ADDENDUM.md S6-S21 and (S3)-(S5) at blob 89cae3a9 (Math- PR34, commit bc7d7545)", + "review_basis": [ + { + "review": "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "ACCEPT", + "depth": "full" + } + ], + "notes": "Single nonauthor provider." + }, + { + "id": "math.align-component.p15-review", + "layer": "D6", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.p15-full-price", + "component_role": "review_record", + "author_provider": "xAI/Grok 4.7 via Cursor (Math- issue 57 lane)", + "fingerprint": "691ea0db5c0f073326c9f7208c9804c6f5d0daf92ba7467529c9d57e08b15eb7", + "source": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md", + "scope": "ACCEPT of Theorem F on the realized disjoint capacity/clutter family (six slices) at blob 582180e4, SHA256 87521901; checked the setwise (P1), (P2), Theorem P2 dependency of P15_REALIZED_COVERS.md", + "review_basis": [ + { + "review": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "ACCEPT Theorem F on the realized family", + "depth": "full" + } + ], + "notes": "Single nonauthor provider; owner reconciliation main#74 5842112010 is in the external record." + }, + { + "id": "math.align-component.p15-realized-covers-setwise", + "layer": "D6", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.p15-full-price", + "component_role": "consumed_interface", + "author_provider": "OpenAI", + "fingerprint": "c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9", + "source": "frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md", + "scope": "setwise interfaces only: realized-family definition (P1), actual local restrictions (P2) and Theorem P2 full-block cover at K >= K_H(d), as consumed by Theorem F with d_i >= 2 (claims/LANDING_CLAIMS.json dependency p15-realized-covers:setwise-interfaces); the note's c_v <= p_v price theorem and any broader P15 statement are not included", + "review_basis": [ + { + "review": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md", + "provider": "xAI/Grok 4.7 via Cursor", + "verdict": "read as the cited setwise dependency and checked for the local restriction and the full-block cover; the c_v <= p_v price theorem not accepted", + "depth": "setwise interfaces" + } + ], + "notes": "The realized-covers claim itself stays HOLD_WITH_DOMAIN in the claim manifest; only the consumed setwise interfaces are represented here, at the D6 review's scope." + } + ], + "proposed_graph_edges": [ + { + "from": "math.rn-count-interface", + "to": "math.align-component.rn-interface-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-count-interface", + "to": "math.uniform-matrix-cap-lifetime", + "required": true, + "relation": "uses_parent_regression_law_and_section9_convention" + }, + { + "from": "math.rn-count-interface", + "to": "math.d1-component.marked-cylinder-cap", + "required": true, + "relation": "consumes_cap_support_scope_section5" + }, + { + "from": "math.lifetime-remainder", + "to": "math.align-component.external-issue-reviews", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.side24-coefficient", + "to": "math.align-component.external-issue-reviews", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.side24-coefficient", + "to": "math.align-component.side24-review-claude", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-fixed-remote-window", + "to": "math.align-component.external-issue-reviews", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.fixed-remote", + "to": "math.rn-fixed-remote-window", + "required": true, + "relation": "covered_by" + }, + { + "from": "math.rn-fixed-annulus-window", + "to": "math.align-component.pr22-annulus-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-fixed-annulus-window", + "to": "math.align-component.two-scale-addendum", + "required": true, + "relation": "consumes_s6_s21_stitching_input" + }, + { + "from": "math.align-component.two-scale-addendum", + "to": "math.align-component.two-scale-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.fixed-annulus-window", + "to": "math.rn-fixed-annulus-window", + "required": true, + "relation": "covered_by" + }, + { + "from": "math.p15-full-price", + "to": "math.align-component.external-issue-reviews", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.p15-full-price", + "to": "math.align-component.p15-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.p15-full-price", + "to": "math.align-component.p15-realized-covers-setwise", + "required": true, + "relation": "consumes_setwise_interfaces_P1_P2_TheoremP2" + }, + { + "from": "math.align-component.p15-realized-covers-setwise", + "to": "math.align-component.p15-review", + "required": true, + "relation": "requires_evidence" + } + ], + "nodes_added": 8, + "edges_added": 16, + "edges_removed": 0, + "external_evidence": { + "file": "reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md", + "sha256": "fa6677f16f688352f3fc9b1703d39324fe57a3b2bf61960591f524f858e99c9d", + "node": "math.align-component.external-issue-reviews", + "status": "mutable external evidence, read back on 2026-09-30 through the session's GitHub connector and transcribed with comment id, URL, posting account, created_at, updated_at and body (Cursor footer HTML omitted); not byte-authenticated offline; OpenAI 5360327058 reports direct authentication of 5841270276, 5841782206, 5841269490 and 5841783172; this is not an additional review and not an additional provider" + } +} diff --git a/reviews/register_alignment_20260930/RECONCILIATION.md b/reviews/register_alignment_20260930/RECONCILIATION.md new file mode 100644 index 000000000..ece9655fd --- /dev/null +++ b/reviews/register_alignment_20260930/RECONCILIATION.md @@ -0,0 +1,270 @@ +# Register alignment — graph nodes for the accepted STATUS rows D2, D3, D4, D5-annulus and D6 + +**Object:** REGISTER-ALIGNMENT-20260930-v1. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. +**Read at:** Math- `main` `aa288385af9392c509233c25a0f4012c9ba023e5` (30 September 2026); main `STATUS.md` at `7bbbdc9`. +Re-verified at `ec6db8c` (Math-#151 merged, 30 September 00:56Z): the ten inventoried sources and the eight live nodes of +§4 are unchanged; the live graph now also carries `math.d5-component.remote-window-proof` (`PROVED_REVIEWED`, fingerprint +`a332bae9…`, `review_basis` "STATUS D4 (main#76, external)") beside the still-`AUTHOR_SIDE_CANDIDATE` +`math.rn-fixed-remote-window`: the D4 dual-node inconsistency is now live, and the `D4_DUPLICATE` check binds it. +**Revision:** v1.1 records the two xAI reads of `670e4ae` and makes the Math-#151-first ordering of the D4 flips +explicit; transitions unchanged. v1.2 re-verifies at `ec6db8c` after Math-#151 merged, binds the now-live duplicate D4 +node (`D4_DUPLICATE` check, `duplicate-drift` mutant) and records that the HOLD's Math-#151 precondition is met; +transitions unchanged. v1.3 answers the two OpenAI AMEND reviews (5360320845, 5360327058) and the five Codex threads +(4139807545–67): review records and consumed sources become fingerprinted component nodes with required edges, regions +gain `covered_by` edges, `apply()` carries scope exclusions and replacement notes, a nine-case `PROPAGATION` check replays +two-commit source mutations through the production reverse-impact rule, the interface's promoted scope is narrowed to +self-contained inequalities plus conditional implications with its consumed premises as edges, the interface evidence +labels separate exact from floating-point checks, and the external issue comments are preserved in `EXTERNAL_REVIEWS.md` +and labelled as mutable evidence; the eight classifications are unchanged. +**Effect:** register reconciliation, **declarative only**. This record introduces **no new mathematical claim**. It binds +five objects that the register already lists as accepted to their exact bytes, quotes the nonauthor verdicts that +accepted them, records that their live graph nodes still carry the pre-review classification, and proposes the node +transitions in machine-readable form (`PROPOSED_TRANSITIONS.json`, `"executed": false`). It carries one bounded +nonauthor review (`RN_COUNT_INTERFACE_REVIEW.md`) of the RN count-interface note, the only unreviewed required premise +in these chains. Execution of any `GRAPH.json` change is a separate act for a **non-Claude lane**. `lemma_closed`, +prizes and premises are untouched. Scientific effect: NONE. + +## 0. Exposure, stated first + +- Every lane shares one GitHub account; no organizational independence is claimed anywhere below. +- I authored none of the six sources inventoried in §2 (all six are OpenAI / ChatGPT notes). I authored + `reviews/d1_chain_reconciliation_20260928/` (Math-#126), the record that moved the D1 parent node + `math.uniform-matrix-cap-lifetime` to `PROVED_REVIEWED`; the D2 and D3 nodes require that node. I authored + `reviews/d5_reconciliation_20260929/` (Math-#138), executed by Math-#151, which handles the mesoscopic, pin-collision + and intermediate regions; none of those is re-proposed here. +- Reviews cited: D2, D4, the D5 fixed annulus and D6 each rest on **one** nonauthor provider, xAI / Grok 4.7 running in + Cursor cloud sessions on 25–26 September. D3 has two providers (xAI; Anthropic, session `01NMeKE…`, in-repo record). + The count-interface review in §3 is by this session. No Cursor agent is restarted or contacted (owner stop of + 27 September); only their published comments are cited, with the byte identities they state. +- **Precedent and caveat.** The D1 node moved to `PROVED_REVIEWED` on xAI plus Anthropic reads. Four of the five flips + proposed here rest on a single provider. The executing lane may ask for a second read before flipping any of them; + this record binds what exists and says so per row. STATUS already records all five as accepted at scope. + +**Verdicts on this record.