diff --git a/.github/workflows/c6-cluster-law.yml b/.github/workflows/c6-cluster-law.yml new file mode 100644 index 000000000..9b053545d --- /dev/null +++ b/.github/workflows/c6-cluster-law.yml @@ -0,0 +1,70 @@ +name: C6 cluster-law checks + +'on': + pull_request: + paths: + - 'frontiers/c6_cluster_law_20260929/**' + - '.github/workflows/c6-cluster-law.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-and-probe: + runs-on: ubuntu-latest + timeout-minutes: 15 + 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/c6_cluster_law_20260929').resolve() + manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files'] + present = sorted(p.name for p in root.iterdir()) + expected_names = sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']) + if present != expected_names: + raise ValueError('packet tree differs from manifest: ' + repr(present)) + for entry in manifest: + path = root / entry['path'] + if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink(): + raise ValueError('flat regular source required: ' + entry['path']) + data = path.read_bytes() + if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']: + raise ValueError('source identity mismatch: ' + entry['path']) + pins = json.loads((root / 'SOURCE_MAP.json').read_text()) + for entry in pins['inputs'] + pins['planar_sources_behind_DL']: + 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 / 'cluster_law_check.py') + expected = (root / 'RESULTS.json').read_bytes() + mutants = ['pins-not-critical', 'wrong-pin-hessian', 'drop-sigma-jacobian', 'three-extra-points', 'shear-drop-cubic', 's-bound-constant', + 'cross-term-not-small', 'window-closed', 'index-sign'] + 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 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', 'unknown'], + capture_output=True, timeout=600).returncode + if rc != 2: + raise RuntimeError('unknown mutant not rejected with rc 2') + print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False})) + PY + git diff --exit-code diff --git a/.github/workflows/c6-marked-fourier.yml b/.github/workflows/c6-marked-fourier.yml new file mode 100644 index 000000000..183d345b5 --- /dev/null +++ b/.github/workflows/c6-marked-fourier.yml @@ -0,0 +1,50 @@ +name: C6 marked Fourier controls + +'on': + pull_request: + paths: + - 'frontiers/c6_marked_fourier_20260929/**' + - '.github/workflows/c6-marked-fourier.yml' + - 'frontiers/c6_fourier_cutoff_20260929/PROOF.md' + - 'frontiers/c6_palm_route_20260929/PROOF.md' + - 'frontiers/c6_palm_route_20260929/SOURCE_MAP.json' + - 'frontiers/d5_dimension_lift_20260929/PROOF.md' + - 'frontiers/elder_dimension_lift_20260928/PROOF.md' + - 'frontiers/c6_rare_cluster_laws_20260929/PROOF.md' + push: + branches: [main] + paths: + - 'frontiers/c6_marked_fourier_20260929/**' + - '.github/workflows/c6-marked-fourier.yml' + - 'frontiers/c6_fourier_cutoff_20260929/PROOF.md' + - 'frontiers/c6_palm_route_20260929/PROOF.md' + - 'frontiers/c6_palm_route_20260929/SOURCE_MAP.json' + - 'frontiers/d5_dimension_lift_20260929/PROOF.md' + - 'frontiers/elder_dimension_lift_20260928/PROOF.md' + - 'frontiers/c6_rare_cluster_laws_20260929/PROOF.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-marked: + runs-on: ubuntu-latest + timeout-minutes: 15 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + fetch-depth: 0 + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Both-mode controls and exact upstream custody + run: | + set -euo pipefail + packet=frontiers/c6_marked_fourier_20260929 + python -B -S "$packet/verify.py" --output "${RUNNER_TEMP}/marked-${GITHUB_RUN_ID}-${GITHUB_RUN_ATTEMPT}" + python -B -S -m unittest discover -s "$packet" -p test_sources.py -v + python -B -O -S -m unittest discover -s "$packet" -p test_sources.py -v + python -B -S "$packet/source_check.py" --repo "${GITHUB_WORKSPACE}" + git diff --exit-code diff --git a/.github/workflows/cluster-exponential-weight.yml b/.github/workflows/cluster-exponential-weight.yml new file mode 100644 index 000000000..b01274852 --- /dev/null +++ b/.github/workflows/cluster-exponential-weight.yml @@ -0,0 +1,40 @@ +name: Cluster exponential weight controls +'on': + pull_request: + paths: + - 'frontiers/cluster_exponential_weight_20260930/**' + - '.github/workflows/cluster-exponential-weight.yml' + push: + branches: [main] + paths: + - 'frontiers/cluster_exponential_weight_20260930/**' + - '.github/workflows/cluster-exponential-weight.yml' + workflow_dispatch: +permissions: + contents: read +jobs: + exact-exponential: + runs-on: ubuntu-latest + timeout-minutes: 15 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + fetch-depth: 0 + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Finite controls and immutable source custody (not proof acceptance) + run: | + set -euo pipefail + packet=frontiers/cluster_exponential_weight_20260930 + python -B -S "$packet/verify_packet.py" + python -B -O -S "$packet/verify_packet.py" + python -B -S "$packet/finite_checks.py" > /tmp/exponential-normal.json + python -B -O -S "$packet/finite_checks.py" > /tmp/exponential-optimized.json + cmp /tmp/exponential-normal.json /tmp/exponential-optimized.json + python -B -S -m unittest discover -s "$packet" -p 'test_*.py' -v + python -B -O -S -m unittest discover -s "$packet" -p 'test_*.py' -v + git fetch --no-tags origin cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b 7a44130b41b827577424a73c4feac4cde20b5568 + python -B -S "$packet/verify_sources.py" --repo-root "$GITHUB_WORKSPACE" + git diff --exit-code diff --git a/.github/workflows/gpdata114-inert-custody.yml b/.github/workflows/gpdata114-inert-custody.yml new file mode 100644 index 000000000..c0b665b88 --- /dev/null +++ b/.github/workflows/gpdata114-inert-custody.yml @@ -0,0 +1,52 @@ +name: GP DATA 114 inert custody +'on': + pull_request: + paths: + - 'imports/gp_data_114_embedded_20260930/**' + - '.github/workflows/gpdata114-inert-custody.yml' + push: + branches: [main] + paths: + - 'imports/gp_data_114_embedded_20260930/**' + - '.github/workflows/gpdata114-inert-custody.yml' + workflow_dispatch: +permissions: + contents: read +jobs: + inert-custody: + runs-on: ubuntu-latest + timeout-minutes: 15 + steps: + - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262 + with: + persist-credentials: false + - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065 + with: + python-version: '3.11.16' + - name: Verify inert bytes and decoder controls only + run: | + set -euo pipefail + packet=imports/gp_data_114_embedded_20260930 + python -B -S - <<'PY' + import hashlib, json, pathlib, urllib.parse, urllib.request + packet = pathlib.Path('imports/gp_data_114_embedded_20260930') + source = json.loads((packet / 'SOURCES.json').read_text())['wrapper'] + url = 'https://raw.githubusercontent.com/' + source['repository'] + '/' + source['commit'] + '/' + urllib.parse.quote(source['path'], safe='/') + raw = urllib.request.urlopen(url, timeout=30).read(65537) + if len(raw) != source['bytes'] or hashlib.sha256(raw).hexdigest() != source['sha256']: + raise SystemExit('exact public wrapper identity mismatch') + pathlib.Path('/tmp/gpdata114-wrapper.txt').write_bytes(raw) + PY + python -B -S "$packet/verify_packet.py" --packet "$packet" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-packet-normal.json + python -B -O -S "$packet/verify_packet.py" --packet "$packet" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-packet-optimized.json + cmp /tmp/gpdata114-packet-normal.json /tmp/gpdata114-packet-optimized.json + python -B -S "$packet/test_verify_packet.py" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-packet-tests-normal.json + python -B -O -S "$packet/test_verify_packet.py" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-packet-tests-optimized.json + cmp /tmp/gpdata114-packet-tests-normal.json /tmp/gpdata114-packet-tests-optimized.json + python -B -S "$packet/inert_decoder.py" --wrapper-input /tmp/gpdata114-wrapper.txt --verify-existing "$packet" > /tmp/gpdata114-normal.json + python -B -O -S "$packet/inert_decoder.py" --wrapper-input /tmp/gpdata114-wrapper.txt --verify-existing "$packet" > /tmp/gpdata114-optimized.json + cmp /tmp/gpdata114-normal.json /tmp/gpdata114-optimized.json + python -B -S "$packet/test_inert_decoder.py" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-tests-normal.json + python -B -O -S "$packet/test_inert_decoder.py" --wrapper-input /tmp/gpdata114-wrapper.txt > /tmp/gpdata114-tests-optimized.json + cmp /tmp/gpdata114-tests-normal.json /tmp/gpdata114-tests-optimized.json + git diff --exit-code diff --git a/.github/workflows/planar-cubic-cluster.yml b/.github/workflows/planar-cubic-cluster.yml new file mode 100644 index 000000000..b1b7d84d3 --- /dev/null +++ b/.github/workflows/planar-cubic-cluster.yml @@ -0,0 +1,31 @@ +name: Planar cubic cluster checks + +'on': + pull_request: + paths: + - 'frontiers/planar_cubic_cluster_20260929/**' + - '.github/workflows/planar-cubic-cluster.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: Verify packet, both sources and both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S frontiers/planar_cubic_cluster_20260929/verify.py + git diff --exit-code diff --git a/.github/workflows/spectral-cluster-closure.yml b/.github/workflows/spectral-cluster-closure.yml new file mode 100644 index 000000000..9fed2e14d --- /dev/null +++ b/.github/workflows/spectral-cluster-closure.yml @@ -0,0 +1,36 @@ +name: Spectral cluster closure checks + +'on': + pull_request: + paths: + - 'frontiers/spectral_cluster_closure_20260929/**' + - '.github/workflows/spectral-cluster-closure.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 non-main source objects + run: | + set -euo pipefail + git fetch --no-tags --depth=1 origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8 + git fetch --no-tags --depth=1 origin f0d11805854b1938a2f89235ec590a37b04a7cb9 + - name: Verify every source and replay both Python modes + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S frontiers/spectral_cluster_closure_20260929/verify.py + git diff --exit-code diff --git a/PROOF_INDEX.md b/PROOF_INDEX.md index dc7cea156..030447a11 100644 --- a/PROOF_INDEX.md +++ b/PROOF_INDEX.md @@ -24,6 +24,11 @@ The shared [public inventory](https://github.com/d6g8k5htny-coder/main/tree/main - D5 two-scale S6-S21: proof [frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md](frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md); review [reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md](reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md). - D5 inner-belt gradient density: proof [frontiers/axial_density_20260925/PROOF.md](frontiers/axial_density_20260925/PROOF.md); review [reviews/replacement_20260925_pr19_pr21/REVIEW.md](reviews/replacement_20260925_pr19_pr21/REVIEW.md) accepts corollary (1) on `|z|<=W r^2`. That acceptance is the density bound only. - D5 fixed-transverse chart: proof [reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md](reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md) (byte-for-byte import of commit `32b80ee085dc6a40113d1e46e333cda50d57ba21`, blob `024d927779f79fadaf932541ea6474f2995f8c50`); review [reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md](reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md). +- D5 planar pin/collar/intermediate first moment: the [source-bound reconciliation](reviews/d5_reconciliation_20260929/RECONCILIATION.md) composes the punctured pin disks, nested microdisks, collar, fixed annulus, window-only intermediate shells, and remote window into D5-a–D5-f. The reviewed scope is fixed `d=2` and `T`, compact birth/gap marks with `k>=k_->0`, all frames and critical indices, existential constants, all heights for D5-a–D5-d and the height window `I_r` for D5-e–D5-f. The exact component nodes and 30 required evidence edges are in [the downstream graph](frontiers/downstream_gate_20260925/GRAPH.json). The xAI planar continuum check is [Math-#141 comment 5894512272](https://github.com/d6g8k5htny-coder/Math-/pull/141#issuecomment-5894512272); it grants no `d>2` transfer and no independent collar proof. The regional shrinking-witness mechanism, numerical constants and RN/24-jet/JETMOD remain outside this planar fold; no elder-selection or other register transition is inferred. +- D5 every fixed dimension (separate packet, not folded into the planar graph here): [full proof](frontiers/d5_dimension_lift_20260929/PROOF.md), [continuum crosswalk](frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md), and its [scope/review