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/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/elder-partner-extremes.yml b/.github/workflows/elder-partner-extremes.yml new file mode 100644 index 000000000..77741c122 --- /dev/null +++ b/.github/workflows/elder-partner-extremes.yml @@ -0,0 +1,39 @@ +name: Elder-selected extreme partner checks + +'on': + pull_request: + paths: + - 'frontiers/elder_partner_extremes_20260930/**' + - '.github/workflows/elder-partner-extremes.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 commits + run: | + set -euo pipefail + git fetch --no-tags origin 80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337 + git fetch --no-tags origin 56a7904f9ad7d409c31e23195c0e7eeab230613a + git fetch --no-tags origin 79f18f0cb318fceccd5bdf4e6dad29815165bc7d + git fetch --no-tags origin 9c6a73489c5ed2465a3cf65dcc950eeeae074b05 + - name: Verify five source identities, packet and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + GIT_NO_REPLACE_OBJECTS: '1' + run: | + set -euo pipefail + python -B -S frontiers/elder_partner_extremes_20260930/verify.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/remainder-vanishing.yml b/.github/workflows/remainder-vanishing.yml new file mode 100644 index 000000000..e046d962d --- /dev/null +++ b/.github/workflows/remainder-vanishing.yml @@ -0,0 +1,73 @@ +name: Remainder vanishing exact controls + +'on': + pull_request: + paths: + - 'frontiers/remainder_vanishing_20260930/**' + - '.github/workflows/remainder-vanishing.yml' + # bound sources (consumed and cited_only in SOURCES.json) + - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md' + - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md' + - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md' + - 'frontiers/c7_total_bounded_20260929/PROOF.md' + - 'frontiers/unrestricted_selection_difference_20260929/PROOF.md' + - 'frontiers/c7_zero_gap_limit_20260929/PROOF.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-controls: + 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('frontiers/remainder_vanishing_20260930').resolve() + manifest = json.loads((root / 'SOURCES.json').read_text()) + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest['files']] + ['SOURCES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest['files']: + path = root / entry['path'] + if path.is_symlink(): + raise ValueError('symlink: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('packet file identity mismatch: ' + entry['path']) + for entry in manifest['consumed'] + manifest['cited_only']: + 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['git_blob']: + raise ValueError('merged source drifted: ' + entry['path']) + script = str(root / 'even_contact_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=600).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=600).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=600).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/CITING.md b/CITING.md new file mode 100644 index 000000000..d597eeca3 --- /dev/null +++ b/CITING.md @@ -0,0 +1,79 @@ +# Citing this research + +Cite the specific proof, calculation, or review you used. This repository brings +together research notes, programs, imported originals and review records with +different recorded contributors. Its hosting account is not a substitute for +the attribution in each artifact. + +## Cite an artifact + +Include its exact title or object ID, path, full commit SHA and a permanent link. +Retain the author or model lane actually recorded in the source; keep reviewer, +recovery and publication-preparer credits separate. If the source leaves an +author unknown, preserve that uncertainty. + +For example, the following identifies a particular source without assigning +authorship to the whole collection: + +> OpenAI / ChatGPT (recorded source author). “SIDE24 coefficient in dimensions +> two and three.” Object `SIDE24-COEFFICIENT-D23-20260924-v1`. +> `coefficients/side24_v1/PROOF.md`, Math-, commit +> `de54d1da2f6cdde59df3c34bb50ecd85c25ca333`. +> [Permanent source link](https://github.com/d6g8k5htny-coder/Math-/blob/de54d1da2f6cdde59df3c34bb50ecd85c25ca333/coefficients/side24_v1/PROOF.md). + +Use the commit you actually consulted. When discussing a reviewed result, cite +the applicable review as well as the source and preserve its hypotheses and +scope. An original file can retain historical review language; later reviews +are separate records. [Proof availability](PROOF_INDEX.md) helps locate them. + +For an imported or recovered artifact, also include its original carrier +identity from the linked manifest or source binding: source repository and +commit/path, or source ID, together with the recorded hash. For example, the +[lifetime-parent manifest](imports/lifetime_parent_20260925/MANIFEST.json) +identifies the original Drive files separately from their repository mirrors. + +## Cite a repository snapshot + +For a reference to the collection rather than an individual result: + +> *Math-: Mathematics — proofs, calculations, and open reviews.* Repository +> snapshot, commit `de54d1da2f6cdde59df3c34bb50ecd85c25ca333`. +> [Permanent snapshot link](https://github.com/d6g8k5htny-coder/Math-/tree/de54d1da2f6cdde59df3c34bb50ecd85c25ca333). + +This snapshot reference supplies no collection-wide author list. A citation, +source hash or repository merge does not broaden a mathematical claim or its +recorded review scope. + +## Provenance and reuse metadata + +Source records distinguish several kinds of material: + +- Research notes and calculations record their own author/model lanes and + dependencies. External mathematical literature cited within them remains + separately attributed; a literature citation alone does not identify a copied + source file. +- Imported originals retain their carrier identities: see the + [hardening-copy ledger](imports/hardening_ebedb780/README.md), + [contact-exposition binding](imports/transverse_contact_library_20260927/SOURCE_BINDING.json) + and [Stage E recovery](imports/upper2d_stage_e_20260926/README.md). +- Some original attribution remains unresolved. The pinned + [GP-DATA-114 source record](https://github.com/d6g8k5htny-coder/Math-/blob/de54d1da2f6cdde59df3c34bb50ecd85c25ca333/imports/gp_data_114_embedded_20260930/SOURCES.json#L6-L8) + records the original author as unknown and names the recovery author + separately. The pinned + [H5 review](https://github.com/d6g8k5htny-coder/Math-/blob/de54d1da2f6cdde59df3c34bb50ecd85c25ca333/imports/upper2d_h5_ledgers_20260926/REVIEW.md#L3-L7) + records unknown original model/provider provenance. + +No project-wide license file is present in the snapshot cited above. The source +records do not provide a complete collection-wide account of creators and reuse +terms. This guide adds citation information and declares no new license terms. + +The [README calculation commands](README.md#run-a-calculation) use the Python +standard library. Historical imported numerical scripts have a different +dependency set: [Stage E](imports/upper2d_stage_e_20260926/README.md#numerical-sources-remain-unverified) +records NumPy, SciPy, mpmath and SymPy without locked versions; the +[GP-DATA-114 recovery note](imports/gp_data_114_embedded_20260930/README.md#historical-scope-and-limits) +also identifies its original program's mpmath dependency. The separate +[Lean package](formal/README.md) preserves originals and documented compiler +repairs, with [Lean](formal/lean-toolchain), [mathlib](formal/lakefile.toml) and +[transitive dependencies](formal/lake-manifest.json) pinned. Those dependency +identities do not supply license terms for this collection's research material. diff --git a/README.md b/README.md index 87a970aa4..75a00d256 100644 --- a/README.md +++ b/README.md @@ -2,7 +2,7 @@ # Mathematics — proofs, calculations, and open reviews -[Research home](https://github.com/d6g8k5htny-coder/main) · [Topic guide](https://github.com/d6g8k5htny-coder/main/blob/main/docs/RESEARCH_INDEX.md) · [Run the checks](https://github.com/d6g8k5htny-coder/main/blob/main/docs/REPRODUCE.md) · [Claim manifest](claims/LANDING_CLAIMS.json) · [Proof availability](PROOF_INDEX.md) · [Work queue](https://github.com/d6g8k5htny-coder/main/issues/86) *(closed downstream record; not a live queue)* +[Research home](https://github.com/d6g8k5htny-coder/main) · [Topic guide](https://github.com/d6g8k5htny-coder/main/blob/main/docs/RESEARCH_INDEX.md) · [Run the checks](https://github.com/d6g8k5htny-coder/main/blob/main/docs/REPRODUCE.md) · [How to cite](CITING.md) · [Claim manifest](claims/LANDING_CLAIMS.json) · [Proof availability](PROOF_INDEX.md) · [Work queue](https://github.com/d6g8k5htny-coder/main/issues/86) *(closed downstream record; not a live queue)* ## Read a result 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": { + "E_F_j_given_level": [ + 0.022087746264714315, + 1.1013906298184966, + 1.079302883553834 + ], + "lambda_inf": 0.08483094979588068, + "lambda_inf_j": [ + 0.0008506175936846622, + 0.042415474897939356, + 0.04156485730425666 + ], + "p_inf": 0.038510836890748947 + } + }, + "grid_convergence": { + "k=1.0,b=0.0": { + "hole_h=0.1,Rc=6": -1.6313351977268296, + "hole_h=0.15,Rc=6": -1.6311476645638006, + "hole_h=0.3,Rc=5": -1.6297553083021523, + "hole_h=0.3,Rc=6": -1.6296607043116234, + "hole_h=0.3,Rc=6,high_order": -1.629660659845337, + "hole_j_h=0.1,Rc=6": [ + -0.31378570687015506, + -1.332596279425655, + 0.015046788568980531 + ], + "lambda_inf_high_order": 0.11609348619092598, + "torus_L=12_approx": 15.086314346932143, + "torus_L=12_direct_h=0.2": 15.086621885475793 + } + }, + "hole": { + "k=0.5,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.084135453753016, + "hole_j": [ + -0.3028235079276278, + -0.8844195042052593, + 0.10310755837987123 + ], + "hole_unweighted": 6.6558328680387655, + "hole_unweighted_j": [ + 1.6907231338002304, + 3.2743866004383944, + 1.6907231338001405 + ], + "lambda_max_on_grid": 0.1867399995903017, + "lambda_min_on_grid": 1.3675393485847551e-126 + }, + "k=0.5,b=1.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -0.7506293381103409, + "hole_j": [ + -0.008308169817322056, + -0.2607974077845683, + -0.4815237605084506 + ], + "hole_unweighted": 6.30991417071404, + "hole_unweighted_j": [ + 0.5638736743837274, + 3.4979879412468344, + 2.2480525550834787 + ], + "lambda_max_on_grid": 0.14996841549563256, + "lambda_min_on_grid": 1.6639091561363451e-127 + }, + "k=1.0,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.6311476645638006, + "hole_j": [ + -0.3137857115939324, + -1.3324986709058282, + 0.015136717935960147 + ], + "hole_unweighted": 1.7266312061970344, + "hole_unweighted_j": [ + 0.5963757449363564, + 0.5338797163242601, + 0.5963757449364179 + ], + "lambda_max_on_grid": 0.17946929470083037, + "lambda_min_on_grid": 0.0 + }, + "k=1.0,b=1.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.184482355875255, + "hole_j": [ + -0.005586604498269274, + -0.46494835381060384, + -0.713947397566382 + ], + "hole_unweighted": 1.850309671857266, + "hole_unweighted_j": [ + 0.23644269280226493, + 1.2280733401923594, + 0.38579363886264156 + ], + "lambda_max_on_grid": 0.13172552616590733, + "lambda_min_on_grid": 0.0 + }, + "k=2.0,b=0.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -2.1382024573116403, + "hole_j": [ + -0.3302820124213753, + -1.7568436474476312, + -0.051076797442633874 + ], + "hole_unweighted": 0.7685792561786426, + "hole_unweighted_j": [ + 0.47398232526493733, + -0.17938539435123557, + 0.4739823252649408 + ], + "lambda_max_on_grid": 0.19321540570437315, + "lambda_min_on_grid": 0.0 + }, + "k=2.0,b=1.0": { + "grid": { + "Rc": 6.0, + "h": 0.15, + "points": 3200, + "symmetry": "z -> -z" + }, + "hole": -1.5681389104271717, + "hole_j": [ + -0.002687318209144709, + -0.6365063443252986, + -0.9289452478927284 + ], + "hole_unweighted": 1.0599352141869065, + "hole_unweighted_j": [ + 0.21808343946028652, + 0.809045285026987, + 0.03280648969963292 + ], + "lambda_max_on_grid": 0.13923606549072168, + "lambda_min_on_grid": 0.0 + } + }, + "near_zero_profile": { + "k=0.5,b=0.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 1.1004555386134373e-07, + "lambda_j": [ + 1.833295387377603e-61, + 1.1004555386134373e-07, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 0.002134947560851226, + "lambda_j": [ + 4.867335485502547e-14, + 0.002133814722278417, + 1.1328385241355547e-06 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 0.00010351960501910217, + "lambda_j": [ + 1.1538890830569103e-14, + 2.4733228443575758e-05, + 7.878637656398753e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 0.0017268476166370293, + "lambda_j": [ + 5.679609634872527e-17, + 9.722830217156197e-07, + 0.001725875333615257 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 1.5109479250080098e-07, + "lambda_j": [ + 0.0, + 6.726828149800317e-23, + 1.510947925008009e-07 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 2.4465748675618857e-07, + "lambda_j": [ + 1.4121881852316648e-19, + 2.446574867560473e-07, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.006639808397580947, + "lambda_j": [ + 2.2246810058068508e-10, + 0.006636884153214578, + 2.924021898268832e-06 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0010080860681121154, + "lambda_j": [ + 7.906936453561304e-12, + 0.0005482721487587344, + 0.0004598139114464446 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.006978048185316661, + "lambda_j": [ + 3.0882611889734727e-14, + 3.225728075402479e-06, + 0.006974822457210376 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 2.3744972247452498e-07, + "lambda_j": [ + 0.0, + 1.5804457804777895e-14, + 2.3744970667006716e-07 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 7.579127842780078e-07, + "lambda_j": [ + 2.863931642068619e-12, + 7.579099203463658e-07, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.013618202308026461, + "lambda_j": [ + 3.692413859844749e-08, + 0.013611984538793328, + 6.180845094534226e-06 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.003402163614855698, + "lambda_j": [ + 1.8765794368147613e-09, + 0.0017711424504387784, + 0.0016310192878374833 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.013104695846141198, + "lambda_j": [ + 6.1196734527672555e-12, + 9.069469804746791e-06, + 0.013095626370216778 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 7.304482397677882e-07, + "lambda_j": [ + 0.0, + 7.078206519941104e-12, + 7.304411615612683e-07 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 7.777866697639638e-06, + "lambda_j": [ + 2.6582147444836883e-10, + 7.77760087616519e-06, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.02767651261013489, + "lambda_j": [ + 1.076438066700652e-06, + 0.02765732481848523, + 1.811135358295865e-05 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.007895409915249209, + "lambda_j": [ + 1.241778010901584e-07, + 0.004304011835059402, + 0.003591273902388717 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.02520173106836129, + "lambda_j": [ + 5.002668219335075e-10, + 4.4380670502617274e-05, + 0.02515734989759185 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 5.524247798601962e-06, + "lambda_j": [ + 0.0, + 9.842349660245204e-10, + 5.523263563635937e-06 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ], + "k=0.5,b=1.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 9.592868095199296e-08, + "lambda_j": [ + 1.902790604784803e-61, + 9.592868095199296e-08, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 0.001546171495926685, + "lambda_j": [ + 1.2987178738954032e-14, + 0.0015450190053427184, + 1.152490570979291e-06 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 9.069941096146156e-05, + "lambda_j": [ + 2.7199580717855453e-15, + 2.3892309257505168e-05, + 6.680710170123643e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 0.0011698486784758868, + "lambda_j": [ + 1.2635802355193506e-17, + 1.1961870908104367e-06, + 0.0011686524913850638 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 1.2182098055323845e-07, + "lambda_j": [ + 0.0, + 1.3040108002122639e-23, + 1.2182098055323845e-07 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 2.0136088041371947e-07, + "lambda_j": [ + 3.3272371468882465e-20, + 2.013608804136862e-07, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.00499415631089021, + "lambda_j": [ + 7.768930026103807e-11, + 0.00499117607627394, + 2.9801569269689147e-06 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0007736062406829406, + "lambda_j": [ + 1.8352209412149486e-12, + 0.00037778386987030065, + 0.000395822368977419 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.005281670828100035, + "lambda_j": [ + 6.337974805859862e-15, + 2.1476300128099494e-06, + 0.005279523198080887 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.859341786723266e-07, + "lambda_j": [ + 0.0, + 3.382737648201086e-15, + 1.8593417528958895e-07 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 6.434443586425052e-07, + "lambda_j": [ + 6.645956674907172e-13, + 6.434436940468377e-07, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.010715824940335128, + "lambda_j": [ + 9.611449894604162e-09, + 0.010708725460507188, + 7.089868378044434e-06 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0027847140995036646, + "lambda_j": [ + 4.141033583565802e-10, + 0.0013181662750302177, + 0.0014665474103700887 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.011190796912326639, + "lambda_j": [ + 1.0297699263006554e-12, + 5.1106644166797555e-06, + 0.01118568624688019 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 5.896808318664907e-07, + "lambda_j": [ + 0.0, + 1.738857818429096e-12, + 5.896790930086724e-07 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 6.6306081413754765e-06, + "lambda_j": [ + 6.211496682960097e-11, + 6.630546026408647e-06, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.020925565258625364, + "lambda_j": [ + 2.3353082589548588e-07, + 0.020897842654748467, + 2.7489073051000613e-05 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.006391708896535126, + "lambda_j": [ + 2.4405130107007963e-08, + 0.002875800496007248, + 0.003515883995397771 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.022424150459999827, + "lambda_j": [ + 6.296803158677452e-11, + 1.7313114522437837e-05, + 0.02240683728250936 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 4.907167592555362e-06, + "lambda_j": [ + 0.0, + 2.422447273352642e-10, + 4.9069253478280265e-06 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ], + "k=1.0,b=0.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 2.1080411912855745e-28, + "lambda_j": [ + 8.62163272136347e-291, + 2.1080411912855745e-28, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 0.0010035014647206176, + "lambda_j": [ + 3.590057018511172e-18, + 0.001003501462854009, + 1.8666049393700075e-12 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 0.00010351960501910048, + "lambda_j": [ + 1.1538890830569043e-14, + 2.473322844357601e-05, + 7.878637656398557e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 0.000736447556744653, + "lambda_j": [ + 1.2752941899337078e-22, + 7.971320953309199e-13, + 0.0007364475559475209 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 1.5175042704749932e-28, + "lambda_j": [ + 0.0, + 6.777344985050669e-44, + 1.5175042704749925e-28 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 6.4087039930494485e-28, + "lambda_j": [ + 7.482288846062231e-58, + 6.4087039930494485e-28, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.0025704808848457618, + "lambda_j": [ + 7.561733334564415e-12, + 0.0025704808730701374, + 4.213891115625729e-12 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0010080860681121113, + "lambda_j": [ + 7.906936453561307e-12, + 0.0005482721487587343, + 0.0004598139114464405 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.0027477128235462544, + "lambda_j": [ + 6.06766812808775e-20, + 7.94301107084324e-12, + 0.002747712815603243 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 2.61285628177819e-28, + "lambda_j": [ + 0.0, + 3.0380051876710584e-35, + 2.6128559779776714e-28 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 9.221098019682842e-27, + "lambda_j": [ + 2.7896581688335026e-33, + 9.221095230024673e-27, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.005194194425041268, + "lambda_j": [ + 2.3016580784159037e-08, + 0.005194171396702473, + 1.1758010765446167e-11 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0034021636148557142, + "lambda_j": [ + 1.8765794368147576e-09, + 0.001771142450438789, + 0.0016310192878374885 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.0050846836857588555, + "lambda_j": [ + 1.2014261385646596e-17, + 5.055149683100958e-08, + 0.005084633134262013 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 6.274845818799208e-27, + "lambda_j": [ + 0.0, + 8.021289091086913e-32, + 6.2747656059082974e-27 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 2.151856922084616e-23, + "lambda_j": [ + 5.722460504428276e-28, + 2.1517996974795717e-23, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.011304121250842698, + "lambda_j": [ + 5.157429191129455e-07, + 0.011303605401517397, + 1.0640618803830285e-10 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.007895409915249217, + "lambda_j": [ + 1.2417780109015846e-07, + 0.004304011835059409, + 0.0035912739023887183 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.010569112630676334, + "lambda_j": [ + 2.6352498253770692e-15, + 1.2610809982661303e-06, + 0.010567851549675433 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.120999565136782e-23, + "lambda_j": [ + 0.0, + 2.800186239519058e-27, + 1.12071954651283e-23 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ], + "k=1.0,b=1.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 1.522276492040333e-28, + "lambda_j": [ + 2.4712162041177822e-288, + 1.522276492040333e-28, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 0.0007101589544587838, + "lambda_j": [ + 9.509579649729554e-19, + 0.0007101589525686068, + 1.8901761042919507e-12 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 9.069941096145963e-05, + "lambda_j": [ + 2.7199580717857666e-15, + 2.389230925750484e-05, + 6.680710170123482e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 0.00042502429251980116, + "lambda_j": [ + 2.6284301305351403e-23, + 7.563605007838872e-13, + 0.00042502429176344064 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 7.005633754908908e-29, + "lambda_j": [ + 0.0, + 1.1701497308425346e-44, + 7.005633754908907e-29 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 4.945854252929738e-28, + "lambda_j": [ + 1.639127578599705e-58, + 4.945854252929738e-28, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.0020530144734832684, + "lambda_j": [ + 2.184575578510603e-12, + 0.0020530144666005075, + 4.698185212868141e-12 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.000773606240682945, + "lambda_j": [ + 1.835220941214936e-12, + 0.00037778386987030396, + 0.0003958223689774201 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.002205886036003058, + "lambda_j": [ + 1.0772978307399943e-20, + 4.048915475195213e-12, + 0.0022058860319541427 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 6.800064836993285e-28, + "lambda_j": [ + 0.0, + 6.763519745506458e-36, + 6.8000647693580875e-28 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 7.640884080017525e-27, + "lambda_j": [ + 6.59203249989553e-34, + 7.640883420814275e-27, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.0041902831765628125, + "lambda_j": [ + 5.424923772567412e-09, + 0.004190277734408652, + 1.723038799339847e-11 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0027847140995036664, + "lambda_j": [ + 4.1410335835665824e-10, + 0.0013181662750302258, + 0.0014665474103700824 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.004267986009122482, + "lambda_j": [ + 1.6015806691238693e-18, + 1.2212726670270489e-08, + 0.00426797379639581 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 6.490050835834233e-27, + "lambda_j": [ + 0.0, + 1.9094750805721791e-32, + 6.490031741083427e-27 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 1.9597753503142905e-23, + "lambda_j": [ + 1.3925142259819909e-28, + 1.9597614251720306e-23, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 0.008913928961954934, + "lambda_j": [ + 1.1802996398141025e-07, + 0.008913810681639087, + 2.5035186640758745e-10 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.006391708896535129, + "lambda_j": [ + 2.4405130107007986e-08, + 0.00287580049600725, + 0.0035158839953977725 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 0.009022106151949533, + "lambda_j": [ + 2.1825854545674902e-16, + 3.1590341643004654e-07, + 0.009021790248532885 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 9.800210691339238e-24, + "lambda_j": [ + 0.0, + 6.78103951267826e-28, + 9.79953258738797e-24 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ], + "k=2.0,b=0.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 1.700854640463175e-112, + "lambda_j": [ + 0.0, + 1.700854640463175e-112, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 2.8027005376563332e-06, + "lambda_j": [ + 1.6990267356486966e-37, + 2.8027005376563332e-06, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 0.00010351960501910388, + "lambda_j": [ + 1.153889083056908e-14, + 2.4733228443577326e-05, + 7.878637656398766e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 1.579044469332315e-06, + "lambda_j": [ + 0.0, + 4.427885001390319e-27, + 1.579044469332315e-06 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 1.1276270966964014e-112, + "lambda_j": [ + 0.0, + 7.218081029813728e-128, + 1.1276270966964008e-112 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 4.52572136447521e-111, + "lambda_j": [ + 2.8697230987881016e-211, + 4.52572136447521e-111, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 6.723284402103214e-06, + "lambda_j": [ + 5.107347803156441e-20, + 6.723284402103163e-06, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0010080860681121223, + "lambda_j": [ + 7.90693645356135e-12, + 0.0005482721487587413, + 0.00045981391144644463 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 8.22618419614483e-06, + "lambda_j": [ + 0.0, + 7.806148711054245e-16, + 8.226184195364214e-06 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.913126768040405e-111, + "lambda_j": [ + 0.0, + 1.7909774515050856e-118, + 1.9131265889426596e-111 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 1.9566775671874284e-107, + "lambda_j": [ + 1.1470532304708705e-118, + 1.956677567175958e-107, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 1.4506344718775665e-05, + "lambda_j": [ + 3.527218989548529e-11, + 1.450630944658577e-05, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.003402163614855708, + "lambda_j": [ + 1.876579436814757e-09, + 0.0017711424504387857, + 0.0016310192878374857 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 1.4503191855802134e-05, + "lambda_j": [ + 0.0, + 9.881223901873418e-11, + 1.4503093043563116e-05 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.336465837825128e-107, + "lambda_j": [ + 0.0, + 2.0918852786396756e-112, + 1.3364449189723417e-107 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 1.042388422743631e-94, + "lambda_j": [ + 9.585499469076187e-100, + 1.0423788372441619e-94, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 4.1855158115202624e-05, + "lambda_j": [ + 2.1398462910532028e-09, + 4.185301826891157e-05, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.007895409915249212, + "lambda_j": [ + 1.2417780109015835e-07, + 0.004304011835059408, + 0.003591273902388713 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 3.892791806960772e-05, + "lambda_j": [ + 0.0, + 3.4673029567213974e-09, + 3.8924450766651e-05 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 2.7316159980880942e-95, + "lambda_j": [ + 0.0, + 1.9533440430832328e-98, + 2.729662654045011e-95 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ], + "k=2.0,b=1.0": [ + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 30, + "exact_regression": true, + "lambda": 1.1257133974630364e-112, + "lambda_j": [ + 0.0, + 1.1257133974630364e-112, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 60, + "exact_regression": true, + "lambda": 1.9124947687286503e-06, + "lambda_j": [ + 5.729491237223246e-38, + 1.9124947687286503e-06, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 90, + "exact_regression": true, + "lambda": 9.069941096146152e-05, + "lambda_j": [ + 2.719958071785549e-15, + 2.389230925750528e-05, + 6.680710170123629e-05 + ], + "r": 0.05 + }, + { + "angle_deg": 120, + "exact_regression": true, + "lambda": 1.8664092965120321e-06, + "lambda_j": [ + 0.0, + 1.735691900669088e-27, + 1.8664092965120321e-06 + ], + "r": 0.05 + }, + { + "angle_deg": 150, + "exact_regression": true, + "lambda": 4.243921997212985e-112, + "lambda_j": [ + 0.0, + 1.253172961470713e-128, + 4.243921997212985e-112 + ], + "r": 0.05 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.05 + }, + { + "angle_deg": 0, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 3.47824398943578e-111, + "lambda_j": [ + 5.620966462066486e-212, + 3.47824398943578e-111, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 5.4717682417127276e-06, + "lambda_j": [ + 1.469748599906681e-20, + 5.471768241712713e-06, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.0007736062406829362, + "lambda_j": [ + 1.8352209412178477e-12, + 0.0003777838698703016, + 0.00039582236897741367 + ], + "r": 0.1 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 5.426687116957742e-06, + "lambda_j": [ + 0.0, + 1.877434092818391e-16, + 5.426687116769999e-06 + ], + "r": 0.1 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.1220594298861341e-111, + "lambda_j": [ + 0.0, + 3.180979303849326e-119, + 1.122059398076341e-111 + ], + "r": 0.1 + }, + { + "angle_deg": 180, + "exact_regression": true, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.1 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 1.7882663314679737e-107, + "lambda_j": [ + 3.2178423176640523e-119, + 1.788266331464756e-107, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 1.1809241761717542e-05, + "lambda_j": [ + 8.269725333334063e-12, + 1.1809233491992208e-05, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.002784714099503671, + "lambda_j": [ + 4.141033583565052e-10, + 0.001318166275030222, + 0.0014665474103700907 + ], + "r": 0.2 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 1.196165491191902e-05, + "lambda_j": [ + 0.0, + 2.370088392382352e-11, + 1.1961631211035097e-05 + ], + "r": 0.2 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 1.1843657859619608e-107, + "lambda_j": [ + 0.0, + 5.663391018186695e-113, + 1.1843601225709425e-107 + ], + "r": 0.2 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.2 + }, + { + "angle_deg": 0, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 30, + "exact_regression": false, + "lambda": 1.0595989301411022e-94, + "lambda_j": [ + 2.4965010725235053e-100, + 1.0595964336400296e-94, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 60, + "exact_regression": false, + "lambda": 3.37878391920449e-05, + "lambda_j": [ + 5.060528857936293e-10, + 3.378733313915911e-05, + 0.0 + ], + "r": 0.4 + }, + { + "angle_deg": 90, + "exact_regression": false, + "lambda": 0.006391708896535132, + "lambda_j": [ + 2.440513010700793e-08, + 0.0028758004960072513, + 0.0035158839953977742 + ], + "r": 0.4 + }, + { + "angle_deg": 120, + "exact_regression": false, + "lambda": 3.245386590834958e-05, + "lambda_j": [ + 0.0, + 8.585213531995287e-10, + 3.2453007386996385e-05 + ], + "r": 0.4 + }, + { + "angle_deg": 150, + "exact_regression": false, + "lambda": 2.2959157703415916e-95, + "lambda_j": [ + 0.0, + 4.725628392771363e-99, + 2.2954432075023144e-95 + ], + "r": 0.4 + }, + { + "angle_deg": 180, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "r": 0.4 + } + ] + }, + "object": "CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1", + "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": { + "a_gauss_legendre_per_side": 24, + "coordinates": "a = tr H / 2, (c, h2) polar; splits at a = 0 and rho = |a|", + "cut_sd": 9.0, + "high_order_check": [ + 32, + 48, + 32 + ], + "max_rel_dev_vs_high_order_at_samples": 0.40391717052519804, + "rho_gauss_legendre_per_side": 24, + "theta_trapezoid": 32 + }, + "ray_profile": { + "k=0.5,b=0.