** xAI inventory read 5901912476 and xAI/Grok (Lucas) review 5360240245, both at `670e4ae`: +**ACCEPT as a declarative inventory of already-accepted STATUS scopes** (classification lag, not new mathematics; no +edge, node, controlling, prize or `lemma_closed` change; single-provider caveat correctly stated; the D3 row is the +two-provider one), with **HOLD on execution** until Math-#151 is on main so that the D4 dual-node inconsistency is +resolved in one direction, and with the single-provider rows recorded as such by the executing lane. Neither read +re-read the count-interface note: that flip rests on this packet's Anthropic review alone. Both say: do not touch +witness-collision here, do not enlarge any scope, do not treat this as a prize or Boolean fold. This record agrees +on every point; v1.1 makes the #151-first ordering explicit in §4 and `execution_order`. Math-#151 merged at `ec6db8c` +(30 September 00:56Z; main#207 closed at 00:57Z), so the first HOLD condition is met; the second, that the executing +lane record the single-provider rows as such, is already stated per row in §4 and in `PROPOSED_TRANSITIONS.json` and is +that lane's to carry out. + +OpenAI source-custody / RN-interface review 5360320845 and OpenAI-Codex engineering review 5360327058 (both at `f2250bf`, +with Codex threads 4139807545/51/58/62/67 at `670e4ae`): **the historical classification lag and the logical-interface +inequalities are sound at their stated scopes; AMEND the executable proposal; HOLD execution.** Their findings: the +acceptance evidence lived only in `review_sources` metadata, which the production loss-only audit +(`git_transition_audit.read_snapshot`, `node.source` only) never sees; a covering-proof change did not reach the promoted +regions; the annulus's `TWO_SCALE_ADDENDUM.md`, D6's `P15_REALIZED_COVERS.md` and the interface's consumed cap / parent +items were not tracked; `apply()` dropped `explicit_limits` and kept stale notes; the (N4)/(N5) checks were floating +point while labelled exact; external issue-comment verdicts were quoted, not preserved. v1.3 answers each (§4, §6; the +eight classifications and scope exclusions unchanged). The region-reporting complaint (4139807545) is superseded on main +since `ec6db8c` (`hard_gate.d4_region_complement` has a `proved_reviewed_regions` bucket) and is not replayed. Neither +OpenAI read is a new mathematical acceptance of the six sources; both keep the single-provider caveat. + +## 1. The lag + +The main `STATUS.md` accepted-scope table (rows D1–D4, D6) and `PROOF_INDEX.md` "Reviewed scoped results" list these +objects as reviewed. The downstream graph does not: + +| Object | STATUS / PROOF_INDEX | Live node | Live classification | Live fingerprint | +|---|---|---|---|---| +| D2 Theorem R | accepted, `O(1)` remainder scope | `math.lifetime-remainder` | `AUTHOR_SIDE_CANDIDATE` (in-node `review_disposition: ACCEPT`) | SHA256 `380b7d0a…` = current | +| D3 SIDE24 coefficient | accepted, `d = 2, 3` arithmetic | `math.side24-coefficient` | `AUTHOR_SIDE_CANDIDATE` (in-node `ACCEPT`) | SHA256 `c06daccc…` = current | +| D4 fixed-remote theorem | accepted, fixed `ρ`, `η` | `math.rn-fixed-remote-window` | `AUTHOR_SIDE_CANDIDATE`; region `math.rn-region.fixed-remote` `COVERED_BY_CANDIDATE` | SHA256 `a332bae9…` = current | +| D5 fixed-annulus window | PROOF_INDEX reviewed row | `math.rn-fixed-annulus-window` | `AUTHOR_SIDE_CANDIDATE` (in-node `ACCEPT`); region `math.rn-region.fixed-annulus-window` `COVERED_BY_CANDIDATE` | git blob `081abc13…` = current | +| D6 P15 full price | accepted, realized family | `math.p15-full-price` | `AUTHOR_SIDE_CANDIDATE`, note "review open" | label `full-price-20260924` | +| (premise) RN count interface | — | `math.rn-count-interface` | `AUTHOR_SIDE_CANDIDATE`, unreviewed | SHA256 `aa993f52…` = current | + +Why it matters: the hard gate's rule is "Only `PROVED_REVIEWED` satisfies a still-required positive premise" +(`frontiers/downstream_gate_20260925/README.md`). As long as these nodes stay author-side, the fixed-remote region is +"covered by candidate" only, and D2, D3, D4 and D6 satisfy no downstream premise, although their STATUS rows are +accepted. The register is also internally inconsistent on D4: since `ec6db8c` the live graph carries +`math.d5-component.remote-window-proof` (Math-#151, executing the D5 proposal of Math-#138) for the same bytes as +`PROVED_REVIEWED`, with `review_basis` "STATUS D4 (main#76, external), nonauthor, ACCEPT, full", beside +`math.rn-fixed-remote-window` at `AUTHOR_SIDE_CANDIDATE`. The same review either suffices for those bytes or it does +not; this record proposes that it does, on the older live node, so that the two agree; collapsing the pair is a +separate one-edge follow-up for the executing lane (§7). The `D4_DUPLICATE` check binds the duplicate's classification, +fingerprint, source and review basis to the D4 transition. + +## 2. Bindings: what each review bound, and what the bytes are now + +All six sources, and the two consumed sources and the addendum's review added in v1.3, are unchanged since their landing +commits. Identities of all fifteen inventoried files (six sources, the D1 parent, four in-repo reviews, two consumed +sources, this packet's two review records) are checked by `alignment_check.py` (`IDENTITIES`); the quoted verdict lines +are checked as substrings where the review is in this repository (`VERDICTS`). The issue-comment reviews are preserved +as a transcribed read-back in `EXTERNAL_REVIEWS.md` (comment id, URL, posting account, timestamps, body) and are +labelled there as mutable external evidence that an offline check cannot re-authenticate; OpenAI 5360327058 reports +having directly authenticated the D2, D3 and D4 comments. + +| Object | Source now | Review binding (quoted) | Verdict (quoted) | +|---|---|---|---| +| D2 | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, 17734 B, SHA256 `380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a`, blob `247b3ecf80bfbe896948d5d489b2d5842a81c481` | main#67 5841270276: "blob `247b3ecf80bfbe896948d5d489b2d5842a81c481`, 17734 bytes, SHA256 `380b7d0a…`"; 5841782206 re-read "and hashed" the same blob | 5841270276: "R1 — ACCEPT … R2 — ACCEPT … R3 — ACCEPT … R4 — ACCEPT for (R12) … R5 — IMPORTED-OPEN … R6 — IMPORTED-OPEN"; 5841782206: "R5 and R6 are **ACCEPT**. Theorem R is accepted at its stated O(1) remainder scope." The imports are mapped to #63 comment 5841570965 (D1-A…E), the review that the D1 reconciliation binds. | +| D3 | `coefficients/side24_v1/PROOF.md`, 10272 B, SHA256 `c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769`, blob `44b66f04f89fcd87383b3603fa69f1feb64cdddd` | main#65 5841269490: "commit `e329fba1…`, directory `coefficients/side24_v1/`", "`PROOF.md` 10272 bytes, SHA256 `c06daccc…`" | "All six coefficient interfaces check out. The disposition is **COEFFICIENT-CALC-REVIEWED / PARENT-IMPORTED-OPEN**." Second provider: `reviews/side24_v1_coefficient_claude_20260929/REVIEW.md` (blob `665d1683f33a7180957923c3cb2abce8ade235d1`): "Overall: **ACCEPT at the arithmetic-enclosure scope**". Owner reconciliation 5841779222 closed the package. The parent import is now the live `PROVED_REVIEWED` D1 node. | +| D4 | `frontiers/remote_window_20260924/PROOF.md`, 18355 B, SHA256 `a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7`, blob `b383bfcc88ec4ad497dff01fb6640e429ba24a84` | main#76 5841783172: "at commit `191ea7d541a486736ba7bbddfd4eac25a6c4567b` … 18355 bytes, SHA256 `a332bae9…`" | "This accepts only the fixed-ρ and fixed-η interfaces below. It does not accept a full RN or 24-jet theorem." Items 1–7: conditional covariance ACCEPT; three determinant factors and `O(k r⁵)` numerator ACCEPT; original endpoint normalizer ACCEPT; contact kernel and `O(k r⁴ ‖E‖)` remainder ACCEPT; fixed-separation factorial moments ACCEPT; selector/window mapping ACCEPT for the §7 implication only; uncovered complement ACCEPT as stated. Owner 5841861362: "returns ACCEPT on all seven fixed-rho/fixed-eta interfaces … Closing #76 as the fixed-remote REVIEW work package only." | +| D5 annulus | `frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md`, 17646 B, SHA256 `1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b`, blob `081abc13c5e66342c13df2af2bd6b114f320486f` | `reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md` (blob `1f153966dae48d10574db0d272f75d08ca3a8b23`): "Immutable commit `2804dc1d…`", "Blob `081abc13…`", "SHA256 `1fd9fe71…`" | rows "Covariance floor … **ACCEPT**", "Cutoff `r^{1/24}` **ACCEPT**", "Rare target in the full joint gradient/height density **ACCEPT**", "Conditioned `C^3` sixth moment **ACCEPT**", at the exact scoped theorem; PR28 and later blobs excluded. | +| D6 | `frontiers/full_price_20260924/PROOF.md`, 11352 B, SHA256 `87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9`, blob `582180e41dca0ad815ad0f18574df42040912149` | `reviews/p15_full_price_nonauthor_20260926/REVIEW.md` (blob `07db19f826763b8faa7414ba2e403d9c45bc0b6e`): "Blob `582180e4…`", "SHA256 `87521901…`" | six rows **ACCEPT** (hazard transform and separate concavity; odd-majority worst case; global hazard direction and independence; full-block cover; sharpness; demand-one boundary). Owner 5842112010: "D6 ANALYTIC REVIEW COMPLETE … Closing #74 as the D6 review work package". | +| premise | `frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md`, 8938 B, SHA256 `aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab`, blob `371fd6d17920f2eb3b1c5ec30297acf6daf1d385` | this packet, `RN_COUNT_INTERFACE_REVIEW.md` | ACCEPT at the logical-interface scope (§3); its consumed formulation was reconstructed inside the D4 review (items 2–3). | +| D1 parent (premise of D2, D3) | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, 40261 B, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7` | live node `math.uniform-matrix-cap-lifetime`, `PROVED_REVIEWED`, four required reading-rule components (Math-#126) | not re-reviewed; checked live (`BASELINE`). | +| consumed