record](frontiers/d5_dimension_lift_20260929/README.md) give Theorems `P_d`, `C_d`, `I_d`, `G_d` for fixed `d>=2`, fixed torus and compact positive-gap marks. OpenAI reviews 5357858391 and 5357882570 accept the continuum rows at their stated source-exposed scope; Math-#141 merged the packet at `e1ca400b3414e5a7bec13b91bc4f1e5293df32be`. This source landing did not itself change `GRAPH.json` or a scientific register. +- C6 planar Fourier upper bound: [full proof](frontiers/c6_fourier_cutoff_20260929/PROOF.md), [source-bound reading note](reviews/c6_fourier_completion_20260929/READING_NOTE.md), and [Anthropic nonauthor review 5356335148](https://github.com/d6g8k5htny-coder/Math-/pull/148#pullrequestreview-5356335148) record `E_{Q_r^W}N(N-1)<=C r^3 log(1/r)` for fixed planar `T`, compact positive-gap marks and existential constants, using the complete planar D5/I5 reading rule. This remains a separately reviewed upper theorem; it is not the later sharp route and does not close the regional witness-collision node. +- C6 sharp Palm route (separate packet, not folded into this planar graph transition): [full proof](frontiers/c6_palm_route_20260929/PROOF.md) and [scope/review record](frontiers/c6_palm_route_20260929/README.md) give `E[(N)_q]<=C_q r^3` for fixed `q>=2` and `E[N(N-1)]=Theta(r^3)` in every fixed `d>=2`, conditional on the exact reviewed first-moment inputs. OpenAI reviews 5356233690 and 5358116559 accept the stated source-exposed sections; Math-#145 merged the packet at `820d4435c5f13c12b9c2f3590e859212d2648dc0`. Its global result does not resolve or move `math.rn-region.witness-collision`. +- C6 rare-cluster consequences (separate conditional packet): [full proof](frontiers/c6_rare_cluster_laws_20260929/PROOF.md), [sharp coefficient addendum](frontiers/c6_rare_cluster_laws_20260929/SHARP_POISSON_COEFFICIENT.md), and [review record](frontiers/c6_rare_cluster_laws_20260929/REVIEW_RECORD.md) state the elementary consequences of the exact D5/C6 count interface, including the sharp ordinary-Poisson obstruction and an `O(r^6)` canonical compound-Poisson approximation with the actual unknown positive-count law. Anthropic review 5358565704 and xAI review 5358623521 accept their respective slices; Math-#153 merged the packet at `190cf2bd4f31c7caf219d5d9b3d1fa468c450ea6`. The limiting cluster law remains unidentified, and no graph/register transition follows. - Cumulative transfer correction: proof [reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md](reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md); review [reviews/d2_cumulative_correction_20260925/REVIEW.md](reviews/d2_cumulative_correction_20260925/REVIEW.md) accepts this correction only. - P15 demand-one extension: exact counterexample [frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md](frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md); review pointer in [claims/LANDING_CLAIMS.json](claims/LANDING_CLAIMS.json) is [main #67](https://github.com/d6g8k5htny-coder/main/issues/67), disposition EXACT_COUNTEREXAMPLE. - D6 P15 full price theorem: proof [frontiers/full_price_20260924/PROOF.md](frontiers/full_price_20260924/PROOF.md); [reviews/p15_full_price_nonauthor_20260926/REVIEW.md](reviews/p15_full_price_nonauthor_20260926/REVIEW.md) records ACCEPT for Theorem F on the stated realized disjoint-capacity/clutter family, with demands at least two, independent probabilities, and the same palette. That review leaves the landing disposition untouched; it does not accept arbitrary downsets, demand-one extension, or an unrestricted prize claim. ASCII `3e-2` in (F1)/(F11) means `3*e-2`; see [ASCII_3E_MINUS_2.md](frontiers/full_price_20260924/ASCII_3E_MINUS_2.md). @@ -40,9 +45,7 @@ The shared [public inventory](https://github.com/d6g8k5htny-coder/main/tree/main ## Open obligations without a complete proof yet -- D5 pin neighborhoods: retained derivations are [reviews/d5_finite_r_hermite_repair_20260925/REPAIR.md](reviews/d5_finite_r_hermite_repair_20260925/REPAIR.md) and [reviews/d5_finite_r_hermite_repair_20260925/C6_REMAINDERS.md](reviews/d5_finite_r_hermite_repair_20260925/C6_REMAINDERS.md), with M1–M7 and S1–S4 accepted in [reviews/replacement_20260925_pr19_pr21/REVIEW.md](reviews/replacement_20260925_pr19_pr21/REVIEW.md). The newer public [PR53 reconnaissance note](https://github.com/d6g8k5htny-coder/Math-/blob/9a6f8a660d4508ec273b67d67e6b3871ddf2aa19/reviews/pin_neighborhood_recon_20260926/NOTE.md) has an [AMEND review in PR55](https://github.com/d6g8k5htny-coder/Math-/blob/4e188e25b1e1ef560f3eeb75c0d354d2ccf0ea22/reviews/d5_pin_neighborhood_20260926/REVIEW.md). These sources are on unmerged branches. **NO COMPLETE PROOF YET:** the inner microdisk bound and the open collar to the reviewed annulus are not supplied by that reconnaissance. [Math #58](https://github.com/d6g8k5htny-coder/Math-/issues/58) tracks the microdisk derivation; the claimed summed pin-neighborhood bound remains AMEND. -- D5 intermediate scale r << |x| << rho: bounded on opposite sides by [frontiers/remote_window_20260924/PROOF.md](frontiers/remote_window_20260924/PROOF.md) and by the reviewed fixed-annulus proofs [frontiers/rn_annulus_bridge_20260925/PROOF.md](frontiers/rn_annulus_bridge_20260925/PROOF.md) and [frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md](frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md). NO COMPLETE PROOF YET: uniform growing-scaled-radius bridge. -- D5 shrinking multiple-witness collision: fixed-separation machinery is [frontiers/remote_window_20260924/PROOF.md](frontiers/remote_window_20260924/PROOF.md) section 6, (15)–(16), at fixed `eta>0`. NO COMPLETE PROOF YET: shrinking-separation factorial-moment/collision estimate. +- D5 shrinking multiple-witness collision: fixed-separation machinery is [frontiers/remote_collision_20260928/PROOF.md](frontiers/remote_collision_20260928/PROOF.md). The reviewed [Fourier upper bound](frontiers/c6_fourier_cutoff_20260929/PROOF.md) and the later merged [sharp Palm route](frontiers/c6_palm_route_20260929/PROOF.md) establish global count statements by routes that do not identify this regional shrinking-separation mechanism. **NO COMPLETE REGIONAL PROOF YET:** `math.rn-region.witness-collision` remains `OPEN_ACTIVE`; the global `Theta(r^3)` result is not a proof of that node. - D0 historical CH-LIFT/Piece-2/24-jet obligations outside reviewed regional bypasses: see [frontiers/downstream_gate_20260925/GRAPH.json](frontiers/downstream_gate_20260925/GRAPH.json). NO COMPLETE PROOF YET where not explicitly superseded; absent historical carriers remain ABSENT. ## Repository rule diff --git a/frontiers/c6_cluster_law_20260929/PROOF.md b/frontiers/c6_cluster_law_20260929/PROOF.md new file mode 100644 index 000000000..ba492c8e6 --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/PROOF.md @@ -0,0 +1,884 @@ +# The cluster law of the C6 window count: `r^(-3) Q_r^W{N = n} -> nu(n)`, with `nu` carried by `{1, 2}` + +**Object:** CL-C6-CLUSTER-LAW-20260929-v1.2. +**Version:** v1.2, 30 September 2026 (v1.0 and v1.1 on 29 September; v1.1 answered the OpenAI Codex review of Math-#159: +spectral chart in place of the shear chart, exclusion lemma in the full transverse space, `K_4 <= kappa` split with a +marked Kac–Rice tail, domination without `D^(-1)`, `n in {1, 2}` in Proposition 4.4, uniform integrability in +Corollary X; v1.2 (and its follow-up on the singular-slice `kappa` mixture and the raw three-site whitening, after +review 5360034762) answers the OpenAI nonauthor review 5359895261 of v1.1: the full normalizer restored in both marked +displays (Lemmas 4.2, 5.2), Lemma 5.1 re-proved by whitening against the midpoint five-jet, the choice of `kappa` by +Fubini on the limit law, an explicit integrable majorant for §6.2 from the new Lemma 3.6 on the sheared cubic, and the +corrected §8 remark: every extra window critical point is a saddle; the Corollary X scope wording follows the OpenAI +follow-up 5359999003). +**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 29 September 2026. +**Disposition:** author-side proof candidate. **Nonauthor analytic review is required.** This note stacks on merged +sources only: [LP], [DL], [C6], [RM], [RC], [EDL], [LM], and it answers the "unknown conditional cluster law" left +open by [RCL] (Math-#153) and the "no positional law of the pairs" non-claim of [C6] §9. +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX` verdict, `GRAPH` node, claim, `lemma_closed` flag, catalog +entry, prize or source body changes. Same GitHub account as every other lane; zero organizational-independence credit. + +## 0. Why this note exists + +[C6] proves `E_(Q_r^W)[(N)_q] <= C_q r^3` for the number `N` of window critical points other than the pins, in every +fixed dimension, and [EDL] proves the matching lower bound `Q_r^W{N >= 2} >= c r^3`. [RCL] turned these bounds into +elementary rare-cluster consequences and stated their essential limitation: the conditional law of `N` given `N > 0`, +that is the measure `nu_r(n) = r^(-3) Q_r^W{N = n}`, is "actual, generally unknown, `r`-dependent", it has weighted-`l1` +subsequential limits (its Theorem S), and the premises alone cannot force a unique limit (its §6 examples). + +This note proves that in the exact Gaussian model the limit exists, identifies it, and shows that it is carried by +`{1, 2}`: at scale `r` around the pins the field has, on the rare event that produces window critical points at all, +the normal form of a planar cubic with the two pins among its critical points; a planar cubic has at most four critical +points; so at most two extra ones exist, and their number in the window is a measurable function of the scaled +third-order jets. Away from the pins the count is the single-point contact intensity of [RM]. The two regions do not +interact at order `r^3`. The mechanism is the one that [EDL] §§3–9 and [LM] §§2–6 used for the lower bound; here it is +run as a dominated-convergence limit over the whole rare configuration set instead of on one open box. + +## 1. Setting and results + +### 1.1 Model and count + +Fix `d >= 2`, `m = d - 1`, `L > 0`. `f` is the centered variance-one Gaussian field on `X = R^d/(L Z^d)` with the exact +normalized periodized Gaussian covariance of [LP] §1. Fix compact `B = [b_-, b_+]` and `K = [k_-, k_+]`, `0 < k_-`. +Frames `R` range over `O(d)`; `u = R e_1` is the axial direction and `y in R^m` the transverse coordinates, with the +midpoint of the pins at the origin. The pins are + + M = -(r/2) u, S = (r/2) u, f(M) = b, f(S) = b - k r^3, grad f(M) = grad f(S) = 0. + +`Q_r` is the Gaussian regression on these `2(d+1)` observations; `F_j(H) = |det H|` if `H` is nonsingular with `j` +negative eigenvalues and `0` otherwise; `W_r = F_d(H_M) F_(d-1)(H_S)`, `Z_r = E_(Q_r) W_r`, `dQ_r^W = (W_r/Z_r) dQ_r`; +`I_r = (b - k r^3, b)`. `N = N_r` is the number of critical points of `f` in `X minus {M, S}` with height in `I_r`, all +indices. For `0 < rho <= 1` and `A >= 4` write + + N_near^A = #{ window critical points, other than the pins, in the ball |X| <= A r }, + N_mid^(A,rho) = #{ ... with A r < |X| < rho }, + N_far^rho = #{ ... with |X| >= rho }, + +distances measured from the midpoint, so that `N = N_near^A + N_mid^(A,rho) + N_far^rho`. Constants `C, c, r_*` +depend on `d, L, B, K` and, where stated, on `A` or `rho`; never on `r, b, k, R`. `K_4` denotes a fixed multiple of +`1 + ||f||_(C^4(X))`. Nothing is uniform as `k -> 0`, as marks grow, or as `d` or `L` vary. No constant is numerical. + +### 1.2 Statements + +Section 3 defines, for every frame `R` and mark pair `(b, k)`, a reduced weight `m_(b,k,R)` on `R^4` (a nonnegative +continuous function of the scaled soft curvature `s` and of the three planar cubic jets `(a_3, beta, c_3)`, obtained +from the `Q_0`-density of the midpoint jets on the singular slice by integrating out the stable directions, (3.11)), +the pin weight + + w_0(s, a_3, beta) = [ -6k(s - beta/2) - a_3^2/4 ]_+ . [ a_3^2/4 - 6k(s + beta/2) ]_+, (1.1) + +the planar cubic + + F_0(X, Z) = 2k X^3 - (3k/2) X - k/2 + (s/2) Z^2 + (a_3/2)(X^2 - 1/4) Z + (beta/2) X Z^2 + (c_3/6) Z^3, (1.2) + +whose critical points other than `(-1/2, 0)` and `(1/2, 0)` number at most two (Lemma 3.2), and the count + + N_oo(s, a_3, beta, c_3) = #{ critical points (X, Z) of F_0, other than (-+1/2, 0), with F_0(X, Z) in (-k, 0) }. + +Let `z_0 = z_0(b, k, R) > 0` be the limiting normalizer of [LP] (5.4), `Lambda_j(x; b, k, u)` the contact kernel of +[RM] (13) and `Lambda = sum_j Lambda_j`. Put, for `n in {1, 2}`, + + nu_near(n) = z_0^(-1) integral_(R^4) m(s, a_3, beta, c_3) w_0(s, a_3, beta) 1{ N_oo = n } ds da_3 dbeta dc_3, (1.3) + + nu(2) = nu_near(2), nu(1) = nu_near(1) + k integral_X Lambda(x; b, k, u) dx. (1.4) + +**Theorem N (cluster law).