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.039490988260693596, + "ratio_unweighted": 63.16746850830896 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.08718208578297156, + "ratio_unweighted": 9.494284065739743 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.1410383214852956, + "ratio_unweighted": 3.449574209574775 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.2081573587558727, + "ratio_unweighted": 1.8922929036340805 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.4032502250014552, + "ratio_unweighted": 1.0829668264222414 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.6796734764925719, + "ratio_unweighted": 0.9536229801411701 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.9193080420260853, + "ratio_unweighted": 0.9692831834619947 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 1.0014637279244158, + "ratio_unweighted": 0.9923682660003456 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.0003721623791384, + "ratio_unweighted": 0.9999486896780728 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.1470057888552203, + "ratio_unweighted": 127.31730271673432 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.3012719494924743, + "ratio_unweighted": 17.139492517290375 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.4672733631091738, + "ratio_unweighted": 5.7542900865743345 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.6476258883836312, + "ratio_unweighted": 2.8972305270680754 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 1.0092769462431534, + "ratio_unweighted": 1.4122132422261242 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 1.1745551028518046, + "ratio_unweighted": 1.0870739712923185 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 1.0931498226541099, + "ratio_unweighted": 1.0059090162305244 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 1.0167466632271587, + "ratio_unweighted": 0.9967020894579159 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 0.999975572733961, + "ratio_unweighted": 0.9999572439947901 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 0.09289674169983599, + "ratio_unweighted": 36.311243176653946 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 0.22186217052024215, + "ratio_unweighted": 5.762957959340962 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 0.43027714112640947, + "ratio_unweighted": 2.4432606595267896 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 0.7477629623898842, + "ratio_unweighted": 1.5918234833864526 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 1.4677489663502015, + "ratio_unweighted": 1.1713421370872288 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 1.5383967630808733, + "ratio_unweighted": 1.057380067510981 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 1.1876291458332473, + "ratio_unweighted": 1.0057473661670722 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.0249948191725586, + "ratio_unweighted": 1.0018271300179102 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 1.0000317438928017, + "ratio_unweighted": 1.0003687231288774 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 1.1323716797741778e-05, + "ratio_unweighted": 0.0011315493169059292 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 0.00023292226647557746, + "ratio_unweighted": 0.0015140500125667095 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 0.004767670553293077, + "ratio_unweighted": 0.007394938967083044 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 0.0508016678151566, + "ratio_unweighted": 0.03779811148859605 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 0.5930390350197559, + "ratio_unweighted": 0.2949378080004211 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 1.059196541105937, + "ratio_unweighted": 0.6545610735726085 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 1.0441338533749804, + "ratio_unweighted": 0.8874737863345261 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 1.0144035103536144, + "ratio_unweighted": 0.9956220541515616 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 1.0010801700126166, + "ratio_unweighted": 1.0014624881536047 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 8.221271971269327e-85, + "ratio_unweighted": 4.466979124611998e-85 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 6.194650535106846e-27, + "ratio_unweighted": 5.064802636612562e-27 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 2.6628488495010084e-05, + "ratio_unweighted": 2.285631609327335e-05 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 0.08331704130263944, + "ratio_unweighted": 0.0786718730295742 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 0.5949140523783994, + "ratio_unweighted": 0.5846942619012991 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 0.9673329862889801, + "ratio_unweighted": 0.9639475613911047 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 1.003538764598419, + "ratio_unweighted": 1.0035168444160498 + } + ], + "k=0.5,b=1.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.04394824928347523, + "ratio_unweighted": 78.41598358314276 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.09570762094163081, + "ratio_unweighted": 11.582997312139367 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.15014313721319839, + "ratio_unweighted": 4.086005447774196 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.2130828818035663, + "ratio_unweighted": 2.1676986752618106 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.3798488219726349, + "ratio_unweighted": 1.1601785742194737 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.6140195790184909, + "ratio_unweighted": 0.9702662424165897 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.8581465414760014, + "ratio_unweighted": 0.962381280209413 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 0.9915281367848631, + "ratio_unweighted": 0.9839832295187693 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.006249568064324, + "ratio_unweighted": 0.9991348984748696 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.15704682922861773, + "ratio_unweighted": 155.37499405267155 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.3046592740342447, + "ratio_unweighted": 20.236107484099122 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.44495068537242854, + "ratio_unweighted": 6.573653633203263 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.5803442484472724, + "ratio_unweighted": 3.2114657256982606 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 0.8278936373071835, + "ratio_unweighted": 1.4941422191225022 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 0.9722244384829813, + "ratio_unweighted": 1.114698926598593 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 0.9929321714206912, + "ratio_unweighted": 1.005217915220107 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 0.9938135784896569, + "ratio_unweighted": 0.9811482160673288 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 0.999743802620589, + "ratio_unweighted": 0.9946356513587027 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 0.10224342953236586, + "ratio_unweighted": 45.22046816869712 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 0.24270134751910086, + "ratio_unweighted": 7.285070271453708 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 0.4725711192601542, + "ratio_unweighted": 3.221752753422984 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 0.8222056004520534, + "ratio_unweighted": 2.195948072906509 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 1.5541448171409233, + "ratio_unweighted": 1.6212598350479626 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 1.5441921785908619, + "ratio_unweighted": 1.3147671938773093 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 1.1794952882669087, + "ratio_unweighted": 1.0989338446408359 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.0143361556167563, + "ratio_unweighted": 1.001350738462004 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 0.9919590571983622, + "ratio_unweighted": 0.9887910067079594 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 1.3317435040219016e-05, + "ratio_unweighted": 0.0014727398557857153 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 0.0002865024291763956, + "ratio_unweighted": 0.0022944910520233904 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 0.006725869287074077, + "ratio_unweighted": 0.013809587090550654 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 0.07818528197785432, + "ratio_unweighted": 0.07883035306431126 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 0.9026916867978404, + "ratio_unweighted": 0.5846628219579861 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 1.4380954211513064, + "ratio_unweighted": 1.0422373862755423 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 1.2202353857941217, + "ratio_unweighted": 1.1034921885156046 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 1.03740771067733, + "ratio_unweighted": 1.027383001739458 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 0.9837835928552275, + "ratio_unweighted": 0.9823220492649262 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 2.375275323493752e-84, + "ratio_unweighted": 2.273077033160661e-84 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 3.7899819078574724e-26, + "ratio_unweighted": 2.7012738643267095e-26 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 7.800773868026436e-05, + "ratio_unweighted": 6.227365346906256e-05 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 0.17495229241250873, + "ratio_unweighted": 0.15973518322083863 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 0.8808634471201148, + "ratio_unweighted": 0.8576923842465419 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 1.057246263472686, + "ratio_unweighted": 1.0515363705643406 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 0.9759421804845596, + "ratio_unweighted": 0.9758240095632997 + } + ], + "k=1.0,b=0.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.039490988260693465, + "ratio_unweighted": 63.16746850830891 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.08718208578297151, + "ratio_unweighted": 9.494284065739745 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.1410383214852956, + "ratio_unweighted": 3.449574209574774 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.20815735875587274, + "ratio_unweighted": 1.8922929036340814 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.4032502250014551, + "ratio_unweighted": 1.0829668264222414 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.6796734764925719, + "ratio_unweighted": 0.9536229801411701 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.9193080420260853, + "ratio_unweighted": 0.9692831834619947 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 1.0014637279244158, + "ratio_unweighted": 0.9923682660003456 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.0003721623791384, + "ratio_unweighted": 0.9999486896780728 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.05692442159554174, + "ratio_unweighted": 49.705878512333015 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.12939808101843256, + "ratio_unweighted": 7.386237144573246 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.23862402067602873, + "ratio_unweighted": 2.8833583592724072 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.4134121746475836, + "ratio_unweighted": 1.745741142416309 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 1.017916297720146, + "ratio_unweighted": 1.2298366863909975 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 1.4788866117538042, + "ratio_unweighted": 1.1545334319654434 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 1.2673775414900035, + "ratio_unweighted": 1.0677253994057987 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 1.0494221655767704, + "ratio_unweighted": 1.0160268638630534 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 1.000102473311175, + "ratio_unweighted": 1.0003102419424925 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 3.0236451379592164e-05, + "ratio_unweighted": 0.011807114643084907 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 0.00016251597971010754, + "ratio_unweighted": 0.004063781466495098 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 0.0010610982385101176, + "ratio_unweighted": 0.005391802677426938 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 0.007560423710252632, + "ratio_unweighted": 0.013378222630051154 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 0.20092666435941745, + "ratio_unweighted": 0.12307554943689902 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 0.9964397398024314, + "ratio_unweighted": 0.528716419747668 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 1.243806787693739, + "ratio_unweighted": 0.9131568384647232 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.0761288060632765, + "ratio_unweighted": 1.0283207179775173 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 1.0016471084088314, + "ratio_unweighted": 1.0024313814077395 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 3.9875523196431946e-25, + "ratio_unweighted": 3.734086414319828e-23 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 2.0309075979121114e-20, + "ratio_unweighted": 1.1307225502312292e-19 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 4.756191498871468e-15, + "ratio_unweighted": 5.556819543196532e-15 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 2.2608509702817752e-10, + "ratio_unweighted": 1.194229248749143e-10 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 0.00017972633491160816, + "ratio_unweighted": 6.196898264769764e-05 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 0.06145082454933448, + "ratio_unweighted": 0.028392328977057995 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 0.5142678330609568, + "ratio_unweighted": 0.3816765792242292 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 0.9775337038827994, + "ratio_unweighted": 0.93490055995845 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 1.0067688578875311, + "ratio_unweighted": 1.0075532747065912 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 1.7533340743566104e-112, + "ratio_unweighted": 1.4991510750309477e-112 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 2.0750446532480025e-22, + "ratio_unweighted": 1.8232309326007793e-22 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 1.5952138227033e-06, + "ratio_unweighted": 1.5299327668704416e-06 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 0.03661185542666673, + "ratio_unweighted": 0.03614166199213023 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 0.6747067855676732, + "ratio_unweighted": 0.6722575316666797 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 1.016845463382698, + "ratio_unweighted": 1.01680240852149 + } + ], + "k=1.0,b=1.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.043948249283475245, + "ratio_unweighted": 78.41598358314282 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.09570762094163077, + "ratio_unweighted": 11.582997312139366 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.15014313721319839, + "ratio_unweighted": 4.086005447774196 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.2130828818035663, + "ratio_unweighted": 2.1676986752618106 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.3798488219726348, + "ratio_unweighted": 1.1601785742194732 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.6140195790184907, + "ratio_unweighted": 0.9702662424165897 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.8581465414760014, + "ratio_unweighted": 0.962381280209413 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 0.9915281367848633, + "ratio_unweighted": 0.9839832295187693 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.006249568064324, + "ratio_unweighted": 0.9991348984748696 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.062463542950728486, + "ratio_unweighted": 61.676049531618965 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.13808468444103122, + "ratio_unweighted": 9.072065420500458 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.24734400066180037, + "ratio_unweighted": 3.5425811380362755 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.41498489460569093, + "ratio_unweighted": 2.160138865697299 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 0.9413614714957307, + "ratio_unweighted": 1.5020731505417482 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 1.2645274223365988, + "ratio_unweighted": 1.2907351621631007 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 1.1158353079347416, + "ratio_unweighted": 1.0860108153620203 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 1.006813404437864, + "ratio_unweighted": 0.9927870476880315 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 0.9962972997286528, + "ratio_unweighted": 0.9913697961901646 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 3.386292089537056e-05, + "ratio_unweighted": 0.014921831094148408 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 0.00018589394208290202, + "ratio_unweighted": 0.005433623500771513 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 0.0012684870922610714, + "ratio_unweighted": 0.008082383426690428 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 0.009529541109407007, + "ratio_unweighted": 0.022708237720117447 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 0.26914760028361945, + "ratio_unweighted": 0.23295993239811416 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 1.2766384950758856, + "ratio_unweighted": 0.8955031761837973 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 1.4118921865099943, + "ratio_unweighted": 1.1953490785156695 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.0839438605960736, + "ratio_unweighted": 1.0651387925877533 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 0.9845744608359893, + "ratio_unweighted": 0.9817735628894225 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 4.702980078075717e-25, + "ratio_unweighted": 5.226886648803399e-23 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 2.796330125110172e-20, + "ratio_unweighted": 2.036049810243635e-19 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 8.45015119266073e-15, + "ratio_unweighted": 1.4182377913543486e-14 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 4.5101624023716076e-10, + "ratio_unweighted": 3.585960480011269e-10 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 0.000392980779124985, + "ratio_unweighted": 0.00019511545884576327 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 0.12580347665686245, + "ratio_unweighted": 0.07345669062471187 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 0.8423037930226176, + "ratio_unweighted": 0.6819004240608795 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 1.1309152804193099, + "ratio_unweighted": 1.1044863047980609 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 0.9737923584022314, + "ratio_unweighted": 0.9726976210713407 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 2.5985985417919607e-111, + "ratio_unweighted": 1.4827757192573578e-111 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 1.3194538782436571e-21, + "ratio_unweighted": 1.0534944610784847e-21 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 6.182737019641913e-06, + "ratio_unweighted": 5.629688820303654e-06 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 0.0867578534241024, + "ratio_unweighted": 0.08418936160188543 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 0.9540155255017941, + "ratio_unweighted": 0.9473122778234625 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 0.9665701626350894, + "ratio_unweighted": 0.9664500332670383 + } + ], + "k=2.0,b=0.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.03949098826069353, + "ratio_unweighted": 63.167468508308936 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.08718208578297146, + "ratio_unweighted": 9.494284065739738 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.14103832148529566, + "ratio_unweighted": 3.449574209574775 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.20815735875587277, + "ratio_unweighted": 1.8922929036340816 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.4032502250014552, + "ratio_unweighted": 1.0829668264222414 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.679673476492572, + "ratio_unweighted": 0.9536229801411701 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.9193080420260852, + "ratio_unweighted": 0.9692831834619947 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 1.0014637279244158, + "ratio_unweighted": 0.9923682660003456 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.0003721623791384, + "ratio_unweighted": 0.9999486896780728 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.00016732906439414897, + "ratio_unweighted": 0.14662230485353844 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.0005918351206295383, + "ratio_unweighted": 0.03335054250125826 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.002221732335366561, + "ratio_unweighted": 0.025452667008775 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.00960744574049454, + "ratio_unweighted": 0.03632812846000721 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 0.17901635780184436, + "ratio_unweighted": 0.17583476242158483 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 1.158012420890866, + "ratio_unweighted": 0.7164680145952175 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 1.5691459915220785, + "ratio_unweighted": 1.1329618380533737 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 1.1481044247123988, + "ratio_unweighted": 1.0800586200218618 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 1.00106205845638, + "ratio_unweighted": 1.0017199286426057 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 4.2570909047678823e-20, + "ratio_unweighted": 1.642768966479709e-17 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 5.561346329111716e-18, + "ratio_unweighted": 1.2691194196274534e-16 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 3.984888782378724e-15, + "ratio_unweighted": 1.638964582966069e-14 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 6.556049861834909e-12, + "ratio_unweighted": 8.581890439582216e-12 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 4.813208563721214e-06, + "ratio_unweighted": 1.9968526250581533e-06 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 0.019625063031415726, + "ratio_unweighted": 0.007070722554295678 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 0.5223976804884817, + "ratio_unweighted": 0.2975552981215637 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.077525956059638, + "ratio_unweighted": 0.9655804975762896 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 1.0089359047973059, + "ratio_unweighted": 1.0105835833727492 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 7.335763014588269e-104, + "ratio_unweighted": 6.0420474215884955e-102 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 9.997887663481414e-86, + "ratio_unweighted": 4.148878367567125e-85 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 2.5023253595985457e-64, + "ratio_unweighted": 1.865109135999556e-64 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 2.6037130967729112e-45, + "ratio_unweighted": 8.273491088006685e-46 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 4.626799481991789e-20, + "ratio_unweighted": 9.127822220680576e-21 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 4.8808936438602504e-08, + "ratio_unweighted": 1.4228092167559723e-08 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 0.006316243491195731, + "ratio_unweighted": 0.0036519722476297677 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 0.4766498380096713, + "ratio_unweighted": 0.42682383681747565 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 1.029474236829988, + "ratio_unweighted": 1.0309718604855411 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 1.0767500563459681e-91, + "ratio_unweighted": 9.551973310530084e-92 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 3.516860377720743e-26, + "ratio_unweighted": 3.40995954725731e-26 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 1.0658185903645112e-07, + "ratio_unweighted": 1.0572969080973156e-07 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 0.058115501006920825, + "ratio_unweighted": 0.057976880282225945 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 1.0652887847290435, + "ratio_unweighted": 1.065208213167798 + } + ], + "k=2.0,b=1.0": [ + { + "angle_deg": 90, + "r": 0.25, + "ratio": 0.04394824928347518, + "ratio_unweighted": 78.41598358314273 + }, + { + "angle_deg": 90, + "r": 0.5, + "ratio": 0.09570762094163085, + "ratio_unweighted": 11.582997312139376 + }, + { + "angle_deg": 90, + "r": 0.75, + "ratio": 0.15014313721319836, + "ratio_unweighted": 4.086005447774195 + }, + { + "angle_deg": 90, + "r": 1.0, + "ratio": 0.21308288180356627, + "ratio_unweighted": 2.1676986752618106 + }, + { + "angle_deg": 90, + "r": 1.5, + "ratio": 0.3798488219726348, + "ratio_unweighted": 1.1601785742194735 + }, + { + "angle_deg": 90, + "r": 2.0, + "ratio": 0.6140195790184907, + "ratio_unweighted": 0.9702662424165897 + }, + { + "angle_deg": 90, + "r": 2.5, + "ratio": 0.8581465414760016, + "ratio_unweighted": 0.962381280209413 + }, + { + "angle_deg": 90, + "r": 3.0, + "ratio": 0.9915281367848633, + "ratio_unweighted": 0.9839832295187693 + }, + { + "angle_deg": 90, + "r": 4.0, + "ratio": 1.006249568064324, + "ratio_unweighted": 0.9991348984748696 + }, + { + "angle_deg": 60, + "r": 0.25, + "ratio": 0.0001861888115879146, + "ratio_unweighted": 0.1835501651650323 + }, + { + "angle_deg": 60, + "r": 0.5, + "ratio": 0.000651748023199925, + "ratio_unweighted": 0.04229439355102498 + }, + { + "angle_deg": 60, + "r": 0.75, + "ratio": 0.0024380120723943265, + "ratio_unweighted": 0.033669785175212026 + }, + { + "angle_deg": 60, + "r": 1.0, + "ratio": 0.010567555681826659, + "ratio_unweighted": 0.05125928172435114 + }, + { + "angle_deg": 60, + "r": 1.5, + "ratio": 0.19778087028706948, + "ratio_unweighted": 0.27376163575799484 + }, + { + "angle_deg": 60, + "r": 2.0, + "ratio": 1.2287991777432359, + "ratio_unweighted": 1.0512695813737116 + }, + { + "angle_deg": 60, + "r": 2.5, + "ratio": 1.5095141927114166, + "ratio_unweighted": 1.3327758266218659 + }, + { + "angle_deg": 60, + "r": 3.0, + "ratio": 1.0811033695053884, + "ratio_unweighted": 1.0625036830807815 + }, + { + "angle_deg": 60, + "r": 4.0, + "ratio": 0.9899710136183282, + "ratio_unweighted": 0.9854229932803231 + }, + { + "angle_deg": 45, + "r": 0.25, + "ratio": 4.8270422463030845e-20, + "ratio_unweighted": 2.114903228591386e-17 + }, + { + "angle_deg": 45, + "r": 0.5, + "ratio": 6.6071609872701334e-18, + "ratio_unweighted": 1.8389431017192102e-16 + }, + { + "angle_deg": 45, + "r": 0.75, + "ratio": 5.213639227127308e-15, + "ratio_unweighted": 2.959200473746951e-14 + }, + { + "angle_deg": 45, + "r": 1.0, + "ratio": 9.74416632554946e-12, + "ratio_unweighted": 1.9256682680611072e-11 + }, + { + "angle_deg": 45, + "r": 1.5, + "ratio": 8.820647482689757e-06, + "ratio_unweighted": 5.688505918521638e-06 + }, + { + "angle_deg": 45, + "r": 2.0, + "ratio": 0.03730919648488384, + "ratio_unweighted": 0.019470572086655903 + }, + { + "angle_deg": 45, + "r": 2.5, + "ratio": 0.8702066369160758, + "ratio_unweighted": 0.6061938152535679 + }, + { + "angle_deg": 45, + "r": 3.0, + "ratio": 1.2787389934620534, + "ratio_unweighted": 1.2100650331311507 + }, + { + "angle_deg": 45, + "r": 4.0, + "ratio": 0.9737283703815497, + "ratio_unweighted": 0.9717355502009581 + }, + { + "angle_deg": 30, + "r": 0.25, + "ratio": 8.932323189309395e-104, + "ratio_unweighted": 9.39490263303482e-102 + }, + { + "angle_deg": 30, + "r": 0.5, + "ratio": 1.5604930503119474e-85, + "ratio_unweighted": 9.673036314668814e-85 + }, + { + "angle_deg": 30, + "r": 0.75, + "ratio": 5.739835217985818e-64, + "ratio_unweighted": 6.944798037637039e-64 + }, + { + "angle_deg": 30, + "r": 1.0, + "ratio": 8.967258562182673e-45, + "ratio_unweighted": 4.536310038439099e-45 + }, + { + "angle_deg": 30, + "r": 1.5, + "ratio": 1.9362101703412645e-19, + "ratio_unweighted": 6.033776429583763e-20 + }, + { + "angle_deg": 30, + "r": 2.0, + "ratio": 1.8989420935492606e-07, + "ratio_unweighted": 7.805839627019295e-08 + }, + { + "angle_deg": 30, + "r": 2.5, + "ratio": 0.018860296106058605, + "ratio_unweighted": 0.012611701203880152 + }, + { + "angle_deg": 30, + "r": 3.0, + "ratio": 0.8072069850735696, + "ratio_unweighted": 0.746540608247414 + }, + { + "angle_deg": 30, + "r": 4.0, + "ratio": 0.9672138168668959, + "ratio_unweighted": 0.9669356714213547 + }, + { + "angle_deg": 0, + "r": 0.25, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.5, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 0.75, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.0, + "ratio": 0.0, + "ratio_unweighted": 0.0 + }, + { + "angle_deg": 0, + "r": 1.5, + "ratio": 3.1486310598395164e-90, + "ratio_unweighted": 2.510132270642916e-90 + }, + { + "angle_deg": 0, + "r": 2.0, + "ratio": 4.294234307952424e-25, + "ratio_unweighted": 3.912279520890651e-25 + }, + { + "angle_deg": 0, + "r": 2.5, + "ratio": 5.736565377884086e-07, + "ratio_unweighted": 5.566094619286819e-07 + }, + { + "angle_deg": 0, + "r": 3.0, + "ratio": 0.1372999295177763, + "ratio_unweighted": 0.1362569068250281 + }, + { + "angle_deg": 0, + "r": 4.0, + "ratio": 0.9802351629287761, + "ratio_unweighted": 0.9800999972460147 + } + ] + }, + "remote_total": { + "k=0.5,b=0.0": { + "L=12": { + "integral_X_Lambda_approx": 15.633326557742928, + "k_integral": 7.816663278871464 + }, + "L=24": { + "integral_X_Lambda_approx": 65.78571259223075, + "k_integral": 32.892856296115376 + }, + "L=6": { + "integral_X_Lambda_approx": 3.09523004912097, + "integral_X_Lambda_periodized_direct": 3.057321620877225, + "k_integral": 1.547615024560485, + "k_integral_direct": 1.5286608104386126 + } + }, + "k=0.5,b=1.0": { + "L=12": { + "integral_X_Lambda_approx": 11.465027432496477, + "k_integral": 5.7325137162482385 + }, + "L=24": { + "integral_X_Lambda_approx": 48.11199774431693, + "k_integral": 24.055998872158465 + }, + "L=6": { + "integral_X_Lambda_approx": 2.3032848545413636, + "integral_X_Lambda_periodized_direct": 2.236984148652817, + "k_integral": 1.1516424272706818, + "k_integral_direct": 1.1184920743264084 + } + }, + "k=1.0,b=0.0": { + "L=12": { + "integral_X_Lambda_approx": 15.086314346932143, + "k_integral": 15.086314346932143 + }, + "L=24": { + "integral_X_Lambda_approx": 65.23870038141997, + "k_integral": 65.23870038141997 + }, + "L=6": { + "integral_X_Lambda_approx": 2.548217838310185, + "integral_X_Lambda_periodized_direct": 2.5420634536888143, + "k_integral": 2.548217838310185, + "k_integral_direct": 2.5420634536888143 + } + }, + "k=1.0,b=1.0": { + "L=12": { + "integral_X_Lambda_approx": 11.031174414731563, + "k_integral": 11.031174414731563 + }, + "L=24": { + "integral_X_Lambda_approx": 47.67814472655202, + "k_integral": 47.67814472655202 + }, + "L=6": { + "integral_X_Lambda_approx": 1.8694318367764495, + "integral_X_Lambda_periodized_direct": 1.812099277399927, + "k_integral": 1.8694318367764495, + "k_integral_direct": 1.812099277399927 + } + }, + "k=2.0,b=0.0": { + "L=12": { + "integral_X_Lambda_approx": 14.579259554184302, + "k_integral": 29.158519108368605 + }, + "L=24": { + "integral_X_Lambda_approx": 64.73164558867214, + "k_integral": 129.46329117734427 + }, + "L=6": { + "integral_X_Lambda_approx": 2.0411630455623455, + "integral_X_Lambda_periodized_direct": 2.1797266766703105, + "k_integral": 4.082326091124691, + "k_integral_direct": 4.359453353340621 + } + }, + "k=2.0,b=1.0": { + "L=12": { + "integral_X_Lambda_approx": 10.647517860179647, + "k_integral": 21.295035720359294 + }, + "L=24": { + "integral_X_Lambda_approx": 47.2944881720001, + "k_integral": 94.5889763440002 + }, + "L=6": { + "integral_X_Lambda_approx": 1.485775282224533, + "integral_X_Lambda_periodized_direct": 1.5301856127083193, + "k_integral": 2.971550564449066, + "k_integral_direct": 3.0603712254166386 + } + } + }, + "samples": { + "k=0.5,b=0.0": [ + { + "coupling": 0.0013347998737805087, + "exact_regression": false, + "lambda": 0.008997161568001377, + "lambda_j": [ + 4.2415324573566405e-08, + 0.008997119151863017, + 8.137870533623332e-13 + ], + "rho": 6.740457311041707, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.23734666626119014, + "exact_regression": false, + "lambda": 0.00018233592692150444, + "lambda_j": [ + 2.350450811356302e-08, + 0.00018231242241339087, + 5.797795764918883e-30 + ], + "rho": 0.0007682261975436862, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.24375768264096007, + "exact_regression": false, + "lambda": 0.05649474402271464, + "lambda_j": [ + 4.657359090868975e-07, + 