by D5 annulus | `frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md`, 15902 B, SHA256 `079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d`, blob `89cae3a9734f2ec7172cd0b6b0b3af3ddd355d73` | `reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md` (10073 B, SHA256 `04cf4c98…`, blob `89882516`): "blob `89cae3a9…`, 15902 bytes, SHA256 `079f9399…`"; the pr22 review: "Read as the imported inner bound, not re-derived: `TWO_SCALE_ADDENDUM.md` blob `89cae3a9…`" | S6–S21 and (S3)–(S5) **ACCEPT** (xAI, Math- PR34); pr22 row "Inner two-scale stitching **ACCEPT**". Component nodes `…two-scale-addendum`, `…two-scale-review` (§4). | +| consumed by D6 | `frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md`, 11467 B, SHA256 `c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9`, blob `173881916ccd0e738bdb41279e835e78e520fcbc` | the p15 review: "Read as the cited setwise dependency, and checked for the local restriction and the full-block cover: … SHA256 `c0dbb821…`. That is the digest named in the proof. The check uses (P1), (P2), and Theorem P2 there. It does not accept that note's `c_v<=p_v` price theorem" | setwise interfaces only, at the D6 review's scope. Component node `…p15-realized-covers-setwise` (§4). | +| external comments (D2, D3, D4, D6) | `reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md` (this packet) | seven comments with id, account, `created_at`, `updated_at`, body; the four `cursor[bot]` reviews and three owner reconciliations | preserved read-back; mutable external evidence, labelled. Component node `…external-issue-reviews` (§4). | + +## 3. The count-interface note + +`RN_COUNT_INTERFACE_REVIEW.md` reviews `RN-COUNT-INTERFACE-20260924-v1` at the scope the note itself states: the Hölder +implications (N1)–(N3), the two counterexamples of its §3 and the layer-cake bound (N5), the marked Kac–Rice formula +(N6)–(N7) as a formula, and the support statement of §5. Verdict: **ACCEPT at the logical-interface scope**; it is not +an expected-count rate and is not treated as one. The finite parts are checked in `INTERFACE` in two labelled parts: exact rational arithmetic (Hölder on finite rational +distributions for `p = 2, 3`; the exponent identity of (N2); the (N3) identity; the single-`p` sharp example at +`r = 2^(−pm)`, where `E N^p = 1` and `E N = r^(3−3/p)` exactly) and floating-point numerical corroborations of arguments +proved in the review (the (N4) example's moments and the located maximum of `(t+1)^p e^(−3t)`; the identity +`∫ min(q, A e^(−t/B)) dt = B q [1 + log(A/q)]` by Simpson quadrature; the uniform tail `P(N_r > t) ≤ e³ e^(−3t)` of the +(N4) example on a grid); OpenAI 5360320845 item 2 asked for exactly this separation. The note's consumed premises are the +parent's regression law and §9 Borel-mark convention (live node `math.uniform-matrix-cap-lifetime`, with its four +reading-rule components) and the cap support source of §5 (live node `math.d1-component.marked-cylinder-cap`, SHA256 +`0bf922b9…`, the digest the note quotes); both are required edges of the proposal, and the promoted scope is narrowed to +(N1)–(N5) with the counterexamples as self-contained statements and (N6)–(N7), §5 as conditional implications at formula +level. A later rigorous statement of (N6) is Lemma 5.1 of `frontiers/c6_palm_route_20260929/PROOF.md` (Math-#145), +cited for the reader and not consumed. + +## 4. Proposed transitions + +Eight node transitions and, since v1.3, the evidence that carries them, represented in the graph itself: eight +fingerprinted review-record / consumed-source component nodes and sixteen required edges (`PROPOSED_TRANSITIONS.json`: +`transitions`, `proposed_graph_nodes`, `proposed_graph_edges`; `edges_unchanged: false`). The change of packaging answers +OpenAI 5360320845 item 1, OpenAI 5360327058 items 1–2 and Codex threads 4139807558/4139807562/4139807567: the production +loss-only audit (`git_transition_audit.read_snapshot`) reads `node.source` only, so acceptance evidence and consumed +sources that live only in `review_sources` metadata would never revalidate a promoted object when they change. The eight +classifications and every scope exclusion are unchanged from v1.0. + +| Node | Current | Proposed | Scope (STATUS row) | Review sources | Required premises after | +|---|---|---|---|---|---| +| `math.rn-count-interface` | `AUTHOR_SIDE_CANDIDATE` | `PROVED_REVIEWED` | (N1)–(N5) and the §3 counterexamples as self-contained probability statements; (N6)–(N7) and §5 as formula-level implications conditional on the parent's regression law / §9 convention and on the cap support source | this packet (`RN_COUNT_INTERFACE_REVIEW.md`, a component node); main#76 items 2–3 | `math.uniform-matrix-cap-lifetime` and `math.d1-component.marked-cylinder-cap` (both live `PROVED_REVIEWED`; the note's consumed premises); `math.align-component.rn-interface-review` | +| `math.lifetime-remainder` | `AUTHOR_SIDE_CANDIDATE` | `PROVED_REVIEWED` | Theorem R, existential `O(1)` remainder | main#67 5841270276, 5841782206 | D1 (live `PROVED_REVIEWED`); `math.align-component.external-issue-reviews` | +| `math.side24-coefficient` | `AUTHOR_SIDE_CANDIDATE` | `PROVED_REVIEWED` | coefficient arithmetic, `d = 2, 3` | main#65 5841269490; in-repo Anthropic record | D1; `…external-issue-reviews`; `…side24-review-claude` | +| `math.rn-fixed-remote-window` | `AUTHOR_SIDE_CANDIDATE` | `PROVED_REVIEWED` | fixed `ρ`, `η`, between-pin window | main#76 5841783172, 5841861362 | `math.rn-count-interface`; `…external-issue-reviews` | +| `math.rn-region.fixed-remote` | `COVERED_BY_CANDIDATE` | `PROVED_REVIEWED` | the D4 region | as the covering node | `math.rn-fixed-remote-window` (new required `covered_by` edge) | +| `math.rn-fixed-annulus-window` | `AUTHOR_SIDE_CANDIDATE` (blob fingerprint) | `PROVED_REVIEWED` (SHA256 fingerprint) | `d = 2` fixed scaled annulus, window | `reviews/pr22_…/REVIEW.md` | `math.rn-count-interface`; `…pr22-annulus-review`; `…two-scale-addendum` (the S6–S21 stitching input, which itself requires `…two-scale-review`) | +| `math.rn-region.fixed-annulus-window` | `COVERED_BY_CANDIDATE` | `PROVED_REVIEWED` | the covering node's scope | as the covering node | `math.rn-fixed-annulus-window` (new required `covered_by` edge) | +| `math.p15-full-price` | `AUTHOR_SIDE_CANDIDATE` (label fingerprint) | `PROVED_REVIEWED` (SHA256 fingerprint) | realized family, `d_i ≥ 2` | `reviews/p15_…/REVIEW.md`; main#74 5842112010 | `…external-issue-reviews`; `…p15-review`; `…p15-realized-covers-setwise` ((P1), (P2), Theorem P2 setwise only, which itself requires `…p15-review`) | + +Component nodes (`kind: reading_rule_component`, `PROVED_REVIEWED`, `controlling: false`, `fingerprint` = SHA256 of the +source; one node per byte identity; none duplicates a source the live graph already carries with a fingerprint): + +| Node | Source | Role | Provider | Consumed by | +|---|---|---|---|---| +| `math.align-component.rn-interface-review` | `reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md` (this packet) | review record | Anthropic (nonauthor) | `math.rn-count-interface` | +| `math.align-component.external-issue-reviews` | `reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md` (this packet: seven issue comments transcribed with id, account, timestamps and body; mutable external evidence, labelled) | external review record | xAI/Grok via Cursor (posted by `cursor[bot]`) and owner; recorded by Anthropic | D2, D3, D4, D6 | +| `math.align-component.side24-review-claude` | `reviews/side24_v1_coefficient_claude_20260929/REVIEW.md` | review record | Anthropic (other session) | D3 | +| `math.align-component.pr22-annulus-review` | `reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md` | review record | xAI | annulus | +| `math.align-component.two-scale-addendum` | `frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md` (`079f9399…`, blob `89cae3a9`) | consumed proof: the S6–S21 stitching input ((S4) on `A ≤ |u| ≤ B`) | OpenAI | annulus | +| `math.align-component.two-scale-review` | `reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md` (`04cf4c98…`, blob `89882516`) | review record of the addendum (S6–S21 and (S3)–(S5) ACCEPT) | xAI | `…two-scale-addendum` | +| `math.align-component.p15-review` | `reviews/p15_full_price_nonauthor_20260926/REVIEW.md` | review record | xAI | D6, `…p15-realized-covers-setwise` | +| `math.align-component.p15-realized-covers-setwise` | `frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md` (`c0dbb821…`, blob `17388191`) | consumed interface: (P1), (P2), Theorem P2 setwise at `K ≥ K_H(d)` only, as the D6 review checked them; the note's `c_v ≤ p_v` price theorem is not included | OpenAI | D6 | + +These are exactly the support dependencies the canonical claim manifest records for the annulus (`two-scale-s6-s21`) +and for the full-price theorem (`p15-realized-covers:setwise-interfaces`). The interface's two consumed premises are +the live D1 node (regression law `Q`, §9 Borel-mark convention, read with its four reading-rule components) and the live +cap component (`0bf922b9…`, the SHA256 the note quotes for its §5 support statement); the (N6) cross-reference to the +C6 Palm route's Lemma 5.1 in the review is for the reader and is not consumed. + +Every proposed node keeps `controlling: false`; each candidate node's `fingerprint` becomes the SHA256 of its source in +§2, and `apply()` carries `scope`, `explicit_limits`, a replacement `notes`, `review_disposition`, `review_sources` and a +structured `review_basis` onto the node, so no promoted node keeps a "review open" or "covered by candidate only" note +(Codex 4139807551; OpenAI 5360327058 item 3). Region nodes keep `coverage_source` and gain a required `covered_by` edge +to their covering node, so that a change to the covering proof reaches the region in `reverse_impact_between` (Codex +4139807558; OpenAI 5360327058 item 1). No `hist.