** All the integrals in (1.3)–(1.4) are finite, the functions `nu(1), nu(2)` are strictly +positive and continuous on `B x K x O(d)`, and, uniformly on that compact set, + + r^(-3) Q_r^W{ N = 1 } -> nu(1), r^(-3) Q_r^W{ N = 2 } -> nu(2), r^(-3) Q_r^W{ N >= 3 } -> 0. (1.5) + +**Corollary Lambda (asymptotic factorial moments).** Uniformly on the same set, for every fixed integer `q >= 1`, + + r^(-3) E_(Q_r^W)[ (N)_q ] -> Lambda_q, Lambda_1 = nu(1) + 2 nu(2), Lambda_2 = 2 nu(2), Lambda_q = 0 (q >= 3). (1.6) + +In particular `E_(Q_r^W) N ~ Lambda_1 r^3`, `E_(Q_r^W)[N(N-1)] ~ 2 nu(2) r^3`, and every factorial moment of order at +least three is `o(r^3)`; the Palm excess of [C6] Corollary P converges: `E_sb[N - 1] -> 2 nu(2)/(nu(1) + 2 nu(2))`. + +**Corollary S (the limit of [RCL] is unique and identified).** With `nu_r(n) = r^(-3) Q_r^W{N = n}`, the weighted-`l1` +convergence of [RCL] Theorem S holds along the full family `r -> 0`, with the limit `nu = nu(1) delta_1 + nu(2) delta_2`; +`lambda = nu(1) + nu(2)`; the positive conditional law `F_r = Law(N | N > 0)` converges in total variation to +`F_oo = (nu(1) delta_1 + nu(2) delta_2)/lambda`; the canonical compound Poissonization `CP(p, F_r)` of [RCL] Theorem C +has the same `r -> 0` limits; and the independent-replica limit of [RCL] §7 is `CP(t nu)` with this explicit `nu`. +None of the three non-implication examples of [RCL] §6 occurs in the model. + +**Corollary X (near point-intensity limit, with a sequential radius limit).** For every bounded continuous `phi` on +`R^2 x R`, uniformly on the compact parameter set, + + r^(-3) E_(Q_r^W) sum_(X in near window critical points) phi( (X . u)/r, |X_perp|/r, (f(X) - b)/r^3 ) + -> z_0^(-1) integral m w_0 sum_(critical points (X,Z) of F_0 in the window) phi( X, |Z|, F_0(X,Z) ), + +where the inner sum runs over the extra critical points of `F_0`, `X` is the scaled axial coordinate and `|Z|` the +scaled transverse distance (§3.2: the rotated chart makes the physical ball a scaled ball). Scope: this is a statement +about the intensity measure of the near points, obtained with `r -> 0` at a fixed near radius `A` first and `A -> oo` +second (§6.5). It does not by itself give convergence of the law of the random configuration (expectations of sums +determine only the intensity), and it concerns near points only: the remote singleton part `k integral Lambda` of +`nu(1)` escapes the `1/r` scaling. + +### 1.3 What the results are not + +Existential constants, fixed `d` and `L`, compact marks. No rate in (1.5)–(1.6); no numerical value of `nu(1)`, `nu(2)` +or `Lambda_q`; no uniformity as `k -> 0`; no statement about the elder selection of the extra points; no spatial +independence between distinct pin pairs; no configuration law for the scaled positions (Corollary X is a near +point-intensity limit with a sequential radius limit); nothing about the remote pair kernel beyond consistency with [RC]. §9 lists +the non-claims. The note is conditional on the consumed sources at the exact bytes of §2. + +### 1.4 The planar case + +For `d = 2` there are no stable directions: `m = 1`, the chart of §3.2 is trivial, `m(s, a_3, beta, c_3)` is the +`Q_0`-density of `(f_zz, f_xxz, f_xzz, f_zzz)(0)` at `f_zz = 0`, evaluated at `(a_3, beta, c_3)`, and every consumed +regional bound is a reviewed planar statement ([PP], [CP], [IW] through [DL], [RM], [RC]). Theorem N at `d = 2` is +therefore conditional on this note's own steps only. + +## 2. Sources consumed, with the exact interface + +Identities (bytes, sha256, git blob, commit on `main`) are in `SOURCE_MAP.json`. + +| Tag | Source | What is consumed | +|---|---|---| +| [LP] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | The model; §2 distinct-site jet rank and positive spectrum; §3 the contact frame `U_r` with target `v_r -> v_0`, the uniform covariance sandwich and the density bound (3.5) for the transverse Hessian; §4 uniform conditional derivative moments; (5.1)–(5.5) endpoint identities, block determinant identity, `Z_r/r^2 -> z_0 > 0`; (13.3) the polynomial target-growth version of the conditional `C^m` moments. | +| [DL] | `frontiers/d5_dimension_lift_20260929/PROOF.md` | Theorem I_d, (1.5)–(1.6): the window shell bound `E_(Q_r^W) N_(r,j)({A_0 r <= |X| <= rho}) <= C r^3 (A_0^(-2) + rho^2)`, uniform in `A_0 >= 4` and `rho <= s_0`; the regime frames of §§4–6 (as re-expressed in [C6] §6) for Lemma 5.2. | +| [C6] | `frontiers/c6_palm_route_20260929/PROOF.md` | Theorem Q (uniform integrability in Corollary Lambda); Lemma 5.1 (marked Kac–Rice for `sigma(f)`-measurable marks, gradient-only kernel); (5.3) the conditional Hölder insertion and the §6 regime table with the squared determinant weight (Lemma 5.3); Lemma 4.1's frame parameter `beta` and the regime bounds `beta <= poly(chi, |v|^(-1), r^(-1), s^(-1))`. | +| [RM] | `frontiers/remote_window_20260924/PROOF.md` | Theorem A (2) with the contact kernel `Lambda_j` of (13), its continuity, positivity and the remote nondegeneracy of §§2–3; the Kac–Rice representation (12). | +| [RC] | `frontiers/remote_collision_20260928/PROOF.md` | Corollary D (fixed-remote second factorial moment `<= C r^5 |E|`) and Corollary E (`Q_r^W{N_j(E) >= 1} = k r^3 integral_E Lambda_j + O(r^4 |E|)`), for `E` contained in `D_rho`, every fixed `d`. | +| [EDL] | `frontiers/elder_dimension_lift_20260928/PROOF.md` | §2 (complete third-order jets and their positive-definite joint covariance with `U_0`), (A11)–(A16) (the rare set and the conditional `C^4` control), (A17) (the normal form on the soft plane; here written in a rotated rather than sheared frame), (A26)–(A28) (the rescaled full gradient and the stable block), §9 (block-triangular stability of the extra critical points). Used as the source of the normal form; the shear chart (A8)–(A10) is replaced by the spectral chart of §3.2; the lower bound (A4) is not consumed. | +| [LM] | `frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md` | (L4)–(L8) the planar normal form and its exact cubic with two extra saddles (used as the positivity witness), (L7) the `C^2` remainder, §6 the exponent arithmetic `r . r^4 / r^2`. | +| [RCL] | `frontiers/c6_rare_cluster_laws_20260929/PROOF.md` | §§2, 4, 6, 7 statements (Theorems C, S, the identity (6.2), the replica limit (7.1)), consumed only in Corollary S as the statements whose hypotheses this note identifies. | + +Nothing else is imported. In particular no elder-selection statement, no numerical enclosure and no Fourier-cutoff +tail is used. For `d = 2` the planar sources behind [DL] are [PP] `reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md`, +[CP] `reviews/d5_local_collar_20260928/COLLAR_PROOF.md`, [IW] `frontiers/intermediate_window_20260928/PROOF.md`, cited +where [DL] cites them. + +## 3. The scaled model near the pins + +### 3.1 Midpoint jets and the contact frame + +Write the transverse coordinates as `y = (z, w)` with `z in R` along the first transverse axis `e_z = R e_2` and +`w in R^n`, `n = d - 2` (empty for `d = 2`). At the midpoint `0` let + + A = D_y^2 f(0) in Sym_m, T = ( partial^alpha f(0) )_(|alpha| = 3, alpha != (3, 0, ..., 0)), + +the transverse Hessian and the third derivatives other than `f_xxx`, and `J = (A, T)`. By [LP] §3 the contact frame +`U_r` is an invertible re-expression of the `2(d+1)` pins with target `v_r -> v_0 = (b, 0, 0, 12k, 0, ..., 0)` and +contact limit `U_0 = (f, f_x, f_xx, f_xxx, f_y, f_xy)(0)`; the entries of `(U_0, J)` are every independent derivative +of order at most three at one site, once each ([EDL] §2), so their covariance is uniformly positive definite on +`O(d)`, and `J` under `Q_r` is a nondegenerate Gaussian vector with means, covariances and inverse covariances +converging, uniformly on the compact parameter set, to those of `J` under `Q_0`, the regression on `U_0 = v_0`. Write +`p_r(j)` for the Lebesgue density of `J` under `Q_r` and `p_0` for its limit; both are bounded by `C exp(-c |j|^2)` +([LP] (3.5) for the `A` block, the same Schur argument for `T`). + +For every target `j` of `J` and every `r`, the regular conditional law of `f` given `(U_r, J) = (v_r, j)` is the Gaussian +regression `f = m_r(j) + Res_r`, with `m_r(j)` affine in `j` with `C^4`-bounded coefficient functions and `Res_r` a +centered Gaussian field independent of `(U_r, J)` whose `C^4(X)` moments of every order are bounded uniformly in `r` +([EDL] (A14)–(A15), [LM] (L12)); by [LP] (13.3), + + E[ K_4^p | U_r = v_r, J = j ] <= C_p (1 + |j|)^p for every fixed p. (3.1) + +Since `Res_r -> Res_0` in `C^4(X)` in every `L^p` (the coefficient functions converge with all derivatives, [LP] §3), +and `m_r(j_r) -> m_0(j)` in `C^4` whenever `j_r -> j`, the conditional laws converge: for every bounded functional `Phi` +of the field that is continuous at `Q_0`-almost every `f` in the `C^4` topology, `E[Phi(f) | (U_r, J) = (v_r, j_r)] -> +E[Phi(f) | (U_0, J) = (v_0, j)]`. This is the device of [LP] §13 and [RM] §5, and it is used below with `Phi` the +indicator that the scaled count equals `n`. + +### 3.2 The spectral chart and the singular slice + +For `d = 2`, `A = f_yy(0)` is a scalar; put `sigma = A`, and there is no `D`, no `U` and no `w`. For `d >= 3` write the +eigenvalues of `A` as `lambda_1, ..., lambda_m` and let `sigma` be the eigenvalue of least modulus; `Q_r`-almost +surely it is unique and simple, since `A` has a Lebesgue density ([LP] (3.5)). Let `u_1` be a unit eigenvector for +`sigma`, `U in O(m)` an orthogonal matrix with first column `u_1`, and `D = diag(lambda_2, ..., lambda_m)` the +remaining eigenvalues in a fixed order, so that + + A = U diag(sigma, D) U^T, |sigma| <= |lambda_j| (j >= 2). (3.2) + +The Weyl integration formula for real symmetric matrices (e.g. Anderson, Guionnet, Zeitouni, *An Introduction to Random +Matrices*, §2.5) writes Lebesgue measure on `Sym_m` in these coordinates as + + dA = c_m prod_(j >= 2) |sigma - lambda_j| . Delta(D) dsigma dD dU, Delta(D) = prod_(2 <= i < j) |lambda_i - lambda_j|, (3.3) + +with `dU` the Haar measure on `O(m)` and `c_m` a fixed positive constant (the restriction `|sigma| <= |lambda_j|` +selects one of the `m` eigenvalue labellings). The density factor is a polynomial in `(sigma, D)`, and it equals +`c_m |det D| Delta(D)` on the singular slice `{sigma = 0}`. + +Rotate the transverse coordinates by `U`: `f~(x, z, w) = f( R diag(1, U) (x, z, w) )`, with `z in R` along `u_1` and +`w in R^n`, `n = m - 1`. Since `diag(1, U)` is orthogonal, `f~` has the same `C^4` norm as `f`, its transverse Hessian +at `0` is `diag(sigma, D)` (so `f~_zw(0) = 0`, `f~_ww(0) = D`), and the physical ball `|X| <= A r` is the scaled ball +`X^2 + Z^2 + |rW|^2 <= A^2` in the coordinates `(x, z, w) = (rX, rZ, r^2 W)` used below. Let `tau` be the third +derivatives of `f~` at `0` other than `f~_xxx`; `tau` is the image of `T` under a fixed orthogonal representation of +`diag(1, U)`, so `|tau| = |T|` and `T -> tau` has Jacobian one for fixed `U`. Write `j = (sigma, U, D, tau)` and +`chart(j)` for the raw jet `(A, T)`. + +The chart replaces the shear of [EDL] (A8)–(A10) by a rotation; no unbounded coefficient enters, at the price of the +polynomial density (3.3). Configurations in which a second eigenvalue is also small (`|lambda_2|` small) are not +excluded from the chart; they are handled by the domination of §4, whose dominating function does not involve +`D^(-1)` (Lemma 4.3), and by the almost-sure nonsingularity of `D` in the pointwise limits. + +### 3.3 The normal form + +Let `F_r(X, Z, W) = [ f~(rX, rZ, r^2 W) - b ] / r^3` and + + s = sigma / r, a_3 = tau_xxz, beta = tau_xzz, c_3 = tau_zzz, Q_j(X, Z) = (tau_xxw_j / 2)(X^2 - 1/4) + tau_xzw_j X Z + (tau_zzw_j / 2) Z^2. + +**Lemma 3.1 (normal form, [EDL] (A17), (A26)–(A28); [LM] (L6)–(L7)).** For every `R_0 >= 1` there is `C = C(d, R_0)` +such that on `{|(X, Z)| <= R_0, |W| <= R_0}`, uniformly in `b, k, R`, `0 < r <= 1`, + + || F_r(., ., 0) - F_0 ||_(C^2) <= C r K_4, (3.4) + r^(-2) grad f~(rX, rZ, r^2 W) = ( partial_X F_0, partial_Z F_0, D W + Q(X, Z) ) + O( r K_4 (1 + |D|) ), (3.5) + +with `F_0` as in (1.2) (the pins are exactly retained: `F_0(-+1/2, 0) = 0, -k` and `grad F_0(-+1/2, 0) = 0`, checked +exactly in the finite control `NF`). Moreover, for `|(X, Z)| <= R_0` and physical `|w| <= R_0 r`, +`|partial_w f~| >= (sigma_min(D) - C r K_4 R_0)|w| - r^2 |Q| - C r^3 K_4`, so if `D` is nonsingular and +`r <= r_0(K_4, R_0, D)`, every critical point of `f~` in the physical ball of radius `R_0 r` has +`|W| <= C |Q(X,Z)| / sigma_min(D) + 1`. + +*Proof.