0.00232023543617016, + 0.054174042850635394 + ], + "rho": 0.23176600388808216, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.2230783664356066, + "exact_regression": false, + "lambda": 7.191585763551508e-28, + "lambda_j": [ + 4.234550533129775e-29, + 6.76813071023853e-28, + 0.0 + ], + "rho": 5.879905949534375e-28, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.11000271594112836, + "exact_regression": false, + "lambda": 0.024165713454268295, + "lambda_j": [ + 1.473127222760222e-05, + 0.014912843942688724, + 0.009238138239351969 + ], + "rho": 0.21968288007726458, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.19403952034402, + "exact_regression": false, + "lambda": 0.13297957632490873, + "lambda_j": [ + 0.013682845859870138, + 0.11545464087023243, + 0.003842089594806162 + ], + "rho": 0.11136949326986714, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.4681640123522247, + "exact_regression": false, + "lambda": 0.00153584335767679, + "lambda_j": [ + 6.512390671793527e-12, + 0.0006691867856965605, + 0.0008666565654678388 + ], + "rho": 0.0032805668892834363, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 0.9998962653416301, + "exact_regression": false, + "lambda": 0.11609460083411186, + "lambda_j": [ + 0.013457536339208605, + 0.08979713535468875, + 0.0128399291402145 + ], + "rho": 0.11610664511727756, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=0.5,b=1.0": [ + { + "coupling": 0.0011816451638475367, + "exact_regression": false, + "lambda": 0.007251517701528527, + "lambda_j": [ + 9.71715596755084e-09, + 0.007251507982692201, + 1.6803582671529624e-12 + ], + "rho": 6.136798019734598, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.19563370398306065, + "exact_regression": false, + "lambda": 0.00017680783359851475, + "lambda_j": [ + 5.4381249056313444e-09, + 0.00017680239547360912, + 3.188833707371104e-28 + ], + "rho": 0.0009037698003909594, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.2955495656568713, + "exact_regression": false, + "lambda": 0.05666077867832258, + "lambda_j": [ + 2.6040919993732018e-08, + 0.0006166792130766919, + 0.056044073424325894 + ], + "rho": 0.19171328691480777, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.4030350487258436, + "exact_regression": false, + "lambda": 3.2150776495275335e-27, + "lambda_j": [ + 2.998040735279339e-29, + 3.18509724217474e-27, + 0.0 + ], + "rho": 2.291516275696237e-27, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09829912442873572, + "exact_regression": false, + "lambda": 0.01807602324863991, + "lambda_j": [ + 2.0645229980440653e-06, + 0.007046349177726891, + 0.011027609547914976 + ], + "rho": 0.18388793749373172, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.1007428796012326, + "exact_regression": false, + "lambda": 0.10472518621398787, + "lambda_j": [ + 0.0019175197320715279, + 0.08891080744153346, + 0.013896859040382887 + ], + "rho": 0.0951404620958591, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.7434247320610644, + "exact_regression": false, + "lambda": 0.0009508748371417395, + "lambda_j": [ + 7.552120071379692e-14, + 0.0001544598011323832, + 0.0007964150359338351 + ], + "rho": 0.0012790465478671152, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.001767535376838, + "exact_regression": false, + "lambda": 0.08442985423318261, + "lambda_j": [ + 0.0008954886875238492, + 0.04295706909710692, + 0.04057729644855183 + ], + "rho": 0.08428088478772908, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=1.0,b=0.0": [ + { + "coupling": 0.0013337511324425409, + "exact_regression": false, + "lambda": 2.708085543210255e-06, + "lambda_j": [ + 1.4604817680949028e-11, + 2.708070938392574e-06, + 0.0 + ], + "rho": 0.0020304279241741686, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.2951302944859689, + "exact_regression": false, + "lambda": 1.1029419098839017e-17, + "lambda_j": [ + 6.013772842640704e-22, + 1.1028817721554752e-17, + 0.0 + ], + "rho": 3.7371355312910374e-17, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.2063740611469615, + "exact_regression": false, + "lambda": 0.008283690554448853, + "lambda_j": [ + 1.3952680692465282e-11, + 0.00019232007669658742, + 0.008091370463799584 + ], + "rho": 0.04013920406668711, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.1695512904330976, + "exact_regression": false, + "lambda": 2.035506651494308e-113, + "lambda_j": [ + 1.3955119259860762e-114, + 1.8959554588957005e-113, + 0.0 + ], + "rho": 1.7404167462724424e-113, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.11000271594112836, + "exact_regression": false, + "lambda": 0.024165713454268295, + "lambda_j": [ + 1.473127222760222e-05, + 0.014912843942688724, + 0.009238138239351969 + ], + "rho": 0.21968288007726458, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.3868030388781087, + "exact_regression": false, + "lambda": 0.1021044225602007, + "lambda_j": [ + 0.011196593733202694, + 0.09066957458517144, + 0.00023825424182657157 + ], + "rho": 0.07362575628821869, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.23360634059314445, + "exact_regression": false, + "lambda": 6.709121442309665e-11, + "lambda_j": [ + 2.289820976759197e-34, + 4.448649044698955e-11, + 2.260472397610711e-11 + ], + "rho": 2.8719774580067866e-10, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 0.9997652205926268, + "exact_regression": false, + "lambda": 0.11614711471938798, + "lambda_j": [ + 0.014005217085133914, + 0.0898231582784744, + 0.012318739355779663 + ], + "rho": 0.11617439007384145, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=1.0,b=1.0": [ + { + "coupling": 0.0011852745548398787, + "exact_regression": false, + "lambda": 2.214975518055656e-06, + "lambda_j": [ + 3.44759304165418e-12, + 2.2149720704626143e-06, + 0.0 + ], + "rho": 0.00186874467945942, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.2157902874249697, + "exact_regression": false, + "lambda": 1.1488932506278306e-17, + "lambda_j": [ + 1.4657489729406693e-22, + 1.1488785931381012e-17, + 0.0 + ], + "rho": 5.324119376908012e-17, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.264998801324006, + "exact_regression": false, + "lambda": 0.007270973301820235, + "lambda_j": [ + 3.2937305623954754e-13, + 5.0501558403101295e-05, + 0.007220471743087761 + ], + "rho": 0.02743775921057936, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.7525229932234514, + "exact_regression": false, + "lambda": 2.2044158243840257e-112, + "lambda_j": [ + 2.7203540887333047e-114, + 2.1772122834966927e-112, + 0.0 + ], + "rho": 1.2578527259887179e-112, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09829912442873572, + "exact_regression": false, + "lambda": 0.01807602324863991, + "lambda_j": [ + 2.0645229980440653e-06, + 0.007046349177726891, + 0.011027609547914976 + ], + "rho": 0.18388793749373172, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.233509563363153, + "exact_regression": false, + "lambda": 0.10333970328413859, + "lambda_j": [ + 0.0021775455319044783, + 0.09971173114849526, + 0.0014504266037388557 + ], + "rho": 0.0837769777823074, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.4035709506744181, + "exact_regression": false, + "lambda": 2.054401073231451e-11, + "lambda_j": [ + 2.065375626345689e-37, + 7.034802327187759e-12, + 1.3509208405126751e-11 + ], + "rho": 5.09055735998413e-11, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0016758359521827, + "exact_regression": false, + "lambda": 0.08402207525361846, + "lambda_j": [ + 0.0009474385223607201, + 0.04357627221188679, + 0.03949836451937094 + ], + "rho": 0.08388150361413874, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=2.0,b=0.0": [ + { + "coupling": 0.0013069758683149615, + "exact_regression": false, + "lambda": 2.7687701498112533e-21, + "lambda_j": [ + 2.0689728873282906e-27, + 2.768768080838366e-21, + 0.0 + ], + "rho": 2.1184554488989396e-18, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.42372525112449744, + "exact_regression": false, + "lambda": 1.0855803610218994e-71, + "lambda_j": [ + 7.590387262964049e-77, + 1.0855727706346365e-71, + 0.0 + ], + "rho": 2.5619911915585558e-71, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.12211928031485988, + "exact_regression": false, + "lambda": 6.391010794773699e-07, + "lambda_j": [ + 4.1282539410900915e-28, + 2.297887135225486e-08, + 6.16122208125115e-07 + ], + "rho": 5.233416687599017e-06, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.0343154814073068, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "rho": 0.0, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.11000271594112836, + "exact_regression": false, + "lambda": 0.024165713454268295, + "lambda_j": [ + 1.473127222760222e-05, + 0.014912843942688724, + 0.009238138239351969 + ], + "rho": 0.21968288007726458, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.816012453345608, + "exact_regression": false, + "lambda": 0.008868968410056192, + "lambda_j": [ + 0.0006676160158203409, + 0.008201335920209833, + 1.6474026017517377e-08 + ], + "rho": 0.0048837596866238685, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.07127840244902967, + "exact_regression": false, + "lambda": 1.0069028958943768e-40, + "lambda_j": [ + 0.0, + 9.573733729955746e-41, + 4.9529522898802227e-42 + ], + "rho": 1.4126339273869123e-39, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 0.9995061393382996, + "exact_regression": false, + "lambda": 0.11638712556143374, + "lambda_j": [ + 0.015137558092887539, + 0.08993422886056755, + 0.011315338607978643 + ], + "rho": 0.11644463298493112, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=2.0,b=1.0": [ + { + "coupling": 0.0011789396796435185, + "exact_regression": false, + "lambda": 2.305551738891681e-21, + "lambda_j": [ + 4.92094898803099e-28, + 2.3055512467967824e-21, + 0.0 + ], + "rho": 1.95561467537408e-18, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.2755841295781233, + "exact_regression": false, + "lambda": 1.3475984017825205e-71, + "lambda_j": [ + 1.728779200220744e-77, + 1.3475966730033203e-71, + 0.0 + ], + "rho": 4.889971000309363e-71, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.19064731192523218, + "exact_regression": false, + "lambda": 4.90214103654052e-07, + "lambda_j": [ + 1.518151932771976e-30, + 5.837755725344078e-09, + 4.843763479287079e-07 + ], + "rho": 2.5713140075444837e-06, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 2.116297477517468, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "rho": 0.0, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09829912442873572, + "exact_regression": false, + "lambda": 0.01807602324863991, + "lambda_j": [ + 2.0645229980440653e-06, + 0.007046349177726891, + 0.011027609547914976 + ], + "rho": 0.18388793749373172, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.5183055627578088, + "exact_regression": false, + "lambda": 0.014886605076929448, + "lambda_j": [ + 0.00020360615405411623, + 0.014682678974993781, + 3.199478815492129e-07 + ], + "rho": 0.009804749084821783, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.11575239500763133, + "exact_regression": false, + "lambda": 8.915112493385315e-42, + "lambda_j": [ + 0.0, + 7.416143163304738e-42, + 1.498969330080577e-42 + ], + "rho": 7.701881669746495e-41, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0014773803550947, + "exact_regression": false, + "lambda": 0.08329489101993598, + "lambda_j": [ + 0.001058099161553433, + 0.04487664547022176, + 0.03736014638816079 + ], + "rho": 0.08317201431988613, + "x": [ + 3.0, + 3.0 + ] + } + ] + }, + "samples_L6": { + "k=0.5,b=0.0": [ + { + "coupling": 0.0013352589970307778, + "exact_regression": false, + "lambda": 0.009000300800690187, + "lambda_j": [ + 4.230624212340141e-08, + 0.009000258493632302, + 8.157609431542862e-13 + ], + "rho": 6.740490661889716, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.23828465890258058, + "exact_regression": false, + "lambda": 0.00018359684099034408, + "lambda_j": [ + 2.318534121104517e-08, + 0.00018357365564913302, + 5.8823203947993176e-30 + ], + "rho": 0.0007704937524551472, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.243694283479726, + "exact_regression": false, + "lambda": 0.056505261476725285, + "lambda_j": [ + 4.663942519359146e-07, + 0.002328927397609625, + 0.054175867684863724 + ], + "rho": 0.2318694581993599, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.2277856239866574, + "exact_regression": false, + "lambda": 2.9186351704542015e-28, + "lambda_j": [ + 1.679383283098468e-29, + 2.7506968421443545e-28, + 0.0 + ], + "rho": 2.377153725727219e-28, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.1105447762765137, + "exact_regression": false, + "lambda": 0.024316082472093964, + "lambda_j": [ + 1.4243157683247784e-05, + 0.014995981530015386, + 0.009305857784395332 + ], + "rho": 0.21996591147165903, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.2066298089169165, + "exact_regression": false, + "lambda": 0.13728850606492624, + "lambda_j": [ + 0.012407671959928847, + 0.12061773264393971, + 0.004263101461057675 + ], + "rho": 0.11377848040084294, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.4435339312049749, + "exact_regression": false, + "lambda": 0.0014132305977552805, + "lambda_j": [ + 1.011140067250916e-11, + 0.0006701395366965291, + 0.0007430910509473508 + ], + "rho": 0.00318629646646396, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0001796681771224, + "exact_regression": false, + "lambda": 0.11602353795434578, + "lambda_j": [ + 0.012257263635414215, + 0.08970371778893044, + 0.014062556530001123 + ], + "rho": 0.11600269596142108, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=0.5,b=1.0": [ + { + "coupling": 0.0011817670071456834, + "exact_regression": false, + "lambda": 0.007252641220824504, + "lambda_j": [ + 9.692188228164635e-09, + 0.0072526315269528935, + 1.6833822216337555e-12 + ], + "rho": 6.137116010999305, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.1958721708453118, + "exact_regression": false, + "lambda": 0.00017761070450846226, + "lambda_j": [ + 5.357023190560863e-09, + 0.0001776053474852717, + 3.223213651819074e-28 + ], + "rho": 0.0009067684487385839, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.2954894243501089, + "exact_regression": false, + "lambda": 0.05664561609144616, + "lambda_j": [ + 2.616130061305281e-08, + 0.0006193203942933197, + 0.05602626953585223 + ], + "rho": 0.19170099307625288, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.42312468525165, + "exact_regression": false, + "lambda": 1.277983386657996e-27, + "lambda_j": [ + 1.1839937696321913e-29, + 1.266143448961674e-27, + 0.0 + ], + "rho": 8.980122401797921e-28, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09856137915405334, + "exact_regression": false, + "lambda": 0.018142141203767663, + "lambda_j": [ + 1.9970278851752472e-06, + 0.007066547115955158, + 0.011073597059927328 + ], + "rho": 0.1840694738596458, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.104243000848708, + "exact_regression": false, + "lambda": 0.10688012373066431, + "lambda_j": [ + 0.0017275589785925784, + 0.09046676395420254, + 0.014685800797869197 + ], + "rho": 0.09679040179427673, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.7076713955852335, + "exact_regression": false, + "lambda": 0.0008630804992395709, + "lambda_j": [ + 1.3586224440589788e-13, + 0.00016318183181683395, + 0.0006998986672868747 + ], + "rho": 0.0012196063096853258, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0073636982372143, + "exact_regression": false, + "lambda": 0.08493602325082197, + "lambda_j": [ + 0.0008311279180490109, + 0.04216619744764276, + 0.0419386978851302 + ], + "rho": 0.08431515191529289, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=1.0,b=0.0": [ + { + "coupling": 0.0013346419825193117, + "exact_regression": false, + "lambda": 2.7095899233224166e-06, + "lambda_j": [ + 1.4527774969604223e-11, + 2.7095753955474468e-06, + 0.0 + ], + "rho": 0.0020301998279776195, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.2969953747999563, + "exact_regression": false, + "lambda": 1.1224037576445477e-17, + "lambda_j": [ + 5.869732556313949e-22, + 1.1223450603189846e-17, + 0.0 + ], + "rho": 3.7791960847893744e-17, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.2060178450562346, + "exact_regression": false, + "lambda": 0.008286051353279226, + "lambda_j": [ + 1.410766466536171e-11, + 0.000193891178795234, + 0.008092160160376328 + ], + "rho": 0.04022006613561785, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.1145089876535326, + "exact_regression": false, + "lambda": 4.796072004560116e-115, + "lambda_j": [ + 3.4911731076275664e-116, + 4.446954693797359e-115, + 0.0 + ], + "rho": 4.303304915160604e-115, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.1105447762765137, + "exact_regression": false, + "lambda": 0.024316082472093964, + "lambda_j": [ + 1.4243157683247784e-05, + 0.014995981530015386, + 0.009305857784395332 + ], + "rho": 0.21996591147165903, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.414012794265677, + "exact_regression": false, + "lambda": 0.11165635583551857, + "lambda_j": [ + 0.009070401466296434, + 0.10229852244076303, + 0.00028743192845910603 + ], + "rho": 0.07896417648293186, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.2177764103830025, + "exact_regression": false, + "lambda": 5.293958951787986e-11, + "lambda_j": [ + 7.856465409282458e-34, + 3.8846869377406346e-11, + 1.4092720140473513e-11 + ], + "rho": 2.4309147820360903e-10, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0001796681771224, + "exact_regression": false, + "lambda": 0.11589987685113023, + "lambda_j": [ + 0.012244199504029468, + 0.0896081090796987, + 0.01404756826740207 + ], + "rho": 0.11587905707217942, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=1.0,b=1.0": [ + { + "coupling": 0.001185582455088569, + "exact_regression": false, + "lambda": 2.2156031987889476e-06, + "lambda_j": [ + 3.428995181999742e-12, + 2.2155997697937655e-06, + 0.0 + ], + "rho": 0.0018687887875528917, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.21630152828091614, + "exact_regression": false, + "lambda": 1.1652541972526321e-17, + "lambda_j": [ + 1.425755148206393e-22, + 1.1652399397011501e-17, + 0.0 + ], + "rho": 5.3871750537947553e-17, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.2647602526319787, + "exact_regression": false, + "lambda": 0.007272452822627791, + "lambda_j": [ + 3.3495494674526707e-13, + 5.09431391428759e-05, + 0.00722150968314996 + ], + "rho": 0.027468068754023377, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.699344381931464, + "exact_regression": false, + "lambda": 5.828525267810146e-114, + "lambda_j": [ + 6.686702677615701e-116, + 5.761658241033989e-114, + 0.0 + ], + "rho": 3.42986702976914e-114, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09856137915405333, + "exact_regression": false, + "lambda": 0.01814214120376766, + "lambda_j": [ + 1.9970278851752477e-06, + 0.007066547115955159, + 0.011073597059927325 + ], + "rho": 0.1840694738596458, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.23758760279945, + "exact_regression": false, + "lambda": 0.11009160491776558, + "lambda_j": [ + 0.0016800177513260261, + 0.10679114245766597, + 0.0016204447087735956 + ], + "rho": 0.08895661581348747, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.36077282204203936, + "exact_regression": false, + "lambda": 1.5648883160661804e-11, + "lambda_j": [ + 9.140474190334415e-37, + 6.678752080597806e-12, + 8.970131080063998e-12 + ], + "rho": 4.337600341424362e-11, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0073636982372143, + "exact_regression": false, + "lambda": 0.08484549608260142, + "lambda_j": [ + 0.0008302420788730016, + 0.042121255545450195, + 0.04189399845827823 + ], + "rho": 0.08422528648895386, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=2.0,b=0.0": [ + { + "coupling": 0.0013086858933194952, + "exact_regression": false, + "lambda": 2.770778712582969e-21, + "lambda_j": [ + 2.0404567457255683e-27, + 2.7707766721262235e-21, + 0.0 + ], + "rho": 2.1172221132107264e-18, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.42701808124108037, + "exact_regression": false, + "lambda": 1.1406299471965422e-71, + "lambda_j": [ + 7.287194153468991e-77, + 1.1406226600023888e-71, + 0.0 + ], + "rho": 2.6711514038970643e-71, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.12154477998515859, + "exact_regression": false, + "lambda": 6.414032765857702e-07, + "lambda_j": [ + 4.335058813161749e-28, + 2.3381153059161878e-08, + 6.180221235266084e-07 + ], + "rho": 5.2770943899366945e-06, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 1.1430168251306654, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "rho": 0.0, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.11054477627651373, + "exact_regression": false, + "lambda": 0.024316082472093967, + "lambda_j": [ + 1.4243157683247784e-05, + 0.014995981530015386, + 0.009305857784395334 + ], + "rho": 0.219965911471659, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.8582064835581518, + "exact_regression": false, + "lambda": 0.011073176601220602, + "lambda_j": [ + 0.0004213885169282436, + 0.010651764953280782, + 2.313101157613296e-08 + ], + "rho": 0.005959066820183158, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.06862800582941338, + "exact_regression": false, + "lambda": 4.112319995533774e-41, + "lambda_j": [ + 0.0, + 4.0154206114458735e-41, + 9.689938408790064e-43 + ], + "rho": 5.992189261269877e-40, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0001796681771224, + "exact_regression": false, + "lambda": 0.11540654904856047, + "lambda_j": [ + 0.012192082071296495, + 0.08922669218137447, + 0.01398777479588952 + ], + "rho": 0.11538581788899459, + "x": [ + 3.0, + 3.0 + ] + } + ], + "k=2.0,b=1.0": [ + { + "coupling": 0.0011796280777567724, + "exact_regression": false, + "lambda": 2.3062138136541946e-21, + "lambda_j": [ + 4.851769018303274e-28, + 2.3062133284772926e-21, + 0.0 + ], + "rho": 1.9550346902896567e-18, + "x": [ + 0.15, + 0.15 + ] + }, + { + "coupling": 0.2770480014866231, + "exact_regression": false, + "lambda": 1.4098910061942384e-71, + "lambda_j": [ + 1.6575310004401293e-77, + 1.409889348663238e-71, + 0.0 + ], + "rho": 5.088977356374516e-71, + "x": [ + 0.5, + 0.3 + ] + }, + { + "coupling": 0.1900764796905154, + "exact_regression": false, + "lambda": 4.921264785333191e-07, + "lambda_j": [ + 1.6121642068146436e-30, + 5.948362819099995e-09, + 4.86178115714219e-07 + ], + "rho": 2.5890971851678063e-06, + "x": [ + -0.7, + 0.9 + ] + }, + { + "coupling": 2.2053144199819084, + "exact_regression": false, + "lambda": 0.0, + "lambda_j": [ + 0.0, + 0.0, + 0.0 + ], + "rho": 0.0, + "x": [ + 1.0, + 0.0 + ] + }, + { + "coupling": 0.09856137915405334, + "exact_regression": false, + "lambda": 0.01814214120376766, + "lambda_j": [ + 1.997027885175247e-06, + 0.007066547115955159, + 0.011073597059927325 + ], + "rho": 0.18406947385964578, + "x": [ + 0.0, + 1.0 + ] + }, + { + "coupling": 1.5240488844580098, + "exact_regression": false, + "lambda": 0.017486965455644866, + "lambda_j": [ + 0.00011220889833009725, + 0.01737436012410728, + 3.964332074908602e-07 + ], + "rho": 0.011474018736520828, + "x": [ + 2.0, + 1.5 + ] + }, + { + "coupling": 0.10866980368422749, + "exact_regression": false, + "lambda": 3.8652595311059566e-42, + "lambda_j": [ + 0.0, + 3.5238285161640815e-42, + 3.4143101494187483e-43 + ], + "rho": 3.5568846174946813e-41, + "x": [ + -1.5, + 0.5 + ] + }, + { + "coupling": 1.0073636982372143, + "exact_regression": false, + "lambda": 0.08448435124554406, + "lambda_j": [ + 0.0008267081536308076, + 0.04194196642966962, + 0.04171567666224363 + ], + "rho": 0.08386678157390745, + "x": [ + 3.0, + 3.0 + ] + } + ] + }, + "scientific_effect": "NONE", + "torus": { + "k=0.5,b=0.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 3.057321620877225, + "integral_j": [ + 0.14801784939882576, + 2.361335154771501, + 0.547968616706899 + ] + } + }, + "k=0.5,b=1.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 2.236984148652817, + "integral_j": [ + 0.015459430315522183, + 1.2312277567375214, + 0.9902969615997732 + ] + } + }, + "k=1.0,b=0.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 2.5420634536888143, + "integral_j": [ + 0.11682034296282702, + 1.984198363640555, + 0.44104474708543256 + ] + } + }, + "k=1.0,b=1.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 1.812099277399927, + "integral_j": [ + 0.011577497666535907, + 1.0306880495051842, + 0.7698337302282068 + ] + } + }, + "k=2.0,b=0.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 2.1797266766703105, + "integral_j": [ + 0.09915874794192943, + 1.7024716491727685, + 0.3780962795556126 + ] + } + }, + "k=2.0,b=1.0": { + "L=6": { + "grid": { + "L": 6.0, + "h": 0.15, + "points": 800, + "symmetry": "z -> -z", + "u": "e_1" + }, + "integral": 1.5301856127083193, + "integral_j": [ + 0.01049099573621032, + 0.9104492495954051, + 0.6092453673767042 + ] + } + } + } +} diff --git a/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..792906abf --- /dev/null +++ b/frontiers/c6_remote_coefficient_numerics_20260930/SOURCE_FILES.json @@ -0,0 +1,56 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "certified": false, + "review_status": "AUTHOR_SIDE_NUMERICAL_NOTE", + "object": "CL-C6-REMOTE-COEFF-NUMERICS-20260930-v1", + "files": [ + { + "path": "NOTE.md", + "bytes": 18777, + "sha256": "aa15838fb0c842f1d24ab5bc883e3546f2ad993b719470df2556b35a9878cf69", + "git_blob": "47fc0fedf10d9e68b6eef69808c3f3f7993c06de" + }, + { + "path": "README.md", + "bytes": 1731, + "sha256": "453e5adc6464cbbe020fcd20d59c3dd91afaaba20c91f96854510abf96b10e97", + "git_blob": "79b0606281822f3ec2809a8b87317dae648508d0" + }, + { + "path": "RESULTS.json", + "bytes": 103899, + "sha256": "e9af3b52761946b57d6f5fa50fdf6ccd8d26610461e471a2e97618da1611e48a", + "git_blob": "3e4a7ddb850818c4477b99f678847e68ab2b92e4" + }, + { + "path": "SOURCE_MAP.json", + "bytes": 6900, + "sha256": "3f3be8f5877c667158aa7aad4249af83d8cbaba37f4e5b06ce7efb3913f84126", + "git_blob": "0d3f8b889d6c8f8e99ac14238d9f09e2889f73ab" + }, + { + "path": "remote.py", + "bytes": 38800, + "sha256": "e31571b6aaa27bbb88de22fe75e31ac23bbf446dbc2ddc521fd8421ae1fe0116", + "git_blob": "9553e3a95484c2dd19b0f83a19a6837cb5c23675" + } + ], + "nonauthor_reads": [ + { + "lane": "OpenAI/GPT-5.6 Sol", + "review_id": 5363739676, + "scope": "Slice A (exact structure) accepted; numerics not certified" + }, + { + "lane": "OpenAI/GPT-6 Astra", + "review_id": 5364077220, + "scope": "Slice B (formulation) accepted with two wording 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/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md b/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md new file mode 100644 index 000000000..7365ed9bc --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md @@ -0,0 +1,254 @@ +# Hard-negative fibre lift of the planar elder barrier + +Author-side deterministic contribution by OpenAI `concave_fibre_geometry`, 30 September 2026. This is a mathematical extension candidate, not a source acceptance decision. It changes no existing source, status, integration disposition or probability/count claim. Its planar input is §§2–6 of `local_elder_geometry_20260930/PROOF.md`, corresponding to the requested PR170 cut. Source-cut verification and acceptance provenance are separate obligations handled by the parent task. + +## 1. The planar input and the additional geometric condition + +Let + +\[ +P(X,Z)=2kX^3-\frac{3k}{2}X-\frac{k}{2} + +\frac{s}{2}Z^2+\frac a2(X^2-1/4)Z + +\frac\beta2XZ^2+\frac c6Z^3, +\] + +with \(k>0\), \(M=(-1/2,0)\), \(S=(1/2,0)\), +\[ +B=\beta-a^2/(12k),\qquad +D=(c-a\beta/(4k)+a^3/(72k^2))/2. +\] +Assume the strict typed condition \(s<-|B|/2\), and exclude +\[ +12kD^2+(s-B)^2(B+2s)=0,\qquad 12kD^2+B^3=0. +\] +Let \(\mathcal Y=\{S\}\cup C(P)\), where \(C(P)\) consists of the additional critical points in \(-kk\), the hard boundary obeys the relative drop estimate + \[ + g_i(x,z)\le G_i(x)-D_0\quad(x\in\overline B_R,\ z\in\partial F_i(x)). \tag{HD} + \] + A graph-centred ellipsoid tube, described below, is a concrete sufficient interpretation of this tube hypothesis. + +No uniform lower bound on the hard Hessian eigenvalues is needed if (HD) is given directly. No control is required outside this tube. Strict negativity by itself, without (HD) or an equivalent closed-barrier condition, does not suffice. + +The hard tube need not cover the whole of \(B_{R+1}\): the reduced height only needs to be defined there for the planar collar construction, whereas the full lift needs a tube over the smaller disk containing the resulting caps. The graph paths also remain in that smaller disk. + +## 2. Deterministic conclusion + +For each \(Y\in\mathcal Y\), let \(Y_i\) be the unique continued critical point of \(G_i\), taking \(S_i=S\), and let +\[ +h_i=\max_{Y\in\mathcal Y}G_i(Y_i),\qquad +\widehat Y_i=\Psi_i(Y_i,\zeta_i(Y_i)),\qquad +\widehat M_i=\Psi_i(M,\zeta_i(M)). +\] +For all sufficiently large \(i\), +\[ +\boxed{\quad d_{f_i}(\widehat M_i)=b_i+r_i^3h_i.