*`, `eng.*` or `regional.*` node moves; no live node other than the +eight changes; no live edge is removed; every added edge leaves one of this record's objects. The order in +`execution_order` creates the component nodes and edges first, then the premise, then each region after its covering +node; the D4 theorem node and its region are executed only after Math-#151 is on main (satisfied at `ec6db8c`): the D4 +bytes are now carried by two live nodes on the same review, and the flip aligns the older one with the newer one; the +executing lane may then collapse `math.d5-component.remote-window-proof` onto the live node or keep both (xAI +5901912476, 5360240245). STATUS, PROOF_INDEX, SELECTOR_REGION and the catalog need no edit for this record: they +already say what the graph is being aligned to. + +`alignment_check.py` applies the proposal to a copy of the live graph and replays it through the hard gate's own +validators (`validate_graph_fail_closed`, `closure_report`, `reverse_impact_between`, imported from +`frontiers/downstream_gate_20260925/hard_gate.py`): the proposed graph is well formed, no node is controlling, +`gate_ok` holds, and the loss-only reverse-impact proposal names exactly the changed nodes and their dependents. +`PROPAGATION` then runs nine two-commit mutations through `reverse_impact_between` with source snapshots in the shape of +`git_transition_audit.read_snapshot` (proof edits reach their regions and consumers; a deleted review record and an +edited external-review record reach the accepted objects and their dependents; consumed-source edits reach their +consumers; an unrelated edit and a no-op reach none of the eight), the guarantee OpenAI 5360320845 asked to be tested +rather than inferred from old-versus-new GRAPH metadata. + +## 5. What is not proposed + +- The mesoscopic, pin-collision and intermediate regions: Math-#151. The witness-collision region and its residual: + Math-#160. `math.rn-mesoscopic-reduction` (legacy PR7 route): unchanged, still `AUTHOR_SIDE_REDUCTION`. +- Any second-provider reads. Where a row rests on one provider the row says so; the lane executing the flip decides + whether that is enough, as it was not for D1. +- Any controlling flag, promotion, `lemma_closed`, prize, STATUS, PROOF_INDEX, catalog or selector edit. + +## 6. Checks + +`alignment_check.py` (stdlib only, run from the repository root): +- **IDENTITIES.** All fifteen inventoried files (the six sources, the D1 parent, the four in-repo reviews, the two + consumed sources, and this packet's two review records) exist as regular files, no symlink on their paths, stated + SHA256 and blob. +- **VERDICTS.** The quoted verdict rows and binding lines of the in-repo reviews (pr22, p15, side24, two-scale), the + object headers of the sources, the D4 sentence that consumes the interface, the interface note's (N1)/(N4)/(N5) + displays, the five PROOF_INDEX rows, the interface review's overall verdict and evidence labels, and the seven comment + ids with their verdict lines and the mutability label in `EXTERNAL_REVIEWS.md` are present as substrings. +- **BASELINE.** Each transition's live node exists with exactly the recorded current classification, fingerprint, + kind and source; the D1 node is `PROVED_REVIEWED` with its four reading-rule components `PROVED_REVIEWED`. +- **TRANSITIONS.** `declarative` true, `executed` false, `edges_unchanged` false; every proposed classification is + `PROVED_REVIEWED` with `controlling: false`, a non-empty scope, explicit limits, replacement notes and at least one + review source; every candidate node's fingerprint equals the SHA256 of its inventoried source; every component node + is a `reading_rule_component`, `PROVED_REVIEWED`, non-controlling, fingerprinted to its inventoried source, with a + role, a provider, a review basis and a consumer, one node per byte identity and none for bytes the live graph already + carries; every added edge is well formed, new, and leaves one of this record's objects; applied to a copy of the live + graph, no live node other than the eight changes, no live edge is removed, every flipped node's required premises (the + live D1 node and cap component, the interface, the component nodes) are `PROVED_REVIEWED`, every region requires its + covering node, and every applied node carries its `explicit_limits`, its replacement `notes` and a `review_basis`. +- **GATE.** The proposed graph passes the hard gate's `validate_graph_fail_closed`; `closure_report` gives `gate_ok` + with no illegal controlling node; `reverse_impact_between(live, proposed)` runs and its impacted set (the eight nodes, + the eight component nodes and their dependents) is recorded. +- **PROPAGATION.** Nine two-commit cases on the proposed graph with source snapshots in the shape of + `git_transition_audit.read_snapshot`: a D4 proof edit reaches the fixed-remote region and the mesoscopic consumer; an + annulus proof edit reaches its region; deleting the interface review record reaches the interface, D4, the annulus and + both regions; editing the external-review record reaches D2, D3, D4, D6 and the fixed-remote region; editing the + two-scale addendum reaches the annulus and its region; editing the realized-covers source reaches D6; a D4 fingerprint + edit reaches the region; an unrelated edit reaches none of the eight; a no-op impacts nothing. +- **INTERFACE.** The finite content of §3, in two labelled parts: exact rational checks (Hölder for `p = 2, 3` on + finite spaces, the (N2) exponent identity, the (N3) identity, the sharp-`p` example at `r = 2^(−pm)`, the integer + monotonicity in (N4)) and floating-point numerical corroborations (the (N4) grid maximum, the (N5) Simpson quadrature + at three parameter points, the (N4) example's uniform tail on a grid). The analytic arguments are in the review; the + numerical part corroborates, it does not prove. +- **NEGATIVES.** Real filesystem faults on a temporary copy of the inventory (delete, change one byte, symlink, symlink + parent of the D4 proof) are rejected. +- **D4_DUPLICATE.** The live graph carries `math.d5-component.remote-window-proof` as `PROVED_REVIEWED`, + non-controlling, with the same SHA256 fingerprint and source as the D4 transition and as the inventory, a `review_basis` + citing main#76 with verdict ACCEPT, `component_of` `math.d5-pin-neighborhood-first-moment`; and the older live node + `math.rn-fixed-remote-window` still carries that fingerprint at `AUTHOR_SIDE_CANDIDATE`, the lag this record aligns. + +Twelve mutants must fail: `allow-symlink`, `no-hash`, `stale-fingerprint`, `drop-review-source`, `controlling-true`, +`executed-flag`, `skip-interface-premise`, `holder-reversed`, `baseline-drift`, `duplicate-drift`, `region-unlinked` +(the `covered_by` edges removed before the propagation replay: proof edits no longer reach the regions) and +`review-metadata-only` (the review-record nodes and their edges removed before the replay, leaving the citations as +metadata only: a deleted or edited review record no longer reaches its accepted object). + +## 7. Relation to other lanes + +- **Math-#151 (Codex, D5 fold; merged at `ec6db8c`, 30 September 00:56Z, final head `609e559`):** landed as proposed, + so the live graph carries `math.d5-component.remote-window-proof` (`PROVED_REVIEWED`, `component_of` + `math.d5-pin-neighborhood-first-moment`, `review_basis` main#76) for the D4 bytes beside `math.rn-fixed-remote-window` + (`AUTHOR_SIDE_CANDIDATE`); the coordination note (comment 5901742635) was not taken up before the merge. This record's + D4 flip aligns the two; collapsing the component node to a `required` edge onto the live node is a one-edge follow-up + for the executing lane, not proposed here. +- **Math-#160 (this session, C6 reconciliation):** already references the live D4 node with supporting edges; nothing + there changes. +- **OpenAI reviews 5360320845 and 5360327058; Codex review 5360253242 (threads 4139807545–67):** answered by v1.3 (§0, + §4, §6); the region-reporting thread is superseded on main since `ec6db8c` and not replayed. The single-provider caveat + and the HOLD on execution stand until the executing lane decides. +- **Math-#163 (navigation refresh):** availability index only; disjoint. +- **main#207 (closed 30 September 00:57Z, after Math-#151 merged):** the delivery note for this record is comment + 5901889845. The executing lane for this record remains non-Claude; any new coordination thread is that lane's to open. diff --git a/reviews/register_alignment_20260930/RESULTS.json b/reviews/register_alignment_20260930/RESULTS.json new file mode 100644 index 000000000..6702db536 --- /dev/null +++ b/reviews/register_alignment_20260930/RESULTS.json @@ -0,0 +1,99 @@ +{ + "checks": { + "BASELINE": true, + "D4_DUPLICATE": true, + "GATE": true, + "IDENTITIES": true, + "INTERFACE": true, + "NEGATIVES": true, + "PROPAGATION": true, + "TRANSITIONS": true, + "VERDICTS": true + }, + "component_nodes": 8, + "edges_added": 16, + "gate": { + "changed_nodes": [ + "math.align-component.external-issue-reviews", + "math.align-component.p15-realized-covers-setwise", + "math.align-component.p15-review", + "math.align-component.pr22-annulus-review", + "math.align-component.rn-interface-review", + "math.align-component.side24-review-claude", + "math.align-component.two-scale-addendum", + "math.align-component.two-scale-review", + "math.lifetime-remainder", + "math.p15-full-price", + "math.rn-count-interface", + "math.rn-fixed-annulus-window", + "math.rn-fixed-remote-window", + "math.rn-region.fixed-annulus-window", + "math.rn-region.fixed-remote", + "math.side24-coefficient" + ], + "impacted": [ + "math.align-component.external-issue-reviews", + "math.align-component.p15-realized-covers-setwise", + "math.align-component.p15-review", + "math.align-component.pr22-annulus-review", + "math.align-component.rn-interface-review", + "math.align-component.side24-review-claude", + "math.align-component.two-scale-addendum", + "math.align-component.two-scale-review", + "math.lifetime-remainder", + "math.p15-full-price", + "math.rn-count-interface", + "math.rn-fixed-annulus-window", + "math.rn-fixed-remote-window", + "math.rn-mesoscopic-reduction", + "math.rn-region.fixed-annulus-window", + "math.rn-region.fixed-remote", + "math.rn-selector-region-crosswalk", + "math.side24-coefficient" + ], + "reverse_impact_keys": [ + "changed_nodes", + "context_changed", + "graph", + "impacted", + "meaning", + "promotion_permission", + "source_snapshots_supplied", + "traversal_edges" + ] + }, + "interface_evidence": { + "exact_rational": [ + "(N1) Holder p=2,3 on finite rational spaces", + "(N2) exponent identity", + "(N3) conditional-mean identity", + "sharp-p example at r=2^-pm", + "(N4) integer monotonicity of ceil(t)" + ], + "numerical_corroboration_floating_point": [ + "(N4) grid maximum of (t+1)^p e^-3t", + "(N5) Simpson quadrature at three parameter points", + "(N4) example uniform tail on a grid" + ], + "results": { + "exact": true, + "numerical_corroboration": true + } + }, + "inventory_files": 15, + "object": "REGISTER-ALIGNMENT-20260930-v1", + "passed": true, + "propagation": { + "annulus-proof-edit-reaches-region": true, + "d4-fingerprint-edit-reaches-region": true, + "d4-proof-edit-reaches-region-and-consumer": true, + "external-review-record-edit-reaches-d2-d3-d4-d6": true, + "interface-review-deleted-reaches-interface-consumers-regions": true, + "no-op-impacts-nothing": true, + "realized-covers-edit-reaches-d6": true, + "two-scale-addendum-edit-reaches-annulus-and-region": true, + "unrelated-edit-leaves-the-eight-untouched": true + }, + "scope": "identity, verdict-row, live-baseline, transition-shape, hard-gate replay, two-commit propagation, interface arithmetic (exact and numerical parts labelled), filesystem-negative and live D4-duplicate checks; no mathematics is re-proved; nothing is written", + "transitions": 8 +} diff --git a/reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md b/reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md new file mode 100644 index 000000000..7e7049e2b --- /dev/null +++ b/reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md @@ -0,0 +1,56 @@ +# Nonauthor review — RN count interface note (D4 premise) + +**Object reviewed:** `frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md` (RN-COUNT-INTERFACE-20260924-v1), author +OpenAI / ChatGPT; on Math- `main` `aa288385af9392c509233c25a0f4012c9ba023e5`; 8938 bytes, SHA256 +`aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab`, git blob `371fd6d17920f2eb3b1c5ec30297acf6daf1d385`; unchanged since its landing commit `8a13ee9` +(24 September 2026). Live graph node `math.rn-count-interface` carries this SHA256 as fingerprint. +**Revision:** v1.3 (30 September 2026) corrects the evidence labels after OpenAI review 5360320845 item 2: the checks +that use floating point are now called numerical corroborations, the rational ones exact; and names the note's consumed +premises (item 3 there; OpenAI 5360327058 item 2). Verdicts unchanged. +**Reviewer:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. Different provider from the +author. Same GitHub account as every lane; no organizational independence is claimed. +**Exposure:** I authored none of the three-fronts notes. I authored the remote-collision packet and the C6L packet, +both downstream of the fixed-remote theorem that consumes this note; neither is an input here. +**Scope:** the logical interface only, as the note itself states: the implications (N1)–(N3), the counterexamples of +§3 with (N4)–(N5), the marked Kac–Rice expression (N6)–(N7) as a formula, and the support statement of §5. No rate, +no RN numerical certificate and no evaluation of the Gaussian triple integral are claimed by the note or accepted +here. Scientific effect: NONE. +**Promoted scope (v1.3):** (N1)–(N5) and the §3 counterexamples as self-contained probability statements; (N6)–(N7) +and §5 as formula-level implications conditional on the note's two consumed premises, the parent's regression law `Q` +with its §9 Borel-mark convention (live node `math.uniform-matrix-cap-lifetime`, read with its four reading-rule +components) and the cap support source of §5 (live node `math.d1-component.marked-cylinder-cap`, SHA256 `0bf922b9…`, the +digest the note quotes); both are required edges in `PROPOSED_TRANSITIONS.json`. Nothing quantitative. + +## Verdicts + +| Item | Statement | Verdict | +|---|---|---| +| (N1) | `E[N 1_E] ≤ (E N^p)^(1/p) P(E)^(1−1/p)` for `p > 1` | **ACCEPT**: Hölder with exponents `p` and `p/(p−1)` applied to `N · 1_E`. Exact rational check for `p = 2` (Cauchy–Schwarz form `(E[N1_E])² ≤ E[N²] P(E)`) and `p = 3` (`(E[N1_E])³ ≤ E[N³] P(E)²`) on finite distributions in `alignment_check.py` (`INTERFACE`). | +| (N2) | `E N^p ≤ A r^(−β)` gives `E[N 1_E] ≤ A^(1/p) C^(1−1/p) r^([3(p−1)−β]/p)` | **ACCEPT**: substitution into (N1); the exponent `3 − 3/p − β/p` is the moment convention, as the note says. | +| (N3) | `E[N 1_E] = q E[N | E]`, `q = P(E) > 0` | **ACCEPT**: definition of conditional expectation; the two remarks (sufficiency of a bounded conditional mean; equivalence only under a lower comparison `q ≥ c r³`) follow. | +| §3 (N4) | `U` uniform, `E_r = {U ≤ r³}`, `N_r = ⌈log(1/r)⌉ 1_{E_r}`: `E N_r / r³ → ∞` while `sup_r E N_r^p < ∞` for every fixed `p` | **ACCEPT**: `P(E_r) = r³`; `E N_r^p = ⌈t⌉^p e^(−3t) ≤ (t+1)^p e^(−3t)` with `t = log(1/r) ≥ 1`, whose supremum over `t ≥ 1` is finite (the logarithmic derivative `−3 + p/(t+1)` is eventually negative). The argument is the log-derivative; `INTERFACE` corroborates it numerically, in floating point, on `r = e^(−t)`, `t = 1, …, 12`, and on a grid against the located maximum (labelled `numerical_corroboration`, not exact). | +| §3 sharp `p` | `N_r = ⌈r^(−3/p)⌉ 1_{E_r}`: `E N_r^p ≤ 2^p` for `r ≤ 1`, `E N_r ≍ r^(3−3/p)` | **ACCEPT**: `r^(−3/p) ≥ 1` gives `⌈r^(−3/p)⌉ ≤ 2 r^(−3/p)`. Checked exactly at `r = 2^(−pm)`, where `r^(−3/p) = 2^(3m)` is an integer, `E N_r^p = 1` and `E N_r = r^(3−3/p)` exactly. | +| (N5) | `P(N > t) ≤ A e^(−t/B)`, `A ≥ 1`, gives `E[N 1_E] ≤ ∫ min(q, A e^(−t/B)) dt = B q [1 + log(A/q)]` | **ACCEPT**: layer cake `E[N 1_E] = ∫_0^∞ P(N > t, E) dt ≤ ∫ min(q, P(N > t)) dt`; the crossover is `t_0 = B log(A/q)` and the two pieces are `q t_0` and `A B e^(−t_0/B) = B q`. The (N4) example has the uniform tail `P(N_r > t) ≤ e³ e^(−3t)` (`A = e³`, `B = 1/3`), so the logarithmic loss in (N5) is not removable from a tail hypothesis alone, as the note says. The identity is proved by the antiderivative; `INTERFACE` corroborates it by floating-point Simpson quadrature at three parameter points and checks the tail comparison on a floating-point grid (numerical corroboration, not exact). | +| §4 (N6) | marked Kac–Rice for `E_Q^W N_B` with the endpoint weight as a field mark and the event indicator inside the conditional expectation | **ACCEPT as a formula at the note's stated level of rigour** (bounded continuous cylinder marks, extension of finite measures, truncation of `W_r`, monotone convergence; compact `B` off the pins, exhaustion), conditional on the note's consumed premise: the parent's regression law `Q` and its §9 Borel-mark convention (live node `math.uniform-matrix-cap-lifetime`, read with its four reading-rule components; a required edge in the proposal). Its fixed-remote instance is the reviewed formula (12)–(14) of the D4 proof, accepted in main#76 comment 5841783172 items 2–3. A later rigorous statement of the same identity with the kernel convention fixed is Lemma 5.1 of `frontiers/c6_palm_route_20260929/PROOF.md` (Math-#145); it is cited for the reader and is not consumed here. Nothing quantitative is asserted by (N6) and nothing quantitative is accepted here. | +| §4 (N7) | an `A r⁵` enclosure of the numerator with `Z_r ≥ z_* r²` gives `E_Q^W N_B ≤ (A/z_*) r³` | **ACCEPT**: division; the note correctly separates this from the cap probability argument (`E_Q[W_r 1_{G^c}]`). | +| §5 | support on the rare event only inside the deterministic cap cylinder; remote regions need their own support statement or the count `N 1_{G^c}` | **ACCEPT as a scope statement**; it consumes the cap source `0bf922b9…` for the containment `{M, S}` on `G_r` (the live node `math.d1-component.marked-cylinder-cap` carries exactly that SHA256; a required edge in the proposal) and adds nothing to it. | +| §6 | completion test: (N7) or a uniform conditional mean with the support relation; finite moments give (N2), tails give (N5) | **ACCEPT** as the logical summary of the above. | + +**Overall: ACCEPT at the logical-interface scope.