* The Taylor expansion of `f~` at the midpoint through order three, with the pin identities of [LM] §3 +(`f~_xxx(0) = 12k + O(r K_4)`, `f~_x(0) = -(r^2/8) f~_xxx(0) + O(r^3 K_4)`, `f~_z(0) = -(r^2/8) f~_xxz(0) + O(r^3 K_4)`, +`f~_xx(0), f~_xz(0) = O(r^2 K_4)`, `f~(0) - b = -(r^3/24) f~_xxx(0) + O(r^4 K_4)`, and, from the two transverse gradient +pins, `f~_w(0) = -(r^2/8) f~_xxw(0) + O(r^3 K_4)`, `f~_xw(0) = O(r^2 K_4)`, while `f~_zw(0) = 0` exactly), is (A17) of +[EDL] on `w = 0` and gives (3.4). Differentiating in `w` and using `D_w^2 f~(0) = D`, `D_(zw) f~(0) = 0` gives (3.5), +the last block being `r^2 (D W + Q) + O(r^3 K_4)` before division. The lower bound on `|partial_w f~|` is (3.5) read at +physical `w = r^2 W` with `|W| <= R_0 / r`: the quadratic remainder `K_4 |w|^2 <= K_4 R_0 r |w|` is relative to the +linear term `D w`. ∎ + +Consequently, when `D` is nonsingular and `r <= r_0(K_4, R_0, D)`, the critical points of `f~` in the physical ball of +radius `R_0 r` are in bijection with the zeros of the reduced planar gradient + + grad G_r(X, Z), G_r(X, Z) = F_r(X, Z, W_*(X, Z)), W_*(X, Z) = the unique solution of the third block of (3.5) = 0, + +`W_* = -D^(-1) Q(X, Z) + O(r K_4 (1 + |D^(-1)|)^2)`, and + + || G_r - F_0 ||_(C^2({|(X,Z)| <= R_0})) <= C r K_4 (1 + |D^(-1)|)^2 (1 + |tau|)^2. (3.6) + +Indeed `F_r(X, Z, W) - F_r(X, Z, 0) = r (Q . W + W^T D W / 2) + O(r^2 K_4)` on bounded `W`, and the implicit function +`W_*` is `C^2` with the stated bound because the third block of (3.5) has derivative `D + O(r K_4)` in `W`. The index of a +critical point of `f~` equals the index of the corresponding critical point of `G_r` plus the index of `D` ([EDL] §9: +the full Hessian is block triangular up to `O(r)` with diagonal blocks the planar Hessian of `G_r` and `D`), and its +height satisfies `(f~ - b)/r^3 = G_r` at the point. The bound (3.6) is used only for the pointwise limits of §4.2, at +fixed jets with `D` nonsingular; it is not used for domination. + +### 3.4 The planar cubic + +**Lemma 3.2 (at most two extra critical points).** For every `(k, s, a_3, beta, c_3)` with `k > 0`, the resultant of +`partial_X F_0` and `partial_Z F_0` with respect to `Z` equals `(X^2 - 1/4) Q_2(X)` identically in +`(X, s, a_3, beta, c_3, k)`, where + + Q_2(X) = c_2 X^2 + s c_1 X + c_0, + c_2 = a_3^3 c_3/4 - 3 a_3^2 beta^2/16 - (9/2) a_3 beta c_3 k + 3 beta^3 k + 9 c_3^2 k^2, + c_1 = -a_3^2 beta/4 - 3 a_3 c_3 k + 6 beta^2 k, + c_0 = 3 s^2 beta k - ( a_3 beta/8 - 3 c_3 k/2 )^2 (3.7) + +(finite control `RS`, exact polynomial identity). Let `E_k` be the set of `(s, a_3, beta, c_3)` for which the two +quadratics `partial_X F_0, partial_Z F_0` have a common factor, or `Q_2` vanishes identically, or some common zero other +than the pins is degenerate, or has height in `{-k, 0}`. `E_k` is a proper algebraic subset of `R^4`, hence +Lebesgue-null, and off `E_k` the critical points of `F_0` are isolated, at most four with multiplicity (Bézout for two +conics), two of them the pins; so the extra real critical points number at most two, they are nondegenerate, and +`N_oo in {0, 1, 2}`. On `E_k` put `N_oo := 0` (the value is irrelevant: `E_k` is null for every `k`). + +*Proof.* The resultant identity is an exact computation (the Sylvester determinant of two quadratics in `Z`), verified +symbolically in `RS`. The pins are the zeros of `partial_X F_0 = 6k X^2 - 3k/2 + a_3 X Z + (beta/2) Z^2` and +`partial_Z F_0 = s Z + (a_3/2)(X^2 - 1/4) + beta X Z + (c_3/2) Z^2` on `Z = 0`, which explains the factor `X^2 - 1/4`. +Off `E_k`, for each root `X_*` of `Q_2` the two quadratics in `Z` are not proportional (they have no common factor), so +they share at most one root `Z`; if they are proportional at `X_*`, the extra points are the at most two zeros of +`partial_X F_0(X_*, .)` (the [LM] configuration, where control `EX` exhibits `X_* = -3/4` as a double root of `Q_2`). +Bézout's theorem bounds the number of isolated common zeros of two conics by four. `E_k` is defined by polynomial +equations in the jets (common factor: vanishing of the resultant as a polynomial in `X`; degeneracy and boundary +heights: vanishing of the Hessian determinant, or of `F_0` or `F_0 + k`, at a common zero, eliminated through the +resultant), and it is proper because the configurations of §7 (exact rational examples with `0`, `1` and `2` extra +critical points in the window, all nondegenerate) lie outside it. The example `(0, 0, 0, 0)`, where the lines +`X = -+1/2` consist of critical points, lies in `E_k`. ∎ + +### 3.5 Large soft curvature excludes near critical points + +**Lemma 3.3 (exclusion, full transverse space).** There are `C = C(d)` and `r_0(kappa, R_0) > 0` such that, if +`K_4 <= kappa`, `r <= r_0(kappa, R_0)`, and the transverse Hessian `A` at the midpoint satisfies `|lambda_j| >= S r` +for every eigenvalue, with + + S >= S_*(tau, R_0, kappa, k) := C (1 + |tau|)^2 (1 + R_0)^3 (1 + kappa)^2 / k^2, (3.8) + +then the only critical points of `f` in the physical ball of radius `R_0 r` about the midpoint are the two pins. +Equivalently, on `{K_4 <= kappa}`, `N^(R_0) >= 1` forces `|s| = |sigma|/r <= S_*(tau, R_0, kappa, k)` for `r <= r_0`. + +*Proof.* Work in the original pin frame (no rotation is needed), with scaled coordinates `x = rX`, `y = rY`, +`|(X, Y)| <= R_0`. From the two transverse gradient pins and the pin identities ([LM] §3, [EDL] (A17), the same +computation as (3.4) in every transverse direction), + + grad_y f(rX, 0) = (r^2/2)(X^2 - 1/4) grad_y f_xx(0) + O(r^3 K_4 (1 + R_0)^3), D_y^2 f(rX, 0) = A + O(r K_4 R_0), + +so at a critical point, `0 = grad_y f(rX, rY) = r^2 [ (1/2)(X^2 - 1/4) grad_y f_xx(0) ] + (A + O(r K_4 R_0)) r Y + O(r^2 K_4 R_0^2)`. +Dividing by `r` and using `|A Y| >= S r |Y|`, + + (S r - C r K_4 R_0) |Y| <= C r (1 + |tau|)(1 + R_0)^2 + C r K_4 R_0^2, i.e. |Y| <= delta := C (1 + |tau| + kappa R_0^2)(1 + R_0)^2 / (S - C kappa R_0), + +which is small for `S` large. In the axial equation, `f_x(rX, rY)/r^2 = 6k X^2 - 3k/2 + X grad_y f_xx(0) . Y + O((1 + |tau|) |Y|^2) + O(r K_4 (1 + R_0)^3)` +([LM] §3, [EDL] (A17): the term `f_xy(0) . Y / r` is `O(r K_4 |Y|)` by the pin identity `f_xy(0) = O(r^2 K_4)`), so +`|6k X^2 - 3k/2| <= C (1 + |tau|) R_0 delta + C (1 + |tau|) delta^2 + C r kappa (1 + R_0)^3`. Since +`|6k X^2 - 3k/2| = 6k |X - 1/2| |X + 1/2|`, for `delta` and `r` small this forces `X` within `C (1 + |tau|) R_0 delta / k` +of `-+1/2`. Hence every critical point in the scaled ball lies within scaled distance +`rho := delta + C (1 + |tau|) R_0 delta / k` of a pin. + +At a pin `P`, `grad f(P) = 0` exactly and, for `|xi| <= 1`, `|grad f(P + r xi)| >= sigma_min(H_P) r |xi| - K_4 r^2 |xi|^2 / 2`. +The Hessian `H_P` is `r` times the block matrix with axial entry `-+6k + O(r K_4)`, mixed axial–transverse row +`O(1 + |tau|)` (the entries `-+(1/2) grad_y f_xx(0) + O(r K_4)`), and transverse block `A/r + O(K_4)` whose eigenvalues +have modulus at least `S - C kappa`. For `S >= C (1 + |tau|)^2 / k + C kappa`, the Schur complement of the transverse +block is `-+6k + O((1 + |tau|)^2 / (S - C kappa))`, so `sigma_min(H_P) >= c k r`, and no other zero of `grad f` lies +within scaled distance `rho_pin := c k / kappa` of `P`. The two conclusions are incompatible as soon as +`rho < rho_pin`, which holds when `S >= S_*` with the constant of (3.8). ∎ + +The lemma uses no planar reduction and no inverse of `D`; the only randomness it sees is through the deterministic +bound `K_4 <= kappa`. Note that on the singular slice `|sigma| <= |lambda_j|` for all `j`, so the hypothesis is exactly +`|s| >= S_*`. + +### 3.6 The weight on the singular slice + +**Lemma 3.4 (pin Hessians).** With `M_2(M) = [[-6k, -a_3/2], [-a_3/2, s - beta/2]]` and `M_2(S) = [[6k, a_3/2], +[a_3/2, s + beta/2]]` (exact second derivatives of `F_0` at the pins, control `NF`), the Hessians of `f~` at the pins are + + H~_i = [[ r M_2(i) + O(r^2 K_4), r c_i + O(r^2 K_4) ], [ r c_i^T + O(r^2 K_4), D + O(r K_4) ]], i in {M, S}, + +with `c_i in R^(2 x n)` linear in `tau` (the entries `-+tau_xxw/2, -+tau_xzw/2`), and + + det H~_i = r^2 det D . det M_2(i) + O( r^3 K_4^d (1 + |tau|)^2 (1 + |s|)^2 ), (3.9) + +exactly as polynomials: the determinant of a matrix whose two soft rows are `O(r)` and whose stable block is `O(1)` is +`r^2` times the product of the soft-block and stable-block determinants plus `r^3` times a polynomial in the entries +(control `BD` checks this for `d = 3` and `d = 4` with exact rational arithmetic in `r`); the remainder involves no +inverse of `D`. The rotation `diag(1, U)` is orthogonal, so `det H_i = det H~_i` and the index is unchanged. Moreover, +by Hadamard's inequality applied to the two soft rows, the row of the least stable eigenvalue `lambda_2` (whose +entries are `O(r K_4)` off the diagonal and `lambda_2 + O(r K_4)` on it) and the remaining rows, + + r^(-4) W_r <= C (1 + |s| + |tau| + K_4)^4 ( |lambda_2| + r K_4 )^2 K_4^(2d - 6) (d >= 3), (3.10) + +and `r^(-4) W_r <= C (1 + |s| + |tau| + K_4)^4` for `d = 2`. + +*Proof.* The entries follow from Lemma 3.1 by differentiating (3.4)–(3.5) once more at the pins: +`partial_X^2 F_0 = 12k X + a_3 Z`, `partial_X partial_Z F_0 = a_3 X + beta Z`, `partial_Z^2 F_0 = s + beta X + c_3 Z`, +evaluated at `(-+1/2, 0)`, and the mixed soft–stable entries are the derivatives of `Q`. The determinant expansion is +the Leibniz formula: every term contains at least two factors from the two soft rows, each `O(r)`; the terms with +exactly two such factors and the identity permutation on the stable block give `r^2 det M_2 det D`; the remaining terms +carry `r^3`. (3.10) is Hadamard's inequality with the stated row norms. ∎ + +**Lemma 3.5 (weight limit).** Write `w_0` as in (1.1). If `D` is nonsingular and `det M_2(M) != 0 != det M_2(S)`, +then under the conditional law of §3.1 with `sigma = r s`, + + r^(-4) W_r -> (det D)^2 w_0(s, a_3, beta) in probability as r -> 0. + +Here `w_0 > 0` exactly when `M_2(M)` is negative definite (index `d` at `M`, given `D < 0`) and `M_2(S)` has one negative +eigenvalue (index `d - 1` at `S`); if `D` is not negative definite the limit weight is `0` because the index conditions +fail. + +*Proof.* (3.9) gives `r^(-2) det H~_M -> det D . det M_2(M)`, and the index of `H~_M` converges to `index(M_2(M)) + +index(D)` because eigenvalues are continuous and the limits are nonsingular. `F_d(H_M) = |det H_M| 1{index = d}` gives +the first factor; the second factor is the same at `S`. The condition `index M_2(M) = 2` is `det M_2(M) > 0` (its +`(1,1)` entry is `-6k < 0`), and `index M_2(S) = 1` is `det M_2(S) < 0` (its `(1,1)` entry is `6k > 0`); these are the +two positive parts in (1.1). ∎ + +### 3.7 The reduced weight + +Let `p_0` be the `Q_0`-density of the raw jet `J = (A, T)` (§3.1). Define + + m(s, a_3, beta, c_3) := integral p_0( chart(0, U, D, tau) ) c_m |det D| Delta(D) (det D)^2 1{D < 0} dU dD dtau_rest, (3.11) + +where `tau_rest` collects the entries of `tau` other than `(a_3, beta, c_3)` (those involving a `w` index), `dU` is the +Haar measure, and `c_m |det D| Delta(D)` is the density (3.3) on the singular slice. The integrand does not depend on +`s`; `m` is a bounded, strictly positive, continuous function of `(a_3, beta, c_3)` with Gaussian decay (the density +`p_0` is Gaussian and nondegenerate, `|chart(0, U, D, tau)| >= c (|D| + |tau|)`, and the Weyl factor is polynomial), and +it depends continuously on `(b, k, R)`. In `d = 2`, `m(s, a_3, beta, c_3) = p_0(0, a_3, beta, c_3)`, the density of +`(f_zz, f_xxz, f_xzz, f_zzz)(0)` at `f_zz = 0`. + +### 3.8 The typed window geometry of the planar cubic + +Put `B = beta - a_3^2/(12k)`, `D = ( c_3 - a_3 beta/(4k) + a_3^3/(72k^2) )/2`, `u = X + a_3 Z/(12k)`, and +`C(u) = 2k u^3 - (3k/2) u - k/2 = (k/2)(u - 1)(2u + 1)^2`. Then, identically in `(X, Z, s, a_3, beta, c_3, k)`, + + F_0(X, Z) = G(u, Z) := C(u) + (s + B u) Z^2 / 2 + (D/3) Z^3 (3.12) + +(control `SH`, as the polynomial identity with `k` cleared). The shear `(X, Z) -> (u, Z)` is linear and unimodular, +so critical points, heights and Hessian determinants of `F_0` and `G` correspond, and the pins are `(u, Z) = (-+1/2, 0)`. +The typed domain of Lemma 3.5 is `{w_0 > 0} = {s < -|B|/2}`, since `det M_2(M) = -6k(s - B/2)` and +`det M_2(S) = 6k(s + B/2)`, and there `w_0 = 9k^2(4s^2 - B^2)`. + +**Lemma 3.6 (saddles only, and the range of `s`).