\quad} \tag{EHD} +\] +Here \(d_f(M)=\sup_{\gamma(0)=M,\ f(\gamma(1))>f(M)}\min_t f(\gamma(t))\) is the ordinary global superlevel maximin. In particular, +\[ +\frac{d_{f_i}(\widehat M_i)-b_i}{r_i^3}\to h_*. +\] +On the actual global Morse distinct-value locus, the unique graph point \(\widehat Y_i\) attaining \(h_i\) is the actual elder merge saddle. If \(C(P)\) is empty, this is exactly the lifted pinned saddle. If \(C(P)\) is nonempty, a continued window saddle has height strictly above \(-k\) and preempts the pinned saddle. Limiting ties do not obstruct the exact statement: one compares the finitely many actual heights at index \(i\). + +The actual lifetime is therefore exactly +\[ +b_i-d_{f_i}(\widehat M_i)=-r_i^3h_i, +\] +and its normalized limit is \(-h_*\). This is a deterministic geometry/lifetime result. It supplies no higher-dimensional tilted-law exhaustion, intensity, coefficient or count statement. + +## 3. Graph regularity and full critical-point indices + +At each fibre stationary point, negative definiteness makes \(g_{zz}\) invertible. The ordinary implicit function argument gives \(\zeta_i\in C^1\) for \(g_i\in C^2\), and +\[ +D\zeta_i=-g_{zz}^{-1}g_{zx},\qquad +\nabla G_i=g_x,\qquad +\nabla^2G_i=g_{xx}-g_{xz}g_{zz}^{-1}g_{zx}. \tag{SC} +\] +Thus \(G_i\in C^2\) even though a \(C^2\) graph is unnecessary. Existence of an interior fibre maximum must be assumed or separately verified; strict concavity alone only gives uniqueness once such a point exists. + +A point of the tube is full critical if and only if it lies on the graph and its soft point is critical for \(G_i\). At such a point, for soft and hard tangent vectors \(v,w\), +\[ +(v,w)^T\nabla^2g_i(v,w) +=v^T\nabla^2G_i v + +(w+g_{zz}^{-1}g_{zx}v)^Tg_{zz}(w+g_{zz}^{-1}g_{zx}v). +\] +This is an explicit invertible triangular change of variables, so full inertia adds \(m\) negative directions to reduced inertia. A reduced maximum lifts to a full maximum; a reduced saddle lifts to a nondegenerate full saddle with one positive direction and \(d-1\) negative directions, the superlevel merge index. No parametric Morse normal-form theorem is invoked. + +## 4. A fixed soft collar works at every level above the actual highest saddle + +For completeness, this records the properties of the planar construction that the lift actually needs. + +The cubic gradient estimate off the second excluded algebraic set gives +\[ +|\nabla P(x)|\ge c_0|x|^2\quad(|x|\ge R_0). +\] +Choose \(R_0\) to contain all window roots, the pins, and a small oval about \(M\). Choose \(R\) sufficiently large to contain all doubled chords and satisfy +\[ +R-R_0>\frac{2k}{c_0R_0^2}+1. +\] +For all sufficiently large \(i\), reduced \(C^1\) convergence on \(\overline B_{R+1}\) gives +\[ +|\nabla G_i(x)|\ge(c_0/2)|x|^2 +\quad(R_0\le|x|\le R+1). +\] +There are no reduced critical points in \((h_i,0)\) inside \(B_{R+1}\). This is elementary finite-root stability: around any nondegenerate root \(Y\), the map +\(x\mapsto x-(\nabla^2P(Y))^{-1}\nabla G_i(x)\) +is a contraction on a sufficiently small closed ball and maps that ball into itself for large \(i\). It has exactly one root there. Outside these disjoint root balls, in a slightly enlarged compact height band containing \([-k,0]\), \(\nabla P\) has a positive lower bound; convergence excludes any other roots. The excluded transition set prevents an additional critical point at \(-k\), and the source proves that the only critical point at or above zero is \(M\). + +Fix \(t_0\in(h_*,0)\) sufficiently close to zero. Uniformly negative Hessians in a small ball about the exact pin give a small disk \(K_{i,0}\) with +\[ +G_i=t_0\text{ on }\partial K_{i,0},\qquad +t_0\le G_i\le0\text{ on }K_{i,0}. +\] +One can construct the disk without a Morse lemma: the radial derivative from \(M\) is strictly negative under the negative Hessian bound, so every ray meets level \(t_0\) once and the implicit function theorem gives its oval. + +For every \(h_ih_i\), choose \(h\in(h_i,\min(t,t_0))\): this is a compact full cap whose boundary is strictly below that target. These direct derivative checks supply all regularity used here. + +The barrel argument already proves global normalized death \(\le h\) for every \(h>h_i\) sufficiently close to \(h_i\). Letting \(h\downarrow h_i\) gives the exact upper bound \(\le h_i\). + +## 6. Exact lower bound on the graph, and actual pairing + +For each \(Y\in\mathcal Y\), use the moving soft critical chord +\[ +x_i(t)=M+t(Y_i-M),\qquad +q_i(t)=G_i(x_i(t)),\qquad 0\le t\le2. +\] +Exact criticality gives \(q_i'(0)=q_i'(1)=0\). Reduced \(C^2\) convergence gives convergence of these restrictions in \(C^2\) to +\[ +q(t)=P(Y)(3t^2-2t^3). +\] +The limit has \(q''(0)=6P(Y)<0\), \(q''(1)=-6P(Y)>0\), negative derivative on \((0,1)\), positive derivative on \((1,2)\), and \(q(2)=-4P(Y)>0\). Second-derivative signs near the two exact derivative zeros, followed by first-derivative convergence away from those zeros, give the same strict monotonicity for \(q_i\) for all sufficiently large \(i\). Therefore its exact minimum is \(G_i(Y_i)\), and its endpoint is older than birth. + +Lift the chord to the actual chart by +\[ +\gamma_{i,Y}(t)=\Psi_i(x_i(t),\zeta_i(x_i(t))). +\] +This path has exactly the same height restriction \(b_i+r_i^3q_i(t)\). The graph may curve sharply and its hard coordinates need not converge; no approximation by a full-dimensional straight chord is made. Its inclusion in the tube is all that is needed. There are finitely many such paths, so one sufficiently large index works for all of them. The path corresponding to an actual maximizer of \(G_i(Y_i)\) gives normalized global death \(\ge h_i\), proving (EHD). + +This also directly identifies the merge at that graph point. Just above \(h_i\), the chord's two sides lead respectively to the birth and to an older endpoint, and they are in different global superlevel components because death is \(h_i\). At level \(h_i\), the chord joins them at \(\widehat Y_i\). On the global distinct-critical-value locus, no other critical point has that same height. The Schur identity proves it is a full merge-index saddle. Hence it is the actual elder saddle under the same ordinary maximin convention as the planar theorem. No Morse–Smale assumption or separatrix identification is needed. + +## 7. A quantitative test for anisotropic hard extent + +A simple sufficient replacement for (HD) is a positive-definite matrix \(A_i(x)\), a fixed depth \(D_0>k\), and the graph-centred ellipsoid tube +\[ +F_i(x)=\{\zeta_i(x)+v:v^TA_i(x)v<2D_0\}. +\] +Assume this compact tube lies in the chart interior and, throughout every fibre, +\[ +g_{zz}(x,z)\preceq-A_i(x). +\] +Twice integrating along the hard segment from the stationary maximum gives +\[ +g_i(x,\zeta_i(x)+v)\le G_i(x)-\frac12v^TA_i(x)v. +\] +On the hard boundary the drop is at least \(D_0\); above a planar cap, \(G_i\le0\), so hard boundary height is at most \(-D_0<-k\). This proves the needed barrier. The theorem only needs this boundary estimate above the cap family; imposing it over all \(B_R\) is a convenient checkable condition. + +The semiaxes are \(\sqrt{2D_0/\lambda_j(A_i(x))}\). They may depend on \(i\), on the base point, and on direction. Thus anisotropic full-chart extent is harmless provided those ellipsoids actually fit. Uniform hard concavity combined with an arbitrarily thin chart is insufficient. In physical coordinates with straight hard fibres (or an affine map in the hard variables), if an unnormalized physical hard Hessian obeys \(f_{yy}\preceq-H_i(x)\), the corresponding physical requirement is a contained ellipsoid +\[ +v^TH_i(x)v\le2D_0r_i^3. +\] +When \(H_i\) has order-one eigenvalues, its physical semiaxes have order \(r_i^{3/2}\). This is a sufficient geometric scale, not an assertion that the actual stochastic model supplies it. + +For an arbitrary nonlinear chart, one must verify the normalized-coordinate hard Hessian/drop directly, or include the chart's extra second-derivative terms; a physical Hessian bound does not automatically transform by congruence away from a full critical point. + +The logically weaker boundary condition is that hard boundary height is at most \(h_i\) over every cap used for levels decreasing to \(h_i\). A fixed margin below \(-k\) avoids depending on the unknown actual winning height and verifies this at once. Some closed full-dimensional barrier condition is indispensable if the exterior is arbitrary; strict hard concavity is one useful means of producing its vertical part. + +## 8. Explicit vertical-escape counterexample without depth + +Choose +\[ +P(X,Z)=2X^3-\frac32X-\frac12-\frac12Z^2+\frac13Z^3. +\] +Here \(k=1,s=-1,a=\beta=0,c=2\), hence \(B=0,D=1\), and the excluded discriminants are \(10\) and \(12\). The four critical points have heights +\[ +P(-1/2,0)=0,\quad P(1/2,0)=-1,\quad +P(-1/2,1)=-1/6,\quad P(1/2,1)=-7/6. +\] +Thus \(h_*=-1/6\). + +Let \(\epsilon_i\downarrow0\), and let \(\chi\in C^\infty(\mathbb R,[0,1])\) be zero on \(( -\infty,1]\) and one on \([2,\infty)\). For definiteness use elder height \(A=\sqrt2\), and set +\[ +H_i(w)=(1-\chi(w/\epsilon_i))(-w^2) + +\chi(w/\epsilon_i)\,[A-(w-3\epsilon_i)^2], +\qquad F_i(X,Z,w)=P(X,Z)+H_i(w). +\] +On the chart \(\mathbb R^2\times(-\epsilon_i,\epsilon_i)\), +\[ +F_i=P-w^2, +\] +so the hard Hessian is the uniformly strictly negative scalar \(-2\), the unique fibre maximum graph is \(w=0\), and its reduced height is exactly \(P\) on every compact soft set. Yet the exterior has an older strict maximum at \((M,3\epsilon_i)\), of height \(A\). Along \(0\le w\le3\epsilon_i\), the first blend term vanishes for \(w\ge2\epsilon_i\) and is otherwise at least \(-4\epsilon_i^2\); the second term is nonnegative for sufficiently small \(\epsilon_i\). Hence +\[ +d_{F_i}(M,0)\ge-4\epsilon_i^2>-1/6. +\] +A maximin beginning at height zero is at most zero, so these actual death levels tend to zero, rather than to \(h_*\). The obstruction is already visible at the hard chart boundary: +\[ +F_i(M,\epsilon_i)=-\epsilon_i^2\to0. +\] +For a compact-manifold version, protect a soft neighbourhood containing the pins, the four critical points and the vertical path; for each \(i\) its normalized radius can be finite but increasing to infinity. Multiply/extend smoothly outside a larger neighbourhood into a coordinate ball of a compact manifold. This retains convergence on every fixed normalized soft disk and changes neither the chart hypotheses nor the exhibited path bound. Additional \(m-1\) hard directions can be appended as \(-|z'|^2\), giving the same failure in every \(d\ge3\). Generic arbitrarily small perturbations away from the protected chart and the protected older maximum can impose the global Morse distinct-value locus while preserving the vertical-path lower bound (with, for example, \(-5\epsilon_i^2\) in place of \(-4\epsilon_i^2\)). The height counterexample itself does not need this extra locus. + +It also realizes genuinely shrinking anisotropic charts. Choose any \(r_i\downarrow0\), take +\(\Psi_i(X,Z,w)=(r_iX,r_iZ,r_i^{3/2}w)\) +in a fixed coordinate ball, and set \(f_i\circ\Psi_i=b_i+r_i^3F_i\) on the protected region. For example, choose its normalized soft radius \(R_i=r_i^{-1/2}\); its physical radius is then \(r_i^{1/2}\), so it fits inside the fixed coordinate ball while containing every fixed normalized soft disk for all large \(i\). Extend outside the protected physical region. On the thin physical hard chart \(|y|\0` followed by the exact bytes. No source file or manifest was changed. + +## Deterministic graph, collar, cap and exact elder saddle + +**Pass, under the displayed fixed-target hypotheses.** SC's complete-jet coupling bounds the actual global C4 norm at a fixed spectral target. With positive hard eigenvalues, the physical hard Hessian stays uniformly negative on each needed soft disk of radius `Rr` and a hard ball of radius `c0r`. The hard-gradient forcing at zero is `O(r^2)`, including the `−tau_xxwj/8` term forced by the endpoint pins. Strict concavity and an inward radial derivative give the interior graph; uniqueness preserves its exact hard-coordinate pins. + +Implicit differentiation gives graph size `O(r^2)`, first soft derivatives `O(r)` and second derivatives `O(1)`. The graph substitution changes the physical potential and its first and second soft derivatives by `O(r^4)`, `O(r^3)` and `O(r^2)`. Rescaling two soft coordinates by `r` and height by `r^3` makes each correction `O(r)` in C2. No inverse hard bound from this pointwise argument is placed in the probability majorant. + +The essential geometric addition is verified: a graph-centered physical hard radius `rho r^(3/2)` fits since `Cr^2+rho r^(3/2)k`. The graph's own `r^2` width would only produce an `r^4` physical drop and would not close the `r^3` death window. + +Off Delta, the cubic gradient grows quadratically at infinity. On a large fixed annulus, C1 convergence retains a gradient floor. Finite-root stability, exact pins and the Sigma exclusion remove all active-strip critical points other than the finite continued set. The cutoff gradient flow therefore transports the small maximum oval to level `h`, and the annulus speed bound prevents its boundary reaching the fixed soft disk boundary within time less than `k`. The same sufficiently large index works for every `h_i=2`, `Lambda` is a retained hard eigenvalue. Holding `J,lambda2,...,lambdam` fixed leaves `U=J+Lambda` fixed during the `lambda1` integration. This does not assume that matrix entries or eigenvalues are independent. + +The actual second soft determinant factor is retained. The inner integral is exactly + +`integral_0^(DrU^2) lambda1(lambda1+CrU) d lambda1 = r^3[(D^3/3)U^6+(CD^2/2)U^5]`. + +Together with the original outer `r^2` weight, this gives an `r^5` unnormalized numerator. Enlarging the ordered `lambda1` interval and dropping its Gaussian factor are valid positive-majorant operations. Division once by the full `Z_r>=z_*r^2` yields the required `r^3` scale. + +Retaining `1{U>A}` in that same integral and using higher uniform residual moments gives the polynomial tail tending to zero. Retaining `1{lambda2<=epsilon}` gives the vanishing hard strip mass by Gaussian polynomial integrability, including higher-corank intersections. There is no inverse eigenvalue or inverse gap factor. The M4 exception has tilted mass `O(r^4)` and vanishes after `r^−3` scaling. + +Weyl's inequality and Hessian Lipschitz control convert endpoint tightness to actual midpoint jet tightness. At fixed `A,epsilon`, a midpoint `h2<=epsilon` on `U<=A` implies endpoint `lambda2<=2epsilon` for small radius. Sending the U tail to zero and then epsilon to zero proves E7, including negative midpoint hard values. This is exhaustion of the full selector-failure measure. No count event, near-occurrence tail, factorial estimate or remote-witness event is substituted. + +## Spectral kernel, null sets and actual marks + +**Pass.** The spectral change contributes one raw soft Jacobian `r`, the actual determinant weight is on the `r^4` soft-sector scale, and the original full normalizer is on the `r^2` scale. After failure rescaling the prefactor is exactly `c_m r^2/Z_r`. The Jacobian's one hard power and determinant weight's two hard powers give the displayed `product h_j^3`; all remaining hard Vandermonde factors are retained. + +The extended sign-frame construction is essential and is stated correctly. Normalized Haar is the reference angular measure with SC's multiplicity constant, while the anisotropic raw Gaussian density stays inside the integral. It is not silently replaced by an invariant or GOE law. Simultaneous changes of O, the complete tensor, and the soft root coordinate leave the physical location unchanged. + +On compact positive hard sectors, the two-short-column bound supplies integrable conditional domination. Exact H1 supplies pointwise convergence of the actual failure indicator on generic typed targets. The determinant-weight limit is continuous and zero on or outside the typed boundary, so no selector convergence is needed there. Sigma, Delta, repeated spectra and the additional position set `D=0` are null sets under the displayed absolute-continuity interface; none is a discarded positive leading mass. Repeated hard spectra need no inverse-gap estimate. + +Compact L1 convergence and E4–E7 give global total variation for the extended jet/failure measure. SC's deterministic integrability bound makes the limiting measure finite; open positive nonempty sectors make its mass positive. The mass is exactly SC's displayed local integral `a1+a2`, with no `beta_far` term. This consumes a deterministic kernel and integral, not SC's global count conclusion. + +The actual partner, normalized death and lifetime marks are claimed only weakly. On generic compact couplings H1 supplies the local winning root and older path; the graph's hard displacement divided by r vanishes. Conditional dominated convergence then gives the joint pushforward, and actual failure exhaustion removes sentinel/essential mass. Weak tightness with converging total masses implies actual failed-partner locality for every deterministic cutoff with `delta_r/r` tending to infinity. No growing-radius root enclosure is needed. + +The fixed-radius Morse null set is used correctly through an arbitrary fixed sequence tending to zero: disintegration and a countable intersection give the relevant joint almost-sure locus along that sequence. The deterministic graph/cap/chord argument then applies. Limits along every such sequence establish the stated limit without an unsupported common null set for uncountably many pinned laws. + +## Compact actual lifetime-density interface and retained boundaries + +**Pass for L1 at its exact population.** P is consumed with ERR's corrected congruence, BOREL's whole-field replacement, and REC's convergence-in-law and embedded-chart reading rules. P's actual radial intensity is `r A_r dr`, with the dimension powers already cancelled. Pushing forward by `ell=kr^3` yields `ell^(-1/3)A_r/(3k^(2/3))`. Multiplication by failure probability `r^3[(1-p_r)/r^3]` gives exactly the density-difference factor `A_r[(1-p_r)/r^3]/(3k^(5/3))` after division by `ell^(2/3)`. + +P supplies the required compact common bounds; these are not inferred from pointwise topology. Dominated convergence therefore gives a finite positive coefficient C_fail for the compact candidate-minus-selected density. The original full z0 cancels against A0 if substituted, with one original pin density remaining. Transverse-frame independence follows from the physical probability and its unique limit, not from false torus rotational invariance or higher-dimensional birth cancellation. + +This density is not a replacement-bar intensity: a maximum may participate in multiple rejected candidates. It is not a standalone second-order selected-density expansion, an unrestricted refined difference, a numerical coefficient, a new count/moment theorem, or a statement uniform in vanishing k, growing dimension, volume or escaping marks. Those boundaries are retained. No numerical or engineering test was treated as mathematical evidence. + +## Recorded disposition + +Initial exact cut: **AMEND**, for the six literal/scope items above. Substantive H/F/M/L1 reasoning: no load-bearing defect found at the stated conditional source scope. All failures and requested corrections are preserved in this report. Publication or integration of a revised cut requires its own exact reread; this report does not accept author changes it has not seen. + +## Intermediate reread: residual coordinate label retained as AMEND + +The first author-corrected PROOF.md had 23415 bytes, SHA256 `fa405e286cb1cc337f57bffa671035a533818a0dc31196874005e1e300c4454c`, Git blob `8c1db93878253a3aa53f48cee1062ce9f9044cb8`. The corrected SOURCE_INTERFACE.md had 18098 bytes, SHA256 `cec41136dfac1fde2247f25dcf7601e1a01a6ec25c4d5f18e8bf1450e8360e48`, Git blob `7eb203bdf1311ff56b312962ba3f5ce6d5dd9766`. I reread both complete bodies and independently checked their identities. HARD_FIBRE_LEMMA.md and SOURCES.json were unchanged. + +The finite-r/contact derivative, exact/limiting lifetime, manifest locator, dependency roles, P lifetime scope, normalized pin arguments and cumulative-sentence removal were corrected. However, PROOF.md line 157 still called the normalized moving soft chord a chord “from M”, after explicitly reserving M for the physical pin. Both my read and the read-only geometric reviewer retained **AMEND** for that remaining instance of the original coordinate finding. The deterministic lemma itself remained correct, and the law reviewer accepted its corrected slice conditionally. This intermediate failed precision outcome is not erased by the final reread. + +## Final exact reread and source-scoped disposition + +The authors changed the remaining soft-chord label to M0. I independently verified and reread the complete final PROOF.md; the complete corrected SOURCE_INTERFACE.md was already reread at its unchanged final identity. The other load-bearing packet files are unchanged. The final reviewed cut is: + +| File | Bytes | Git blob | SHA256 | +|---|---:|---|---| +| PROOF.md | 23416 | `923d3236b5291a29bd162937f172190db84dcef5` | `5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e` | +| HARD_FIBRE_LEMMA.md | 18907 | `7365ed9bcf145c832d1c09f0b685fbe8d6695050` | `434ffd30ac96eb635211ea627459b78c1d39806f75b0634e0f4876dce0765b33` | +| SOURCE_INTERFACE.md | 18098 | `7eb203bdf1311ff56b312962ba3f5ce6d5dd9766` | `cec41136dfac1fde2247f25dcf7601e1a01a6ec25c4d5f18e8bf1450e8360e48` | +| SOURCES.json | 7186 | `f072e1bb5c79456be3c79451b6df5b287807edfa` | `b97a29ea09f8e8270bfddda8f364f0d61e431e30c1681af4b019f3fb48709a4d` | + +**Final disposition: ACCEPT at the displayed conditional source scope.** All initial literal precision findings and the intermediate residual label are resolved in this exact cut. No remaining mathematical or packet-precision AMEND was found. I did not edit any packet file, author any amendment to its proof, or make a remote write. + +This acceptance covers the contained negative hard-tube lift and its thin-domain falsifier; exact identification of the actual highest continued elder saddle with arbitrary exterior; full retained endpoint-matrix failure exhaustion and hard-boundary removal; the extended spectral jet/failure law in total variation with coefficient a1+a2; the weak actual joint partner/death/lifetime law and actual failed-partner locality; and the positive ell^(2/3) compact candidate-minus-selected lifetime-density coefficient L1 at its exact population. + +The hypotheses remain conditions. In particular the Gaussian/regression/cap/normalizer/Borel interfaces are consumed at the immutable identities above, with ERR, BOREL and REC's reading rules. LOCAL remains the exact public draft source identified by the manifest; its prior same-provider noncontributor review is not an external verdict on it or on this successor. SC's displayed analytic and deterministic spectral integral interfaces are consumed without promoting its global count theorem. Source verification, publication and finite checks supply no mathematical acceptance by themselves. + +The final acceptance supplies no replacement-bar intensity, standalone count or moment extension, register/status action, numerical enclosure, unrestricted refined lifetime difference, separate selected-density second-order expansion, or new parameter uniformity. It carries zero organizational-independence credit. The initial and intermediate AMEND records above remain part of the honest review history. diff --git a/frontiers/concave_fibre_elder_20260930/PROOF.md b/frontiers/concave_fibre_elder_20260930/PROOF.md new file mode 100644 index 000000000..923d3236b --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/PROOF.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/concave_fibre_elder_20260930/README.md b/frontiers/concave_fibre_elder_20260930/README.md new file mode 100644 index 000000000..cd78f41c8 --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/README.md @@ -0,0 +1,43 @@ +# From the soft plane to the actual elder partner in fixed d>=3 + +[PROOF.md](PROOF.md) extends the planar actual-pair theorem through negative +hard fibres and identifies the full higher-dimensional selector-failure law. +This is an additive mathematical candidate, AUTHOR_SIDE / HOLD, conditional +on the exact public source interfaces and pending qualified nonauthor review. + +The geometric step is a contained hard tube of width r^(3/2): its height drop +matches the r^3 death window and closes every transverse escape. The actual +critical graph itself has hard displacement O(r^2), which is too narrow to +provide that barrier. The full proof and an explicit thin-tube falsifier are +in [HARD_FIBRE_LEMMA.md](HARD_FIBRE_LEMMA.md). Graph-lifted critical chords and +thickened compact planar caps identify the actual global merge saddle as the +highest continued local window saddle for all sufficiently small r. + +Retaining the residual/hard-jet tail mark in the parent's endpoint-matrix +integral exhausts the ACTUAL failure measure. No count event or factorial bound +is substituted. It gives the full extended spectral-jet failure measure in +total variation, coefficient (1-p_r)/r^3 -> a1+a2, and the weak actual +replacement-location/death/lifetime law. The remote singleton count coefficient +does not enter. Actual failed partners localize inside every shrinking cutoff +whose ratio to r tends to infinity. + +The exact compact birth/gap candidate-minus-selected density consequently has +an identified positive ell^(2/3) coefficient in every fixed dimension. This is +the parent's exact universal radial ledger, not a new dimension-dependent +power, a replacement-bar counting formula, or a standalone second-order +selected-density expansion. + +[SOURCE_INTERFACE.md](SOURCE_INTERFACE.md) displays the analytic graph and full +matrix-tail derivations. [SOURCES.json](SOURCES.json) links all 11 full public +sources with immutable identities and separates required inputs from context. +[RECONNAISSANCE.md](RECONNAISSANCE.md) records the limited primary-source +comparison without a novelty claim. [NONAUTHOR_REVIEW.md](NONAUTHOR_REVIEW.md) +preserves the initial and intermediate AMEND findings and the final conditional +source-scoped ACCEPT of the corrected exact prose. It supplies zero +organizational-independence credit; qualified external review remains pending. + +All current contributors are OpenAI, with zero organizational-independence +credit. No existing proof, count packet, register, graph, source status, prize, +Boolean, workflow or active peer branch is changed. Uniformity in varying d, +volume or k->0, unrestricted refined lifetime differences and numerical +coefficients remain outside this note. diff --git a/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md b/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..b6511fb8a --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/RECONNAISSANCE.md @@ -0,0 +1,32 @@ +# Primary-source reconnaissance for the hard-fibre elder step + +30 September 2026, OpenAI author-side reconnaissance. No novelty or literature +priority claim. This is evidence of the actual access depth, not acceptance. + +The classical comparison is reduction along nondegenerate Hessian directions. +The primary abstract of Guangcun Lu, *The splitting lemmas for nonsmooth +functionals on Hilbert spaces I*, arXiv:1211.2127v2, +https://arxiv.org/abs/1211.2127v2, was read. It discusses a splitting theorem +and critical-group results near a critical point under weaker smoothness. Only +the author abstract and bibliographic metadata were inspected; its 68-page +proof was not read and no theorem from it is used in this packet. + +Searches also exposed Golubitsky–Marsden's *The Morse Lemma in Infinite +Dimensions via Singularity Theory*. The Ohio State reprint URL and Caltech +author-record URL returned restricted-fetch errors in the available public +retrieval service. No full-source exposure or proof consumption is claimed for +those attempts, and no alternate credential or access path was used. + +The inference guiding this packet is narrower than a splitting theorem: +critical-point reduction alone does not determine which labelled maximum +merges in the arbitrary exterior. The new argument proves its own Schur +completion, fibre concavity drop, compact full-dimensional cap and graph path. +HARD_FIBRE_LEMMA §8 explicitly falsifies elder transfer from reduced graph +convergence alone. No black-box splitting lemma supplies the hard-side depth. + +All mathematical dependencies actually consumed are the full public repository +sources in SOURCES.json, read at their exact immutable identities. SC supplies +the local analytic graph but explicitly excludes elder selection; the new +barrier is therefore a separately reviewed consumer step. The retained matrix +tail and actual-pair lifetime pushforward are proved in PROOF.md and +SOURCE_INTERFACE.md rather than inferred from a count theorem or a paper title. diff --git a/frontiers/concave_fibre_elder_20260930/SOURCES.json b/frontiers/concave_fibre_elder_20260930/SOURCES.json new file mode 100644 index 000000000..f072e1bb5 --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/SOURCES.json @@ -0,0 +1,121 @@ +{ + "repository": "d6g8k5htny-coder/Math-", + "public_main": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "planar_pr": 170, + "planar_head": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "scientific_effect": "NONE", + "review_independence_credit": 0, + "sources": [ + { + "id": "SC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "role": "required: \u00a7\u00a72,5,6 raw/full jet disintegration, orthogonal spectral Jacobian, actual hard graph C2 convergence, continuous typed weight, deterministic integral; global count law NOT consumed" + }, + { + "id": "CUB", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "role": "required: displayed original planar cubic, exact classifier and polynomial null sets; count expectations NOT consumed" + }, + { + "id": "EDL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_dimension_lift_20260928/PROOF.md", + "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_dimension_lift_20260928/PROOF.md", + "bytes": 23971, + "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", + "role": "context only: preceding actual root/index interface; no count or pairing conclusion consumed" + }, + { + "id": "DL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/d5_dimension_lift_20260929/PROOF.md", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "role": "scope audit only: dimension-lift count result excludes elder selection; no count conclusion consumed" + }, + { + "id": "LOCAL", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "role": "required: Theorem E compact planar collar and moving critical chords; position uniqueness off D=0; draft at exact stated source scope" + }, + { + "id": "P", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "role": "required: endpoint matrix density/residual, double soft-factor bound, full Z, global cap-failure inclusion, genericity/Borel and compact lifetime ledger" + }, + { + "id": "ELDERD", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_lower_all_d_20260929/PROOF.md", + "git_blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_lower_all_d_20260929/PROOF.md", + "bytes": 16485, + "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5", + "role": "context only: preceding elder-failure lower event; not used as a coefficient or upper barrier" + }, + { + "id": "ERR", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "role": "required reading rule: endpoint congruence correction" + }, + { + "id": "BOREL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "role": "required reading rule: whole-field Borel weighted Kac-Rice repair" + }, + { + "id": "REC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "role": "required reading rules and reviewed source scope: W1 convergence-in-law and embedded-chart radius" + }, + { + "id": "CAP", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "role": "required: deterministic global good-cap selection implication" + } + ], + "source_exposure": "Full public immutable sources. Required interfaces remain conditional on their displayed assumptions and actual review scope. Context sources are not mathematical premises." +} diff --git a/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md b/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md new file mode 100644 index 000000000..7eb203bdf --- /dev/null +++ b/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md @@ -0,0 +1,191 @@ +# Source-interface audit for a higher-dimensional elder bridge + +Read-only source audit, 30 September 2026. Same-provider OpenAI contributor; zero organizational-independence credit. No mathematical acceptance, scientific-status transition, existing-proof modification, PR action, or count-law promotion is asserted. The analytic lemmas reconstructed below are new author-side consumer steps requiring source-bound review. + +## 1. Exact public source cut + +Public `Math-/main` readback was `ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f`. Public PR170 is an open draft at head `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`, branch `codex/local-planar-elder-f68038be`. All listed files were fetched in full from the exact public commit. The scratch copies in `sources/` have been checked against the public Git blob identities, including UTF-8 byte lengths. Existing mutable source maps were not changed. + +| Tag | Exact path | Git blob | Bytes | Interface consumed | +|---|---|---|---:|---| +| SC | `frontiers/spectral_cluster_closure_20260929/PROOF.md` | `16c56821b52fd76b0be791622b9c3809eafde75a` | 25006 | §§2,5,6: raw midpoint regression, orthogonal spectral chart, physical anisotropic normal form, hard slaving, full weight, local spectral kernel and its integrability. Its count exhaustions are not premises. | +| CUB | `frontiers/planar_cubic_cluster_20260929/PROOF.md` | `bb446d08db8a944537a743ad550b88c1c2ad5758` | 19889 | C1–C15, Theorem C and Lemma S: original-coordinate cubic, classifier, stable roots, shear only for algebra. | +| EDL | `frontiers/elder_dimension_lift_20260928/PROOF.md` | `7303bd791a68a1139251f0f6e403a9f7cc89b006` | 23971 | Explanatory context only: A7–A28 compare physical jets, bounded shear charts and actual root/index constructions. SC supplies the orthogonal graph interface used here; no EDL count or pairing conclusion is a premise. | +| ELDERD | `frontiers/elder_lower_all_d_20260929/PROOF.md` | `7f41c9e315be0a7985d4c770a11a2cdf717bd46a` | 16485 | Context only: soft eigenplane path gives an elder-failure lower event, without an upper barrier or actual replacement partner. | +| DL | `frontiers/d5_dimension_lift_20260929/PROOF.md` | `9d82c707fdb17d3072a8930f26dabedf59e456fc` | 53727 | Scope audit only: §1.3 expressly excludes elder selection. No count/first-moment conclusion is consumed. | +| P | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | `dfed3b8d318a3ab1950957f393307733a4bef3f2` | 40261 | §§3–8: endpoint matrix density, independent full-field residual, original weight and full normalizer, cap-failure inclusion, genericity and maximin; §§10–12: exact compact radial/lifetime ledger and its domination interfaces. | +| CAP | `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` | `0633aca3c2a2882b0de4399da0a75d64c2e6b2e1` | 15160 | §§1–5: deterministic all-dimensional good-cap implication. | +| ERR | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` | `213594d6ca6a86fb938110f4d166d9ce275a02d0` | 1782 | Corrected endpoint congruence in P §5. | +| BOREL | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` | `fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a` | 9062 | Required whole-field Borel marked Kac–Rice replacement for P §9. | +| REC | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` | `75da2597971510f843f8d90c743950cb8c177342` | 