** The note proves exactly what it claims and no more: correct +inequalities and counterexamples, and the form of the full-pin Kac–Rice expression that the fixed-remote theorem later +evaluated. It is not an expected-count rate and is not treated as one. Evidence labels: the (N1)–(N3) and sharp-`p` +checks in `INTERFACE` are exact rational arithmetic; the (N4) grid, the (N5) quadrature and the tail grid are floating-point +numerical corroborations of arguments proved above, not exact checks. + +## Relation to its consumers +- `math.rn-fixed-remote-window` (D4) reads the interface as the formulation it completes: PROOF §1 "identified a + three-determinant integral whose unnormalized numerator must be O(r^5), but did not bound it. This note supplies + that bound"; §7 "Equation (14) now supplies it for every fixed remote region and the entire BETWEEN-PIN HEIGHT WINDOW". + The D4 review reconstructed the three determinant factors, the `O(k r⁵)` numerator and the original normalizer + (items 2–3), i.e. the consumed content of (N6)–(N7). +- `math.rn-fixed-annulus-window` (D5 fixed annulus) uses the same normalized three-determinant intensity with the + original normalizer (its (A8) and §5 ledger), reviewed in `reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md`. + +## Not established here +No rate, no RN or 24-jet certificate, no `ENV-RESCOV` or historical carrier, no register transition. This file is a +review record; the proposal that consumes it is `RECONCILIATION.md` in this directory. diff --git a/reviews/register_alignment_20260930/SOURCE_FILES.json b/reviews/register_alignment_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..e49e54aee --- /dev/null +++ b/reviews/register_alignment_20260930/SOURCE_FILES.json @@ -0,0 +1,47 @@ +{ + "schema": 1, + "object": "REGISTER-ALIGNMENT-20260930-v1", + "scientific_effect": "NONE", + "disposition": "PROPOSAL_FOR_A_NON_CLAUDE_LANE", + "declarative": true, + "executed": false, + "base": "aa288385af9392c509233c25a0f4012c9ba023e5", + "files": [ + { + "path": "PROPOSED_TRANSITIONS.json", + "bytes": 40189, + "sha256": "dad87926726d8a86d5ff33d1d03eecafa9b05bf9e4c0eaeb08dd67104efc695d", + "git_blob": "1c1bea9ba98ca212159fa907e2bdeda1ff863d36" + }, + { + "path": "RECONCILIATION.md", + "bytes": 32725, + "sha256": "0baf1c4e3c48ca3e2bb4eaeb419ee8ba368f082447f38f638fd10ce023c5249f", + "git_blob": "ece9655fd80ea51c18d7b1b2410d4c073854db89" + }, + { + "path": "RN_COUNT_INTERFACE_REVIEW.md", + "bytes": 8115, + "sha256": "52d9038a3f27335d3c219babec785984e00b77f28d7cd7b390477c0bc6a749d1", + "git_blob": "7e7049e2b5e96e51394d44683d61c9d0e03b9323" + }, + { + "path": "RESULTS.json", + "bytes": 3452, + "sha256": "cbbff25994c22c55fd49c59bc2748af1c6ef128793a34880613cf7cb1d9162df", + "git_blob": "6702db536f00ec5689ab398c003f46a91b645d7b" + }, + { + "path": "alignment_check.py", + "bytes": 31556, + "sha256": "eb1888ea57416f87fbfa4692b6c07b20261c97bd08ce926aee7ec6fca5486502", + "git_blob": "cd110b0cfb2cfaf6f11e7405d99e0070c816a224" + }, + { + "path": "EXTERNAL_REVIEWS.md", + "bytes": 29877, + "sha256": "fa6677f16f688352f3fc9b1703d39324fe57a3b2bf61960591f524f858e99c9d", + "git_blob": "2be80e35bce1ffc90d1f7a0675ad5e8830ddbd28" + } + ] +} diff --git a/reviews/register_alignment_20260930/alignment_check.py b/reviews/register_alignment_20260930/alignment_check.py new file mode 100644 index 000000000..cd110b0cf --- /dev/null +++ b/reviews/register_alignment_20260930/alignment_check.py @@ -0,0 +1,585 @@ +"""Checks for REGISTER-ALIGNMENT-20260930-v1 (RECONCILIATION.md, PROPOSED_TRANSITIONS.json, RN_COUNT_INTERFACE_REVIEW.md). + +Standard library only. Run from the repository root: python -B -S reviews/register_alignment_20260930/alignment_check.py +Checks: IDENTITIES, VERDICTS, BASELINE, TRANSITIONS, GATE, PROPAGATION, INTERFACE, NEGATIVES, D4_DUPLICATE. No mathematics +is re-proved; the proposed graph is replayed through the downstream hard gate's own validators, and two-commit source +mutations through its reverse-impact rule. Nothing is written to the repository. +""" +import argparse +import copy +import hashlib +import importlib.util +import json +import math +import os +import pathlib +import shutil +import sys +import tempfile +from fractions import Fraction as F + +MUTANTS = ("allow-symlink", "no-hash", "stale-fingerprint", "drop-review-source", "controlling-true", "executed-flag", + "skip-interface-premise", "holder-reversed", "baseline-drift", "duplicate-drift", "region-unlinked", + "review-metadata-only") +MUT = None +HERE = "reviews/register_alignment_20260930" +GRAPH = "frontiers/downstream_gate_20260925/GRAPH.json" +HARD_GATE = "frontiers/downstream_gate_20260925/hard_gate.py" +D1 = "math.uniform-matrix-cap-lifetime" +D4_DUP = "math.d5-component.remote-window-proof" +D4_NODE = "math.rn-fixed-remote-window" +IFACE = "math.rn-count-interface" +D4 = "frontiers/remote_window_20260924/PROOF.md" +INVENTORY = { + "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md": [ + 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"reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md": [ + "29a0c6d00ce739baf1d29a3163977d8e707e2079e216fba4fee3669cebec127d", + "1f153966dae48d10574db0d272f75d08ca3a8b23" + ], + "reviews/p15_full_price_nonauthor_20260926/REVIEW.md": [ + "691ea0db5c0f073326c9f7208c9804c6f5d0daf92ba7467529c9d57e08b15eb7", + "07db19f826763b8faa7414ba2e403d9c45bc0b6e" + ], + "reviews/side24_v1_coefficient_claude_20260929/REVIEW.md": [ + "258c414c46be93142ee552cfa4c32c03dd89b6bf7b0b470334f152e1e0a57eef", + "665d1683f33a7180957923c3cb2abce8ade235d1" + ], + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md": [ + "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "dfed3b8d318a3ab1950957f393307733a4bef3f2" + ], + "frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md": [ + "079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d", + "89cae3a9734f2ec7172cd0b6b0b3af3ddd355d73" + ], + "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md": [ + "04cf4c986f29ac7ac33b3cc87feac23f3cf494e0c1b3bf4476e4e6666de46574", + "8988251631cab8e65eb2b01a08f7fc593b018570" + ], + "frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md": [ + "c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9", + "173881916ccd0e738bdb41279e835e78e520fcbc" + ], + "reviews/register_alignment_20260930/RN_COUNT_INTERFACE_REVIEW.md": [ + "52d9038a3f27335d3c219babec785984e00b77f28d7cd7b390477c0bc6a749d1", + "7e7049e2b5e96e51394d44683d61c9d0e03b9323" + ], + "reviews/register_alignment_20260930/EXTERNAL_REVIEWS.md": [ + "fa6677f16f688352f3fc9b1703d39324fe57a3b2bf61960591f524f858e99c9d", + "2be80e35bce1ffc90d1f7a0675ad5e8830ddbd28" + ] +} +VERDICTS = { + "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md": ["**Object:** LIFETIME-BOUNDED-REMAINDER-20260924-v1."], + "frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md": [ + "**Object:** RN-COUNT-INTERFACE-20260924-v1.", + "E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p).", + "N_r=ceil(log(1/r)) 1_Er.", + "= B q[1+log(A/q)]."], + "coefficients/side24_v1/PROOF.md": ["Object: SIDE24-COEFFICIENT-D23-20260924-v1."], + "frontiers/remote_window_20260924/PROOF.md": [ + "**Object:** RN-FIXED-REMOTE-WINDOW-20260924-v1.", + "Equation (14) now supplies it for every fixed remote region and the entire BETWEEN-PIN HEIGHT WINDOW."], + "frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md": ["**Object:** D5-FIXED-ANNULUS-STITCH-20260925-v1."], + "frontiers/full_price_20260924/PROOF.md": ["**Object:** P15-FULL-TRANSFORMED-PRICE-20260924-v1."], + "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md": [ + "| Blob | `081abc13c5e66342c13df2af2bd6b114f320486f` |", + "| SHA256 | `1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b` |", + "| Cutoff `r^{1/24}` | **ACCEPT** |", + "| Inner two-scale stitching | **ACCEPT** |", + "`TWO_SCALE_ADDENDUM.md` blob `89cae3a9734f2ec7172cd0b6b0b3af3ddd355d73`, 15902 bytes"], + "reviews/p15_full_price_nonauthor_20260926/REVIEW.md": [ + "| Blob | `582180e41dca0ad815ad0f18574df42040912149` |", + "| SHA256 | `87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9` |", + "| 1. Coordinatewise hazard transform, separate concavity, chord, `p=0,1` | **ACCEPT** |", + "SHA256 `c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9`. That is the digest named in the proof.", + "| 4. Full-block cover, same palette, realized family | **ACCEPT** |"], + "reviews/side24_v1_coefficient_claude_20260929/REVIEW.md": [ + "c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769", + "Overall: **ACCEPT at the arithmetic-enclosure scope**"], + "PROOF_INDEX.md": [ + "- D2 lifetime remainder: proof [frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md]", + "- D3 SIDE24 coefficient: proof [coefficients/side24_v1/PROOF.md]", + "- D4 fixed-remote RN theorem: proof [frontiers/remote_window_20260924/PROOF.md]", + "- D5 fixed-annulus height-window fallback: proof [frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md]", + "- D6 P15 full price theorem: proof [frontiers/full_price_20260924/PROOF.md]"], + "frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md": ["**Object:** D5-TWO-SCALE-20260925-v1."], + "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md": [ + "blob `89cae3a9734f2ec7172cd0b6b0b3af3ddd355d73`, 15902 bytes, SHA256 `079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d`", + "| S6 | **ACCEPT** |"], + "frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md": ["**Object:** P15-REALIZED-COVERS-20260924-v1."], + HERE + "/RN_COUNT_INTERFACE_REVIEW.md": [ + "**Overall: ACCEPT at the logical-interface scope.