** Let `s < -|B|/2` and let `(X, Z)` be a critical point of `F_0` +other than the pins with `F_0(X, Z) in (-k, 0)`. Then `Z != 0`, the planar Hessian of `F_0` at `(X, Z)` has determinant +`-3k(B + 4 s u) < 0`, so the point is a nondegenerate saddle (index `1` in the plane, hence index `d - 1` in the full +model when the stable block is negative definite), and + + -s <= 2|B| + (32 k D^2)^(1/3). (3.13) + +In particular, on `{w_0 > 0}`, `N_oo >= 1` forces `|s| <= 2|B| + (32 k D^2)^(1/3) <= C_k (1 + |a_3| + |beta| + |c_3|)^2`, +with `C_k` bounded for `k` in a compact subset of `(0, oo)`, and every counted point is nondegenerate (the degeneracy +clause in the definition of `E_k` is vacuous on the typed window). Both statements are exercised on an exact rational +family in control `TW`. + +*Proof.* Write `x = -s > |B|/2 >= 0` and `w = s + B u`. The gradient of `G` is `partial_u G = 6k u^2 - 3k/2 + (B/2) Z^2`, +`partial_Z G = Z (w + D Z)`, and its Hessian determinant is `12 k u (w + 2 D Z) - B^2 Z^2`. A critical point with +`Z = 0` has `6k u^2 = 3k/2`, so it is a pin; hence `Z != 0` and `w + D Z = 0`. + +*Case `D != 0`.* Then `Z = -w/D`, and the first gradient equation reads `B w^2 / D^2 = 3k (1 - 4u^2)`. The height is +`G = C(u) + w Z^2/2 + (D/3) Z^3 = C(u) + w^3/(6 D^2)`, and the Hessian determinant is +`12 k u (w - 2w) - B^2 w^2/D^2 = -12 k u w - 3k B (1 - 4u^2) = -3k (B + 4 u (w - B u)) = -3k (B + 4 s u)`. + +(a) `B = 0`: `u = -+1/2`. At `u = 1/2`, `w = s` and `G = -k + s^3/(6D^2) < -k`, outside the window. At `u = -1/2`, +`w = s` and `G = -x^3/(6 D^2)`, in the window iff `x^3 < 6 k D^2`, which implies (3.13); the determinant is +`-3k . 4 s (-1/2) = 6 k s < 0`. + +(b) `B > 0`: `1 - 4u^2 = B w^2/(3k D^2) >= 0` gives `|u| <= 1/2`, and `|u| = 1/2` forces `w = 0`, a pin; so `|u| < 1/2`. +Then `w = -x + B u <= -x + B/2 < 0` and `C(u) <= 0`, so `G > -k` gives `|w|^3/(6D^2) < k + C(u) <= k`, i.e. +`|w| < (6 k D^2)^(1/3)`; with `|w| = x - B u >= x - B/2` this is `x < B/2 + (6 k D^2)^(1/3)`, which implies (3.13). +For the sign, write `|w|^3/(6 D^2) = |w| k (1 - 4u^2)/(2B)` and `k + C(u) = (k/2)(u + 1)(2u - 1)^2` (the identity +`2 + (u - 1)(2u + 1)^2 = (u + 1)(2u - 1)^2`): the window inequality `G > -k` becomes `|w| (1 + 2u) < B (u + 1)(1 - 2u)`, +i.e. `x < B u + B (u + 1)(1 - 2u)/(1 + 2u) = B/(1 + 2u)`. Hence `B + 4 s u = B - 4 x u > B (1 - 2u)/(1 + 2u) > 0` when +`u > 0`, and `B - 4 x u >= B > 0` when `u <= 0`: the determinant is negative. + +(c) `B < 0`: `4u^2 - 1 = |B| w^2/(3k D^2) >= 0` gives `|u| >= 1/2`, and `|u| = 1/2` forces `w = 0`, a pin; so `|u| > 1/2`. +Now `w^3/(6D^2) = w k (4u^2 - 1)/(2|B|)` with `w = -x - |B| u`, and expanding, + + G = (k/2)(2u + 1) [ -1 - (x/|B|)(2u - 1) ]. + +For `u > 1/2` both factors give `G <= -k`: outside the window. For `u < -1/2` put `t = -u > 1/2`, so +`G = -(k/2)(2t - 1)( x (2t + 1)/|B| - 1 )`; since `x > |B|/2` the bracket is positive, `G < 0`, and `G > -k` reads +`(2t - 1)( x (2t + 1) - |B| ) < 2|B|`. The determinant is `-3k( -|B| + 4 x t ) < 0` because `4 x t > 2 x > |B|`. For (3.13) +assume `x >= 2|B|` (otherwise it holds) and put `delta = 2t - 1 > 0`: the window inequality gives `delta (2x - |B|) < 2|B|`, +hence `delta < 4|B|/(3x) <= 2/3` and `t < 5/6`; then `w^2 = (3 k D^2/|B|) delta (delta + 2) < 32 k D^2/(3x)`, while +`|w| = x - |B| t >= 7x/12`; so `x^3 < (144/49)(32/3) k D^2 < 32 k D^2`. + +*Case `D = 0`, `B != 0`.* Then `w = 0`, i.e. `u = x/B`, and `Z^2 = 3k (1 - 4u^2)/B`. The height is `G = C(u)`, which lies +in `(-k, 0)` iff `u in (-1, 1) \ {-+1/2}` (`C(-1) = -k`, `C(-1/2) = 0`, `C(1/2) = -k`, `C(1) = 0`, `C` decreasing on +`[-1/2, 1/2]`, `k + C(u) = (k/2)(u + 1)(2u - 1)^2`), so `x = |B| |u| < |B|`; the Hessian determinant is `-B^2 Z^2 < 0`. + +*Case `D = 0`, `B = 0`.* `partial_Z G = s Z` with `s != 0`: there is no critical point off `Z = 0`. ∎ + +## 4. The near limit + +### 4.1 Scaled representation + +Fix `A >= 4` and write `N^A = N_near^A`. For every Borel set `E` of field configurations, + + Q_r^W(E) = Z_r^(-1) integral p_r(j) E[ W_r 1_E | U_r = v_r, J = j ] dj, (4.1) + +by the tower property with the regular conditional law of §3.1. Change variables to the spectral coordinates +`(sigma, U, D, tau)` with the density (3.3) and then `sigma = r s`: + + r^(-3) Q_r^W(E) = (r^2 / Z_r) integral ds dU dD dtau p_r( chart(rs, U, D, tau) ) c_m prod_(j>=2) |rs - lambda_j| Delta(D) r^(-4) E[ W_r 1_E | U_r = v_r, J = chart(rs, U, D, tau) ], (4.2) + +the `s`-integral being over `{ |rs| <= min_j |lambda_j| }`. The factor `r` from `dsigma = r ds` and the factor `r^(-4)` +from the weight combine with `r^2/Z_r -> 1/z_0` ([LP] (5.4)) to the exponent `1 + 4 - 2 = 3` of [LM] §6. + +### 4.2 Convergence of the scaled count + +**Lemma 4.1.** Fix `A >= 4`, fix `kappa > 0` outside the countable set `Kap_0` defined after the proof, and fix `(s, U, D, tau)` with `D` nonsingular and `(s, a_3, beta, c_3)` outside `E_k` and outside +the null set of jets for which an extra critical point of `F_0` lies on the circle `X^2 + Z^2 = A^2`. Let +`j_r = chart(rs, U, D, tau)` and `j_0 = chart(0, U, D, tau)`. Then, under the conditional laws `Q_(r, j_r)` of §3.1, + + 1{ N^A = n } 1{K_4 <= kappa} -> 1{ N_oo^A = n } 1{K_4 <= kappa} in probability, + N_oo^A := #{ extra critical points (X, Z) of F_0 with X^2 + Z^2 < A^2 and F_0(X, Z) in (-k, 0) }, + +and likewise for the counts restricted to any fixed index, and for the scaled positions and heights (Corollary X). + +*Proof.* Couple the conditional laws as in §3.1: `f = m_r(j_r) + Res_r` with `Res_r -> Res_0` in `C^4` in probability +and `m_r(j_r) -> m_0(j_0)` in `C^4`. Fix a realization with `K_4 < oo` along which the convergence holds; `K_4(f_r) -> K_4(f_0)` for it, so `1{K_4 <= kappa}` +converges whenever `K_4(f_0) != kappa`, an event of full `Q_(0, j_0)`-probability for the jets admitted below. By Lemma 3.1 (applicable since `D` is nonsingular and `r -> 0`), every +critical point of `f~` in the physical ball `X^2 + Z^2 + |rW|^2 <= A^2` has bounded `W`, so its `(X, Z)` lies in the disc +of radius `A + O(r)`, and by (3.6) `G_r -> F_0` in `C^2` on the disc of radius `A + 1`. The extra critical points of `F_0` +there are finitely many and nondegenerate, none lies on the circle of radius `A`, and none has height in `{-k, 0}`. +Ordinary `C^1` stability (the inverse function theorem on disjoint small discs around the nondegenerate zeros, plus +`|grad F_0| >= delta > 0` on the complement of those discs in the disc of radius `A + 1`, which `C^1`-closeness +preserves for small `r`) gives: for small `r`, `grad G_r` has exactly one zero in each small disc and none elsewhere; +each converges to the corresponding zero of `grad F_0`, its `G_r`-height converges (so its window membership +stabilizes), its planar Hessian converges (so its index stabilizes), and by the bijection of §3.3 these are exactly +the critical points of `f~` in the physical ball other than the pins, which are the two fixed nondegenerate zeros +`(-+1/2, 0)` of `grad F_0` and attract no other zero. Hence `N^A -> N_oo^A` for every such realization; convergence in +probability follows, together with the convergence of scaled positions and heights. ∎ + +**Choice of `kappa`.** Only the limit law enters the argument above, and Proposition 4.4 integrates it over the +singular slice, not over full jet space: the limit targets are `j_0 = chart(0, U, D, tau)`, and the coordinate `s` +does not enter `Q_(0, j_0)`. Let `mu_l` be the finite measure `dU ⊗ Leb(dD) ⊗ Leb(dtau) ⊗ Q_(0, chart(0, U, D, tau))` +on `{ |D| + |tau| <= l } x C^4(X)`, `l in N`, with `dU` the Haar measure; this is exactly the measure on the slice that +(4.2) reduces to at `r = 0`. The `K_4`-marginal of each `mu_l` has countably many atoms; let `Kap_0` be the countable +union of these atom sets over `l`. For `kappa` outside `Kap_0`, Fubini on `mu_l` gives +`integral Q_(0, chart(0, U, D, tau)){K_4 = kappa} dU dD dtau = mu_l{K_4 = kappa} = 0` for every `l`, hence +`Q_(0, chart(0, U, D, tau)){K_4 = kappa} = 0` for `(dU ⊗ Leb ⊗ Leb)`-almost every `(U, D, tau)`, and therefore for +almost every `(s, U, D, tau)` in the domain of (4.2) and (4.4). These are the jets admitted in Lemma 4.1 and all that +Proposition 4.4 integrates over. No union of atom sets over uncountably many conditional laws is used, and no +full-dimensional null set is invoked on the codimension-one slice. The complement of `Kap_0` is unbounded, and +`kappa -> oo` in Proposition 4.4 and §6.5 is taken along it. (For the uniformity in the parameters, `Kap_0` is formed +at the limit parameter of the sequential argument, which is the only place where the pointwise statement is used.) + +### 4.3 The large-`K_4` tail + +**Lemma 4.2.** For every `A >= 4` and every fixed `p >= 1` there is `C(A, p)` such that for `0 < r <= r_*` and +`kappa >= 1`, + + r^(-3) Q_r^W{ N^A >= 1, K_4 > kappa } <= C(A, p) kappa^(-p/2). (4.3) + +*Proof.* `Q_r^W{N^A >= 1, K_4 > kappa} <= E_(Q_r^W)[N^A 1{K_4 > kappa}]`, and the mark `W_r 1{K_4 > kappa}` is +`sigma(f)`-measurable and nonnegative, so [C6] Lemma 5.1 under `Q_r` (gradient-only kernel, all heights, which dominates +the windowed count), divided once by the full normalizer, gives + + E_(Q_r^W)[ N^A 1{K_4 > kappa} ] <= Z_r^(-1) sum_j integral_(|X| <= A r) p_(grad f(X) | U_r)(0) E_(Q_r)[ W_r F_j(H_X) 1{K_4 > kappa} | U_r, grad f(X) = 0 ] dX. + +The left side is under the weighted law; the right side is under `Q_r`, with the original weight `W_r` inside the +conditional expectation and `Z_r^(-1)` outside, once each (this is the form of [C6] (6.3) and of [DL] (4.10); the +unnormalized integral is `Z_r` times the left side, not the left side). + +Cauchy–Schwarz on the conditional expectation and Markov's inequality with the conditional moments +`E[K_4^p | U_r, grad f(X) = 0] <= C_p (1 + beta_X)^p` ([C6] Lemma 4.4 with the frame parameter `beta_X` of [C6] Lemma 4.1) +bound the integrand by `p_(grad f(X)|U_r)(0) E[(W_r F_j(H_X))^2 | .]^(1/2) (C_p (1 + beta_X)^p)^(1/2) kappa^(-p/2)`. By +[C6] (5.3) and the §6 regime table, the density times the square root of the squared-weight conditional moment is +bounded, in every regime, by the source's unmarked intensity bound times `(1 + beta_X)^(3d)`, and the regime's +Gaussian penalty absorbs any fixed power of `(1 + beta_X)` by raising the index of the exponential series in [C6] §6.2 +(the recorded ledgers `2n >= 6 + 3d`, `2n/3 >= 15 + 39d` are the choices for [C6]'s degree); the regional ledgers of +[C6] §6, which are stated for the normalized quantity `Z_r^(-1) E_(Q_r) N^(W_r Xi)`, then give +`Z_r^(-1) integral_(|X| <= A r) (...) dX <= C(A, p) r^3`. ∎ + +### 4.4 Domination on `{K_4 <= kappa}` + +**Lemma 4.3.** Fix `A >= 4` and `kappa >= 1`. There is `r_0(kappa, A) > 0` and an integrable function +`g_(A, kappa)(s, U, D, tau)` on `R x O(m) x R^n x R^(dim tau)`, not depending on `r`, such that for `0 < r <= r_0` and all +`(s, U, D, tau)` in the domain of (4.2), + + p_r( chart(rs, U, D, tau) ) c_m prod_(j>=2) |rs - lambda_j| Delta(D) r^(-4) E[ W_r 1{ N^A >= 1 } 1{K_4 <= kappa} | U_r = v_r, J = chart(rs, U, D, tau) ] <= g_(A, kappa)(s, U, D, tau), + +and the same without the indicator `1{N^A >= 1}`, with `integral_(|s| <= S) g < oo` for every `S`. + +*Proof.* Write `j = chart(rs, U, D, tau)`. On `{K_4 <= kappa}`, Lemma 3.3 gives `1{N^A >= 1} <= 1{ |s| <= S_*(tau, A, kappa, k) }`, +a deterministic indicator with `S_*` polynomial in `|tau|` by (3.8); note that it does not involve `D`. On the same +event, (3.10) gives `r^(-4) W_r <= C (1 + |s| + |tau| + kappa)^4 (|lambda_2| + r kappa)^2 kappa^(2d - 6)`, again with no +inverse of `D`. The density satisfies `p_r(j) <= C exp(-c |j|^2) <= C exp(-c (|D|^2 + |tau|^2))` (the chart is an +isometry in `T` and `|A| >= |D|`), and the Weyl factor is bounded by `C (1 + |s| r + |D|)^(m(m-1)/2)`. Hence the left +side is at most + + g_(A, kappa) := C(A, kappa) exp(-c (|D|^2 + |tau|^2)) (1 + |D|)^(m(m-1)/2) (1 + |s| + |tau|)^4 (|lambda_2| + 1)^2 1{ |s| <= S_*(tau, A, kappa, k) }, + +which is integrable: bounded `s`-range for each `tau`, polynomial growth in `|tau|` against the Gaussian factor +(`S_*` is polynomial in `|tau|`), Gaussian decay in `D`, and finite Haar measure in `U`. Without the count indicator, +the bound is the same without `1{|s| <= S_*}`, integrable over bounded `s`. ∎ + +Remark (nearly singular stable block). Nothing in `g_(A, kappa)` blows up as `lambda_2 -> 0`: the factor +`(|lambda_2| + r kappa)^2` in (3.10) only helps, and the exclusion range `S_*` is independent of `D`. The inverse +`D^(-1)` enters only the pointwise limits of Lemma 4.1, at fixed jets with `D` nonsingular, which is `Q_0`-almost +every jet. + +### 4.5 The near limit for fixed `A` + +**Proposition 4.4.