23312 | §1 reading rule; §5 W1 convergence-in-law clarification and embedding radius. | +| LOCAL | `frontiers/local_elder_geometry_20260930/PROOF.md` at PR170 head | `ef2aa57959ea9f721bbf2316ce94cf616c1c9113` | 39722 | Theorem E, §§3–6: planar compact collar and exact highest continued partner; §8 scalar failure exhaustion. SHA256 `f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b`. | + +For the first ten rows the immutable URL is `https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/` followed by the path. For LOCAL replace the commit with `79f18f0cb318fceccd5bdf4e6dad29815165bc7d`. `SOURCES.json` gives complete URLs and SHA256 values. + +## 2. The supplied analytic graph interface + +Fix d≥3, m=d−1, one original axial frame, b∈R, k>0, and a midpoint spectral target. In SC's orthogonal coordinates + + A_mid = O diag(rs,−h2,…,−hm) Oᵀ, + 00 and fixed coupling C4 bound K, Taylor gives + + f_ww ≤ −h_min I/2, + |f_w(t,0)| ≤ C r², + |f_tw(t,w)| ≤ C r. + +The graph follows from strict concavity and the inward hard radial derivative. Implicit differentiation gives `|zeta_r|≤Cr²`, `|D_t zeta_r|≤Cr`, and `|D_t² zeta_r|≤C`. For `psi_r(t)=f(t,zeta_r(t))`, define `delta_r=psi_r(t)−f(t,0)`. Then + + |delta_r|≤Cr⁴, + |D_t delta_r|≤Cr³, + |D_t² delta_r|≤Cr². + +The Hessian identity used in the last bound is + + D_t² psi_r = f_tt−f_tw f_ww^(−1) f_wt + +evaluated on the graph. Each rescaled soft derivative contributes one factor r. Hence `(psi_r(rX,rZ)−b)/r³−(f(rX,rZ,0)−b)/r³=O(r)` in C². Inverse-hard constants are allowed at a fixed positive hard target; they must not be placed in the failure-tail majorant below. + +Since the original hard gradients vanish at each pin and the hard graph is unique, `zeta_r(M)=zeta_r(S)=0`. The reduced potential retains the exact critical pins and values 0,−k. Nondegenerate reduced critical points are actual full roots on the graph. Block congruence gives + + index(H_full)=(d−2)+index(H_reduced). + +Thus CUB's window saddles lift to actual index-(d−1) roots, and their hard coordinates are r² times `diag(h)^(−1)Q+o(1)`. Their physical coordinates divided by r tend to `(X,O(Z,0))` in the original axial/transverse frame. + +EDL's distinct construction uses the bounded determinant-one shear `y=(z,w−qz)` with q=D^(−1)v. Its original physical r-scaled location is `(X,Z,−qZ)` after the axial frame, not `(X,Z,0)`. In that chart sigma, not the raw a entry of A, is the soft curvature. Both SC's orthogonal chart and EDL's determinant-one shear preserve criticality, inertia and Hessian determinants; a general invertible linear change would introduce its squared determinant factor. SC's orthogonal chart is the appropriate global norm-controlled chart here. CUB's separate planar shear `u=X+aZ/(12k)` is only an algebraic change; reported physical X must be recovered as `u−aZ/(12k)`. + +## 3. Exact missing topological consumer step + +LOCAL Theorem E is stated on surfaces. It cannot simply be cited as a theorem about the full d-dimensional field. The new consumer needs to thicken its compact planar collar and control every hard boundary exit. + +The graph profile alone on an r²V box is insufficient. Hard side boundaries at that scale have a height drop of order r⁴, smaller than the r³ death window. A correct thickening uses a larger hard radius, for example `rho_r=A r^(3/2)` centered on zeta_r. Strict hard concavity yields + + f(t,w) ≤ psi_r(t)−(h_min/4)|w−zeta_r(t)|². + +Choose A with `h_min A²/4>k`. For every planar collar disk K_(r,h) supplied by the LOCAL §§4–6 construction for the actual reduced profile, psi_r≤b inside and psi_r=b+r³h on its boundary. The graph-centered hard tube over K_(r,h), of radius rho_r, is contained in the local chart for small r. Its soft boundary has height ≤b+r³h and its hard side boundary has height 0`, a witness event, or any remote-count estimate. + +Let `F_r={actual global ordinary partner of M is not S}` and let `B=−D_y²f(M)>0` on typed support. Order its eigenvalues + + 0[4/(3k)]r M3², r M4≤3k/10}. + +The fourth-derivative exception has tilted mass O(r⁴) by P (7.7) and the full floor (5.5), hence has zero r^(−3) limit. On the remaining depth branch, + + lambda1≤D rU². + +P (6.2), with its second soft factor retained, gives + + W_r ≤ C r² U^(2m) lambda1(lambda1+C rU). + +The endpoint matrix density has the Gaussian bound P (3.5). Its spectral Jacobian is bounded by `C Lambda^v`; the hard angular volume is finite. For fixed J and lambda2,…,lambdam, U is independent of lambda1. Thus for any retained event E depending only on J and the hard eigenvalues, + + E_Q[W_r;depth,E] + ≤ C r² E_J ∫ e^(−c sum_hard lambda_j²) Lambda^v U^(2m) + 1_E ∫_0^(DrU²) lambda1(lambda1+C rU) d lambda1 d lambda_hard + ≤ C r⁵ E_J ∫ e^(−c sum_hard lambda_j²) + Lambda^v U^(2m+6) 1_E d lambda_hard. (T1) + +The positive majorant permits extension of the lambda1 domain beyond its actual ordered interval; the Gaussian lambda1 factor can be dropped. The integral is exactly the r³ soft integral before the outer r² weight. Divide by the original full `Z_r≥z_*r²` once. + +With `E={U>A}`, every higher J moment and Gaussian polynomial integrability give, for every fixed p>0, + + limsup_(r→0) r^(−3) Q_r^W(F_r,U>A) ≤ C_p A^(−p). (T2) + +This follows directly from `U^q 1{U>A}≤A^(−p)U^(q+p)`, q=2m+6. It requires no common residual law for every r; the uniform moment bounds suffice. + +With `E={lambda2≤eta}`, averaging J's moments leaves an ordinary integrable Gaussian polynomial over the ordered positive hard eigenvalues. Its mass on the lambda2 strip tends to zero (a bound C eta for eta≤1 suffices). Consequently, + + lim_(eta→0) limsup_(r→0) + r^(−3) Q_r^W(F_r,lambda2≤eta)=0. (T3) + +This handles all higher-corank intersections in measure and never divides by a hard eigenvalue. It is a failure-measure statement, stronger for this purpose than SC §4's near-occurrence tail. + +## 5. Endpoint-to-midpoint conversion + +Let `mu1≤…≤mum` be the signed eigenvalues of `−D_y²f(0)`. Hessian Lipschitz control and Weyl's inequality give + + |muj−lambdaj|≤r M3/2≤C rU. + +With SC's midpoint variables `s=−mu1/r`, `h_j=mu_j` for j≥2 and rotated complete tensor tau, the depth branch satisfies + + |s|≤D U²+C U, + |h|+|tau|+K≤C U. (T4) + +T2 therefore implies full failure tightness in the actual midpoint spectral jets and in K. No replacement of the endpoint density by a midpoint density occurred in this tail argument. + +On U≤A, midpoint `h2≤eta` implies endpoint `lambda2≤2eta` for all sufficiently small r at each fixed eta,A. T3 then gives + + lim_(eta→0) limsup_(r→0) + r^(−3) Q_r^W(F_r,h2≤eta)=0. (T5) + +The error from U>A is sent to zero by T2 and the M4 exception is o(r³). In particular midpoint hard eigenvalues that are negative can carry no unidentified leading failure mass. Repeated hard eigenvalues need no inverse-gap estimates: the spectral chart boundary is null; on each compact sector its Jacobian is bounded and vanishes at a collision. Positive hard lower bounds are used only for pointwise graph convergence and removed using T5. + +For d=2/m=1, retain LOCAL §8's existing scalar near/far split and tail proof; T1 cannot be used with Lambda fixed because Lambda=lambda1. + +## 6. Exact resulting marked-law interface, conditional on the new geometry + +SC §2 disintegrates the raw physical midpoint (A,T) law, with density h_r→h_0. Its ordered orthogonal spectral transformation contributes `r c_m J_r`, where + + J_r=prod_(j≥2)(h_j+rs) prod_(2≤i0 the original typed weight has + + W_r/r⁴ → prod_(j≥2)h_j² w_+(s,a,beta), + w_+=9k²[B−2s]_+[−B−2s]_+, B=beta−a²/(12k). + +The continuous determinant-weight limit is zero outside the typed domain and on its boundary; no selector convergence is needed there. The full normalizer is P (5.4), with ERR and REC's W1 reading rule. For any bounded failure multiplier the exact scaled disintegration is + + (c_m r²/Z_r) J_r h_r(O diag(rs,−h)Oᵀ,Rot_O tau) + E[(W_r/r⁴) multiplier]. (T6) + +There is one raw soft-coordinate Jacobian r, one original W_r, and one full Z_r. On compact jets SC (6) and the two short-column determinant bound SC (15) provide an integrable coupling envelope without using a count event. Conditional on the deterministic fibre bridge, pointwise failure convergence is `1_F→1{n>0}` off LOCAL's polynomial null sets. The actual death and location converge to h_* and the winning physical root. Dominated convergence proves the compact marked failure law. T2–T5 exhaust the actual failure measure, so no near-occurrence exhaustion is substituted. + +The candidate finite limiting failure measure is + + dmu_fail = 1{n(s,a,beta,c)>0} (c_m/z0) + h_0(O diag(0,−h)Oᵀ,Rot_O tau) + prod_(j≥2)h_j³ prod_(2≤i=2, L>0, b real, k>0 and an axial/transverse orthonormal frame. Use the +exact variance-one periodized field, original endpoint observations and the +one full determinant-tilted pair-Palm law Q_r^W of the sources. Write F_r for +failure of the prescribed saddle to be the ordinary elder partner of the +prescribed maximum. Let T_r be the ACTUAL replacement-partner displacement +from the midpoint divided by r and let + + lambda_r=(b-d_f(M))/(k r^3). (1) + +Out-of-chart or essential partners may be given an isolated sentinel. The +source interface says their probability on F_r is o(r^3). We do not identify +an arbitrary witness or farthest root with an elder partner. + +The two exact input types are deliberately separate: + +- **ELDER2:** PR170 at79f18f0cb318fceccd5bdf4e6dad29815165bc7d, + PROOF.md, Theorem E and Sections8.6-8.8. In d=2 it supplies the actual highest + local saddle, the full failure measure and its weak location/lifetime marks. +- **ELDERD:** PR175 at9c6a73489c5ed2465a3cf65dcc950eeeae074b05, + PROOF.md, H1/S1-S3/M1. In d>=3 it supplies the hard-negative-fibre bridge, + the extended spectral failure measure, and the actual weak marked law. +- **R:** PR166 RADIAL_TAIL.md at80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337, + R4-R13, supplies the exact root-resolved density of the limiting microscopic + measure and its Gaussian majorant. This is used to integrate the already + identified selector, NOT to establish actual elder pairing from counts. +- **C:** PR169 PROOF.md at56a7904f9ad7d409c31e23195c0e7eeab230613a, + E10-E25, supplies the deterministic companion/height map, inner/outer sectors + and exact shape integrals. Its whole-cluster maximum theorem is not used as + an elder theorem. +- **D:** COMPANION_RATIO.md at that same PR169 head, D1-D7, supplies the exact + unit-square density/height formulas. Only Section6 below needs this input. + +All complete commit/path/blob/SHA256 bindings are in SOURCES.json. ELDER2 and +ELDERD are distinct author-side source candidates with source-bound reviews; +this consumer is conditional on their displayed mathematical interfaces. It +neither promotes their status nor discharges a missing premise by citation. +The prior source proofs and review bodies remain unchanged. + +The common limiting microscopic measure is denoted M. Its restriction to +n>0, where n is the number of extra strict-window cubic saddles, is the finite +failure measure mu_F with mass a_F=a1+a2>0. It is NOT the global nonempty count +measure: remote singleton mass is absent. With theta=(s,h,O,tau), + + dM=(c_m/z0) H(h) h0(A0,Rot_O tau) + 9k^2(4s^2-B^2) ds dh dO dtau, (2) + +on the positive hard/typed domain. Here H is the ordered hard spectral +polynomial, A0=O diag(0,-h)O^T, B=beta-a^2/(12k), and s<-|B|/2. +The scalar case has empty hard products; a normalized soft sign-frame +augmentation changes no physical integral. There is one full z0 and no +additional pin density or root multiplicity. + +The actual interface is finite-measure weak convergence, with converging masses, + + r^-3 Q_r^W((T_r,lambda_r) in dot,F_r) + -> pushforward of mu_F by (T_*,eta_*), (3) + +where T_* is the root with GREATEST cubic height P, and eta_*=-P(T_*)/k. +Thus eta_* is the SMALLEST downward height among the window roots. +The assertion that this is the actual elder partner is precisely the elder +source premise, not a conclusion of the polynomial computations below. + +**Order:** first r->0. Then the radius threshold t->infinity inside (3)'s limit. +Any further small-lifetime or large-companion limit is taken third. No uniform +simultaneous limit, finite-r inverse moment, or full lifetime-intensity claim +is made. + +## 2. A once-per-elder selector with a fixed shape domain + +Use the root variables of R/C: + + u=X+aZ/(12k), b0=3-12u^2, v=(-s)Z^2/k, + S={-3/20 on S. (4) + +A counted companion is given, off an algebraic null set, by + + Lc=4v^2-8b0uv+b0^2, + t_c=(4v^2-b0^2)/Lc, + u_c=[2b0v-u(4v^2+b0^2)]/Lc, v_c=v t_c^2. (5) + +C proves that the map is an involution, t_c<0 on doublets, and that for an +outer-|Z| root eta_ct). Then + + **Theorem 1.** + G_eld(t) ~ C_cusp E0 t^-11, + mu_F(n=2,|T_*|>t) ~ C_cusp D t^-11. (9) + +For every fixed t>0 the radius event is a continuity set of the limiting +selected measure, and therefore + + lim_(t->infinity) t^11 lim_(r->0) + r^-3 Q_r^W(F_r, |T_r|>t, T_r finite)=C_cusp E0. (10) + +Conditional on a distant actual failed partner, in that same iterated order, +its lifetime fraction and selected shape have the universal laws in Section4. +This is candidate pair-Palm sampling given failure; it does NOT count each +replacement persistence bar once across different rejected candidates. + +Proof: inserting (7) into R's root integral gives exactly + + (108 k^7 c_m/z0) Q(u,v)1_E 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. (11) + +The selector 1_E is independent of Z. The radius is still +|T_*|^2=(u-aZ/(12k))^2+Z^2. Under Z=t z it tends, after division by t, to +gamma(a)|z|. For t>=3 the radius condition implies |z|>1/(2gamma(a)). +The source majorant is a polynomial in h times + + Q 1_E |z|^-12 exp[-c(|h|^2+a^2+|xi|^2)] + 1{|z|>1/(2gamma(a))}. + +Integrating z costs only a factor proportional to gamma^11; the remaining +integral is finite. Dominated convergence yields the common C_cusp times the +shape mass E0. Restriction to counted doublets replaces E by the inner sector +and gives D. No hard-eigenvalue inverse or unproved radius ordering enters. + +For each t>0 and fixed u,a, the radius equation is a nonconstant quadratic in +Z with at most two solutions, hence null in (11). Thus |T_*| has no positive +atoms. Finite-mass weak convergence in (3), including negligible sentinels, +allows the unbounded exterior-radius indicator by Portmanteau. A bounded +continuous function of the other marks can be inserted as well, since its +only new discontinuity is that sphere. The denominator G_eld(t)>0 for all t, +by the positive tail at large t and monotonicity. This proves the fixed-t +conditional transfer and then (10). No uniformity in a moving t(r) is used. +QED. + +Some exact coefficient ratios, independent of the model amplitude, are + + G_eld(t)/F_point(t) -> E0/I=296269/657408, + G_eld(t)/G_max(t) -> E0/(I-D), + mu_F(n=2,|T_*|>t)/G_eld(t) -> D/E0. (12) + +They are approximately0.450662298,0.457796706,0.0345807201, respectively. +These are three different denominators. The final doublet probability concerns +the local cubic configuration of an extreme elder, not the global torus count. +The role of n here is entirely inside the limiting measure; no global-count +moment theorem is needed to identify the actual partner. + +For nonnegative powers, the radius of this selected limiting law has moment +threshold11, and + + integral min(|T_*|,t)^11 dmu_F ~11 C_cusp E0 log t. (13) + +The proof is Tonelli applied to (9). It is not uniform integrability of the +finite-r spatial moments in (3). + +## 4. An explicit universal lifetime-fraction density + +In the extreme-selected limit, the shape density is Q1_E/E0. The normalized +radius q=|T_*|/t is independently Pareto(11), with density11q^-12 on q>1; +its sign is symmetric and the remaining cusp parameters have their original +H gamma^11 h0/J_cusp density. This follows from (11) with q=gamma(a)|z|, +using the same majorant. Projecting to physical spatial marks gives WEAK +convergence. No assertion of spatial total variation is required. + +**Theorem 2.** Let e in(0,1). Define + + p_lo(e)=sqrt(1-e)-1/2, + p_hi(e)=cos((1/3) arccos(2e-1)). (14) + +Equivalently p_hi is the unique root in(1/2,1) of +2p^3-(3/2)p+1/2=e. The limiting fraction lambda=(b-death)/(k r^3) of a distant +failed elder partner has density + + f_eld(e)=(6/E0) integral_(p_lo(e))^(p_hi(e)) + Q(-p,6(e+p-1/2)) dp. (15) + +The integrand is a polynomial, so (15) is an explicit algebraic density with +one radical and one specified cubic root. It is strictly positive on(0,1). +This is a density of the LIMITING conditional mark, not an assertion that the +original fixed-r persistence intensity has this density or converges in TV. + +Proof: eta=e sets v=6(e+p-1/2), dv=6 de. The full shape region at fixed e gives + + sqrt(1-e)-1/2 < p < 1/2+sqrt(e). + +For p>1/2 write x=p-1/2. The lower shape condition becomes x^2=1 the entire +shape interval is outer. It follows that O2 removes exactly + + p_hi(e)1/2; +4cos^3(theta)-3cos(theta)=cos(3theta) gives (14). Nonnegativity, normalization +and strict positivity follow from the original selected shape integral. + +Every nonnegative integer moment is rational. To see this directly, integrate +Q(u,v)(u+1/2+v/6)^j over S and subtract the same polynomial integral over the +two strips (6), all of which have polynomial bounds and rational endpoints. +The first three unnormalized moments are + + E0=888807/280, + E1=3524743221/1361360, + E2=190490806059/87127040. + +Thus + + E[lambda]=1174914407/1440459878, + E[lambda^2]=63496935353/92189432192. (16) + +They are approximately0.8156522961481611 and0.6887658795940618. +Neither is the point-intensity nor the maximum-anchor lifetime mean. All +shape factors are universal in the fixed model parameters; directions and +C_cusp need not be universal. QED. + +For reproducible evaluation, with x=p-1/2 the unnormalized density integrand is + + 10368(e-x^2)(e+x^2+2x)(e+2ex+x^2). (17) + +Its integration endpoints are sqrt(1-e)-1 and the nonnegative root of +3x^2+2x^3=e. The companion uses rational bisection plus interval polynomial +evaluation to enclose (15), not unconstrained floating quadrature. + +## 5. A fourth-power small-lifetime law and its inverse-moment threshold + +**Theorem 3.** In the limiting probability law of Section4, + + f_eld(e) ~ (3328/E0)e^3, + P(lambda<=e) ~ (832/E0)e^4=(232960/888807)e^4, + f_eld(1-) =1312200/296269. (18) + +Consequently E[lambda^-p] is finite exactly for nonnegative p<4. At the critical +order, the capped inverse moment has asymptotic + + E[min(lambda^-1,T)^4] ~ (3328/E0)log T. (19) + +All three limits occur INSIDE the already-formed extreme-elder mark law: r->0, +then distant-partner radius->infinity, then e->0 (or T->infinity). Weak marked +convergence alone would not justify analogous finite-r inverse moments. + +Proof: use (17), put x=sqrt(e)z and divide the density numerator by e^3. The +lower endpoint tends to0, the upper to1/sqrt(3), and the integrand tends to + + 10368*2z(1-z^4). + +The scaled intervals lie in a fixed compact set for all small e and the +integrands have a common polynomial bound. Hence their limit integral is +10368(1/3-1/81)=3328. Integrating this density asymptotic gives the CDF formula. +At e=1 the integral of (17) over x in[-1,1/2] is98415/7; division by E0 gives +the endpoint in (18). Continuity and strict positivity away from zero then +show that only the lower endpoint controls inverse-moment integrability. +Finally lambda^-1 has tail coefficient832/E0 at exponent4; Tonelli gives +4*(832/E0)log T in (19). QED. + +## 6. The more distant companion has a thirteenth-power ratio tail + +This section conditions the limiting extreme-ELDER law further on a doublet. +Let rho=|Z_inner/Z_outer| in(0,1) and z in(0,1) be D's unit-square coordinates +of the OUTER root. Write L(rho,z) for D2's unnormalized density: + + L=165888 rho z^7(z+1)^4 (rho z+1)^4(rho z+rho+1)^7 + /[(rho+1)^5(2rho z+rho+1)^9]. (20) + +Its total mass is J. The elder is the inner root. C E18-E19 give the deterministic +change-of-variables identity + + Q(T)|det DT|=Q|t_c|^11, integral rho^11 L =D. + +**Theorem 4.** Under the limiting extreme elder law given a doublet, (rho,z) +has density + + rho^11 L(rho,z)/D. (21) + +It is NOT L/J. The scalar radial ratio q=|T_*|/threshold remains independent +Pareto(11). In physical positions divided by that threshold the configuration is + + { epsilon q e_*, -epsilon q rho^-1 e_* }, (22) + +where the first point is the actual limiting elder and the second the farther, +lower saddle. Its heights are respectively + + eta_eld=rho^3 z^2(z+1)(rho z+rho+2) + /[(rho+1)(2rho z+rho+1)], + eta_other=z^2(rho z+1)(rho z+2rho+1) + /[(rho+1)(2rho z+rho+1)]. (23) + +Let V=R_other/R_eld=1/rho in this limit, and H0=6401/3960. Then + + P(V>T | extreme elder, doublet) ~ C13 T^-13, + C13=165888 H0/(13D)>0. (24) + +Its nonnegative moments are finite exactly below13. Conditional further on +V>T, (T/V,z) converges in total variation on the fixed scalar-coordinate +unit square to independent densities13y^12 and z^7(z+1)^4/H0. In particular + + eta_eld V^3 -> 2z^2(z+1), eta_other -> z^2 (25) + +as a further weak-mark consequence. No spatial TV is claimed for (22). + +Proof: the inner-selected shape measure is Q on the inner sector. Pull it +back to the outer sector through the companion involution. The weighted +Jacobian gives exactly the factor rho^11, and then D's unit-square bijection +gives (21). Equivalently the physical inner-radius cutoff has asymptotic +rho gamma(a)|Z_outer|>threshold, giving the same rho^11 after radial integration. +It is not legitimate to retain the old maximum-conditioned doublet density. +The linear transformation of the sign and the radial change q=rho gamma|Z| +yield independence and (22)-(23). + +The rational factor multiplying165888 rho z^7(z+1)^4 in (20) converges uniformly +to1 as rho->0 and is bounded on the closed square. Thus the numerator of (21) +is asymptotic to165888 rho^12 z^7(z+1)^4, whose integral over rho<1/T gives +165888 H0/(13D) T^-13. Substituting rho=y/T and dividing by that mass proves +the normalized L1/TV claim. The uniform height limits of (23) give (25). +Moment integrability follows since V>=1 and the tail coefficient is positive. +The density L/J from maximum selection instead has a second-power tail for +1/rho; that is a different measure, not a contradiction. QED. + +## 7. Review obligations and excluded conclusions + +A: actual-selector composition (1)-(11), matching ELDER2/ELDERD's failure +measure to the root density without a count substitute; no lost sign/Haar/ +normalizer; fixed-t continuity and the iterated-limit order. +B: selected shape interval, all polynomial integrals and moments, density +(15), fourth-power endpoint and inverse-moment threshold (18)-(19). +C: rho^11 reweighting, thirteenth-power companion ratio, scalar TV versus +moving spatial marks, and the independence/height statements (21)-(25). + +The tests corroborate exact algebra, root domains and interval arithmetic. +They do not prove the Gaussian or actual-pairing sources, grant the new +consumer independent acceptance, or formalize these theorems in Lean. +No original r-limit rate, growing-volume/vanishing-k uniformity, finite-r +inverse moments, unrestricted lifetime density, replacement-bar intensity, +remote singleton elder coefficient or across-pair independence is claimed. +The finite physical-radius ordering counterexample in C is retained: the +elder selector is based on height before taking any radius limit. diff --git a/frontiers/elder_partner_extremes_20260930/README.md b/frontiers/elder_partner_extremes_20260930/README.md new file mode 100644 index 000000000..8c3d59674 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/README.md @@ -0,0 +1,73 @@ +# Distant elder partners: a selected lifetime law, not a maximum-cluster law + +OpenAI / GPT-6 Astra Pro. **Author-side conditional candidate; new nonauthor +review OPEN. Scientific effect NONE. No self-merge.** + +The actual elder rule selects the HIGHEST saddle. On a cubic doublet that is +the inner-|Z| saddle, the opposite selector to the extreme-cluster maximum +anchor. This packet composes the actual failure/partner interfaces of PR170 +(d=2) and PR175 (d>=3) with the exact microscopic root measure of PR166/169. +It does not establish actual pairing merely from a window-count law. + +## Four new conditional results + +1. The limiting actual selected partner has radius tail + `C_cusp (888807/280) t^-11`. Its coefficient is I-J, not the point coefficient + I or the maximum-cluster coefficient I-D. Only about3.458% of these distant + selected partners belong to doublets. The nonempty failure measure excludes + the remote singleton coefficient. +2. Its extreme-selected lifetime fraction e has an explicit algebraic density: + + f(e)=(6/E0) integral_[sqrt(1-e)-1/2]^[p_hi(e)] + Q(-p,6(e+p-1/2)) dp, + 2p_hi^3-(3/2)p_hi+1/2=e, p_hi in(1/2,1), E0=888807/280. + + All nonnegative integer moments are rational. The mean is exactly + 1174914407/1440459878, about0.8156522961481611. +3. The small-lifetime CDF is asymptotic to `(232960/888807)e^4`, so inverse + lifetime moments have threshold4 in THIS limiting probability law. No + finite-r inverse-moment convergence is inferred from weak convergence. +4. Given a doublet, selecting for large elder radius reweights the former + companion-ratio square density by rho^11. The other/elder radius ratio has + tail exponent13, rather than exponent2 under maximum-cluster selection. + +PROOF.md gives the full arguments, exact scope and fixed-threshold transfer. +The order is **first r->0, then the selected microscopic radius t->infinity**; +small-lifetime and relative-companion limits occur inside that second limit. +No uniform simultaneous limit, unrestricted persistence-lifetime density, +replacement-bar intensity, original finite-r rate or Gaussian coefficient +numerical enclosure is claimed. + +## Source and review boundary + +SOURCES.json binds five complete historical commit/path/Git-blob/SHA256 identities. +ELDER2/ELDERD remain separate author-side source candidates with their own +source-scoped reviews; this new consumer is conditional on their stated actual +selector/failure-measure interfaces. Linking them neither promotes their status +nor revalidates their Gaussian/topological proof. C and D were authored by this +conversation. Same account/provider is not independent scientific acceptance. + +Requested scopes: A actual-elder source composition and selected tail; B explicit +lifetime density/moments/endpoints; C rho^11 reweighting and exponent13. No new +analytic review is inferred from a code pass or an old source review. + +## Replay + +In a full checkout containing the pinned source objects: + + python -B -S frontiers/elder_partner_extremes_20260930/verify.py + +Standalone packet: + + python -B -S verify.py --local-only + +The latter reports source_pins_checked=false. The hosted workflow must fetch and +verify all five actual sources. It checks declared sizes where supplied, all +SHA256/blob identities and regular Git path modes, with replacements disabled. +Local Git fixtures test behavior; they do not authenticate project sources. + +The standard-library suite has41 methods, run under ordinary and optimized +Python. It includes893 exact rational root-shape cases, direct unsheared cubic +checks,12 independent inner-root Jacobians, polynomial interpolation, rational +root-isolation density intervals and13 real-Git/inventory tests. Six semantic +mutants must reject in each mode. No Lean formalization of these new theorems. diff --git a/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md b/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..8a4f041bc --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/RECONNAISSANCE.md @@ -0,0 +1,33 @@ +# Narrow primary-source reconnaissance — 30 September 2026 + +Actual web queries in this pass included: +- site.arxiv.org persistence elder rule Gaussian random fields critical points +- site.arxiv.org Palm theory extremes stationary regularly varying random fields anchored clusters Planinic +- site.arxiv.org "elder rule" "Gaussian" persistence + +Primary abstract/metadata pages opened: + +1. Hrvoje Planinic, Palm theory for extremes of stationary regularly varying + time series and random fields, arXiv:2104.03810. + https://arxiv.org/abs/2104.03810 + Existing framework: typical exceedances, typical clusters and their different + biases. This packet does not claim to invent anchored/Palm sampling. + +2. Bojan Basrak, Ilya Molchanov and Hrvoje Planinic, Extremal behavior of + stationary marked point processes, arXiv:2210.01474. + https://arxiv.org/abs/2210.01474 + Related marked extreme-process framework. Its hypotheses are not silently + imported as a theorem about our Gaussian pin pairs. + +Access depth: abstract/metadata only, not full-text theorem comparison. No +exhaustive novelty/priority claim or absence-from-literature conclusion is made. +No result from these pages is a missing proof step. The actual mathematical +inputs are the exact source-bound elder and root-density interfaces in SOURCES. + +The new point is source-specific: highest-height selection removes the OUTER +root sector, rather than the inner sector removed by maximum-radius anchoring. +The explicit lifetime density and companion exponent follow by recalculating +that selector's measure, not by renaming the maximum-cluster distribution. + +Consensus was not invoked in this pass (the current request names GitHub and +Superpowers). No changed quota or connector result is claimed. diff --git a/frontiers/elder_partner_extremes_20260930/RESULTS.json b/frontiers/elder_partner_extremes_20260930/RESULTS.json new file mode 100644 index 000000000..7c4015952 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/RESULTS.json @@ -0,0 +1,62 @@ +{ + "companion_over_elder_moment_threshold": 13, + "constants": { + "double_given_extreme_elder": { + "lower": "0.034580720107731639", + "upper": "0.034580720107731640" + }, + "elder_to_maximum_tail_ratio": { + "lower": "0.457796705679599976", + "upper": "0.457796705679599977" + }, + "elder_to_point_tail_ratio": { + "lower": "0.450662297994548286", + "upper": "0.450662297994548287" + }, + "mean_lifetime_fraction": { + "lower": "0.815652296148161094", + "upper": "0.815652296148161095" + }, + "outer_over_elder_ratio_tail_constant": { + "lower": "187.906062596204997257", + "upper": "187.906062596204997258" + }, + "second_lifetime_fraction_moment": { + "lower": "0.688765879594061834", + "upper": "0.688765879594061835" + } + }, + "height_density_values": { + "1/100": { + "lower": "0.000001295538683393", + "upper": "0.000001295538683394" + }, + "1/2": { + "lower": "0.411333720068101484", + "upper": "0.411333720068101485" + }, + "1/4": { + "lower": "0.039478044779395017", + "upper": "0.039478044779395018" + }, + "3/4": { + "lower": "1.649750418945205199", + "upper": "1.649750418945205200" + }, + "99/100": { + "lower": "4.279300396681162358", + "upper": "4.279300396681162359" + } + }, + "inverse_lifetime_moment_threshold": 4, + "mathematical_acceptance": false, + "rational_shape_cases": 893, + "scientific_effect": "NONE", + "scope": "finite algebra, not validation of Gaussian or elder premises", + "selected_shape_moments": { + "0": "888807/280", + "1": "3524743221/1361360", + "2": "190490806059/87127040" + }, + "small_lifetime_cdf_coefficient": "232960/888807" +} diff --git a/frontiers/elder_partner_extremes_20260930/SOURCES.json b/frontiers/elder_partner_extremes_20260930/SOURCES.json new file mode 100644 index 000000000..e32cf6444 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/SOURCES.json @@ -0,0 +1,50 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "scope": "Conditional exact interfaces; no parent revalidation or source-status change", + "sources": [ + { + "id": "R", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "use": "R4-R13 exact root integral and Gaussian majorant; not pairing from counts." + }, + { + "id": "C", + "commit": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "bytes": 21891, + "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", + "sha256": "b90cf745e2e67bc9b501c481552f7080d27c28245580668cf72aefa5b7f04715", + "use": "E10-E25 companion/height map, sectors, exact integrals. Author exposure." + }, + { + "id": "D", + "commit": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/COMPANION_RATIO.md", + "bytes": 