**", + "numerical corroborations of arguments proved above, not exact checks"], + HERE + "/EXTERNAL_REVIEWS.md": [ + "issuecomment-5841270276", "issuecomment-5841782206", "issuecomment-5841269490", "issuecomment-5841779222", + "issuecomment-5841783172", "issuecomment-5841861362", "issuecomment-5842112010", + "R5 and R6 are **ACCEPT**. Theorem R is accepted at its stated O(1) remainder scope.", + "All six coefficient interfaces check out. The disposition is **COEFFICIENT-CALC-REVIEWED / PARENT-IMPORTED-OPEN**.", + "This accepts only the fixed-ρ and fixed-η interfaces below. It does not accept a full RN or 24-jet theorem.", + "D6 ANALYTIC REVIEW COMPLETE", + "An issue comment is a mutable external object"], +} + + +def git_blob(data): + return hashlib.sha1(b"blob %d\0" % len(data) + data).hexdigest() + + +def no_symlink_on_path(root, path): + parts = pathlib.PurePosixPath(path).parts + return not any((root / pathlib.PurePosixPath(*parts[:i])).is_symlink() for i in range(1, len(parts) + 1)) + + +def identities_ok(root): + for path, (sha, blob) in INVENTORY.items(): + p = root / path + if not os.path.lexists(p): + return False + if MUT != "allow-symlink" and not no_symlink_on_path(root, path): + return False + if not p.is_file(): + return False + data = p.read_bytes() + if MUT != "no-hash" and (hashlib.sha256(data).hexdigest() != sha or git_blob(data) != blob): + return False + return True + + +def check_verdicts(root): + for path, needles in VERDICTS.items(): + text = (root / path).read_text(encoding="utf-8") + if not all(n in text for n in needles): + return False + return True + + +def load_spec(root): + spec = json.loads((root / HERE / "PROPOSED_TRANSITIONS.json").read_text(encoding="utf-8")) + if MUT == "executed-flag": + spec["executed"] = True + if MUT == "skip-interface-premise": + spec["transitions"] = [t for t in spec["transitions"] if t["node"] != IFACE] + if MUT == "drop-review-source": + for t in spec["transitions"]: + if t["node"] == D4_NODE: + t["proposed"]["review_sources"] = [] + if MUT == "controlling-true": + for t in spec["transitions"]: + if t["node"] == D4_NODE: + t["proposed"]["controlling"] = True + if MUT == "stale-fingerprint": + for t in spec["transitions"]: + if t["node"] == D4_NODE: + t["proposed"]["fingerprint"] = "0" * 64 + if MUT == "baseline-drift": + for t in spec["transitions"]: + if t["node"] == D4_NODE: + t["current"]["classification"] = "PROVED_REVIEWED" + return spec + + +def check_baseline(root, spec): + live = json.loads((root / GRAPH).read_text(encoding="utf-8"))["nodes"] + ok = True + for t in spec["transitions"]: + n = live.get(t["node"]) + if not isinstance(n, dict): + return False + ok &= n.get("classification") == t["current"]["classification"] + ok &= n.get("fingerprint") == t["current"]["fingerprint"] + ok &= n.get("kind") == t["kind"] and n.get("controlling") is False + if t["kind"] == "candidate": + ok &= n.get("source") == t["proposed"]["source"] + d1 = live.get(D1, {}) + ok &= d1.get("classification") == "PROVED_REVIEWED" and d1.get("fingerprint") == INVENTORY[ + "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md"][0] + rr = d1.get("reading_rule", []) + ok &= len(rr) == 4 and all(live.get(x, {}).get("classification") == "PROVED_REVIEWED" for x in rr) + return bool(ok) + + +def apply(spec, graph): + """The proposal applied to a copy of the live graph: the eight flips with their scope exclusions and replacement + notes, plus the review-record / consumed-source component nodes and the required edges that carry the evidence.""" + new = copy.deepcopy(graph) + for t in spec["transitions"]: + n = new["nodes"][t["node"]] + p = t["proposed"] + n["classification"] = p["classification"] + n["controlling"] = p["controlling"] + n["fingerprint"] = p["fingerprint"] + if p.get("source"): + n["source"] = p["source"] + n["scope"] = p["scope"] + n["explicit_limits"] = p["explicit_limits"] + n["notes"] = p["notes"] + n["review_disposition"] = p["review_disposition"] + n["review_sources"] = [r["ref"] for r in p["review_sources"]] + n["review_basis"] = [{"review": r["ref"], "provider": r.get("provider", "unstated"), "verdict": r.get("verdict", "")} + for r in p["review_sources"]] + if p.get("coverage_source"): + n["coverage_source"] = p["coverage_source"] + for c in spec["proposed_graph_nodes"]: + new["nodes"][c["id"]] = {k: v for k, v in c.items() if k != "id"} + new["edges"] = list(new["edges"]) + [dict(e) for e in spec["proposed_graph_edges"]] + return new + +def check_transitions(root, spec): + graph = json.loads((root / GRAPH).read_text(encoding="utf-8")) + ok = spec.get("declarative") is True and spec.get("executed") is False and spec.get("edges_unchanged") is False + ids = [t["node"] for t in spec["transitions"]] + ok &= len(ids) == len(set(ids)) and len(ids) >= 1 + for t in spec["transitions"]: + p = t["proposed"] + ok &= p["classification"] == "PROVED_REVIEWED" and p["controlling"] is False + ok &= bool(p.get("scope")) and bool(p.get("explicit_limits")) and bool(p.get("notes")) + ok &= len(p.get("review_sources", [])) >= 1 + if t["kind"] == "candidate": + ok &= p.get("source") in INVENTORY and p["fingerprint"] == INVENTORY[p["source"]][0] + else: + ok &= t["kind"] == "region" and p.get("coverage_source") in ids + ok &= not t["node"].startswith(("hist.", "eng.", "regional.")) + # component nodes: one per byte identity, fingerprinted to the inventory, PROVED_REVIEWED evidence records, + # none for bytes the live graph already carries with a fingerprint (Codex 4139312869 on Math-#160) + comps = spec["proposed_graph_nodes"] + cids = [c["id"] for c in comps] + live_fp_sources = {n.get("source") for n in graph["nodes"].values() + if isinstance(n, dict) and n.get("fingerprint") and n.get("source")} + ok &= len(cids) == len(set(cids)) and not (set(cids) & set(graph["nodes"])) and len(comps) == spec["nodes_added"] + ok &= len({c["source"] for c in comps}) == len(comps) + for c in comps: + ok &= c["kind"] == "reading_rule_component" and c["classification"] == "PROVED_REVIEWED" and c["controlling"] is False + ok &= c["source"] in INVENTORY and c["fingerprint"] == INVENTORY[c["source"]][0] + ok &= c["source"] not in live_fp_sources + ok &= bool(c.get("component_role")) and bool(c.get("author_provider")) and len(c.get("review_basis", [])) >= 1 + consumers = c.get("components_of") or [c.get("component_of")] + ok &= all(x in ids + cids for x in consumers) and len(consumers) >= 1 + edges_new = spec["proposed_graph_edges"] + known = set(graph["nodes"]) | set(cids) + live_keys = {(e["from"], e["to"], e["relation"]) for e in graph["edges"]} + seen = set() + for e in edges_new: + ok &= {"from", "to", "required", "relation"} <= set(e) and type(e["required"]) is bool + ok &= e["from"] in known and e["to"] in known and bool(e["relation"].strip()) + key = (e["from"], e["to"], e["relation"]) + ok &= key not in seen and key not in live_keys + seen.add(key) + ok &= e["from"] in ids or e["from"] in cids # every added edge leaves one of this record's objects + ok &= len(edges_new) == spec["edges_added"] + if not ok: + return False, None + new = apply(spec, graph) + nodes = new["nodes"] + edges = new["edges"] + ok &= edges[:len(graph["edges"])] == graph["edges"] and len(edges) == len(graph["edges"]) + len(edges_new) + ok &= all(graph["nodes"][k] == nodes[k] for k in graph["nodes"] if k not in ids) # only the eight live nodes change + for t in spec["transitions"]: + nid = t["node"] + req = [e["to"] for e in edges if e["from"] == nid and e["required"] is True] + ok &= all(nodes[d]["classification"] == "PROVED_REVIEWED" for d in req) + ok &= sorted(req) == sorted(t["required_premises_after"]) + if t["kind"] == "region": + ok &= t["proposed"]["coverage_source"] in req # a region depends on its covering node + ok &= nodes[t["proposed"]["coverage_source"]]["classification"] == "PROVED_REVIEWED" + n = nodes[nid] + ok &= n.get("explicit_limits") == t["proposed"]["explicit_limits"] and n.get("notes") == t["proposed"]["notes"] + ok &= "review open" not in n["notes"] and "candidate only" not in n["notes"] + ok &= isinstance(n.get("review_basis"), list) and len(n["review_basis"]) >= 1 + for c in comps: # component nodes' own required premises + req = [e["to"] for e in edges if e["from"] == c["id"] and e["required"] is True] + ok &= all(nodes[d]["classification"] == "PROVED_REVIEWED" for d in req) + return bool(ok), new + +def load_hard_gate(root): + hg_spec = importlib.util.spec_from_file_location("hard_gate", root / HARD_GATE) + hg = importlib.util.module_from_spec(hg_spec) + hg_spec.loader.exec_module(hg) + return hg + + +def check_gate(root, spec, new): + if new is None: + return False, {} + hg = load_hard_gate(root) + old = json.loads((root / GRAPH).read_text(encoding="utf-8")) + try: + hg.validate_graph_fail_closed(new) + rep_new = hg.closure_report(new) + rep_old = hg.closure_report(old) + ri = hg.reverse_impact_between(old, new) + except Exception: + return False, {} + ok = rep_new.get("gate_ok") is True and not rep_new.get("illegal_controlling") + ok &= rep_old.get("gate_ok") is True + ok &= not any(n.get("controlling") for n in new["nodes"].values()) + changed = sorted([t["node"] for t in spec["transitions"]] + [c["id"] for c in spec["proposed_graph_nodes"]]) + imp = ri.get("impacted") if isinstance(ri, dict) else None + names = [] + if isinstance(imp, list): + for x in imp: + names.append(x if isinstance(x, str) else str(x.get("node") or x.get("id") or x)) + # loss-only rule: every changed node is itself in the impacted set + ok &= isinstance(ri, dict) and set(changed) <= set(names) + return bool(ok), {"changed_nodes": changed, "impacted": sorted(set(names)), "reverse_impact_keys": sorted(ri.keys()) if isinstance(ri, dict) else []} + + +def check_interface(): + """Finite content of RN_COUNT_INTERFACE_REVIEW.md: exact rational checks, and floating-point corroborations + labelled as such (OpenAI 5360320845 item 2).""" + ok = True + num = True + le = (lambda a, b: a >= b) if MUT == "holder-reversed" else (lambda a, b: a <= b) + # (N1) on finite rational spaces, p = 2 and p = 3 (Cauchy-Schwarz / Holder in power form) + spaces = [([F(1, 4)] * 4, [0, 1, 3, 7], [1, 0, 1, 0]), ([F(1, 2), F(1, 3), F(1, 6)], [2, 5, 11], [1, 1, 0]), + ([F(1, 8), F(3, 8), F(1, 2)], [0, 0, 9], [0, 0, 1]), ([F(9, 10), F(1, 10)], [0, 13], [0, 1])] + for probs, N, E in spaces: + assert sum(probs) == 1 + ene = sum(p * n * e for p, n, e in zip(probs, N, E)) + pe = sum(p * e for p, e in zip(probs, E)) + m2 = sum(p * n * n for p, n in zip(probs, N)) + m3 = sum(p * n ** 3 for p, n in zip(probs, N)) + ok &= le(ene ** 2, m2 * pe) and le(ene ** 3, m3 * pe ** 2) + if pe > 0: # (N3) + ok &= ene == pe * (ene / pe) + # (N2) exponent: (3(p-1) - beta)/p == 3 - 3/p - beta/p + for p in (2, 3, 5): + for beta in (F(0), F(1, 2), F(3)): + ok &= F(3 * (p - 1), 1) / p - beta / p == 3 - F(3, p) - beta / p + # (N4), NUMERICAL (floating point): r = e^-t, t = 1..12: E N / r^3 = ceil(t) increases; ceil(t)^p e^-3t <= (t+1)^p e^-3t, + # and the grid maximum of (t+1)^p e^-3t stays below the located critical value. The proof is the log-derivative argument. + for p in (1, 2, 4, 7): + vals = [math.ceil(t) ** p * math.exp(-3 * t) for t in range(1, 13)] + bound = [(t + 1) ** p * math.exp(-3 * t) for t in range(1, 13)] + num &= all(v <= b + 1e-15 for v, b in