** For every `A >= 4` and `n in {1, 2}`, uniformly on the compact parameter set, + + r^(-3) Q_r^W{ N^A = n } -> nu_near^A(n) := z_0^(-1) integral m w_0 1{ N_oo^A = n } ds da_3 dbeta dc_3, (4.4) + +and `r^(-3) Q_r^W{ N^A >= 3 } -> 0`. The integrals are finite. Moreover `nu_near^A(n) -> nu_near(n)` as `A -> oo`, and +`integral m w_0 N_oo ds da_3 dbeta dc_3 < oo`. + +*Proof.* Fix `kappa` as in Lemma 4.1 and split `1 = 1{K_4 <= kappa} + 1{K_4 > kappa}`. By Lemma 4.2 the second part +contributes at most `C(A, 2) kappa^(-1)` to `r^(-3) Q{N^A = n}` for `n >= 1`, uniformly in `r`. For the first part apply +(4.2) with `E = {N^A = n, K_4 <= kappa}`. The integrand converges pointwise, for `Q_0`-almost every `(s, U, D, tau)` +(`D` nonsingular, jets off `E_k` and off the circle-null set), to +`p_0(chart(0, U, D, tau)) c_m |det D| Delta(D) (det D)^2 w_0(s, a_3, beta) E_0[1{K_4 <= kappa} | j_0] 1{N_oo^A = n}`, by +Lemma 4.1, Lemma 3.5 (convergence in probability of `r^(-4) W_r` together with the uniform `L^2` bound (3.1) gives +convergence of the conditional expectation of its product with a bounded functional) and the continuity of the +density and of the Weyl factor. Lemma 4.3 dominates uniformly in `r`. The prefactor `r^2/Z_r -> z_0^(-1)`. Hence + + limsup / liminf r^(-3) Q{N^A = n} are within C(A, 2)/kappa of z_0^(-1) integral (...) E_0[1{K_4 <= kappa} | j_0] 1{N_oo^A = n}; + +let `kappa -> oo` along the complement of `Kap_0` (monotone convergence on the right, `E_0[1{K_4 <= kappa}|j_0] -> 1`) and integrate out `(U, D, tau_rest)` +to obtain (4.4) with the reduced weight (3.11). For `n >= 3` the pointwise limit of the indicator is zero by Lemma 3.2. +Finiteness (see also the explicit majorant of §6.2, from Lemma 3.6): by Fatou's lemma applied along `r -> 0` to the `K_4 <= kappa` part of the representation (4.2) with `E = {N^A >= 1}` (the integrand converges pointwise to the (4.4)-integrand times `E_0[1{K_4 <= kappa}|j_0]`, and the latter tends to `1` as `kappa -> oo`), `integral m w_0 N_oo^A <= liminf r^(-3) E N^A <= C`, where the last bound is [DL] Theorems P_d, C_d and I_d (pin balls and collar of scaled radius `4`, shells from `4r` to `A r`: `C r^3 (1 + sum_j (r/s_j)^2 + s_j^2) <= C r^3` uniformly in `A <= 1/r`). The `A -> oo` limit: `N_oo^A` is nondecreasing in `A` with limit `N_oo`, so +`integral m w_0 N_oo < oo` by monotone convergence and the uniform bound; then `1{N_oo^A = n} -> 1{N_oo = n}` +pointwise (for every jet, once `A` exceeds the radius of the finitely many extra critical points) and is dominated by +`N_oo`, so dominated convergence gives `nu_near^A(n) -> nu_near(n)`. Uniformity on the compact parameter set: the +dominating functions and all limits depend continuously on `(b, k, R)`, so a sequence of parameters along which the +convergence fails would have a convergent subsequence contradicting the pointwise statement at the limit parameter, +exactly as in [LP] §5. ∎ + +## 5. The middle and far regions, and the absence of cross terms + +### 5.1 Shells + +By [DL] Theorem I_d, (1.6), summed over the `d + 1` indices, for `A >= 4` and `rho <= s_0`, + + E_(Q_r^W) N_mid^(A, rho) <= C r^3 ( A^(-2) + rho^2 ), (5.1) + +with `C` independent of `A` and `rho`. + +### 5.2 The far region + +By [RC] Corollary E applied to `E = D_rho = {|x| >= rho}` and summed over indices, and by [RC] Corollary D, + + Q_r^W{ N_far^rho >= 1 } = k r^3 integral_(D_rho) Lambda dx + O(r^4), Q_r^W{ N_far^rho >= 2 } <= C r^5, (5.2) + +with constants depending on `rho`. The function `rho -> k integral_(D_rho) Lambda` is nondecreasing as `rho` decreases +and bounded: indeed, for `rho' < rho`, `k integral_(rho' <= |x| < rho) Lambda = lim_r r^(-3) E N_(r)({rho' <= |x| < rho}) +<= C rho^2` by (5.2) applied on the annulus and (5.1). So `k integral_X Lambda dx := lim_(rho -> 0)` exists and is +finite, and `k integral_(D_rho) Lambda -> k integral_X Lambda` with error at most `C rho^2`. + +### 5.3 The witness-conditioned far first moment + +**Lemma 5.1.** Let `X` be a point of the near region `|X| <= A r` and `Q_X` the regression of `Q_r` on the additional +observation `grad f(X) = 0` (the gradient-only kernel of [C6] Lemma 5.1, defined for every `X`). There are +`r_0(A, rho) > 0` and `C(rho)` such that, for `r <= r_0` and `rho <= s_0`, with `Y_x = (grad f(x), f(x))`, + + Cov_(Q_X)( Y_x ) >= (gamma_rho/4) I for every x in D_rho, (5.3a) + E_(Q_X)[ N_far^rho ] <= C(rho) k r^3 (1 + beta_X)^(2d), (5.3) + +where `gamma_rho > 0` is the `r`-independent floor (5.3b) and `beta_X` is the frame parameter of [C6] Lemma 4.1 for the +regime containing `X` (bounded by a polynomial in `chi`, `|v|^(-1)`, `r^(-1)` as recorded in the [C6] §6 table). + +*Proof.* Two facts. Let `J_5` be the complete midpoint Taylor jet through order five (the `C(d+5, 5)` coefficients +`D^alpha f(0)/alpha!`), a nondegenerate Gaussian vector that does not depend on `r`. + +(i) By the distinct-site jet rank of [LP] §2, `(J_5, Y_x)` is a list of distinct derivative functionals at the two +distinct sites `0` and `x`, so its covariance is positive definite; the Schur complement is continuous in `(x, R)` and +`{x in D_rho} x O(d)` is compact, so + + Cov( Y_x | J_5 ) >= gamma_rho I, Cov( Y_x ) <= M I, (5.3b) + +with `gamma_rho > 0` and `M` independent of `r`. + +(ii) The vector to whiten is chosen regime by regime so that its whole remainder is small relative to its floor. +In the collar (`|X| >= r/4` in the scaled pin-ball coordinates, [DL] §5), let `V` be the raw three-site scaled vector +of [DL] §5.2, `V = ( f(ra), r grad f(ra), f(rb), r grad f(rb), f(rc), r grad f(rc) )`, with `a, b` the pins and +`c = X/r`. Taylor's formula through total degree five gives `V = B_c D_r J_5 + R_V` with `||R_V||_(L^2) <= C r^6`, where +`B_c` has a uniform row singular-value floor on the collar ([DL] §5.1, Lemma) and `D_r = diag(r^|alpha|) >= r^5 I`; with +`Cov(J_5) >= c I` this gives `Cov(V) >= c r^10 I` for small `r` (the remainder is absorbed exactly as in [DL] (5.1)). In +the pin ball let `V = (U_r, Y^perp)`, where `U_r` is the contact frame of [LP] §3, `U_r = T_U J_5 + R_U` with +`||R_U||_(L^2) <= C r` and `Cov(U_r) >= c I`, and `Y^perp := Y - E[Y | U_r]` is the affine prior-regression residual of +the normalized gradient frame `Y` of [DL] (4.2). `Y^perp` is independent of `U_r`, `Cov(Y^perp) = Cov_(Q_r)(Y) >= c_0 I` +([DL] (4.5)), so `Cov(V)` is block diagonal with `lambda_min Cov(V) >= min(c, c_0)` (the floors of `U_r` and of `Y` +given `U_r` are not combined through a raw joint covariance, which could be small along a direction mixing the two +blocks). Residualizing [DL] (4.2), `Y^perp = B (J - E[J | U_r]) + R^perp` with `||R^perp||_(L^2) <= C r`; the midpoint +jets `J` are linear images of `J_5` and the regression coefficient `Cov(J, U_r) Cov(U_r)^(-1)` is bounded, so with +`U_r = T_U J_5 + R_U` both blocks of `V` are `T_V J_5 + O_(L^2)(r)`. The σ-algebra of `(U_r, Y^perp)` is that of +`(U_r, grad f(X))`. In both cases + + V = T_V J_5 + R_V, ||Cov(V)^(-1/2)|| ||R_V||_(L^2) <= eta_r := C r, (5.3c) + +by `C r^(-5) . C r^6` in the collar and `C . C r` in the pin ball, uniformly in `X` in the near region. The σ-algebra +of `Q_X`, generated by `(U_r, grad f(X))`, is contained in that of `V`: equal in the pin ball (`Y` is an invertible +image of `grad f(X)` given `U_r`), and strictly smaller in the collar, where `V` also carries the witness value `f(X)`. + +Whitening. Put `Ut := Cov(V)^(-1/2) (V - E V)`, so `Cov(Ut) = I` and `Ut = T J~_5 + E` with `J~_5` the centred jet and +`||E||_(L^2) <= eta_r` by (5.3c). The σ-algebra of `V` is that of `Ut`. For a unit vector `e`, +`Var(e . Y_x | V) = min_a Var(e . Y_x - a . Ut)`, attained at `a = Cov(e . Y_x, Ut)` with `|a|^2 <= Var(e . Y_x) <= M`. +Since `e . Y_x - a . T J~_5` is `e . Y_x` minus a linear function of `J_5`, its variance is at least `Var(e . Y_x | J_5) +>= gamma_rho` by (5.3b). Therefore + + std( e . Y_x - a . Ut ) >= std( e . Y_x - a . T J~_5 ) - |a| ||E||_(L^2) >= gamma_rho^(1/2) - M^(1/2) eta_r >= gamma_rho^(1/2)/2 + +for `r <= r_0(A, rho)`, i.e. `Cov(Y_x | V) >= (gamma_rho/4) I`, uniformly in `X` and in the regime. Conditional +covariances of a Gaussian vector increase under coarsening of the conditioning (`Cov(Y | G) = Cov(Y | F) + Cov(E[Y | F] | G)` +for `G` contained in `F`), so `Cov_(Q_X)(Y_x) = Cov(Y_x | U_r, grad f(X)) >= Cov(Y_x | V) >= (gamma_rho/4) I`. This is +(5.3a); no witness value is pinned in the kernel `Q_X`, it is only used inside the whitened vector. + +Consequences. By (5.3a) the conditional density of `Y_x` at `(0, t)` under `Q_X` is at most `C(rho)`. For the +conditional mean of `H_x` condition transitively under `Q_r`. First condition on the regime frame `Lambda = tau` of [C6] +Lemma 4.1 (an invertible image of `grad f(X)` given `U_r`): that lemma bounds the conditional means of `H_x` and of +`Y_x` by `C (1 + beta_X)` and their conditional covariances by their prior bounds. Then fix `Y_x = (0, t)`: the +conditional mean of `H_x` becomes `E[H_x | Lambda] + Cov(H_x, Y_x | Lambda) Cov(Y_x | Lambda)^(-1) ((0, t) - E[Y_x | Lambda])`, +and `Cov(Y_x | Lambda) = Cov_(Q_X)(Y_x) >= (gamma_rho/4) I` by (5.3a), so `|E[H_x | Lambda, Y_x = (0, t)]| <= C_rho (1 + beta_X + |t|)`; +the conditional covariance of `H_x` is bounded. No joint floor of `(U_r, Lambda)` is asserted. Hence `E[ F_j(H_x) | Lambda, Y_x = (0, t) ] <= C E[(1 + |H_x|)^d | ...] <= C_rho (1 + beta_X)^d (1 + |t|)^d`. +Apply the Kac–Rice formula of [RM] (12) under `Q_X` to `D_rho` and the window of length `k r^3`, on which `|t|` is +bounded: (5.3) follows, with the exponent `2d >= d` (the weaker displayed exponent is kept). ∎ + +**Lemma 5.2 (no near–far cross term).** For `A >= 4` and `rho <= s_0`, + + Q_r^W{ N_near^A >= 1, N_far^rho >= 1 } <= E_(Q_r^W)[ N_near^A 1{ N_far^rho >= 1 } ] <= C(A, rho) r^(9/2). (5.4) + +*Proof.* The mark `W_r 1{N_far^rho >= 1}` is `sigma(f)`-measurable and nonnegative, so [C6] Lemma 5.1 under `Q_r` +(gradient-only kernel, all heights, which dominates the windowed near count), divided once by the full normalizer, gives + + E_(Q_r^W)[ N_near^A 1{N_far >= 1} ] <= Z_r^(-1) sum_j integral_(|X| <= A r) p_(grad f(X) | U_r)(0) E_(Q_r)[ W_r F_j(H_X) 1{N_far >= 1} | U_r, grad f(X) = 0 ] dX, + +the left side under the weighted law and the right side under `Q_r` with `W_r` inside and `Z_r^(-1)` outside, once +each. Cauchy–Schwarz on the conditional expectation (which is `E_(Q_X)` at the target), `1{N_far >= 1} <= N_far`, and +(5.3) bound the integrand by + + p_(grad f(X)|U_r)(0) E_(Q_X)[ (W_r F_j(H_X))^2 ]^(1/2) ( C k r^3 (1 + beta_X)^(2d) )^(1/2). + +By [C6] (5.3) and the §6 regime table, the density times the square root of the squared-weight conditional moment is +bounded, in every regime, by the source's unmarked intensity bound times `(1 + beta_X)^(3d)`, and the regime's +Gaussian penalty absorbs the additional `(1 + beta_X)^d` as it absorbs [C6]'s marked factor: in every regime of +[C6] §6.2 the absorption of a power of `beta_X` is by an exponential factor in the degeneracy variable, and any fixed +polynomial degree is absorbed by raising the index of the exponential series there (the ledgers `2n >= 6 + 3d`, +`2n/3 >= 15 + 39d` are the recorded choices for [C6]'s degree; a larger `n` serves here at the cost of the constant). The +regional ledgers of [C6] §6, stated for the normalized quantity, then give `Z_r^(-1) integral_(|X| <= A r) (...) dX <= C(A) r^3`. +Multiplying by `(C k r^3)^(1/2)` gives (5.4). ∎ + +Remark. Only `o(r^3)` is needed; the exponent `9/2` is what Cauchy–Schwarz delivers and is not claimed sharp. + +## 6. Proof of Theorem N and of the corollaries + +### 6.1 The sandwich + +Fix `A >= 4` and `rho <= s_0`. Write `N_rest = N_mid^(A,rho) + N_far^rho`. From the inclusions + + {N = 2} subset {N^A = 2} cup {N^A = 1, N_rest >= 1} cup {N_rest >= 2}, + {N^A = 2} subset {N = 2} cup {N^A >= 1, N_rest >= 1}, + +and the same with `2` replaced by `n >= 3` (upper inclusion only), + + | Q{N = 2} - Q{N^A = 2} | <= Q{N^A >= 1, N_mid >= 1} + Q{N^A >= 1, N_far >= 1} + Q{N_mid >= 1} + Q{N_far >= 2} + <= 2 C r^3 (A^(-2) + rho^2) + C(A, rho) r^(9/2) + C(rho) r^5, + +by (5.1), (5.4) and (5.2); similarly `Q{N >= 3} <= Q{N^A >= 3} + (the same error)`. For `n = 1`, + + Q{N = 1} = Q{N^A = 1, N_rest = 0} + Q{N^A = 0, N_rest = 1}, + | Q{N^A = 1, N_rest = 0} - Q{N^A = 1} | <= Q{N^A >= 1, N_rest >= 1}, + | Q{N^A = 0, N_rest = 1} - Q{N_far^rho = 1} | <= Q{N^A >= 1, N_rest >= 1} + Q{N_mid >= 1}, + +and `Q{N_far^rho = 1} = k r^3 integral_(D_rho) Lambda + O(r^4) + O(r^5)` by (5.2). Divide by `r^3`, let `r -> 0` using +Proposition 4.4, then let `A -> oo` and `rho -> 0` using Proposition 4.4's `A -> oo` limit and §5.2: + + lim r^(-3) Q{N = 2} = nu_near(2), lim r^(-3) Q{N = 1} = nu_near(1) + k integral_X Lambda, lim r^(-3) Q{N >= 3} = 0, + +with the error after `r -> 0` bounded by `C(A^(-2) + rho^2)`, which is what makes the two outer limits exist. The +limits exist as full limits (not along subsequences) because the sandwich bounds `liminf` and `limsup` between +quantities that differ by `C(A^(-2) + rho^2)`. Uniformity on the compact parameter set is inherited from Proposition +4.3, (5.1), (5.2) and (5.4), whose constants are uniform. This proves (1.5) modulo positivity and continuity. + +### 6.2 Positivity and continuity + +`nu(2) >= nu_near(2) > 0`: the [LM] cubic (L4), that is `(s, a_3, beta, c_3) = (-3k/2, 0, -2k, 0)`, has two extra +critical points `P_+-` at height `-7k/32` in the window and pin weight `w_0 = 3k^2 . 