10032, + "git_blob": "56348f6765793cd8115041ba367b821eb173f695", + "sha256": "74ff3af5ec071f57d40938fbeca6bca1c189272d5c1bc0440e41cbe764c04d23", + "use": "D1-D7 exact square-coordinate density and heights only; reweighted anew here. Author exposure." + }, + { + "id": "ELDER2", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "use": "d=2 Theorem E, failure measure and actual weak partner/lifetime law in Sections8.6-8.8. Conditional candidate interface, not promoted." + }, + { + "id": "ELDERD", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "use": "d>=3 H1/S1-S3/M1 actual selector/failure measure/weak marks. Conditional candidate interface, not promoted." + } + ] +} diff --git a/frontiers/elder_partner_extremes_20260930/VALIDATION.md b/frontiers/elder_partner_extremes_20260930/VALIDATION.md new file mode 100644 index 000000000..938346370 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/VALIDATION.md @@ -0,0 +1,45 @@ +# Author-side validation, not analytic acceptance + +The initial eight-method mathematics scaffold failed all eight assertions. +After implementation those eight passed. Expanded mathematics tests now total28. +The separate13-method source/inventory scaffold produced20 failing assertions, +including subtests. A local attempt to reuse an older verifier found no mounted +file at that path and left the scaffold unchanged; its failed replay was not +called success. The completed verifier was then written and all13 tests passed. + +Fresh final baseline:41 tests PASS in normal Python and41 PASS with -O, both +also under the hosted GIT_NO_REPLACE_OBJECTS=1 environment. Each mode executes +all six named semantic mutants (return1) and an unknown label (return2), with +identical deterministic baseline stdout and no stderr. Mathematical core: +-893 rational shape/selector/companion checks; +-40 direct original-cubic gradient/value/Hessian cases; +-12 independent exact inner-root Jacobians from the square coordinates; +-independent dense polynomial interpolation of moment integrals; +-rational bisection/interval enclosures for the algebraic height density; +-small-height and small-ratio finite diagnostics, strict boundaries and ties; +-the exact finite physical-radius-order counterexample retained from PR169. + +Custody tests use real temporary Git repositories: full identity fields, +missing/unsafe paths, omitted/duplicated IDs, noncommit tree/blob objects, +symlink payloads, historical replacement forgery and packet drift. These are +simulated failures, not observed tampering of the user's sources. + +All code is Python standard library. Runtime Python3.13.5; the target workflow +uses Python3.11.16 as in the existing repository. Exploratory symbolic algebra +was used privately to cross-check strip constants, not as a shipped dependency. +The final manuscript's rounded second-moment decimal was corrected against the +exact rational result before publication; no source file or earlier review changed. + +No complete project checkout could be obtained locally (the public raw-host +request failed DNS). Hence local-only replay explicitly reports +source_pins_checked=false. Historical authentication of the five ACTUAL source +objects remains the hosted requirement. The two elder source bindings supply +SHA256 and Git blob; the other three also include declared byte size. No size +was invented for an unread full-byte copy. A matching content hash is not a +proof that the source's mathematics or external review is correct. + +The new actual-source composition, infinite-dimensional domination and weak +transfer require source-bound nonauthor analytic review. Neither these finite +tests nor existing Lean workflow success formalizes this manuscript. Any +hosted success must be recorded against its actual head/run, not anticipated +in this author-time record. No self-merge or scientific register transition. diff --git a/frontiers/elder_partner_extremes_20260930/elder.py b/frontiers/elder_partner_extremes_20260930/elder.py new file mode 100644 index 000000000..9a97c39e6 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/elder.py @@ -0,0 +1,250 @@ +"""Exact controls for elder-selected microscopic extreme laws. + +Standard library only. This is finite algebra and interval arithmetic, not a +proof of the actual Gaussian elder-selector interface or the limiting theorem. +""" +import argparse +from fractions import Fraction as F +import json +from math import comb + +MUTANTS=('choose-outer','count-both','drop-half','wrong-lifetime-sign', + 'forget-radius-bias','wrong-height-power') +_MUTANT=None + + +def rational(x): + if type(x) not in (int,F): raise TypeError('exact int/Fraction required') + return F(x) + + +def b0(u): return 3-12*u*u + +def height(u,v): return u+F(1,2)+v/6 + +def shape(u,v): + return -F(3,2) 10368 int_0^(1/sqrt(3)) 2z(1-z^4) dz. + return F(0) if _MUTANT=='wrong-height-power' else F(10368)*(F(1,3)-F(1,81)) + + +def outer_over_elder_tail_exponent(): + return 2 if _MUTANT=='forget-radius-bias' else 1+11+1 + + +def imul(a,b): + vals=[x*y for x in a for y in b] + return min(vals),max(vals) + +def iadd(a,b):return a[0]+b[0],a[1]+b[1] + +def ipoly(coefs,x): + ans=(F(0),F(0)) + for j in range(max(coefs),-1,-1): + c=coefs.get(j,F(0));ans=iadd(imul(ans,x),(c,c)) + return ans + + +def _isolate(target,func,left,right,bits): + for _ in range(bits): + mid=(left+right)/2;value=func(mid) + if value==target:return mid,mid + if value=hi:continue + for j in range(1,20): + v=lo+(hi-lo)*F(j,20);sel=selection(u,v);pair=companion(u,v) + if pair and shape(pair[0],pair[1]): + if sel+selection(pair[0],pair[1])!=1:raise ValueError('one partner per configuration') + if sel in (0,1) and sel!=1-int(outer(u,v)):raise ValueError('height/strip selector mismatch') + elif sel!=1:raise ValueError('singleton must select its one root') + n+=1 + return n + + +def result(): + n=checks() + return {'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'finite algebra, not validation of Gaussian or elder premises', + 'rational_shape_cases':n, + 'selected_shape_moments':{str(j):str(moment(j)) for j in range(3)}, + 'small_lifetime_cdf_coefficient':str(F(832)/moment(0)), + 'inverse_lifetime_moment_threshold':4, + 'companion_over_elder_moment_threshold':13, + 'constants':{key:decimal_interval(val) for key,val in constants().items()}, + 'height_density_values':{str(e):decimal_interval(height_density_bounds(e)) for e in (F(1,100),F(1,4),F(1,2),F(3,4),F(99,100))}} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--mutant',choices=MUTANTS);args=parser.parse_args() + global _MUTANT + _MUTANT=args.mutant + try:ans=result() + 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/elder_partner_extremes_20260930/test_elder.py b/frontiers/elder_partner_extremes_20260930/test_elder.py new file mode 100644 index 000000000..6f9670e88 --- /dev/null +++ b/frontiers/elder_partner_extremes_20260930/test_elder.py @@ -0,0 +1,265 @@ +from fractions import Fraction as F +import unittest +import elder + +class InitialTests(unittest.TestCase): + def test_selected_mass(self): + self.assertEqual(elder.moment(0), F(888807,280)) + def test_selected_mean(self): + self.assertEqual(elder.moment(1)/elder.moment(0), F(1174914407,1440459878)) + def test_selected_second(self): + self.assertEqual(elder.moment(2)/elder.moment(0), F(63496935353,92189432192)) + def test_outer_not_elder(self): + self.assertEqual(elder.selection(F(-1), F(6)), 0) + def test_inner_is_elder(self): + self.assertEqual(elder.selection(F(-13,23), F(294,529)), 1) + def test_near_zero_density(self): + self.assertEqual(elder.small_height_density_coefficient(),F(3328)) + def test_upper_density(self): + lo,hi=elder.height_density_bounds(F(1)) + self.assertEqual((lo,hi),(F(1312200,296269),F(1312200,296269))) + def test_new_ratio_exponent(self): + self.assertEqual(elder.outer_over_elder_tail_exponent(),13) + + + +class GeometryTests(unittest.TestCase): + def test_full_shape_grid(self): + self.assertEqual(elder.checks(),893) + + def test_selector_matches_cubic_boundary(self): + for ip in range(-7,24): + p=F(ip,16) + for j in range(1,15): + e=F(j,15); v=6*(e+p-F(1,2)); u=-p + if not elder.shape(u,v):continue + boundary=2*p**3-F(3,2)*p+F(1,2) + expected=F(1) if p<=F(1,2) or e>boundary else F(1,2) if e==boundary else F(0) + self.assertEqual(elder.selection(u,v),expected) + + def test_actual_cubic_gradient_height_and_type(self): + # Directly differentiate the unsheared cubic, independently of selector. + cases=0 + for p in (F(1,4),F(3,5),F(3,4),F(1),F(5,4)): + u=-p; b=3-12*u*u;lo,hi=abs(b)/2,3-6*u + for j in range(1,5): + v=lo+(hi-lo)*F(j,5) + for k,a,Z in ((F(1),F(-12),F(1)),(F(3,2),F(4),F(-2))): + s=-k*v/Z**2; B=k*b/Z**2; Dc=k*(v-b*u)/Z**3 + beta=B+a*a/(12*k);c=2*Dc+a*B/(4*k)+a**3/(144*k*k) + roots=[(u,v,F(1))] + other=elder.companion(u,v) + if other and elder.shape(other[0],other[1]):roots.append(other) + es=[] + for uu,vv,t in roots: + z=t*Z;x=uu-a*z/(12*k) + gx=6*k*x*x-F(3,2)*k+a*x*z+beta*z*z/2 + gz=s*z+a*(x*x-F(1,4))/2+beta*x*z+c*z*z/2 + val=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 + hxx=12*k*x+a*z;hzz=s+beta*x+c*z;hxz=a*x+beta*z + self.assertEqual((gx,gz),(0,0)) + self.assertEqual(-val/k,elder.height(uu,vv)) + self.assertTrue(-k1/2 and ell>1/2 imply 4p ell>1. The denominator in candidate (6) is a nonzero square on the open generic doublet sector. Thus sign(eta_c−eta)=sign(ell−p). Outer seeds have vAp and their companions have larger eta. Consequently the inner seed is the highest-P root. Equal-height v=Ap and the degree-drop/boundary curves are null. The companion involution exchanges the open outer and inner sectors. Singletons have no second qualifying root. Therefore S_E=S\O2 contains exactly one selected root of every generic nonempty configuration. + +The RADIAL R4–R10 map is locally a diffeomorphism with the exact root multiplicity. Restricting that root integral to S_E counts each nonempty configuration once. It introduces neither a factor two nor a restricted jet normalizer. Restricting further to I2 gives the selected doublet event; removing both doublet sectors gives the selected singleton event. These are exact cubic selections before the radius limit, independent of a and Z. This is the substantive new selector operation. + +## Tail and marked-law audit + +The root density (8) matches RADIAL R11 term by term: raw jet Jacobian 2, stationary-point Jacobian 6k^3, typed weight 9k^4, physical X=u−aZ/(12k), full tensor, normalized Haar, ordered hard spectrum, original c_m/z0. The cusp entries beta=a^2/(12k), c=a^3/(144k^2) and physical shear factor gamma are retained. + +After Z=t z, the power is t^-11. The exact physical threshold implies |z|>1/(2gamma) for t>=3. SC (4),(10) gives Gaussian control in raw orthogonally rotated coordinates; dropping beta,c leaves a valid upper bound in h,a,xi. Integrating |z|^-12 on that threshold majorant costs only a constant times gamma^11. The bounded shape region/Q and polynomial hard factor are then Gaussian-integrable. This is sufficient domination on all retained variables and avoids inverse-hard constants. The pointwise limiting threshold boundary gamma|z|=1 is null. + +The two signs contribute 2gamma^11/11, giving exactly C0. The selected shape integrals are integral_(S\O2)Q=I−J, integral_I2 Q=D, and the singleton integral I−J−D. The imported values are assigned to the correct sectors. Positivity follows analytically from open positive-density inner-doublet and singleton sectors. The singleton example near u=0,v=2 is valid because B>0 excludes a doublet. Hence (10) and the selected doublet frequency D/(I−J) are correct at this scope. + +For bounded continuous marked tests the same majorant applies. The q=gamma|z| substitution factors the limit into shape Q/(I−J) on S_E, Pareto(11) radius, uniform sign and the cusp parameter law. The height fraction is the pushforward eta=u+1/2+v/6 of this selected shape law, independent of q and latent parameters. It is not the point-selected or maximum-anchor height law. + +On I2, t_c<0 and |t_c|>1. Thus the selected extreme position tends to epsilon q e_* and its deeper companion to −epsilon q |t_c| e_*. The rational companion map may diverge near exceptional boundaries, but it is continuous on compact regular subsets and the limiting shape density is integrable. Exhausting those subsets proves the claimed marked weak convergence. No spatial total-variation transfer is made. + +In the outer chart, the involution identity Q(T)|det DT|=Q rho^11 yields the selected-doublet shape density Q rho^11/D. Comparison with Q/J for a maximum-radius outer anchor gives exactly the claimed additional rho^11 bias. This is a deterministic coordinate transformation, not independence or temporal Palm sampling. + +## Exact physical-order counterexample + +Candidate (12) is consistent with (2)–(4). For k=1, a=−12 and Z=1, B=−15/4, s=−14/5 satisfies s<−|B|/2. The seed has b0=−15/4, L=−7031/400 and t_c=−6919/7031. Its physical X=u+Z=1/4 gives R^2=17/16. The companion has X_c=u_c+Z_c=−48583/28124, so + + R_c^2=(48583^2+27676^2)/28124^2 + =3126268865/790959376. + +Its eta_c is the stated rational value and is less than 13/60; both shapes lie in the strict window. Hence the companion is selected and is physically farther in this finite cubic example. Thresholds strictly between the two radii falsify replacement of the selected event by the finite-t minimum-radius event. The asymptotic coincidence of their leading doublet coefficient does not repair that false finite-t identity. + +## Actual elder interface and sequential quantifiers + +PR175 S1 is exactly 1{n>0}dM: same raw Gaussian density, c_m, full z0, normalized Haar/sign extension, ordered hard eigenvalues, full cubic tensor, H and w on the typed domain. Its M1 mark is the highest-P additional saddle on failure, with death h_*=−k eta_E. The original-coordinate location agrees with (2). No count conclusion, remote singleton coefficient, new adjacency weight or good-event normalizer is used. + +The imported theorem is explicitly at fixed d>=3 in its original pair-Palm model. Its contained hard-depth tube, graph/pinned C2 interface, actual global genericity/Borel repair, endpoint-residual/full-weight estimates, full normalization and actual failure exhaustion remain hypotheses. Reading its full companions confirms that these are the stated source obligations. This review does not establish them independently. The d=2 microscopic calculation therefore does not acquire a planar finite-r identification from this citation. + +For fixed t>0, the limiting selected radius has no atom at t: at fixed other root coordinates R^2 is a quadratic in Z with positive leading coefficient gamma^2, so R=t has finitely many Z solutions. Absolute continuity of the restricted coarea integral makes that boundary null. The isolated sentinel is excluded from the real-radius event; its mass is negligible by the explicitly imported exhaustion. Finite-measure weak convergence and Portmanteau consequently apply to the radius event and bounded continuous joint tests restricted to it. E(t)>0 follows from open singleton shapes and arbitrarily large |Z| with positive raw Gaussian density. Normalization of the restricted measures is legitimate. + +Thus the r->0 conditional limit at each fixed t follows from M1, and only then does Theorem 2 give t->infinity. Equation (13) is an iterated limit. The argument supplies no t(r), finite-r uniform tail, spatial moment uniform integrability, essential-class probability, or replacement-bar intensity. Retained shape marks are obtained on regular jet sectors where root selection is continuous; transition/tie sets are null and compact exhaustion removes the remainder. + +## Review boundary + +No mandatory amendment or obstruction was found for the conditional implications in the exact candidate. The briefly missing SOURCES.json at the start of the review was supplied without changing the proof; the final source map was read and bound above, so that initial packaging gap is resolved. + +This acceptance is source-bound analytic assessment by a same-provider nonauthor. It is not source acceptance, organizational independence, a scientific-status promotion or proof by hashes/tests. No finite-r lifetime-moment or essential-probability claim was reviewed; those remain outside this assignment. diff --git a/frontiers/elder_selected_extremes_20260930/PROOF.md b/frontiers/elder_selected_extremes_20260930/PROOF.md new file mode 100644 index 000000000..1310dc11b --- /dev/null +++ b/frontiers/elder_selected_extremes_20260930/PROOF.md @@ -0,0 +1,311 @@ +# The elder-selected microscopic radius–lifetime law + +OpenAI author-side candidate, 30 September 2026. **AUTHOR_SIDE / HOLD.** +Scientific effect NONE. This is a new selector consumer, not another count +estimate. The author contributed PR170/175 and reviewed PR169 Slice A. A +same-provider noncontributor review gives no organizational-independence credit. +All mathematical implications below retain their exact imported hypotheses. + +## 1. Interfaces and scope + +Fix d>=2, k>0 and the fixed model parameters of the sources. Work first with +the already-formed microscopic finite measure on nonempty configurations. +[RADIAL] R4–R12 supplies the root-resolved density and Gaussian domination; +[CUB] supplies the typed strict-window classifier; [EXT] E12–E19 supplies the +companion map, height ordering and shape partition, and E20–E26 its integrals. +[MICRO] M1–M2 and [SC] supply the original-frame measure and embedding only. +Their global count/moment conclusions are not used. Full immutable public +texts and byte/blob identities are in [SOURCES.json](SOURCES.json). + +Only §6 uses an actual finite-r elder theorem: [ELDER] S3 and M1 at its exact +fixed-d>=3 original pair-Palm scope. Its deterministic contained barrier, +whole-field genericity/Borel repair, original full normalization and exhaustion +remain hypotheses of this consumer; linking it does not accept its proof. +This consumer does not independently close PR170/175's composition review. +The d=2 microscopic conclusion does not automatically identify the planar +finite-r law: an exact frame/normalization interface is needed for that use. +No uniformity in dimension, k, hard gaps, t or a simultaneous r,t limit is claimed. + +Write theta=(s,h,O,tau). The ordered hard eigenvalues are +00) is finite and strictly positive. +The probability used below is pi=1{n>0}dM/A_micro. + +The soft cubic, original position and downward height 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=X e_axis+Z e_soft(O), R=|z|, eta=-P(X,Z)/k. (2) + +Every additional counted root has limiting full index d-1 and 00. (3) + +For a doublet define + + L=4v^2-8b0uv+b0^2, + t_c=(4v^2-b0^2)/L, + u_c=[2b0v-u(4v^2+b0^2)]/L, + v_c=v t_c^2, Z_c=t_c Z. (4) + +This map preserves the jet, swaps the two roots and is involutive off the +null exceptional sets. [EXT] proves its determinant/intensity identity + + Q(u_c,v_c)|det D(u_c,v_c)/D(u,v)|=Q(u,v)|t_c|^11. (5) + +For completeness, the explicit doublet partition needed here is as follows. +Put p=-u, A=12p^2-3 and + + V_+(p)=[36p^2-12p-3+3sqrt((2p-1)^3(18p+7))]/4. + +The outer sector O2 has .51. + +One can verify the partition directly from the strict doublet inequality: + + (v-A)^2(2v+A)-12(v-Ap)^2=(3+6p-v) H_p(v), + H_p(v)=-2v^2+(36p^2-12p-3)v-72p^3+36p^2+18p-9. + +H_p(A/2)>0, its upper root is V_+, and for p<1 its zero lies +between Ap and A; for p>=1 the height cap 3+6p lies below A. +The two admissible roots are on the negative branch of the same hyperbola. +Strict convexity of line-minus-branch, negative at Z=0, puts zero between +the roots, hence t_c<0. Substitution in (4) yields the |t_c| split at v=Ap. +These are the deterministic interfaces of [EXT] E14–E16, not a stochastic +assumption about two independent witnesses. + +On a doublet let ell=v/A. Exact substitution gives + + eta_c-eta=2(ell-p)(4p ell-1)^3/(4ell^2-8p ell+1)^2. (6) + +Here p>1/2 and ell>1/2, so 4p ell>1. Consequently the outer seed +has eta_ceta and +IS selected. The equal-height curve v=Ap is null. Therefore the exact +highest-height seed domain is + + S_E=S minus O2, (7) + +containing all singleton seeds and precisely the inner seed of every doublet. +This equality holds at every nondegenerate cubic jet, before any large-radius +limit, and is independent of a and Z. Coarea restricted to S_E consequently +counts one selected root per nonempty configuration. It does not renormalize +the jet measure, double a bar, or equate point intensity with cluster intensity. + +## 3. Full root density and selected-tail coefficients + +The original density in the root coordinates of [RADIAL] is + + K0 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, + K0=108 k^7 c_m/z0. (8) + +The physical radius is exactly sqrt((u-aZ/(12k))^2+Z^2). +Multiplying (8) by 1{(u,v) in S_E} and 1{R>t} is an EXACT +representation of E(t)=M(n>0,R_E>t), since (7) selects precisely one seed. +The doublet version E2(t)=M(n=2,R_E>t) restricts the shape to I2. +The singleton version E1(t) restricts it to S minus (O2 union I2). + +Define + + 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, + C0=(216/11) k^7(c_m/z0) J_cusp, + I=246528/35, J=1083417/280, + D=27066286003/223205220-(79298560/4782969) log(2). (9) + +[EXT] E20–E26 gives integral_S Q=I, integral_O2 Q=J, +integral_I2 Q=D. The last integral is the inner-root integral, not a hard +curvature or the cubic coefficient sometimes denoted D. These exact +integrals are imported, not inferred from a numeric PASS label. + +**Theorem 1.** For the fixed microscopic measure (1), + + E(t) ~ C0(I-J)t^-11, + E2(t) ~ C0 D t^-11, + E1(t) ~ C0(I-J-D)t^-11. (10) + +All three coefficients are strictly positive. In particular, conditional on +R_E>t under pi, the doublet probability tends to D/(I-J). This differs from +the cluster-MAXIMUM doublet frequency J/(I-D). + +Proof. Put Z=t z in (8), retaining the selector (7). The Jacobian contributes +t^-11 |z|^-12 dz. At each fixed nonzero z, beta and c tend to the cusp +entries in (9), and R/t tends to gamma(a)|z|. The shape selector remains +exactly S_E; it need not converge from a radius-based selector. +For t>=3, R>t and |u|<1.5 imply |z|>1/(2gamma(a)) by +t1/(2gamma) costs C gamma^11. S is bounded, Q is bounded on S, +H is a nonnegative polynomial and Haar measure is finite. Thus the remaining +Gaussian integral is finite. This proves domination over all unbounded +retained variables without an inverse hard eigenvalue. The limiting integral +on each sign is gamma^11/11. Both signs and K0 give C0 times the shape +integral I-J, D or I-J-D. This proves (10). +Strict positivity follows from open inner-doublet and singleton sectors, +Q>0 and positive raw Gaussian density. For example a neighborhood of +u=0, v=2 is a singleton sector: b0>0 excludes a doublet. J_cusp is positive +and finite by the same Gaussian argument. QED. + +## 4. The selected joint radius, location and lifetime-fraction law + +**Theorem 2.** Under pi conditional on R_E>t, retain the selected seed +shape (u,v), q=R_E/t, epsilon=sign Z and (a,h,O,xi). Their weak limit is +the product probability + + Q(u,v)1{S_E}/(I-J) du dv; + 11q^-12 dq on q>1; + uniform epsilon in {-1,1}; + H(h)gamma(a)^11 h0(A0,Rot_O tau_cusp)/J_cusp + da dxi dh dO. (11) + +The selected height fraction is eta_E=u+1/2+v/6. Its limiting law is the +pushforward of the FIRST factor in (11), independent of q and the remaining +factors. Thus (11) specifies the selected lifetime-fraction distribution, +without substituting the point-selected or maximum-radius height law. + +With e_*=[-a e_axis/(12k)+e_soft(O)]/gamma(a), the selected position +divided by t tends to epsilon q e_*. For doublets the companion position +divided by t tends to -epsilon q |t_c| e_*, where |t_c|>1 on I2; +its height fraction eta_c exceeds eta_E by (6). The selected root is therefore +asymptotically the higher, nearer root of this extreme doublet, while the +other root is deeper and farther. These are SECOND limits of the microscopic +geometry, not finite-r collinearity or independence of two roots. + +Proof. Insert a bounded continuous test function in the exact selected +integral (8). The majorant in Theorem 1 is unchanged. Dominated convergence +and q=gamma(a)|z| give gamma^11 q^-12 dq; normalization by (10) +produces (11). The position map is continuous away from already null +exceptional sets and has the limits just displayed. One may exhaust compact +shape/latent subsets to avoid the rational companion denominators on the +boundary, then let the excluded mass tend to zero under the integrable +limit density. This proves joint weak convergence of the marked finite +configurations, not moving-map TV convergence of positions. QED. + +An equivalent description of the doublet shape is useful for consumers of +the outer-root chart. If rho=|t_c|<1 on O2, swapping inner to outer in (5) +gives outer density Q rho^11/D. The ordinary maximum-radius outer density +is Q/J. Hence elder-selected extreme doublets have an additional rho^11 +size bias. This is a deterministic root-swap identity, not a temporal Palm +or independent-root hypothesis. It also makes clear why the maximum-radius +law cannot be copied as the actual elder-selected law. + +## 5. Physical-order falsifier retained at finite radius + +Do not replace E2(t) by the doublet minimum-physical-radius tail at finite t. +They have the same leading coefficient C0 D, but their selectors can differ. +Here is an exact typed cubic example, k=1: + + Z=1, a=-12, s=-14/5, beta=33/4, c=-31/40, + u=-3/4, v=14/5, eta=13/60, R^2=17/16, + u_c=-20907/28124, Z_c=-6919/7031, + v_c=670215854/247174805, + eta_c=618523783/2966097660, + R_c^2=3126268865/790959376. (12) + +Both roots are strict-window saddles. eta_cR. Substitution in (2)–(4) verifies every +entry. The typed inequality holds since B=-15/4 and s=-14/5<-15/8; +the seed is in O2 and its companion in I2. For thresholds between these +two radii, the selected doublet exceeds the threshold while the minimum +radius does not. Thus a finite-t physical-minimum identity is false. +Only after Z=t z and t->infinity does the bounded u offset vanish. A +proof that silently identifies the selectors before that limit is invalid. + +## 6. Sequential consequence for the actual elder partner + +Fix d>=3 and the exact source parameters and original Q_r^W of [ELDER]. +Let F_r be failure of pairing M with S; T_r the actual partner displacement +divided by r, and L_r=(b-d_f(M))/r^3. As in [ELDER], chart/essential +exceptions use an isolated sentinel; only REAL T_r enter a radius event. +Assume [ELDER] S3/M1, including its full exhaustion and sentinel exclusion. +Its limit measure is exactly 1{n>0}dM in (1), with selected position z_E +and death value -k eta_E. The equality of measures is termwise: original +c_m/z0, normalized Haar, ordered hard spectrum, full raw tensor and w on +the same typed domain. No new normalization or point-count factor occurs. + +**Corollary 3 (sequential actual selector/lifetime law).** For every fixed t>0, + + lim_(r->0) r^-3 Q_r^W(F_r, T_r real, |T_r|>t)=E(t). + +Consequently + + lim_(t->infinity) t^11 lim_(r->0) + r^-3 Q_r^W(F_r,T_r real,|T_r|>t)=C0(I-J). (13) + +Conditional on this actual event, first let r->0 at fixed t, then let +t->infinity. The limit of (T_r/t,L_r/k), and the retained microscopic +marks when desired, is the selected law of Theorem 2: position epsilon +q e_* and lifetime fraction eta_E with the FIRST shape factor of (11). + +Proof. [ELDER] M1 is weak convergence of finite measures, with mass limit +from S3. The boundary {|z_E|=t} has limiting measure zero. Indeed, in (8) +restricted to S_E, for fixed other coordinates R^2=(u-aZ/(12k))^2+Z^2 +is a quadratic in Z with strictly positive leading coefficient gamma^2. +The equation R=t has finitely many Z solutions and hence zero mass in the +absolutely continuous root integral. Portmanteau therefore gives the +fixed-t event mass and bounded continuous marked tests restricted to that +event. E(t)>0: an open singleton shape and arbitrarily large |Z| in (8) +have positive density. Divide by E(t) to obtain the first conditional limit, +then apply Theorems 1–2 for the second. Sentinel mass is o(r^3) by the +imported exhaustion. No moment uniform integrability is used. QED. + +Equation (13) is an ITERATED limit. It proves neither a uniform finite-r +radius bound nor a limit for t=t(r), nor finite-r lifetime moments, essential +class probabilities, unconditioned bar intensity, or identification outside +the exact imported elder interface. The active lifetime-moment review is +not consumed by this proof. The present step advances which saddle pairs +and its lifetime mark; the original source composition remains conditional. + +## 7. Falsifiers and remaining acceptance obligations + +This consumer fails if: (6) selects the wrong height root on an open doublet +set; S_E does not contain exactly one seed; the original unsheared density +(8) loses a Jacobian or type weight; the Gaussian majorant is unavailable +for the fixed source; or [ELDER]'s selected limit/normalization does not +match (1). Failure of [ELDER] invalidates §6, not the separately formed +microscopic implications of §§2–5. Example (12) falsifies the tempting +finite physical-minimum substitution, not (10). + +A qualified source-bound noncontributor review of this new composition is +still required. Same-provider review, arithmetic checks, hashes, publication, +CI and engineering merge do not establish organizationally independent +mathematical acceptance. No scientific Boolean, predecessor disposition or +prize is changed. The next unresolved issue for this candidate is review of +the exact elder-selected interface, not another count packet. diff --git a/frontiers/elder_selected_extremes_20260930/README.md b/frontiers/elder_selected_extremes_20260930/README.md new file mode 100644 index 000000000..c87b4f94e --- /dev/null +++ b/frontiers/elder_selected_extremes_20260930/README.md @@ -0,0 +1,18 @@ +# Elder-selected radius and lifetime + +[Full proof](PROOF.md): the higher additional saddle, selected before any +extreme limit, has microscopic tail coefficient `C0(I-J)` and an explicit +joint radius/lifetime-fraction law. On doublets its seed is the inner soft +root; a finite physical-radius counterexample is retained in proof §5. + +This is an author-side conditional candidate. The actual finite-r conclusion +uses PR175's exact fixed-d>=3 elder interface and only the sequential limit +`r->0` followed by `t->infinity`. No additional count estimate is asserted. + +- [Immutable full source cut](SOURCES.json), including the complete elder + dependency cut and all direct byte/blob identities. +- [Full noncontributor review](NONAUTHOR_REVIEW.md), with source exposure and + zero same-provider organizational-independence credit stated explicitly. + +Qualified source-bound composition review remains required. No merge, test, +hash or publication changes a scientific disposition. diff --git a/frontiers/elder_selected_extremes_20260930/SOURCES.json b/frontiers/elder_selected_extremes_20260930/SOURCES.json new file mode 100644 index 000000000..7d56f30a3 --- /dev/null +++ b/frontiers/elder_selected_extremes_20260930/SOURCES.json @@ -0,0 +1,219 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "new_consumer_review": "AUTHOR_SIDE/HOLD", + "sources": [ + { + "id": "RADIAL", + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "use": "R4-R12 exact original-frame root density and Gaussian domination only.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337/frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md" + }, + { + "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.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md" + }, + { + "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.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf/frontiers/spectral_cluster_closure_20260929/PROOF.md" + }, + { + "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.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8/frontiers/planar_cubic_cluster_20260929/PROOF.md" + }, + { + "id": "EXT", + "commit": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "bytes": 21891, + "sha256": "b90cf745e2e67bc9b501c481552f7080d27c28245580668cf72aefa5b7f04715", + "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", + "use": "E12-E19 deterministic companion/height identities and partition; E20-E26 exact shape integrals. Not the maximum-selector law.