zip(vals, bound)) + tmax = max(1.0, p / 3 - 1) # critical point of (t+1)^p e^{-3t} + sup = (tmax + 1) ** p * math.exp(-3 * tmax) + grid = [(t / 100 + 1) ** p * math.exp(-3 * (t / 100)) for t in range(100, 400000, 7)] + num &= max(grid) <= sup + 1e-9 and math.isfinite(sup) + ok &= all(math.ceil(t) > math.ceil(t - 1) for t in range(2, 13)) # E N_r / r^3 -> infinity along the grid (exact integers) + # sharp p: r = 2^{-p m}: E N^p = 1, E N = r^{3-3/p} exactly + for p in (2, 3, 4): + for m in (1, 2, 3): + r = F(1, 2 ** (p * m)) + n_r = 2 ** (3 * m) # = ceil(r^{-3/p}) exactly + ok &= n_r ** p * r ** 3 == 1 and le(F(1), F(2 ** p)) and n_r * r ** 3 == F(1, 2 ** (3 * m * (p - 1))) + ok &= n_r * r ** 3 == r ** 3 / r ** F(3, 1) * F(1, 2 ** (3 * m * (p - 1))) # r^{3-3/p} = 2^{-3m(p-1)} + # (N5), NUMERICAL (floating point): int_0^inf min(q, A e^{-t/B}) dt = B q (1 + log(A/q)), q <= A; Simpson quadrature at + # three parameter points corroborates the antiderivative identity proved in the review + for (A, B, q) in ((1.0, 1.0, 0.1), (20.0, 0.5, 0.001), (3.0, 2.0, 2.0)): + t0 = B * math.log(A / q) + n = 200000 + T = t0 + 60 * B + h = T / n + s = 0.0 + for i in range(n + 1): + t = i * h + w = 1 if i in (0, n) else (4 if i % 2 else 2) + s += w * min(q, A * math.exp(-t / B)) + quad = s * h / 3 + num &= abs(quad - B * q * (1 + math.log(A / q))) <= 1e-6 * B * q * (1 + math.log(A / q)) + # uniform tail of the (N4) example, NUMERICAL (floating point): P(N_r > t) = r^3 1{t < ceil(log 1/r)} <= e^3 e^{-3t} + for s_ in range(1, 15): + r3 = math.exp(-3 * s_) + for t10 in range(0, 10 * s_ + 30): + t = t10 / 10 + p_tail = r3 if t < math.ceil(s_) else 0.0 + num &= le(p_tail, math.exp(3) * math.exp(-3 * t) + 1e-300) + return {"exact": bool(ok), "numerical_corroboration": bool(num)} + + +def check_negatives(src): + rejected = [] + for fault in ("delete", "change", "symlink", "symlink-parent"): + with tempfile.TemporaryDirectory() as tmp: + root = pathlib.Path(tmp) + for path in INVENTORY: + dst = root / path + dst.parent.mkdir(parents=True, exist_ok=True) + shutil.copyfile(src / path, dst) + target = root / D4 + if fault == "delete": + target.unlink() + elif fault == "change": + target.write_bytes(target.read_bytes() + b"\n") + elif fault == "symlink": + moved = root / "elsewhere" / "PROOF.md" + moved.parent.mkdir() + shutil.move(str(target), str(moved)) + os.symlink(moved, target) + else: + moved = root / "elsewhere_dir" + shutil.move(str(target.parent), str(moved)) + os.symlink(moved, target.parent) + rejected.append(not identities_ok(root)) + return all(rejected) + + +def check_d4_duplicate(root, spec): + """Since ec6db8c (Math-#151) the live graph carries the D4 bytes twice; bind the duplicate to the D4 transition.""" + live = json.loads((root / GRAPH).read_text(encoding="utf-8"))["nodes"] + dup = live.get(D4_DUP) + old = live.get(D4_NODE) + t = next(t for t in spec["transitions"] if t["node"] == D4_NODE) + want = spec["live_duplicate"] + fp = t["proposed"]["fingerprint"] + if MUT == "duplicate-drift": + fp = "1" * 64 + if not isinstance(dup, dict) or not isinstance(old, dict): + return False + ok = dup.get("classification") == "PROVED_REVIEWED" and dup.get("controlling") is False + ok &= dup.get("fingerprint") == fp == want["fingerprint"] == INVENTORY[t["proposed"]["source"]][0] + ok &= dup.get("source") == t["proposed"]["source"] == want["source"] + ok &= dup.get("component_of") == want["component_of"] + ok &= any("main#76" in str(b.get("review")) and b.get("verdict") == "ACCEPT" for b in dup.get("review_basis", [])) + ok &= old.get("fingerprint") == dup.get("fingerprint") and old.get("classification") == t["current"]["classification"] + return bool(ok) + + +def snapshot(root, graph): + """Source snapshot in the shape of git_transition_audit.read_snapshot, taken from the working tree.""" + out = {} + for nid, node in graph["nodes"].items(): + ref = node.get("source") + if ref is None: + out[nid] = {"kind": "record_only"} + continue + if ref.startswith(("https://", "http://", "external:")): + out[nid] = {"kind": "external_unresolved", "reference": ref} + continue + p = root / ref.rstrip("/") + if p.is_file(): + data = p.read_bytes() + out[nid] = {"kind": "blob", "reference": ref, "bytes": len(data), "sha256": hashlib.sha256(data).hexdigest()} + elif p.is_dir(): + h = hashlib.sha256() + for f in sorted(x for x in p.rglob("*") if x.is_file() and "__pycache__" not in x.parts): + h.update(str(f.relative_to(p)).encode("utf-8") + b"\0" + hashlib.sha256(f.read_bytes()).digest()) + out[nid] = {"kind": "tree", "reference": ref, "sha256": h.hexdigest()} + else: + out[nid] = {"kind": "missing", "reference": ref} + return out + + +def check_propagation(root, spec, new): + """Two-commit mutations on the proposed graph through the production loss-only rule (reverse_impact_between with + source snapshots): evidence and consumed-source changes must reach the objects they support (OpenAI 5360320845, + 5360327058; Codex 4139807558, 4139807567). Nothing scientific is decided here.""" + if new is None: + return False, {} + hg = load_hard_gate(root) + g = copy.deepcopy(new) + if MUT == "region-unlinked": + g["edges"] = [e for e in g["edges"] if e.get("relation") != "covered_by"] + if MUT == "review-metadata-only": + drop = {c["id"] for c in spec["proposed_graph_nodes"] if str(c.get("component_role", "")).startswith("review_record")} + g["nodes"] = {k: v for k, v in g["nodes"].items() if k not in drop} + g["edges"] = [e for e in g["edges"] if e["from"] not in drop and e["to"] not in drop] + base = snapshot(root, g) + by_source = {} + for nid, n in g["nodes"].items(): + if n.get("source"): + by_source.setdefault(n["source"], []).append(nid) + + def run(edit): + g2, s2 = copy.deepcopy(g), copy.deepcopy(base) + edit(g2, s2) + try: + return set(hg.reverse_impact_between(g, g2, old_sources=base, new_sources=s2)["impacted"]) + except Exception: + return None + + def source_edit(path, kind="blob"): + def f(g2, s2): + for nid in by_source.get(path, []): + s2[nid] = ({"kind": "blob", "reference": path, "bytes": 1, "sha256": "1" * 64} if kind == "blob" + else {"kind": "missing", "reference": path}) + return f + + def fp_edit(g2, s2): + g2["nodes"][D4_NODE]["fingerprint"] = "2" * 64 + + eight = {t["node"] for t in spec["transitions"]} + cases = { + "d4-proof-edit-reaches-region-and-consumer": (source_edit(D4), {D4_NODE, "math.rn-region.fixed-remote", "math.rn-mesoscopic-reduction"}), + "annulus-proof-edit-reaches-region": (source_edit("frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md"), + {"math.rn-fixed-annulus-window", "math.rn-region.fixed-annulus-window"}), + "interface-review-deleted-reaches-interface-consumers-regions": (source_edit(HERE + "/RN_COUNT_INTERFACE_REVIEW.md", "missing"), + {IFACE, D4_NODE, "math.rn-fixed-annulus-window", "math.rn-region.fixed-remote", "math.rn-region.fixed-annulus-window"}), + "external-review-record-edit-reaches-d2-d3-d4-d6": (source_edit(HERE + "/EXTERNAL_REVIEWS.md"), + {"math.lifetime-remainder", "math.side24-coefficient", D4_NODE, "math.p15-full-price", "math.rn-region.fixed-remote"}), + "two-scale-addendum-edit-reaches-annulus-and-region": (source_edit("frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md"), + {"math.rn-fixed-annulus-window", "math.rn-region.fixed-annulus-window"}), + "realized-covers-edit-reaches-d6": (source_edit("frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md"), {"math.p15-full-price"}), + "d4-fingerprint-edit-reaches-region": (fp_edit, {D4_NODE, "math.rn-region.fixed-remote"}), + } + results = {} + for name, (edit, must) in cases.items(): + imp = run(edit) + results[name] = imp is not None and must <= imp + imp = run(source_edit("reviews/d5_collar_count_20260928/REVIEW.md")) + results["unrelated-edit-leaves-the-eight-untouched"] = imp is not None and not (imp & eight) + imp = run(lambda g2, s2: None) + results["no-op-impacts-nothing"] = imp == set() + return bool(all(results.values()) and len(results) == 9), results + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument("--mutant", choices=MUTANTS) + MUT = ap.parse_args().mutant + root = pathlib.Path(".").resolve() + spec = load_spec(root) + ident = identities_ok(root) + trans_ok, new = check_transitions(root, spec) + gate_ok, gate = check_gate(root, spec, new) + prop_ok, prop = check_propagation(root, spec, new) + iface = check_interface() + checks = {"IDENTITIES": ident, "VERDICTS": ident and check_verdicts(root), "BASELINE": check_baseline(root, spec), + "TRANSITIONS": trans_ok, "GATE": gate_ok, "PROPAGATION": prop_ok, + "INTERFACE": iface["exact"] and iface["numerical_corroboration"], "NEGATIVES": check_negatives(root), + "D4_DUPLICATE": check_d4_duplicate(root, spec)} + passed = all(checks.values()) and len(checks) == 9 + print(json.dumps({"object": "REGISTER-ALIGNMENT-20260930-v1", "checks": checks, "passed": passed, + "inventory_files": len(INVENTORY), "transitions": len(spec["transitions"]), + "component_nodes": len(spec["proposed_graph_nodes"]), "edges_added": len(spec["proposed_graph_edges"]), + "gate": gate, "propagation": prop, + "interface_evidence": {"exact_rational": ["(N1) Holder p=2,3 on finite rational spaces", "(N2) exponent identity", + "(N3) conditional-mean identity", "sharp-p example at r=2^-pm", + "(N4) integer monotonicity of ceil(t)"], + "numerical_corroboration_floating_point": ["(N4) grid maximum of (t+1)^p e^-3t", + "(N5) Simpson quadrature at three parameter points", + "(N4) example uniform tail on a grid"], + "results": iface}, + "scope": "identity, verdict-row, live-baseline, transition-shape, hard-gate replay, two-commit propagation, " + "interface arithmetic (exact and numerical parts labelled), filesystem-negative and live D4-duplicate " + "checks; no mathematics is re-proved; nothing is written"}, indent=2, sort_keys=True)) + return 0 if passed else 1 + + +if __name__ == "__main__": + sys.exit(main())