15k^2 = 45 k^4 > 0` (control `EX`, +matching [LM] (L5)); by Lemma 3.2 these properties persist on an open neighbourhood of that jet, on which `m > 0`, so +the integral (1.3) over that neighbourhood is positive. `nu_near(1) > 0`: the exact jet `(s, a_3, beta, c_3) = +(-1/2, 3, 1/3, 0)` at `k = 1` (rescaled by `k` in general) has extra critical points `(-3/2, 3)` at height `-1/2` (in the +window) and `(45/22, -357/11)` at height `-9947/121` (outside), with pin weights `7/4` and `17/4` (control `EX`); the +same open-neighbourhood argument applies. `nu(1) >= k integral_X Lambda > 0` by the positivity of `Lambda_j` in [RM]. +Finiteness and continuity of (1.3), with an explicit majorant. On `{w_0 > 0}` the pin weight is +`w_0 = 9k^2 (4 s^2 - B^2) <= 36 k^2 s^2`, and by Lemma 3.6 the indicator `1{N_oo >= 1}` is supported on +`|B|/2 < -s <= 2|B| + (32 k D^2)^(1/3)`. Hence, for every `n >= 1`, + + m w_0 1{N_oo = n} <= 36 k^2 s^2 m 1{ |B|/2 < -s <= 2|B| + (32 k D^2)^(1/3) }, + +and integrating in `s` gives at most `12 k^2 m ( 2|B| + (32 k D^2)^(1/3) )^3 <= C_K m (1 + |a_3| + |beta| + |c_3|)^6` for +parameters in a compact set `K` (the constant depends on `K` through `k` only). Since `m <= C_K exp(-c_K (a_3^2 + +beta^2 + c_3^2))` uniformly on `K` (§3.7), this majorant is integrable uniformly on `K`. It proves the finiteness of +(1.3) a second time (independently of the Fatou argument of Proposition 4.4), and it does not depend on `A`, so it also +dominates the `A -> oo` step of Proposition 4.4. Continuity in `(b, k, R)`: `m`, `w_0`, `z_0`, `F_0` and `Lambda` depend +continuously on the parameters, the indicator `1{N_oo = n}` is continuous off the null set `E_k` (every counted point +is a nondegenerate saddle by Lemma 3.6, and the boundary heights are excluded off `E_k`), and the integrand of (1.3) is +dominated uniformly on `K` by the integrable majorant above; dominated convergence in the parameters gives continuity. + +### 6.3 Corollary Lambda + +`(N)_q = sum_n (n)_q 1{N = n}`. For `M >= q + 1`, `sum_(n > M) (n)_q Q{N = n} <= E[(N)_(q+1)]/(M - q) <= C_(q+1) r^3 / +(M - q)` by [C6] Theorem Q. Hence `r^(-3) E[(N)_q]` is within `C_(q+1)/(M - q)` of `sum_(n <= M) (n)_q r^(-3) Q{N = n}`, +which converges to `(1)_q nu(1) + (2)_q nu(2)` by (1.5); let `M -> oo`. This gives (1.6). The Palm statement is the +quotient of `Lambda_2` by `Lambda_1`. + +### 6.4 Corollary S + +[RCL] Theorem S extracts subsequential limits of `nu_r`; (1.5) identifies every subsequential limit with `nu`, so the +full family converges in the weighted `l1` sense of [RCL] (6.1) (the tail bound of [RCL] is uniform, so coordinatewise +convergence upgrades to weighted-`l1` convergence exactly as in its proof). The statements about `F_r`, `CP(p, F_r)` +and `CP(t nu)` are [RCL] Theorems C, S and (7.1) with the limit inserted. The three examples of [RCL] §6 have, +respectively, a non-convergent `nu_r`, a limit carried by `{2}` alone (`nu(1) = 0`), and a limit law with unbounded +support; each is excluded by (1.5) (existence of the limit, `nu(1) > 0`, `nu(n) = 0` for `n >= 3`). + +### 6.5 Corollary X + +Same proof as Proposition 4.4 with the functional `Phi^A = sum_(extra near critical points) phi(scaled data)` in place of +the indicator. Two adjustments. First, `Phi^A` is not bounded (`|Phi^A| <= sup|phi| N^A`), so before applying (4.2) +truncate at `N^A <= M`: by [C6] Theorem Q, `r^(-3) E[N^A 1{N^A > M}] <= r^(-3) E[(N)_2]/(M - 1) <= C_2/(M - 1)` +uniformly in `r`, which is the uniform integrability needed; on `{N^A <= M}` the functional is bounded, Lemma 4.1 +gives the convergence of the scaled positions and heights, and Lemma 4.3 dominates with `M sup|phi|`. Let +`kappa -> oo`, then `M -> oo`, then `A -> oo` (the `A -> oo` step as in Proposition 4.4, dominated by `sup|phi| N_oo`). +Second, the transverse scaled distance: in the rotated chart of §3.2 the physical transverse displacement of a +critical point at scaled coordinates `(X, Z, W)` is `r (Z, rW)` with `|W|` bounded by Lemma 3.1, so `|X_perp|/r = |Z| + O(r)`. +Third, the index: by Lemma 3.6 every limit point in the sum is a nondegenerate saddle of `F_0`, of index `d - 1` in +the full model (Lemma 3.1, stable block negative definite on `{w_0 > 0}`), so the index marginal of the near law is +concentrated on `d - 1` and `nu_near(n)` is carried by index-`(d - 1)` points; the remote singleton part of `nu(1)` +carries the index-specific kernels `Lambda_j` of [RM], all `j`. + +## 7. The reduced integral, and exact configurations + +### 7.1 Structure of `nu_near` + +The integrand of (1.3) depends on the full jet only through `(s, a_3, beta, c_3)`: the count `N_oo` and the pin weight +`w_0` are functions of the planar cubic (1.2) alone, and the stable directions enter only through `m`. Hence `nu_near(n)` +is a four-dimensional integral for every `d`: + + nu_near(n) = z_0^(-1) integral_(R^4) m(s, a_3, beta, c_3) w_0(s, a_3, beta) 1{ N_oo(s, a_3, beta, c_3) = n } ds da_3 dbeta dc_3. + +The domain of integration is bounded in `s` from above by the index conditions (`s < beta/2 - a_3^2/(24k)` and +`s < a_3^2/(24k) - beta/2`) and, for the extra critical points to lie in the window, from below: for `|s| -> oo` with +`(a_3, beta, c_3)` fixed, the extra critical points of `F_0` are either the pins' neighbours excluded by Lemma 3.3 or +lie at scaled distance of order `|s|`, where the Euler identity for the cubic part gives `F_0 = s^3 zeta^2 / 6 + O(s^2)` +along the rescaled solutions `(X, Z) = s (xi, zeta)` of the limiting system `6k xi^2 + a_3 xi zeta + (beta/2) zeta^2 = 0`, +`zeta (1 + beta xi + c_3 zeta / 2) + (a_3/2) xi^2 = 0` (control `EU`); `zeta = 0` forces `xi = 0`, so on nontrivial +solutions `|zeta|` is bounded below and the height leaves the window `(-k, 0)` for large `|s|`. This is the reason the +near count carries no logarithm although the shells' all-height count would (compare [DL] §8: the height window is +load-bearing for the shells), and it is the analytic content of Lemma 3.3 and Lemma 4.3. + +### 7.2 Exact configurations (finite control `EX`) + +With `k = 1` and jets `(s, a_3, beta, c_3)`: + +| Jets | Extra critical points `(X, Z)` | Heights | In window | Pin weights `(w_M, w_S)` | `N_oo` | +|---|---|---|---|---|---| +| `(-3/2, 0, -2, 0)` ([LM] (L4)) | `(-3/4, +-sqrt(15/8))` | `-7/32, -7/32` | yes, yes | `(3, 15)` | 2 | +| `(-55/48, 5/2, -5/3, -5/2)` | `(-2, 3)`, `(-783/1616, -129/1616)` | `-27/32`, `-4735/83566592` | yes, yes | `(5/16, 215/16)` | 2 | +| `(-1/2, 3, 1/3, 0)` | `(-3/2, 3)`, `(45/22, -357/11)` | `-1/2`, `-9947/121` | yes, no | `(7/4, 17/4)` | 1 | +| `(-33/10, 3, -12/5, -3)` | `(-2, 5/2)`, `(-282/541, -1495/1082)` | `-41/16`, `-3199921/4682896` | no, yes | `(207/20, 117/4)` | 1 | +| `(-396, -3, -204, -3)` | `(-2, 1/2)`, `(-263314/140087, -115623/280174)` | `-119/8`, `< -k` | no, no | `(7047/4, 11961/4)` | 0 | +| `(1000, 2, 1, -1)` | none real (`Q_2` has negative discriminant) | — | — | `w_0 = 0` (index conditions fail) | 0 | + +All entries are exact rationals or exact quadratic irrationals verified by the control. The scaling `k -> 1` is the +substitution `(s, a_3, beta, c_3, X, Z) -> (k s, k a_3, k beta, k c_3, X, Z)`, under which `F_0` scales by `k`. + +## 8. Remarks on the mechanism + +- The cubic (1.2) is the two-parameter family that the four pins leave free at order three on the soft plane; its + extra critical points appear and disappear in pairs through fold points (the discriminant of `Q_2`), which is why + `N_oo` takes the values `0, 1, 2` and why `nu(1)` has a near contribution at all: a pair with one member above the + upper window edge or below the lower one. No statement about elder selection is made. By Lemma 3.6 every extra + critical point with height in the window is a nondegenerate saddle of the planar cubic, of index `d - 1` in the full + model; maxima or degenerate critical points of `F_0` occur only outside the window (control `TW`). This does not + affect the count coefficients, but it is the correct input for any positional or index-law consumer (Corollary X). +- The `r^3` is `r` (the soft transverse curvature must be `O(r)`, one Lebesgue direction) times `r^4/r^2` (the pin + weight on a soft configuration relative to the normalizer). Nothing is small in the third-order jets. +- The far contribution to `nu(1)` is the integral of the reviewed contact kernel; its integrability near the pins is a + consequence of the shell bound, not a new estimate. +- The near–far cross term is bounded by Cauchy–Schwarz inside the marked Kac–Rice integrand (Lemma 5.2); it is the + only place where a witness-conditioned bound on the remote count is needed, and only its order `o(r^3)` is used. + +## 9. What this note does not establish + +No rate of convergence in (1.5)–(1.6); no numerical value or enclosure of `nu(1)`, `nu(2)`, `z_0`, `m` or +`integral Lambda`; no uniformity in `k`, `L` or `d`; no elder-selection statement about the extra points; no joint +law of the clusters of distinct pin pairs; no all-height near count (the window is load-bearing, §7.1); no register or +catalog change. The catalog entry C6 and the GRAPH node `math.rn-region.witness-collision` are not moved by this packet. + +## 10. Finite controls + +`cluster_law_check.py` (standard library, exact rationals and exact polynomial arithmetic over `Q`) verifies: + +- `NF`: the pin conditions of `F_0` (values `0` and `-k`, vanishing gradients at `(-+1/2, 0)`) as polynomial identities in + `(s, a_3, beta, c_3, k)`; the pin Hessian blocks `M_2(M), M_2(S)` of Lemma 3.4. +- `RS`: the resultant of `partial_X F_0, partial_Z F_0` in `Z` equals `(X^2 - 1/4) Q_2(X)` with the coefficients (3.7), + as an identity in `Q[X, s, a_3, beta, c_3, k]`. +- `EX`: the six configurations of §7.2: exact critical points, heights, window membership, pin weights, `N_oo`; the + [LM] weight `45 k^4`; the [LM] double root of `Q_2`. +- `BD`: the block-determinant expansion (3.9) for `d = 3` and `d = 4` as an exact identity in `r` (the `r^2` + coefficient is `det D . det M_2`, the remainder is divisible by `r^3`). +- `EU`: the Euler identity `xi G_xi + zeta G_zeta = 3 G_3 + zeta^2` for the limiting cubic-plus-quadratic `G`, hence + `G = zeta^2/6` at its critical points, and `zeta = 0 => xi = 0`. +- `SH`: the shear identity (3.12), `F_0(X, Z) = C(u) + (s + Bu) Z^2/2 + (D/3) Z^3` with `u = X + a_3 Z/(12k)`, as the + polynomial identity in `Q[u, Z, s, a_3, beta, c_3, k]` obtained by clearing `k` (`(12k)^3 F_0` with `12k X = 12k u - a_3 Z`). +- `TW`: on an exact rational family of critical points of `G` (parametrized by `(u, Z, D, k)`, `B` and `s` solved + exactly), the Hessian-determinant identity `det = -3k(B + 4su)` at every critical point, the height identity + `G = C(u) + w^3/(6D^2)`, and, on every typed in-window sample, `det < 0` and `(-s - 2|B|)_+^3 <= 32 k D^2` (Lemma 3.6); + the number of typed in-window samples is recorded. +- `LG`: the exponent ledgers: `1 + 4 - 2 = 3`; the cross term `3 + 3/2 > 3`; the shell error `A^(-2) + rho^2`; the + Hadamard exponent of (3.10) (`4 + 2 + (2d - 6) = 2d` soft-and-stable row factors for `d = 3, 4`); the large-`K_4` + tail `kappa^(-p/2) -> 0`; the uniform-integrability step of §6.3 and §6.5 (`1/(M - 1) -> 0`). + +Nine mutants (`pins-not-critical`, `wrong-pin-hessian`, `drop-sigma-jacobian`, `three-extra-points`, +`cross-term-not-small`, `window-closed`, `index-sign`, `shear-drop-cubic`, `s-bound-constant`) each exit `1`. The controls verify identities, exact +configurations and exponent bookkeeping only. They do not verify the Gaussian regression, the dominated convergence, +the stability argument or the Kac–Rice steps; those are the written arguments and the cited sources. + +## 11. Attribution and reconnaissance + +The mechanism of §§3–4 is the rare-set construction of [EDL] §§3–9 and [LM] §§2–6, used here as a limit theorem +rather than as a lower bound. The dominated-convergence device is that of [LP] §13 and [RM] §5. The local normal form +is a cusp-type unfolding of a planar cubic; no novelty is claimed for that algebra, which is classical (V. I. Arnold, +*Catastrophe Theory*, 3rd ed., Springer 1992, for the fold and cusp normal forms; the specific pinned cubic is the +one written in [LM] (L6) and [EDL] (A17)). `RECONNAISSANCE.md` records that the external literature check remained +blocked by the Consensus monthly quota and that no citation was invented. diff --git a/frontiers/c6_cluster_law_20260929/README.md b/frontiers/c6_cluster_law_20260929/README.md new file mode 100644 index 000000000..1166376f7 --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/README.md @@ -0,0 +1,90 @@ +# C6 cluster law (CL-C6-CLUSTER-LAW-20260929-v1.2) + +Author-side proof candidate by Anthropic Claude. **Nonauthor analytic review required.