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/56a7904f9ad7d409c31e23195c0e7eeab230613a/frontiers/extreme_cluster_sampling_20260930/PROOF.md" + }, + { + "id": "ELDER", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "bytes": 23416, + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "use": "S3/M1 actual elder-failure finite measure and selected marked weak limit, fixed d>=3, conditional for Corollary3 only; AUTHOR_SIDE/HOLD.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/PROOF.md" + }, + { + "id": "ELDER_HARD", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md", + "bytes": 18907, + "sha256": "434ffd30ac96eb635211ea627459b78c1d39806f75b0634e0f4876dce0765b33", + "git_blob": "7365ed9bcf145c832d1c09f0b685fbe8d6695050", + "use": "Full companion proof of ELDER contained hard-negative-fibre barrier.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/HARD_FIBRE_LEMMA.md" + }, + { + "id": "ELDER_INTERFACE", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md", + "bytes": 18098, + "sha256": "cec41136dfac1fde2247f25dcf7601e1a01a6ec25c4d5f18e8bf1450e8360e48", + "git_blob": "7eb203bdf1311ff56b312962ba3f5ce6d5dd9766", + "use": "Full original-field analytic interface of ELDER.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/SOURCE_INTERFACE.md" + }, + { + "id": "ELDER_CUT", + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "path": "frontiers/concave_fibre_elder_20260930/SOURCES.json", + "bytes": 7186, + "sha256": "b97a29ea09f8e8270bfddda8f364f0d61e431e30c1681af4b019f3fb48709a4d", + "git_blob": "f072e1bb5c79456be3c79451b6df5b287807edfa", + "use": "Complete required ELDER dependency cut and reading rules, reproduced as elder_dependency_cut below.", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/SOURCES.json" + } + ], + "elder_dependency_cut": { + "repository": "d6g8k5htny-coder/Math-", + "public_main": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "planar_pr": 170, + "planar_head": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "scientific_effect": "NONE", + "review_independence_credit": 0, + "sources": [ + { + "id": "SC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "role": "required: §§2,5,6 raw/full jet disintegration, orthogonal spectral Jacobian, actual hard graph C2 convergence, continuous typed weight, deterministic integral; global count law NOT consumed" + }, + { + "id": "CUB", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "role": "required: displayed original planar cubic, exact classifier and polynomial null sets; count expectations NOT consumed" + }, + { + "id": "EDL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_dimension_lift_20260928/PROOF.md", + "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_dimension_lift_20260928/PROOF.md", + "bytes": 23971, + "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", + "role": "context only: preceding actual root/index interface; no count or pairing conclusion consumed" + }, + { + "id": "DL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/d5_dimension_lift_20260929/PROOF.md", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "role": "scope audit only: dimension-lift count result excludes elder selection; no count conclusion consumed" + }, + { + "id": "LOCAL", + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md", + "bytes": 39722, + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "role": "required: Theorem E compact planar collar and moving critical chords; position uniqueness off D=0; draft at exact stated source scope" + }, + { + "id": "P", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "role": "required: endpoint matrix density/residual, double soft-factor bound, full Z, global cap-failure inclusion, genericity/Borel and compact lifetime ledger" + }, + { + "id": "ELDERD", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "frontiers/elder_lower_all_d_20260929/PROOF.md", + "git_blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/frontiers/elder_lower_all_d_20260929/PROOF.md", + "bytes": 16485, + "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5", + "role": "context only: preceding elder-failure lower event; not used as a coefficient or upper barrier" + }, + { + "id": "ERR", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "role": "required reading rule: endpoint congruence correction" + }, + { + "id": "BOREL", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "role": "required reading rule: whole-field Borel weighted Kac-Rice repair" + }, + { + "id": "REC", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "role": "required reading rules and reviewed source scope: W1 convergence-in-law and embedded-chart radius" + }, + { + "id": "CAP", + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "role": "required: deterministic global good-cap selection implication" + } + ], + "source_exposure": "Full public immutable sources. Required interfaces remain conditional on their displayed assumptions and actual review scope. Context sources are not mathematical premises." + } +} 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. Existing marked-law results are explicitly retained premises.", + "sources": [ + { + "bytes": 40261, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "P", + "local_path": "sources/P.md", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "role": "Required \u00a7\u00a72,4\u20135,7\u20138: positive Fourier spectrum and pin rank; all-order global conditional moments; W/r\u00b2 moments and full Z floor; failure O(r\u00b3); essential-death definition and almost-sure Morse locus. \u00a712 finite-bar intensity moments are not the present estimand.", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md" + }, + { + "bytes": 1782, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "id": "ERR", + "local_path": "sources/ERR.md", + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "role": "required reading rule: endpoint congruence correction", + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md" + }, + { + "bytes": 9062, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "id": "BOREL", + "local_path": "sources/BOREL.md", + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "role": "required reading rule: whole-field Borel weighted Kac-Rice repair", + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_section9_borel_repair_20260925/REPAIR.md" + }, + { + "bytes": 23312, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "id": "REC", + "local_path": "sources/REC.md", + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "role": "required reading rules and reviewed source scope: W1 convergence-in-law and embedded-chart radius", + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md" + }, + { + "bytes": 23416, + "commit": "9c6a73489c5ed2465a3cf65dcc950eeeae074b05", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "id": "MARK_D", + "local_path": "sources/MARK_D.md", + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "role": "Required fixed-dimensional stated failure-mark limit #175 S3/M1, consumed as a premise; this note does not re-prove it.", + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e", + "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/9c6a73489c5ed2465a3cf65dcc950eeeae074b05/frontiers/concave_fibre_elder_20260930/PROOF.md" + }, + { + "bytes": 39722, + "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "id": "MARK_2", + "local_path": "sources/MARK_2.md", + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "role": "Required planar stated failure-mark limit #170 S2/S3 and \u00a78.8, consumed as a premise; this note does not re-prove it.", + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b", + "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/79f18f0cb318fceccd5bdf4e6dad29815165bc7d/frontiers/local_elder_geometry_20260930/PROOF.md" + }, + { + "bytes": 15160, + "commit": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f", + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "id": "CAP", + "local_path": "sources/CAP.md", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "role": "Retained parent-selection dependency: CAP \u00a7\u00a71\u20135, consumed through the reconciled P failure O(r\u00b3) premise; 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. A review using another OpenAI agent has zero organizational-independence credit. diff --git a/frontiers/local_elder_geometry_20260930/SOURCES.json b/frontiers/local_elder_geometry_20260930/SOURCES.json new file mode 100644 index 000000000..d77728d1c --- /dev/null +++ b/frontiers/local_elder_geometry_20260930/SOURCES.json @@ -0,0 +1,134 @@ +{ + "repository": "d6g8k5htny-coder/Math-", + "source_cut": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "sources": [ + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "role": "required: exact pinned law, full normalizer, scalar cap-failure integration, Borel maximin, compact lifetime pushforward" + }, + { + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "bytes": 15160, + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "role": "required: sufficient deterministic cap event implies global elder selection" + }, + { + "path": "frontiers/elder_lower_all_d_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/elder_lower_all_d_20260929/PROOF.md", + "bytes": 16485, + "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5", + "role": "context only: preceding elder-failure lower-bound theorem" + }, + { + "path": "frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "aaefc8da9668e819591dd9db0b34442394f2362e", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md", + "bytes": 10229, + "sha256": "8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c", + "role": "context only: preceding lifetime-density distinction" + }, + { + "path": "frontiers/unrestricted_selection_difference_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "5a55b179a974a90c65d257f3f93765cc6fb30bb8", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/unrestricted_selection_difference_20260929/PROOF.md", + "bytes": 13028, + "sha256": "93a63002d32fd5c35ae994fe28b1205d8098026d6ddde19c67d4977dcf5817a9", + "role": "context only: unrestricted rejected candidate density differs from compact-mark density" + }, + { + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "role": "required: exact planar classifier, pinned C2 convergence, Gaussian physical-jet disintegration and domination; G11-G13 integral finiteness" + }, + { + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "75da2597971510f843f8d90c743950cb8c177342", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "role": "required reading rule and acceptance scope: W1, cap chart embedding, separate leading/full-pair law" + }, + { + "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "ref": "be9b7fa95de9b7e59a67347640e157ed3027f624", + "blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/be9b7fa95de9b7e59a67347640e157ed3027f624/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md", + "bytes": 17139, + "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769", + "role": "context and MIXED_SCALES only: conditional shrinking-cutoff factorial localization; not consumed by elder theorem" + }, + { + "path": "frontiers/c6_palm_route_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/c6_palm_route_20260929/PROOF.md", + "bytes": 53364, + "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "role": "MIXED_SCALES only: factorial upper bound; not consumed by elder theorem" + }, + { + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "ref": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf/frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "role": "MIXED_SCALES only: compact/origin-jet mixed estimate; not consumed by elder theorem" + }, + { + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "role": "required reading rule: endpoint congruence direction" + }, + { + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "role": "required reading rule: source-bound global Borel marked Kac-Rice interface" + }, + { + "path": "frontiers/c6_cluster_law_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/c6_cluster_law_20260929/PROOF.md", + "bytes": 70131, + "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "role": "MIXED_SCALES only: repaired main fixed physical remote estimate" + }, + { + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "ref": "98fdb54954a2d9ac0b8ba8ca03e279701f70d751", + "blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "url": "https://github.com/d6g8k5htny-coder/Math-/blob/98fdb54954a2d9ac0b8ba8ca03e279701f70d751/frontiers/d5_dimension_lift_20260929/PROOF.md", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "role": "MIXED_SCALES only: intermediate shell estimate" + } + ], + "source_exposure": "Public full texts linked by immutable commits. 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 j 0`, the [P] field on `X = R^d/(LZ^d)`; near pins `M = −ru/2`, `S = ru/2` with heights `b`, +`b − kr³` and zero gradients; the observation rows `U_r`, target `v_r`, density `π_r`, the regression law `Q`, +the coupling `F_r` of [R] (R3)–(R4), `m = d − 1`, `P = 1 + |b| + k`; the scaled endpoint Hessians of [R] §4 +(*scaling dictionary*, quoted): "`alpha_i=F_r,xx(i)/r` and `beta_i=grad_y F_r,x(i)/r`. Let `D_r=diag(sqrt(r),I)`. +Then `K_i=D_r^(-1) H_i D_r^(-1) =[[alpha_i,sqrt(r) beta_i^T],[sqrt(r) beta_i,A_i]]`, `|det H_i|/r=|det K_i|`" — so +`H_i` (the full Hessian of `F_r` at the pin `i`, in the coordinates `(x, y)`) has entries `rα_i`, `rβ_i`, `A_i`, and +`K_i` has the scalar `α_i`, the off-diagonal `√r β_i` and the transverse block `A_i = D_y²F_r(i)` (`i = M, S`); the +filtered determinants `F_j`; `Z_r/r² = E_Q[F_d(K_M)F_{d−1}(K_S)]`; `A_0 = D_y²F_0(0)`, +`z_0 = E[(6k)²(det A_0)²1{A_0 < 0}]`; `A_r = 12π_r(v_r)Z_r/r²`, `A_0 = 12π_0(v_0)z_0`; the majorant +`H(b,k) = CP^Ne^{−c(b²+k²)}` (`N` enlarged when needed); the near/far split at a separation `r_0`, `a = ℓ/r_0³`; and +`c = c_{d,L}` of [P] (15.2). Admissible radii: `r_0 ≤ r_0^* := min{r_0^{[R]}, L/(4√2), r_0^{[K]}, r_0^{[Z]}}` ([R]'s compactness band on +which (R2)–(R5) hold, the cap-cylinder embedding of [REC] §5, and the radii of [C7-K] (K2) and [Z] §4). [R] proves +`|A_r − A_0| ≤ r(k + r)H` ((R11), first order in `r`) and from it Theorem R, `|ν_eld(ℓ) − cℓ^{−1/3}| ≤ C`, and states +that neither convergence of the remainder nor a second coefficient is identified. + +From [Z] (`frontiers/c7_zero_gap_limit_20260929/PROOF.md`, blob `5b6328ea`; Anthropic A/B/C reads on Math- #155): +the unmarked candidate kernel and the equal-height kernel + + Ψ_ℓ^{cand}(b, y) = p_y(v_{b,ℓ}) E_{Q_{y,b,ℓ}}[W], Ψ_0(b, y) = p_y(v_{b,0}) E_{Q_{y,b,0}}[W] ((Z2)), + +`B_{d,L} = ∫_{X∖{0}}∫_R Ψ_0(b, y) db dy ∈ (0, ∞)` ((Z3)), and Theorem Z (Z4): the rejected density +`ρ_rej(ℓ) = ν_cand(ℓ) − ν_eld(ℓ) → B_{d,L}` as `ℓ ↓ 0`. + +**Lemma E (even contact expansion).** There are `C, N` depending on `d, L` only such that for all `b ∈ R`, `k > 0` +and `0 < r ≤ r_0^*` + + |Z_r/r² − z_0(b, k, u)| ≤ C r² P^N, |A_r − A_0| ≤ r² H(b, k). (E.1) + +The first-order term of (R11) is absent: the two endpoint corrections enter the typed product with opposite signs +and cancel identically (§1, Steps 1–3), and on the inertia-flip layers the product of the two typed determinants is +`O(r²)` (Step 4). + +**Theorem R+.** For every fixed `d ≥ 2`, `L > 0`, as `ℓ ↓ 0`, + + ν_cand(ℓ) = c ℓ^{−1/3} + B_{d,L} + o(1), (R+.1) + ν_eld(ℓ) = c ℓ^{−1/3} + o(1), i.e. ν_eld(ℓ) = c ℓ^{−1/3} (1 + o(ℓ^{1/3})). (R+.2) + +So the bounded remainder of Theorem R has the constant term `B_{d,L}` (the equal-height kernel mass of [Z]) for the +candidate density and `0` for the elder density. Moreover, for every `r_0 ∈ (0, r_0^*]`, + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} → ∫_{0=1 and each +finite derivative order q**, `||F_r-F_0||_{L^p(C^q)} <= C_{p,q} r^2 P`, `||1+||F_r||_{C^q}+||F_0||_{C^q}||_p <= C_{p,q} P`, +`P=1+|b|+k`" — the second line with `q = 7` bounds the `C^7` norms in `L^p`, the first with `q = 2` bounds +`r^{−2}‖F_r − F_0‖_{C^2}` in `L^p`, so `T := 1 + k + ‖F_r‖_{C^7} + ‖F_0‖_{C^7} + r^{−2}‖F_r − F_0‖_{C^2}` has +`‖T‖_p ≤ C_pP` for every `p` (`[R]` §4 chose `q = 4` for its `T` because (R8)–(R11) need four derivatives; the +`C^q` moments exist for every `q` because `F_r` is the Gaussian regression of the smooth [P] field on finitely many +rows, [R] §2). Write `f = F_r`, `f_j = ∂_x^jf(0)`, and `O(x)` for a quantity bounded by `C|x|` with `C = C(d)`. + +*Step 1 (axial pins to second order).* The third and fourth rows of `U_r` at their pinned values `0` and `12k` +give, by Taylor's formula with integral remainder (the rules are centred, so the odd orders cancel), + + (f_x(S) − f_x(M))/r = f_2 + (r²/24) f_4 + O(r⁴T) = 0, + (6/r²)[f_x(M) + f_x(S) − 2(f(S) − f(M))/r] = f_3 + (r²/40) f_5 + O(r⁴T) = 12k. + +Taylor expansion of `f_xx` at `∓r/2` to second order, `f_xx(∓r/2) = f_2 ∓ (r/2)f_3 + (r²/8)f_4 + O(r³T)`, and +substitution of the two identities give + + α_M = −6k + (r/12) f_4 + ρ_M, α_S = 6k + (r/12) f_4 + ρ_S, |ρ_M| + |ρ_S| ≤ C r² T. (1.1) + +(For a polynomial profile of degree five the expansions are exact, `α_{M,S} = ∓6k + (r/12)f_4 ∓ (r²/120)f_5`; +checker E1.) The first-order coefficients coincide, so + + α_M α_S = −36k² + 6k(ρ_M − ρ_S) + r²(f_4/12)² + (rf_4/12)(ρ_M + ρ_S) + ρ_Mρ_S = −36k² + O(r²(1 + k)T²). (1.2) + +*Step 2 (transverse blocks).* With `B := ∂_xD_y²f(0)` and `‖D_y²f(0) − A_0‖ ≤ r²T` (coupling), + + A_M = A_0 − (r/2)B + O(r²T), A_S = A_0 + (r/2)B + O(r²T), (1.3) + +and `det(A_0 + X) = det A_0 + tr(adj(A_0)X) + O(‖X‖²T^{m−2})`, so + + det A_M det A_S = (det A_0)² + det A_0 · [−(r/2) + (r/2)] tr(adj(A_0)B) + O(r²T^{2m}) = (det A_0)² + O(r²T^{2m}). (1.4) + +The transverse difference row `(f_y(S) − f_y(M))/r = f_{xy}(0) + (r²/24)f_{xxxy}(0) + O(r⁴T) = 0` gives +`f_{xy}(0) = O(r²T)`, hence with `γ := ∇_yf_{xx}(0)`, + + β_M = −γ/2 + O(rT), β_S = γ/2 + O(rT), q_i := β_iᵀadj(A_i)β_i = q + O(rT^{m+1}), q := γᵀadj(A_0)γ/4. (1.5) + +(The sign of `β_i` enters `q_i` quadratically. That `f_{xy}(0)` is `O(r²)` and not `O(r)` is essential: an `O(r)` +value would shift `β_M` and `β_S` by a common `O(1)` and make `q_M − q_S = O(1)`.) + +*Step 3 (the product of determinants).* [R] Lemma R3.2, quoted: "For a symmetric (n-1) by (n-1) matrix A, scalar +alpha, vector beta and t>=0, set `K_t=[[alpha,sqrt(t) beta^T],[sqrt(t) beta,A]]`. … Indeed +`det K_t=alpha det A-t beta^T adj(A) beta`, INCLUDING singular A." With `t = r` and `(alpha, beta, A) = (α_i, β_i, A_i)`, +`K_r` is the scaled Hessian `K_i` of the dictionary in §0 (off-diagonal entries `√r β_i`), so the exact identity is + + det K_i = α_i det A_i − r β_iᵀadj(A_i)β_i = α_i det A_i − r q_i, + +`q_i` as in (1.5): the parameter multiplying `q_i` is the first power of `r` because the off-diagonal entries of `K_i` +carry `√r` (with `β_i = ∇_yF_{r,x}(i)/r` already containing the `1/r` of the dictionary; in terms of the unscaled +Hessian, `det H_i = r det K_i = rα_i det A_i − r²q_i`). Hence + + det K_M det K_S = α_Mα_S det A_M det A_S − r(α_M det A_M q_S + α_S det A_S q_M) + r² q_M q_S. + +By (1.2), (1.4) the first term is `−36k²(det A_0)² + O(r²(1 + k)²T^{N_1})`. In the second, by (1.1), (1.3), (1.5), +`α_M det A_M q_S = −6k det A_0 q + O(r(1 + k)T^{N_1})` and `α_S det A_S q_M = +6k det A_0 q + O(r(1 + k)T^{N_1})`, so +the bracket is `O(r(1 + k)T^{N_1})` and the term is `O(r²(1 + k)T^{N_1})`. The third is `O(r²T^{N_1})`. Hence + + | det K_M det K_S + 36k²(det A_0)² | ≤ C r² (1 + k)² T^{N_1}. (1.6) + +Structurally, `det K_M = −6k det A_0 + rY + O(r²)` and `det K_S = +6k det A_0 + rY + O(r²)` with the **same** +`Y = (f_4/12)det A_0 + 3k tr(adj(A_0)B) − q`; the product's `r`-term is `6k det A_0(Y − Y) = 0`. This is the +reflection symmetry of the pinned family: reversing the axis exchanges `M` and `S`, and the rows of `U_r` are even +in `r` (checker E2 verifies, on a general pinned quintic in two variables, the exact identity +`det H_M(−r) = det H_S(r)`, hence that `det H_M det H_S/r²` is an even polynomial in `r` with constant term +`−36k²A_0²`, while each single factor is not even). + +*Step 4 (inertia).* For `A_i` invertible, Haynsworth's inertia additivity for the block matrix `K_i` gives +`inertia(K_i) = inertia(A_i) + inertia(σ_i)`, `σ_i := α_i − rβ_iᵀA_i^{−1}β_i`. So `index K_M = d` iff +`A_M < 0` and `σ_M < 0`; `index K_S = d − 1` iff (`A_S < 0` and `σ_S > 0`) or (`index A_S = m − 1` and `σ_S < 0`). +Put `u := C_1 r T (1 + T/k)` with `C_1 = C_1(d) ≥ 2` large, and the good event + + G := { λ_max(A_0) ≤ −u } ∩ { k ≥ C_1 r T }. + +On `G`: `‖A_i − A_0‖ ≤ rT ≤ u/C_1`, so `A_M, A_S < 0`; `α_S ≥ 6k − CrT > 0` and `rβ_Sᵀ(−A_S)^{−1}β_S ≥ 0`, so +`σ_S > 0`; `α_M ≤ −6k + CrT ≤ −5k` and `rβ_Mᵀ(−A_M)^{−1}β_M ≤ rT²/λ_min(−A_M) ≤ 2rT²/u ≤ 2k/C_1`, so `σ_M < 0`. Hence +on `G` the indices are `d` and `d − 1`, `1{A_0 < 0} = 1`, and + + F_d(K_M)F_{d−1}(K_S) − 36k²(det A_0)²1{A_0<0} = |det K_M det K_S| − 36k²(det A_0)², + +whose absolute value is at most the left side of (1.6). Off `G` there are three cases, on each of which both typed +products are `O(r²)` times a polynomial: + +- on `{k < C_1rT}`: `F_d(K_M)F_{d−1}(K_S) ≤ C(k + r)²T^{2d} ≤ Cr²T^{2d+2}` ([R] §4's bound `F_j(K_i) ≤ C(k + r)T^d`, + `T ≥ 1`) and `36k²(det A_0)² ≤ Cr²T^{2m+2}`; +- on `{λ_max(A_0) ≥ u} ∩ {k ≥ C_1rT}`: `λ_max(A_M) ≥ u − rT > 0`, so `K_M` is not negative definite, `F_d(K_M) = 0`, + and `1{A_0 < 0} = 0` — both products vanish (this case also contains the pattern `index A_S = m − 1`, `σ_S < 0`, + which `F_d(K_M) = 0` kills); +- on `{|λ_max(A_0)| < u} ∩ {k ≥ C_1rT}`: by Weyl, `|det A_i| ≤ (|λ_max(A_0)| + rT)T^{m−1} ≤ 2uT^{m−1}`, so + `|det K_i| ≤ (6k + CrT)·2uT^{m−1} + rT^{m+1} ≤ C(ku + rT²)T^{m−1} ≤ CkuT^{m−1}` (as `rT² ≤ ku/C_1`, i.e. + `T ≤ k + T`), hence `F_d(K_M)F_{d−1}(K_S) ≤ Ck²u²T^{2m−2}`, and likewise `36k²(det A_0)²1{A_0<0} ≤ 36k²u²T^{2m−2}`; + with `k²u² = C_1²r²T²(k + T)²` both are `≤ Cr²(k + T)²T^{2m}`. + +No smallness of the layers' probability is needed: each typed determinant on the last layer is `O(r)` and their +product `O(r²)`. + +*Step 5 (assembly).* Taking expectations, `|Z_r/r² − z_0| ≤ E[Cr²(1 + k)²T^{N_1} + Cr²(k + T)²T^{2m+2}] ≤ Cr²P^N`. +For `A_r − A_0 = 12[π_r(v_r) − π_0(v_0)]Z_r/r² + 12π_0(v_0)[Z_r/r² − z_0]`, use (R5) (`|π_r(v_r) − π_0(v_0)| ≤ +Cr²P²e^{−c(b²+k²)}`, `π_0(v_0) ≤ Ce^{−c(b²+k²)}`), (R10) (`Z_r/r² ≤ C(k + r)²P^N`) and the first bound; every term is +`r²` times a polynomial in `P` times the Gaussian factor, i.e. `≤ r²H`. ∎ + +## 2. Proof of Theorem R+ + +*The separation variable.* As in [R] §6 and [Z] (Z14), with `r = (ℓ/k)^{1/3}`, `k = ℓ/r³`, `dk = 3ℓr^{−4}dr`, + + ν_cand^{near,r_0}(ℓ) = ∫_0^{r_0}∫_R∫_{S^{d−1}} r^{−2} A_r(b, ℓ/r³, u) dσ(u) db dr, + c ℓ^{−1/3} = ∫_0^{∞}∫_R∫_{S^{d−1}} r^{−2} A_0(b, ℓ/r³, u) dσ(u) db dr, + +and, by the pin-density and determinant normalizations of [P] (10.1)–(11.1) exactly as in (Z14) but without the +mark, `r^{−2}A_r(b, ℓ/r³, u) = r^{d−1}Ψ_ℓ^{cand}(b, ru)` (the polar factor `r^{d−1}`, the pin density +`12r^{−(d+3)}π_r(v_r)` and the determinant scale `r²`). Hence + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} = ∫_0^{r_0}∫∫ r^{−2}(A_r − A_0)(b, ℓ/r³, u) − ∫_{r_0}^{∞}∫∫ r^{−2}A_0(b, ℓ/r³, u). (2.1) + +*Domination.* By (E.1), `|r^{−2}(A_r − A_0)(b, ℓ/r³, u)| ≤ H(b, ℓ/r³) ≤ Ce^{−cb²}`, an integrable majorant on +`(0, r_0) × R × S^{d−1}` **independent of `ℓ`** — the near-diagonal domination that [Z] §5 records as unavailable +for candidates from (R11) alone (with `r(k + r)H` the integrand is only `≤ (1 + ℓ/r⁴)H`). The second term of (2.1) +is at most `∫_{r_0}^∞ r^{−2}(ℓ/r³)²H ≤ Cℓ²r_0^{−7}` (`A_0 ≤ k²H`). + +*Pointwise limit.* Fix `r, b, u`. As `ℓ ↓ 0`, `k = ℓ/r³ ↓ 0` and `A_r(b, k, u) → A_r(b, 0, u)`: the pinned law +`Q_{r,b,k}` is the regression law with target `v_r(b,k)`, affine in the target with the `k`-free covariance of +[R] §2, so `F_r` depends continuously on `k` in `C²`, the typed product is continuous off the type boundaries +(Gaussian-null by nondegeneracy, [Z] §2), and it is dominated by a polynomial in the endpoint Hessians; `π_r(v_r)` +is continuous in the target. Also `A_0(b, k, u) → 0` (`A_0 ≤ k²H`). Since `r^{−2}A_r(b, 0, u) = r^{d−1}Ψ_0(b, ru)` +((Z2) at `y = ru`), dominated convergence in (2.1) gives + + ν_cand^{near,r_0}(ℓ) − cℓ^{−1/3} → ∫_0^{r_0}∫∫ r^{d−1}Ψ_0(b, ru) dσ db dr = ∫_{0 0`, follows as + soon as the far constant is known to be polynomial in `1/ρ` (take `ρ = ℓ^{θ}`, `θ` small, `N` larger than + `θ` times that degree); the constant's `ρ`-dependence enters through the far-pair nondegeneracy at separation + `ρ` and, for #188, the `N`-site conditional density and `ℓ_1 = ρ²/(64N²)`. None of this is proved here. +3. **Consistency.** (R+.1) with (Z4) is consistent with [C7-K] (T2) and [U] (U1) (`ρ_rej = Θ(1)`); (R+.3) with + [P] (14.1) (candidate density `O(1)` at fixed separation) and with [Z] (Z18) (rejected mass below separation `δ` + is `≤ Cδ`); Lemma E with the remark in Math- #186 ("Where the `O(1)` comes from": at fixed marks the contact error + is `O(r²)`; that remark also noted that this alone does not improve Theorem R, which is why Theorem R+ needs the + domination-plus-limit argument of §2 rather than the ledger of (R16)). The `2r_0` of #186's C7 model check is + the (R17)–(R18) loss term, i.e. the rejected mass at separations `≍ r_0`, in agreement with Remark 1. +4. **For the manuscript.** Once reviewed, (R+.2) supports reading the abstract's `(1 + o(1))` as + `(1 + o(ℓ^{1/3}))` and, for the candidate density, `cℓ^{−1/3} + B_{3,24} + o(1)`. Until then E10 of the V3 edit + pack (`(1 + O(ℓ^{1/3}))`, which rests on Theorem R and its two reads, the second still unmerged in Math- #186) + is the safe statement, and E3's "do not write an `o(1)` remainder for `ν_eld` alone" and E10's "not known to be + improvable" would be revised by this note when it lands. No `+ c_1 + o(1)` with `c_1 ≠ 0` is claimed for `ν_eld`; + the claim is precisely that its constant term is `0`. + +## 4. Sources (exact identities in `SOURCES.json`) + +| Tag | Path | Role | +|---|---|---| +| [R] | `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (blob `247b3ecf`) | Theorem R; §§2–6: coupling (R2)–(R5), Lemma R3.2, (R8)–(R11), the separation-variable identity — consumed | +| [Z] | `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (blob `5b6328ea`) | (Z2)–(Z4), (Z10)–(Z12), (Z14), (Z16)–(Z18) — consumed | +| [C7-K] | `frontiers/c7_total_bounded_20260929/PROOF.md` (blob `28748b08`) | §4 (T1) proof with cutoff `ρ`, (T2) — Remark 2, Remark 3 | +| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`) | §§1–4, 7, 9 (as [E2]), (10.1), (11.1), 14, (15.2) — through [R], [Z] | +| [E1] | `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md` (blob `213594d6`) | reading rule | +| [E2] | `reviews/d1_section9_borel_repair_20260925/REPAIR.md` (blob `fe9b9ce4`) | §9 replacement | +| [REC] | `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md` (blob `75da2597`) | W1, embedding radius, consumption contract | +| [U] | `frontiers/unrestricted_selection_difference_20260929/PROOF.md` (blob `5a55b179`) | (U1) — cited only (Remark 3) | +| #187, #188 | `frontiers/far_elder_rate_20260930/PROOF.md`, `frontiers/far_elder_flat_ridge_20260930/PROOF.md` (unmerged) | far elder rates — cited only (Remark 2) | +| #186 | `reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md` (unmerged) | the review remark that motivated Lemma E — cited only | + +## 5. Exact controls (`even_contact_check.py`; stdlib; exact rationals; byte-identical under `-O`) + +E1 the axial pin identities on a quintic profile: with `f_2 = −(r²/24)f_4`, `f_3 = 12k − (r²/40)f_5` the pins +`f(∓r/2) = b, b − kr³`, `f_x(∓r/2) = 0` hold exactly and `α_M = −6k + (r/12)f_4 − (r²/120)f_5`, +`α_S = 6k + (r/12)f_4 + (r²/120)f_5`, `α_Mα_S = −36k² + r²(f_4²/144 − kf_5/10) − r⁴f_5²/14400`, at rational +`(r, k, f_4, f_5, b)` — the coefficients `1/24, 1/40, 1/12, 1/120` are the content of Step 1; mutant M1 (drop the +`(r²/40)f_5` pin term) fails. +E2 the reflection symmetry on a general pinned quintic in two variables (six pinned coefficients solved exactly, +fifteen free ones fixed at rationals): `det H_M(−r) = det H_S(r)` exactly at rational `r`; consequently +`det H_M det H_S/r²`, recovered as the polynomial of degree ten in `r` by exact interpolation at eleven rational +`r`, has only even powers, constant term `−36k²A_0²` (`A_0 = f_yy(0)`), and the recorded `r²` coefficient. The parity +is forced by the symmetry, so this is a check of the mechanism of Step 3, not of its bookkeeping; mutant M2 +applies the parity test to the single factor `det H_M/r`, which has `r¹, r³, r⁵` terms, and fails. +E3 the block identity `det K = α det A − r βᵀadj(A)β` for `K = [[α, √r βᵀ],[√r β, A]]` (`r` a rational square, +`s = √r` rational, entries `s·β` — [R] Lemma R3.2 with `t = r`, the dictionary of §0) and Haynsworth's inertia count +on rational `3×3` and `4×4` examples with `A` negative definite, of index `m − 1`, and indefinite, and the scalar +Schur complement of both signs (inertia by exact congruence diagonalisation); the Step 4 case split at rational +points: `F_d(K_M) = 0` when `λ_max(A_M) > 0`, and the layer inequality `|det A| ≤ (|λ_max(A)|)·‖A‖^{m−1}` for +`|λ_max(A)|` small. +E4 the ledger of §2 and Remark 2 by exact monomial substitution `r = (ℓ/k)^{1/3}`: `r^{−2}(ℓ/r³)² = ℓ²r^{−8}` +(so `∫_{r_0}^∞ ≍ ℓ²r_0^{−7}`), `k^{−2/3}r^{3}/k = ℓk^{−8/3}`, `ℓκ^{−5/3} = κ^{7/3} = ℓ^{7/12}` at `κ = ℓ^{1/4}`, +`ℓ^{2/3}a^{−1/3} = r_0ℓ^{1/3}`, and `r ≤ k ⇔ ℓ ≤ k⁴`; mutant M3 (claims the loss exponent `1/3`) fails. +E5 the layer algebra of Step 4: `k²u² = C_1²r²T²(k + T)²` and `rT² ≤ ku/C_1` for `u = C_1rT(1 + T/k)`, exactly. +Mutants `M1`–`M3` exit 1; an unknown label exits 2. + +**What the controls do not test.** They check exact identities and finitely many rational instances of Steps 1–3 +(E1, E2), of the block/inertia facts and the case split of Step 4 (E3), of the ledger (E4) and of the layer algebra +(E5). They do **not** test Step 4 as a statement about the random matrices — the Weyl and Hadamard bounds on the +layer `{|λ_max(A_0)| < u}`, the choice of `C_1`, and the claim that the product of the two typed determinants is +`O(r²)` times a polynomial in `T` there — nor Step 5 (the `C^7` moment import from (R4) and the passage to +expectations), nor anything in §2 (the majorant, the pointwise limit, the far limit, the use of (Z4)). Those are +proved in prose only and are review slice A (Steps 4–5) and slice B (§2). The controls are not acceptance. + +## 6. Review slices + +A: Lemma E — Steps 1–3 (the expansions and the product cancellation) and Step 4 (Haynsworth, the good event, the + three layers), Step 5. +B: §2 — the separation-variable identity against [R] §6 / [Z] (Z14), the `ℓ`-uniform majorant from (E.1), the + pointwise limit `A_r(b,k,u) → A_r(b,0,u)`, the far candidate limit from [Z], and the use of (Z4). +C: the statement's scope, Remarks 1–4, and the sources' consumption (in particular that nothing unmerged is used). diff --git a/frontiers/remainder_vanishing_20260930/README.md b/frontiers/remainder_vanishing_20260930/README.md new file mode 100644 index 000000000..097ed0e6e --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/README.md @@ -0,0 +1,52 @@ +# The remainder of Theorem R: `B_{d,L}` for candidates, `0` for elder pairs (CL-D2-REMAINDER-VANISHING-20260930-v1.1) + +Author-side proof candidate, Anthropic Claude, 30 September 2026. Scientific effect NONE. Nonauthor review required. +Relative to merged, reviewed sources only ([R] Theorem R, [Z] Theorem Z, [C7-K], the reconciled [P] chain); +Math- #187/#188 are cited only, for a conditional rate remark. + +**Statement (Theorem R+).** For the periodized Gaussian field on the fixed torus (every `d ≥ 2`, `L > 0`), as +`ℓ ↓ 0`, + + ν_cand(ℓ) = cℓ^{−1/3} + B_{d,L} + o(1), ν_eld(ℓ) = cℓ^{−1/3} + o(1), + +where `B_{d,L} = ∫∫Ψ_0` is the equal-height kernel mass of Theorem Z. Theorem R's bounded remainder is thus +identified: its constant term is the rejected mass `B_{d,L}` for candidate pairs and `0` for elder pairs, so +`ν_eld = cℓ^{−1/3}(1 + o(ℓ^{1/3}))`. Also `ν_cand^{near,r_0} − cℓ^{−1/3} → ∫_{|y|=1 and each finite derivative order q" quoted at the definition +of `T`; B2 the signed near bound proved in §2 as a separate remark (from `A_r ≥ 0` for every `r` and `A_0 ≤ k²H`), +not part of (R+.3); B1 the not-claimed list extended (no RN/24-jet closure; R+ does not enlarge (Z4)). Lemma E, +Theorem R+ and (R+.1)–(R+.3) unchanged. + +Review slices (§6): A Lemma E; B the separation-variable limit argument and the use of Theorem Z; C scope and +remarks. diff --git a/frontiers/remainder_vanishing_20260930/RESULTS.json b/frontiers/remainder_vanishing_20260930/RESULTS.json new file mode 100644 index 000000000..4d636012e --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/RESULTS.json @@ -0,0 +1,85 @@ +{ + "checks": { + "E1_axial_pins_quintic": { + "info": { + "alpha_M": "-6k + (r/12) f4 - (r^2/120) f5", + "points": 3, + "product": "-36k^2 + r^2 (f4^2/144 - k f5/10) - r^4 f5^2/14400" + }, + "passed": true + }, + "E2_parity_pinned_quintic_2d": { + "info": { + "constant": "-1296", + "degrees_present": [ + 0, + 2, + 4, + 6, + 8, + 10 + ], + "r2_coefficient": "-3851051/2025", + "reflection_identity": "det H_M(-r) = det H_S(r) at 3 rational r" + }, + "passed": true + }, + "E3_block_and_inertia": { + "info": { + "cases": 64, + "positive_eigenvalue_cases": 12 + }, + "passed": true + }, + "E4_ledger": { + "info": { + "a^7/3 ell^-1/3": [ + "2", + "0", + "-7" + ], + "ell kappa^-5/3": [ + "7/12", + "0", + "0" + ], + "ell^2/3 a^-1/3": [ + "1/3", + "0", + "1" + ], + "k^-2/3 r^2": [ + "2/3", + "-4/3", + "0" + ], + "k^-2/3 r^3/k": [ + "1", + "-8/3", + "0" + ], + "kappa^7/3": [ + "7/12", + "0", + "0" + ], + "r^-2 (ell/r^3)^2 = ell^2 r^-8": [ + "2", + "0", + "-8" + ] + }, + "passed": true + }, + "E5_layer_algebra": { + "info": { + "identity": "k^2 u^2 = C1^2 r^2 T^2 (k+T)^2", + "inequality": "r T^2 <= k u / C1" + }, + "passed": true + } + }, + "object": "CL-D2-REMAINDER-VANISHING-20260930-v1.1", + "passed": true, + "scientific_effect": "NONE" +} diff --git a/frontiers/remainder_vanishing_20260930/SOURCES.json b/frontiers/remainder_vanishing_20260930/SOURCES.json new file mode 100644 index 000000000..73732cec0 --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/SOURCES.json @@ -0,0 +1,142 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "object": "CL-D2-REMAINDER-VANISHING-20260930-v1.1", + "kind": "author-side proof candidate: Lemma E (even contact expansion) and Theorem R+ (nu_cand = c ell^-1/3 + B_{d,L} + o(1), nu_eld = c ell^-1/3 + o(1)) relative to the merged reviewed sources [R], [Z], [C7-K] and the reconciled [P] chain; nothing unmerged consumed; nonauthor analytic review required", + "author": "Anthropic Claude", + "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU", + "read_at": { + "branch_base": "3e0a91b69892b9717564c592e7a966492a81566a" + }, + "consumed": [ + { + "path": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md", + "bytes": 17734, + "sha256": "380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a", + "git_blob": "247b3ecf80bfbe896948d5d489b2d5842a81c481", + "tag": "R", + "use": "Theorem R; \u00a7\u00a72-7: the coupling (R2)-(R5), Lemma R3.2, (R8)-(R13), the ledgers (R16)-(R18); the near/far decomposition" + }, + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "tag": "P", + "use": "\u00a7\u00a71-4, 7, 9 (as [E2]), 14, (15.2) - through [R]" + }, + { + "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "bytes": 1782, + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "tag": "E1", + "use": "reading rule" + }, + { + "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "bytes": 9062, + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "tag": "E2", + "use": "\u00a79 replacement" + }, + { + "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "bytes": 23312, + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da", + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "tag": "REC", + "use": "W1; consumption contract (D2 row)" + }, + { + "path": "frontiers/c7_total_bounded_20260929/PROOF.md", + "bytes": 14982, + "sha256": "16c0efe372c1b1f26460aaa109287fed78e6fbd2dd00a0530998db296278dd01", + "git_blob": "28748b086ec6761ef467dc67cba475fbdf8b7455", + "tag": "C7-K", + "use": "\u00a74 proof of (T1) with cutoff rho (Remark 2), (T2) (Remark 3)" + }, + { + "path": "frontiers/c7_zero_gap_limit_20260929/PROOF.md", + "bytes": 17592, + "sha256": "f0e563c45c41333f2bb6e5bf5cb98f0d3b592e2abf2af1c1b098c21ad77cb777", + "git_blob": "5b6328ea3d6afcbeb731698b3834e536f72d1bcb", + "tag": "Z", + "use": "(Z2)-(Z4), (Z10)-(Z12), (Z14), (Z16)-(Z18): the equal-height kernel, B_{d,L}, Theorem Z (Z4), the far-set domination and pointwise limits - consumed; OpenAI author, Anthropic A/B/C reads on Math-#155 (5359005993, 5359022219)" + } + ], + "cited_only": [ + { + "path": "frontiers/unrestricted_selection_difference_20260929/PROOF.md", + "bytes": 13028, + "sha256": "93a63002d32fd5c35ae994fe28b1205d8098026d6ddde19c67d4977dcf5817a9", + "git_blob": "5a55b179a974a90c65d257f3f93765cc6fb30bb8", + "tag": "U", + "use": "(U1) - Remark 3; OpenAI review reviews/c7_nonvanishing_openai_20260929" + } + ], + "cited_unmerged": [ + { + "path": "frontiers/far_elder_flat_ridge_20260930/PROOF.md", + "pr": "d6g8k5htny-coder/Math-#188", + "commit": "85f0586a468c2b2f73af0e340d251e33a52caade", + "git_blob": "50cfbf4fcb9a3526f979eaedf885d42824a3c3a3", + "bytes": 22806, + "sha256": "957f9c5e6557a558b2c055fa2a1f11ba301581995b8958658fad2edf47136f57", + "tag": "188", + "use": "cited only (Remark 2, conditional rate): far elder density O(ell^N); not consumed, not verified by the workflow" + }, + { + "path": "frontiers/far_elder_rate_20260930/PROOF.md", + "pr": "d6g8k5htny-coder/Math-#187", + "commit": "ecd47e7b1d447eefec963b1fa8395777db09a190", + "git_blob": "37dcf6ef933251c8890ab653cafd648610779342", + "bytes": 16413, + "sha256": "67f606327b66a2b4f1b8c67a57d119ac9ddfd2ce827ca8cb0c4daefa31904a3b", + "tag": "187", + "use": "cited only (Remark 2, conditional rate): far elder density O(ell^(2/3)); not consumed" + }, + { + "path": "reviews/d2_remainder_full_depth_claude_20260930/REVIEW.md", + "pr": "d6g8k5htny-coder/Math-#186", + "commit": "ace171cb0cee416d9253405781e3e843b0711b16", + "git_blob": "529c5264ed790ab1df36c14f567f460e2a5ed974", + "bytes": 20595, + "sha256": "8679afd19ad313ade3602bba48b744183fdbefc9596a7793cfe7f921988a5dfd", + "tag": "186", + "use": "cited only: the review remark on the O(r^2) contact error that motivated Lemma E" + } + ], + "mutants": [ + "M1", + "M2", + "M3" + ], + "referee": { + "kind": "clean-context Anthropic Claude subagent before landing (same provider, same account; not a nonauthor review)", + "findings": "no blocking; applied: Step 4 layer inequality false on the indefinite part (three-case split: lambda_max(A_0) >= u kills F_d(K_M); |lambda_max| < u via Weyl); Theorem R+ restated without any dependence on Math-#188 (Theorem Z (Z4) plus the ell-uniform candidate majorant from Lemma E identify the constant B_{d,L}); Proposition L removed as a proposition (it is [C7-K] section 4), Proposition C folded into (R+.3) with its limit identified; T dominates C^7; admissible radii stated; remarks corrected (attribution of the #186 model check, the rate remark, the manuscript remark, no-second-coefficient wording); controls: E2 recorded as symmetry-forced with the exact reflection identity added, E3 exercises the indefinite layer, E4 tautology removed" + }, + "files": [ + { + "path": "PROOF.md", + "bytes": 25353, + "sha256": "0cb5b949261476f10e539677e2744e3288752474393af25ba787e67b1350f32a" + }, + { + "path": "even_contact_check.py", + "bytes": 20220, + "sha256": "473ae49f2d4b3a319b5bd90a28ef34c7c9ae8a6f7af953d75ebc39969ea02f74" + }, + { + "path": "RESULTS.json", + "bytes": 1393, + "sha256": "b58b5e90419415c69aa16f1a13e09920147db9e24b131bd53864beb67fb6e428" + }, + { + "path": "README.md", + "bytes": 4393, + "sha256": "f3078a42edc8a0dbe88b6cca645ddff1518366756c2423b3ffc898bb283cd02f" + } + ] +} diff --git a/frontiers/remainder_vanishing_20260930/even_contact_check.py b/frontiers/remainder_vanishing_20260930/even_contact_check.py new file mode 100644 index 000000000..5c1cd64ae --- /dev/null +++ b/frontiers/remainder_vanishing_20260930/even_contact_check.py @@ -0,0 +1,422 @@ +#!/usr/bin/env python3 +"""Exact finite controls for PROOF.md (CL-D2-REMAINDER-VANISHING-20260930-v1.1). Standard library only; exact rationals. + + E1 axial pin identities on a quintic profile f(x) = b0 + f1 x + f2 x^2/2 + f3 x^3/6 + f4 x^4/24 + f5 x^5/120: with + b0, f1, f2, f3 solved from the four pins f(-r/2) = b, f(r/2) = b - k r^3, f'(-r/2) = f'(r/2) = 0 one finds + f2 = -(r^2/24) f4, f3 = 12k - (r^2/40) f5, and alpha_M = f''(-r/2)/r = -6k + (r/12) f4 - (r^2/120) f5, + alpha_S = f''(r/2)/r = 6k + (r/12) f4 + (r^2/120) f5, alpha_M alpha_S = -36k^2 + r^2 (f4^2/144 - k f5/10) + - r^4 f5^2/14400 -- all exact at rational (r, k, f4, f5, b); mutant M1 drops the (r^2/40) f5 term; + E2 reflection symmetry: for a general quintic in (x, y) with fifteen free rational coefficients, the six pinned + coefficients (c00, c10, c20, c30, c01, c11) are solved exactly at rational r and at -r from f(M) = b, + f(S) = b - k r^3, grad f(M) = grad f(S) = 0, M = (-r/2, 0), S = (r/2, 0); the identity det H_M(-r) = det H_S(r) + holds exactly; consequently det H_M det H_S / r^2, recovered by exact Lagrange interpolation at eleven rational r + as the polynomial of degree <= 10 in r, has only even powers, constant term -36 k^2 A0^2 (A0 = f_yy(0) = 2 c02) + and the recorded r^2 coefficient (the parity is forced by the symmetry: a check of the mechanism, not of the + bookkeeping); mutant M2 applies the parity test to the single factor det H_M / r, which is not even; + E3 the block identity det K = alpha det A - r beta^T adj(A) beta for K = [[alpha, sqrt(r) beta^T], [sqrt(r) beta, A]] + ([R] Lemma R3.2 with t = r; r a rational square, entries sqrt(r) beta), and Haynsworth's inertia additivity + inertia(K) = inertia(A) + inertia(alpha - r beta^T A^{-1} beta), on rational 2x2/3x3 blocks A that are negative + definite, of index m - 1, and indefinite, with the scalar Schur complement of both signs (inertia by exact + congruence diagonalisation); the Step 4 case split: F_d(K_M) = 0 whenever lambda_max(A_M) > 0 (A_M has a + positive eigenvalue), and |det A| <= |lambda_max(A)| * ||A||_op^(m-1) when |lambda_max(A)| is the smallest + absolute eigenvalue, at rational points; + E4 the ledger of section 2 and Remark 2 by exact monomial substitution r = (ell/k)^(1/3): r^-2 (ell/r^3)^2 = + ell^2 r^-8 (so the far-cutoff tail is ell^2 r0^-7), k^(-2/3) r^3 / k = ell k^(-8/3), ell kappa^(-5/3) = kappa^(7/3) + = ell^(7/12) at kappa = ell^(1/4), ell^(2/3) a^(-1/3) = r0 ell^(1/3) for a = ell r0^-3, and r <= k iff ell <= k^4 at + rational (k, r) with ell = k r^3; mutant M3 claims the loss exponent 1/3; + E5 the layer algebra of Step 4: k^2 u^2 = C1^2 r^2 T^2 (k + T)^2 and r T^2 <= k u / C1 for u = C1 r T (1 + T/k), exactly. + M mutants (--mutant M1|M2|M3) exit 1; an unknown label exits 2. + Scope: E1-E5 check exact identities and finitely many rational instances. They do NOT test Step 4 of PROOF.md as a + statement about random matrices (the Weyl/Hadamard layer bound, the choice of C1, the O(r^2) product on the flip + layer), nor Step 5 (the C^7 moment import from [R] (R4) and the expectations), nor section 2 (majorant, pointwise + and far limits, the use of (Z4)); those are proved in prose only. The controls are not acceptance. +""" +import argparse +import json +import sys +from fractions import Fraction as F + + +# ---------------------------------------------------------------------------------------------------- linear algebra +def solve(Mx, rhs): + n = len(Mx) + A = [list(r) + [b] for r, b in zip(Mx, rhs)] + for col in range(n): + piv = next((r for r in range(col, n) if A[r][col] != 0), None) + if piv is None: + return None + A[col], A[piv] = A[piv], A[col] + inv = 1/A[col][col] + A[col] = [v*inv for v in A[col]] + for r in range(n): + if r != col and A[r][col] != 0: + f = A[r][col] + A[r] = [a - f*b for a, b in zip(A[r], A[col])] + return [A[r][n] for r in range(n)] + + +def det(M): + n = len(M) + if n == 1: + return M[0][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 adj(M): + n = len(M) + out = [[F(0)]*n for _ in range(n)] + for i in range(n): + for j in range(n): + minor = [row[:j] + row[j+1:] for r_, row in enumerate(M) if r_ != i] + out[j][i] = (-1)**(i+j)*det(minor) if n > 1 else F(1) + return out + + +def inertia(M): + """(negatives, zeros, positives) of a symmetric rational matrix by congruence diagonalisation.""" + n = len(M) + A = [row[:] for row in M] + neg = zero = pos = 0 + for i in range(n): + if A[i][i] == 0: + piv = next((j for j in range(i + 1, n) if A[j][j] != 0), None) + if piv is not None: + A[i], A[piv] = A[piv], A[i] + for row in A: + row[i], row[piv] = row[piv], row[i] + else: + j = next((j for j in range(i + 1, n) if A[i][j] != 0), None) + if j is None: + zero += 1 + continue + # row_i += row_j, col_i += col_j (congruence) makes the (i,i) entry 2 A[i][j] + A[j][j] = 2 A[i][j] + for c in range(n): + A[i][c] += A[j][c] + for r_ in range(n): + A[r_][i] += A[r_][j] + p = A[i][i] + if p == 0: + zero += 1 + continue + if p < 0: + neg += 1 + else: + pos += 1 + for r_ in range(i + 1, n): + if A[r_][i] != 0: + f = A[r_][i]/p + for c in range(n): + A[r_][c] -= f*A[i][c] + for c in range(n): + A[c][r_] -= f*A[c][i] + return neg, zero, pos + + +# ---------------------------------------------------------------------------------------------------- E1 +def check_E1(mutant): + n = 0 + for (r, k, f4, f5, b) in ((F(1, 10), F(2), F(3), F(-7), F(1, 3)), (F(1, 3), F(1, 5), F(-11), F(13), F(0)), + (F(2, 7), F(4), F(1, 2), F(1, 3), F(-5))): + f2 = -(r*r/24)*f4 + f3 = 12*k - (r*r/40)*f5 + if mutant == 'M1': + f3 = 12*k + # f1, b0 from the gradient and height pins + def fx(x, f1): + return f1 + f2*x + f3*x*x/2 + f4*x**3/6 + f5*x**4/24 + # f'(-r/2) = 0 gives f1 + f1 = -(f2*(-r/2) + f3*(r*r/4)/2 + f4*(-r/2)**3/6 + f5*(r/2)**4/24) + if fx(r/2, f1) != 0: + return False, 'gradient pin at S fails' + def f(x, b0): + return b0 + f1*x + f2*x*x/2 + f3*x**3/6 + f4*x**4/24 + f5*x**5/120 + b0 = b - (f(-r/2, F(0))) + if f(-r/2, b0) != b or f(r/2, b0) != b - k*r**3: + return False, 'height pins fail' + def fxx(x): + return f2 + f3*x + f4*x*x/2 + f5*x**3/6 + aM, aS = fxx(-r/2)/r, fxx(r/2)/r + if aM != -6*k + (r/12)*f4 - (r*r/120)*f5 or aS != 6*k + (r/12)*f4 + (r*r/120)*f5: + return False, 'alpha expansions' + if aM*aS != -36*k*k + r*r*(f4*f4/144 - k*f5/10) - r**4*f5*f5/14400: + return False, 'alpha product' + n += 1 + return True, {'points': n, 'alpha_M': '-6k + (r/12) f4 - (r^2/120) f5', 'product': '-36k^2 + r^2 (f4^2/144 - k f5/10) - r^4 f5^2/14400'} + + +# ---------------------------------------------------------------------------------------------------- E2 +MONOS = [(i, j) for i in range(6) for j in range(6 - i)] # 21 monomials x^i y^j, i + j <= 5 +PINNED = [(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1)] +FREE = [m for m in MONOS if m not in PINNED] + + +def d_coef(i, j, dx, dy, x, y): + """value of d^dx_x d^dy_y (x^i y^j) at (x, y).""" + c = F(1) + for t in range(dx): + if i - t <= 0: + return F(0) + c *= (i - t) + for t in range(dy): + if j - t <= 0: + return F(0) + c *= (j - t) + return c*x**(i - dx)*y**(j - dy) + + +def pinned_poly(r, k, b, free, sign): + """Return the full coefficient dict of the pinned quintic (sign = +1: f(S) = b - k r^3; sign = -1: b + k r^3).""" + M = (-r/2, F(0)); S = (r/2, F(0)) + conds = [((0, 0), M, b), ((0, 0), S, b - sign*k*r**3), ((1, 0), M, F(0)), ((1, 0), S, F(0)), ((0, 1), M, F(0)), ((0, 1), S, F(0))] + rows, rhs = [], [] + for (dx, dy), pt, val in conds: + rows.append([d_coef(i, j, dx, dy, *pt) for (i, j) in PINNED]) + rhs.append(val - sum(free[m]*d_coef(m[0], m[1], dx, dy, *pt) for m in FREE)) + sol = solve(rows, rhs) + coef = dict(free) + for m, v in zip(PINNED, sol): + coef[m] = v + return coef + + +def hess(coef, x, y): + fxx = sum(c*d_coef(i, j, 2, 0, x, y) for (i, j), c in coef.items()) + fxy = sum(c*d_coef(i, j, 1, 1, x, y) for (i, j), c in coef.items()) + fyy = sum(c*d_coef(i, j, 0, 2, x, y) for (i, j), c in coef.items()) + return fxx, fxy, fyy + + +def interpolate(xs, ys): + """Coefficients (low to high) of the unique polynomial of degree < len(xs) through (xs, ys): exact Lagrange.""" + n = len(xs) + coeffs = [F(0)]*n + for i in range(n): + # basis polynomial l_i + num = [F(1)] + denom = F(1) + for j in range(n): + if j != i: + num = [F(0)] + num + for t in range(len(num) - 1): + num[t] -= xs[j]*num[t + 1] + denom *= (xs[i] - xs[j]) + for t in range(n): + coeffs[t] += ys[i]*num[t]/denom + return coeffs + + +def check_E2(mutant): + k, b = F(3, 2), F(-1, 4) + free = {} + vals = [F(2), F(-1), F(1, 2), F(3), F(-2, 3), F(5), F(-1, 5), F(7, 3), F(1), F(-4), F(2, 7), F(-3, 2), F(4, 5), F(-1, 7), F(6)] + for m, v in zip(FREE, vals): + free[m] = v + sign = 1 + def dets(r): + coef = pinned_poly(r, k, b, free, sign) + fxxM, fxyM, fyyM = hess(coef, -r/2, F(0)) + fxxS, fxyS, fyyS = hess(coef, r/2, F(0)) + return fxxM*fyyM - fxyM*fxyM, fxxS*fyyS - fxyS*fxyS + # the reflection symmetry: the pinned family at -r has M and S exchanged, so det H_M(-r) = det H_S(r) + for r in (F(1, 10), F(3, 7), F(1)): + dMm, dSm = dets(-r) + dMp, dSp = dets(r) + if dMm != dSp or dSm != dMp: + return False, 'reflection identity det H_M(-r) = det H_S(r) fails' + def value(r): + dM, dS = dets(r) + if mutant == 'M2': + return dM/r # a single endpoint: NOT even in r + return dM*dS/(r*r) + xs = [F(n, 10) for n in range(1, 12)] # eleven rational r values + ys = [value(r) for r in xs] + poly = interpolate(xs, ys) + # verify at a twelfth value that the degree-10 interpolant is the polynomial (degree <= 10 in r) + r12 = F(13, 10) + if sum(c*r12**t for t, c in enumerate(poly)) != value(r12): + return False, 'not a polynomial of degree <= 10 in r' + odd = [t for t in range(1, 11, 2) if poly[t] != 0] + if odd: + return False, 'odd powers present: %s' % odd + A0 = 2*free[(0, 2)] + if poly[0] != -36*k*k*A0*A0: + return False, 'constant term is not -36 k^2 A0^2' + c02, c12, c21, c22, c31, c40, c50 = (free[(0, 2)], free[(1, 2)], free[(2, 1)], free[(2, 2)], free[(3, 1)], free[(4, 0)], free[(5, 0)]) + r2 = (16*c02**2*c40**2 - 48*c02**2*c50*k - 8*c02*c21**2*c40 + 24*c02*c21*c31*k - 72*c02*c22*k*k + + 36*c12**2*k*k - 12*c12*c21**2*k + c21**4) + if poly[2] != r2: + return False, 'r^2 coefficient differs from the displayed polynomial' + return True, {'reflection_identity': 'det H_M(-r) = det H_S(r) at 3 rational r', 'degrees_present': [t for t in range(11) if poly[t] != 0], + 'constant': str(poly[0]), 'r2_coefficient': str(poly[2])} + + +# ---------------------------------------------------------------------------------------------------- E3 +def block(alpha, beta, A, r): + n = len(A) + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(len(A)): + K[0][i+1] = K[i+1][0] = r*beta[i] # sqrt(r) beta entries squared give r; with r rational we use t = r directly + for j in range(len(A)): + K[i+1][j+1] = A[i][j] + return K + + +def check_E3(mutant): + # [R] Lemma R3.2 with t = r: K = [[alpha, sqrt(r) beta^T], [sqrt(r) beta, A]] and det K = alpha det A - r beta^T adj(A) beta. + # Exact rationals: r is chosen a rational square, s = sqrt(r) rational, and the off-diagonal entries are s*beta, + # so the identity tested is det K = alpha det A - s^2 q = alpha det A - r q with q = beta^T adj(A) beta. + cases = 0 + A_list = [ + [[F(-2), F(1, 2)], [F(1, 2), F(-3)]], # negative definite + [[F(-2), F(0)], [F(0), F(1)]], # index m-1 = 1 + [[F(-3), F(1), F(0)], [F(1), F(-2), F(1, 2)], [F(0), F(1, 2), F(-4)]], # negative definite 3x3 + [[F(-3), F(1), F(0)], [F(1), F(2), F(1, 2)], [F(0), F(1, 2), F(-4)]], # indefinite 3x3 + ] + for A in A_list: + m = len(A) + adjA = adj(A); dA = det(A) + for beta in ([F(1), F(-2)] if m == 2 else [F(1), F(-2), F(1, 3)], [F(0)]*m): + for alpha in (F(-5), F(-1, 10), F(1, 10), F(5)): + for s in (F(1, 2), F(2)): # sqrt(r) = s, r = s^2 + r = s*s + n = m + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(m): + K[0][i+1] = K[i+1][0] = s*beta[i] + for j in range(m): + K[i+1][j+1] = A[i][j] + q = sum(beta[i]*sum(adjA[i][j]*beta[j] for j in range(m)) for i in range(m)) + if det(K) != alpha*dA - r*q: + return False, 'block determinant identity' + if dA != 0: + sigma = alpha - r*q/dA # alpha - r beta^T A^{-1} beta (A^{-1} = adj/det) + negK = inertia(K)[0] + negA = inertia(A)[0] + if negK != negA + (1 if sigma < 0 else 0): + return False, 'Haynsworth inertia additivity fails' + cases += 1 + # good-event inequalities: A_S negative definite and alpha_S > 0 give sigma_S > 0 + A = A_list[0]; adjA = adj(A); dA = det(A) + for beta in ([F(1), F(-2)], [F(3), F(1, 7)]): + q = sum(beta[i]*sum(adjA[i][j]*beta[j] for j in range(2)) for i in range(2)) + for alpha in (F(1, 100), F(3)): + for r in (F(1, 10), F(1)): + if not (alpha - r*q/dA > 0): + return False, 'sigma_S > 0 fails' + # sigma_M < 0 under the margins: alpha_M <= -5k, r beta^T(-A)^{-1}beta <= 2rT^2/u <= 2k/C1 + k, T, C1, r = F(1), F(4), F(100), F(1, 1000) + u = C1*r*T*(1 + T/k) + if not (2*r*T*T/u <= 2*k/C1 and -5*k + 2*k/C1 < 0): + return False, 'sigma_M margin' + # Step 4, second layer: A_M with a positive eigenvalue => K_M has a positive eigenvalue => F_d(K_M) = 0 + layer2 = 0 + for A in ([[F(-2), F(0)], [F(0), F(1, 50)]], [[F(-3), F(1)], [F(1), F(1, 10)]], [[F(-3), F(1), F(0)], [F(1), F(1, 20), F(1, 2)], [F(0), F(1, 2), F(-4)]]): + m = len(A) + if inertia(A)[2] == 0: + return False, 'test matrix has no positive eigenvalue' + for alpha in (F(-5), F(-1, 10)): + for s_ in (F(1, 2), F(1, 10)): + beta = [F(1)]*m + n = m + 1 + K = [[F(0)]*n for _ in range(n)] + K[0][0] = alpha + for i in range(m): + K[0][i+1] = K[i+1][0] = s_*beta[i] + for j in range(m): + K[i+1][j+1] = A[i][j] + if inertia(K)[2] == 0: # K negative semidefinite would give index d + return False, 'K_M negative definite although A_M has a positive eigenvalue' + layer2 += 1 + # Step 4, third layer: |det A| <= |lambda_max(A)| ||A||_op^(m-1) when lambda_max is the eigenvalue of least modulus: + # diagonal examples, exact + for diag in ([F(-3), F(-1, 100)], [F(-5), F(-2), F(1, 1000)], [F(-4), F(-3), F(-1, 10)]): + A = [[F(0)]*len(diag) for _ in diag] + for i, dv in enumerate(diag): + A[i][i] = dv + lam_max = max(diag) + opnorm = max(abs(v) for v in diag) + if not (abs(det(A)) <= abs(lam_max)*opnorm**(len(diag) - 1)): + return False, 'layer determinant bound' + return True, {'cases': cases, 'positive_eigenvalue_cases': layer2} + + +# ---------------------------------------------------------------------------------------------------- E4 +class Mono: + def __init__(self, ell=F(0), k=F(0), r0=F(0)): + self.e = (F(ell), F(k), F(r0)) + def __mul__(self, o): + return Mono(*[a + b for a, b in zip(self.e, o.e)]) + def __pow__(self, p): + p = F(p); return Mono(*[a*p for a in self.e]) + + +def check_E4(mutant): + ell, k, r0 = Mono(1, 0, 0), Mono(0, 1, 0), Mono(0, 0, 1) + r = (ell*k**(-1))**F(1, 3) + a = ell*r0**(-3) + kappa = ell**F(1, 4) + rr = Mono(0, 0, 1) # a separation variable, reusing the third slot + claims = { + 'r^-2 (ell/r^3)^2 = ell^2 r^-8': (((rr**(-2))*(ell*rr**(-3))**2).e, (F(2), F(0), F(-8))), + 'k^-2/3 r^2': ((k**F(-2, 3)*r**2).e, (F(2, 3), F(-4, 3), F(0))), + 'k^-2/3 r^3/k': ((k**F(-2, 3)*r**3*k**(-1)).e, (F(1), F(-8, 3), F(0))), + 'ell^2/3 a^-1/3': (((ell**F(2, 3))*a**F(-1, 3)).e, (F(1, 3), F(0), F(1))), + 'ell kappa^-5/3': ((ell*kappa**F(-5, 3)).e, (F(7, 12), F(0), F(0))), + 'kappa^7/3': ((kappa**F(7, 3)).e, (F(7, 12), F(0), F(0))), + 'a^7/3 ell^-1/3': ((a**F(7, 3)*ell**F(-1, 3)).e, (F(2), F(0), F(-7))), + } + for name, (got, want) in claims.items(): + if got != want: + return False, 'monomial claim fails: %s got %s' % (name, got) + # absolute loss exponent: ell^(7/12) scaled -> ell^(7/12 - 1/3) = ell^(1/4) + loss = F(7, 12) - F(1, 3) + if mutant == 'M3': + loss = F(1, 3) + if loss != F(1, 4): + return False, 'loss exponent' + # r <= k <=> ell <= k^4, with r = (ell/k)^(1/3) computed exactly where ell/k is a rational cube + for (K, rv) in ((F(1, 10), F(1, 10)), (F(1, 10), F(1, 5)), (F(1, 2), F(1, 4)), (F(1, 16), F(1, 8))): + L = K*rv**3 # ell = k r^3, so r = (ell/k)^(1/3) exactly + if (rv <= K) != (L <= K**4): + return False, 'threshold r <= k <=> ell <= k^4 fails' + return True, {name: [str(x) for x in got] for name, (got, want) in claims.items()} + + +# ---------------------------------------------------------------------------------------------------- E5 +def check_E5(mutant): + for (k, T, C1, r) in ((F(1, 3), F(5), F(50), F(1, 100)), (F(4), F(2), F(10), F(1, 7)), (F(1, 100), F(30), F(1000), F(1, 10**6))): + u = C1*r*T*(1 + T/k) + if k*k*u*u != C1*C1*r*r*T*T*(k + T)**2: + return False, 'k^2 u^2 identity' + if not (r*T*T <= k*u/C1): + return False, 'rT^2 <= ku/C1' + return True, {'identity': 'k^2 u^2 = C1^2 r^2 T^2 (k+T)^2', 'inequality': 'r T^2 <= k u / C1'} + + +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'): + sys.stderr.write('unknown mutant label\n'); sys.exit(2) + results = {} + ok_all = True + for name, fn in (('E1_axial_pins_quintic', check_E1), ('E2_parity_pinned_quintic_2d', check_E2), ('E3_block_and_inertia', check_E3), + ('E4_ledger', check_E4), ('E5_layer_algebra', check_E5)): + ok, info = fn(mutant) + results[name] = {'passed': bool(ok), 'info': info} + ok_all = ok_all and ok + out = {'object': 'CL-D2-REMAINDER-VANISHING-20260930-v1.1', '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/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; header: nonauthor analytic review required) \u2014 cited in a non-blocking remark only, not relied on", + "sha256": "93a63002d32fd5c35ae994fe28b1205d8098026d6ddde19c67d4977dcf5817a9" + }, + { + "blob": "64794e5801b70d950c7e1202f7b6b7d07055cbb6", + "bytes": 10807, + "path": "reviews/c7_nonvanishing_openai_20260929/REVIEW.md", + "role": "nonauthor OpenAI review of Theorem U (ACCEPT at blob 5a55b179) \u2014 the review binding for the Theorem U citation", + "sha256": "68b6cd43351706745992c4af242abdc5bdabd46ab5ac99347f2c12b505e1a15d" + }, + { + "blob": "5b6328ea3d6afcbeb731698b3834e536f72d1bcb", + "bytes": 17592, + "path": "frontiers/c7_zero_gap_limit_20260929/PROOF.md", + "role": "Theorem Z (OpenAI candidate; header: non-OpenAI review required) \u2014 named in the scope statement only, not relied on", + "sha256": "f0e563c45c41333f2bb6e5bf5cb98f0d3b592e2abf2af1c1b098c21ad77cb777" + } + ], + "delivered_under": "owner instruction in claude.ai session 2026-09-30 (close open mathematical items; 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": "ca7298f97379d48ee717c1efa4570fd3bc91eded22be0e16c501702dd1415874" + }, + { + "bytes": 1666, + "path": "moment-checks-normal.log", + "sha256": "29e85d8acec68f817812b97b51986e8b86ab5dce2266ca0c3367f72815bb9e7b" + }, + { + "bytes": 1219, + "path": "moment-checks-optimized.json", + "sha256": "ca7298f97379d48ee717c1efa4570fd3bc91eded22be0e16c501702dd1415874" + }, + { + "bytes": 1666, + "path": "moment-checks-optimized.log", + "sha256": "29e85d8acec68f817812b97b51986e8b86ab5dce2266ca0c3367f72815bb9e7b" + }, + { + "bytes": 1896, + "path": "AUTHOR_COMMENT_5907459466.json", + "sha256": "835a557cdd953cccaf6b1e15cd08e4facddc0e476d11e2c2d9ecfd414b524c68" + }, + { + "bytes": 1221, + "path": "AUTHOR_COMMENT_5903080058.json", + "sha256": "a0918eb36d54a2d78133bb7f01e66c5dc89c8e7248f10940067ca4a09aa8da74" + }, + { + "bytes": 1396, + "path": "SOURCES.json", + "sha256": "22b3064a278faff1d915dfb658e9d7e41ec5c4ee69f31e944d47a699a82f8a07" + } + ], + "claim_id": "13dc84b1-bc12-42df-8c7f-96f23cb1dd7d", + "portable_copy_of_sha256": "b532794b61e6ffd5539ce9f7dc5509b44d04d2c238a14f33e214c8e387532a64", + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "bytes": 15755, + "commit": "71772a8906cf6663d16be2cd5f63e5d4ebdcee2b", + "git_blob": "65f24b1f9a7b80522a2ee059f5f6877c9f970f0d", + "id": "PR176", + "local_path": "PROOF.md", + "repository_path": "frontiers/radial_rate_20260930/PROOF.md", + "sha256": "a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710" + }, + { + "bytes": 12533, + "commit": "80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "id": "R", + "local_path": "R.md", + "repository_path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973" + }, + { + "bytes": 25006, + "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "id": "SC", + "local_path": "SC.md", + "repository_path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb" + }, + { + "bytes": 40261, + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "P", + "local_path": "P.md", + "repository_path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7" + } + ] +} diff --git a/reviews/radial_moment_corollaries_20260930/IDENTITY_AND_CONTROL_OBSERVATIONS.json b/reviews/radial_moment_corollaries_20260930/IDENTITY_AND_CONTROL_OBSERVATIONS.json new file mode 100644 index 000000000..a2aff201b --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/IDENTITY_AND_CONTROL_OBSERVATIONS.json @@ -0,0 +1,198 @@ +{ + "c28": { + "PROOF.md": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/continuation28/payload/proof/PROOF.md", + "bytes": 13952, + "sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "git_blob": "d5ee8abee120a8bec2f6c470e517ff9db4645183" + }, + "SOURCES.json": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/continuation28/payload/proof/SOURCES.json", + "bytes": 5148, + "sha256": "45d36269416ba0033edf184a540c32b3007e91fb2031ab8b77ae89eb1942db2e", + "git_blob": "46a791e3313df1954a332db2da7d3489d29bc48c" + }, + "check_exact.py": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/continuation28/payload/proof/check_exact.py", + "bytes": 4445, + "sha256": "2d8b4a2cc61908412574734c0266e5b568c32a653d19eb70edd1c0e46a3c8cab", + "git_blob": "a15e2b4df65966769f7cd347453e4e660ae12da5" + }, + "observed_control_replay": { + "good_quartic_cases": 54, + "normalization_exponent_cases": 12, + "outside_hypothesis_axial_negative_controls": 18, + "proof_sha256": "3c69e20a332457211b8b1182a4819ce9b1dda824801f3a90157751f41b8fedf8", + "runner_sha256": "2d8b4a2cc61908412574734c0266e5b568c32a653d19eb70edd1c0e46a3c8cab", + "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": [ + { + "bytes": 40261, + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "P", + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7" + }, + { + "bytes": 1782, + "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0", + "id": "ERR", + "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028" + }, + { + "bytes": 9062, + "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a", + "id": "BOREL", + "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f" + }, + { + "bytes": 23312, + "git_blob": "75da2597971510f843f8d90c743950cb8c177342", + "id": "REC", + "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da" + }, + { + "bytes": 23416, + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "id": "MARK_D", + "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e" + }, + { + "bytes": 39722, + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "id": "MARK_2", + "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b" + }, + { + "bytes": 15160, + "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1", + "id": "CAP", + "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc" + } + ], + "sources_sha256": "45d36269416ba0033edf184a540c32b3007e91fb2031ab8b77ae89eb1942db2e", + "status": "PASS" + } + }, + "c27": { + "MOMENT_CONSEQUENCE.md": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/coordination/math-advance-20260930/pr176/MOMENT_CONSEQUENCE.md", + "bytes": 11110, + "sha256": "29bba56b56fd3223a2ea71d012f9abe04eb30240bc5bbfe8f5b131e0bba6f4dd", + "git_blob": "fb3bef3a9a46cc0135a88bae6c26e1bade4e7384" + }, + "COROLLARY_CUSTODY.json": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/coordination/math-advance-20260930/pr176/COROLLARY_CUSTODY.json", + "bytes": 3536, + "sha256": "b532794b61e6ffd5539ce9f7dc5509b44d04d2c238a14f33e214c8e387532a64", + "git_blob": "f957db01bba3dc8ca73120ded1557d1d81c27259" + }, + "moment_consequence_checks.py": { + "path": "/Users/dylanroy/Documents/Codex/2026-09-19/connect-research-repository/work/coordination/math-advance-20260930/pr176/moment_consequence_checks.py", + "bytes": 8259, + "sha256": "0c35140179e0345eb03764bf29c3c81453ea278ce1a6319ab33edf6a548e03e6", + "git_blob": "95275971632ea61c3d2cd06e2f73f0836c1d7b83" + }, + "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. Source preservation is mandatory. diff --git a/reviews/radial_moment_corollaries_20260930/SOURCES.json b/reviews/radial_moment_corollaries_20260930/SOURCES.json new file mode 100644 index 000000000..37f3dc7b5 --- /dev/null +++ b/reviews/radial_moment_corollaries_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/reviews/radial_moment_corollaries_20260930/moment-checks-normal.json b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.json new file mode 100644 index 000000000..d311bc520 --- /dev/null +++ b/reviews/radial_moment_corollaries_20260930/moment-checks-normal.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-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))