** Scientific effect: NONE. +**Stacked candidate:** consumes only merged sources at exact bytes (`SOURCE_MAP.json`): [LP], [DL], [C6], [RM], [RC], +[EDL], [LM], and, for the consumer corollary only, [RCL] (Math-#153). + +## What it proves, if the arguments and the inputs hold + +For the exact pinned model of `imports/lifetime_parent_20260925` with compact marks, the original endpoint weight and +the full normalizer, in every fixed `d >= 2`, with `N` the window critical-point count other than the pins: + +| Result | Statement | Answers | +|---|---|---| +| Theorem N | `r^(-3) Q_r^W{N = n} -> nu(n)` with `nu(1), nu(2) > 0` explicit and `nu(n) = 0` for `n >= 3` | the "unknown conditional cluster law" of Math-#153 | +| Corollary Lambda | `r^(-3) E[(N)_q] -> Lambda_q`: `Lambda_1 = nu(1) + 2nu(2)`, `Lambda_2 = 2nu(2)`, `Lambda_q = 0` for `q >= 3` | sharpens [C6] Theorem Q to asymptotics; Palm excess converges | +| Corollary S | the limit `nu` of Math-#153's `nu_r` exists (no subsequence), `F_oo = (nu(1) delta_1 + nu(2) delta_2)/(nu(1)+nu(2))`, `CP(t nu)` explicit | uniqueness and identification of the compound-Poisson limits | +| Corollary X | the near point-intensity measure of the scaled positions and heights of the extra window critical points converges (`r -> 0` at fixed near radius, then radius `-> oo`) to that of the critical points of a planar cubic under the weighted jet measure; not a configuration law | the "positional law of the pairs" non-claim of [C6] §9, at intensity level | + +`nu(2) = nu_near(2)`, `nu(1) = nu_near(1) + k integral_X Lambda`, where `nu_near(n)` is a four-dimensional Gaussian +integral over the scaled soft transverse curvature `s` and the three planar cubic jets `(a_3, beta, c_3)` of the +midpoint (PROOF.md (1.3)), with the pin weight `w_0 = [-6k(s - beta/2) - a_3^2/4]_+ [a_3^2/4 - 6k(s + beta/2)]_+`, and +`Lambda` is the reviewed remote contact kernel of [RM]. + +**Planar case.** At `d = 2` the chart is the identity and every consumed regional bound is a reviewed planar statement, +so Theorem N at `d = 2` is conditional on this note's own steps only (PROOF.md §1.4). + +## Mechanism (PROOF.md §§3–6) + +1. Near the pins, one transverse curvature of order `r` (one Lebesgue direction: the factor `r`) puts the field into the + normal form of a planar cubic `F_0` on the soft plane, with the two pins among its critical points ([EDL] (A17), + [LM] (L6)); the stable directions are slaved (Lemma 3.1). A planar cubic has at most four critical points, so at most + two extra ones (Lemma 3.2: the resultant is `(X^2 - 1/4) Q_2(X)` with `Q_2` quadratic, an exact identity). In the + sheared form `C(u) + (s + Bu) Z^2/2 + (D/3) Z^3` (Lemma 3.6, exact identity) every extra window critical point is a + nondegenerate saddle and the typed window forces `-s <= 2|B| + (32 k D^2)^(1/3)`, the integrable majorant of §6.2. +2. The pin weight on such a configuration is `r^4 (det D)^2 w_0` and the normalizer is `r^2 z_0` (Lemmas 3.4–3.5), + giving the exponent `1 + 4 - 2 = 3`. +3. The scaled representation (4.2) in the spectral chart of §3.2, a `K_4 <= kappa` split (Lemma 4.2: the large-`K_4` + part is `O(kappa^(-p/2)) r^3` by marked Kac–Rice) and dominated convergence on `{K_4 <= kappa}` (Lemma 4.3, with the + full-space exclusion Lemma 3.3 for large soft curvature) give the near limits for every scaled radius `A`; the counts + converge by `C^1` stability (Lemma 4.1). +4. The shells contribute `C r^3 (A^(-2) + rho^2)` ([DL] Theorem I_d), the remote region `k r^3 integral Lambda + O(r^4)` + ([RC] Corollary E), a second remote point `O(r^5)` ([RC] Corollary D), and a near point together with a remote one + `O(r^(9/2))` (Lemma 5.2: Cauchy–Schwarz inside [C6]'s marked Kac–Rice formula with a witness-conditioned remote first + moment, Lemma 5.1). The sandwich of §6.1 then closes with `A -> oo`, `rho -> 0`. + +## Run + + python -B -S cluster_law_check.py # rc 0, output = RESULTS.json + python -B -O -S cluster_law_check.py # rc 0, byte-identical + python -B -S cluster_law_check.py --mutant NAME # rc 1 for each of the nine names in the docstring + +Eight exact groups (`NF RS EX BD EU SH TW LG`): the pin conditions and pin Hessians of the cubic as polynomial identities; the +resultant factorization and the explicit `Q_2`; six exact configurations with `0`, `1`, `2` extra critical points in the +window and their pin weights (including the [LM] cubic with weight `45 k^4`); the block-determinant expansion in `r` +for `d = 3, 4`; the Euler identity behind the large-curvature exclusion; the shear identity (3.12); the saddle sign, +height identity and `s`-range of Lemma 3.6 on an exact rational family of critical points; the exponent ledgers (soft slice, cross term, +shell error, Hadamard row factors of (3.10), the large-`K_4` tail and the uniform-integrability tail). They verify identities, +exact configurations and exponent bookkeeping only. + +## Review requests (exact interfaces) + +1. Lemma 3.1: the normal form in the rotated chart and the stable slaving, in particular the bound on `|W|` for critical + points in the physical ball and the `C^2` remainder (3.6) with its `(1 + |D^(-1)|)` growth (used only pointwise). +2. Lemma 3.3: the exclusion of near critical points for large scaled soft curvature, proved in the full transverse + space on `{K_4 <= kappa}`; the threshold (3.8) and the pin nondegeneracy radius. +3. Lemma 4.1: the coupling of the conditional laws and the `C^1` stability count; the null sets (`E_k`, the circle, the + atoms of `K_4`). +4. Lemma 4.2: the large-`K_4` tail through [C6] Lemma 5.1 and the regime ledgers; Lemma 4.3: the dominating function + built from (3.8) and the Hadamard bound (3.10), with no inverse of `D`; the Weyl density (3.3). +5. Lemma 5.1: the uniform conditional covariance floor (5.3a) of the remote jets given the pins and a near witness, + proved by whitening the joint frame against the midpoint five-jet ((5.3b), (5.3c)) with the [LP] §2 distinct-site + floor and the [DL] frame remainders; the full normalizer `Z_r^(-1)` once in each marked display (Lemmas 4.2, 5.2); Lemma 5.2: the Cauchy–Schwarz insertion and the absorption of `(1 + beta)^(d/2)`. +6. Lemma 3.6: the sheared cubic, the saddle sign `-3k(B + 4su) < 0` and the range (3.13); the choice of `kappa` by + Fubini on the limit law (after Lemma 4.1); §6.1: the sandwich and the order of the limits `r -> 0`, `A -> oo`, `rho -> 0`; §6.2 positivity and the explicit majorant; §6.3 uniform + integrability from [C6] Theorem Q. +7. That every step cited from the seven consumed proof sources is used at its stated scope and exact bytes. + +## What this does not do + +No rate, no numerical value of `nu(1)`, `nu(2)`, `Lambda_q`, no uniformity in `k`, `L` or `d`, no elder-selection +statement about the extra points, no joint law of clusters of distinct pin pairs, no all-height near count, no register +or catalog change. The catalog entry C6 and the GRAPH node `math.rn-region.witness-collision` are not moved by this +packet. + +## Provenance + +Consumed sources and their exact identities on `main` `7a1cb09`: `SOURCE_MAP.json`. This packet's inventory: +`SOURCE_FILES.json`. External check: `RECONNAISSANCE.md`. Same GitHub account as every lane; zero +organizational-independence credit; [C6], [DL] and this note are Claude-authored; [EDL], [LM], [RM], [RC], [RCL] and [LP] +are OpenAI-authored. diff --git a/frontiers/c6_cluster_law_20260929/RECONNAISSANCE.md b/frontiers/c6_cluster_law_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..ca622d39d --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/RECONNAISSANCE.md @@ -0,0 +1,62 @@ +# Limited external reconnaissance — 29 September 2026 + +Scope: whether a scaling limit for the number of critical points of a Gaussian field near a pinned degenerate +configuration (a "cluster law" for window critical points at the scale of the pins) is already in the literature. Not a +novelty, priority or exhaustiveness claim. + +## What was attempted + +The Consensus connector returned its monthly quota error on every call made from this account today ("You've used all +30 searches this month; resets on October 1st"), as recorded in `frontiers/c6_palm_route_20260929/RECONNAISSANCE.md` +and `frontiers/d5_dimension_lift_20260929/RECONNAISSANCE.md`. One further call was made on 29 September 2026 at about +22:35 UTC with the query "critical points Gaussian random field near degenerate critical point scaling limit cluster +count cubic normal form fold bifurcation persistence pairs"; it returned the same quota error. No result was invented, +and no web search was performed. + +## Records already cited by the consumed sources + +Read from the reconnaissance sections of the consumed sources, not re-inspected here; none is imported as a theorem: + +- D. Armentano, J.-M. Azaïs, J. R. León, *On a general Kac-Rice formula for the measure of a level set*, + arXiv:2304.07424v3 — the marked Kac–Rice framework used by [C6] Lemma 5.1 and by [RM]. +- L. Gass, M. Stecconi, *The number of critical points of a Gaussian field: finiteness of moments*, arXiv:2305.17586 — + finiteness of moments for a fixed nondegenerate field; not uniform in the pinned family; not consumed. +- L. H. Y. Chen, A. Xia, *Stein's method, Palm theory and Poisson process approximation*, Ann. Probab. 32 (2004) — + cited by [RCL]; not consumed. + +## Classical facts used with a citation in the text + +Bézout's theorem for two plane conics (at most four isolated common zeros); the Sylvester resultant of two quadratics; +the inverse function theorem and `C^1` stability of nondegenerate zeros; dominated convergence; Gaussian regression and +the Schur complement; the fold and cusp normal forms of a planar cubic (V. I. Arnold, *Catastrophe Theory*, 3rd ed., +Springer 1992), cited only for the classical algebra of the unfolding, not as a probabilistic input. + +## Assessment + +The consumed candidates [EDL] and [LM] contain the rare-set construction and the cubic normal form as a lower-bound +device; [RCL] states the cluster law as unknown. The contribution here is to run the same construction as a +dominated-convergence limit over the whole configuration set, together with the exclusion of large soft curvature and +the separation of the near and remote regions. No claim of first discovery is made for any component. + +## Addendum, 30 September 2026 (00:35Z): external check completed through web search + +The Consensus connector still returned the exhausted-monthly-quota error at 00:33Z–00:35Z on 30 September (four +attempts; one returned a transient rate-limit message instead), after the owner reported that the quota had been +increased. The check was therefore made through general web search (three extended queries) and direct reads of the +arXiv abstract pages listed below. No full texts were read; no citation below is used as a source of any step of +PROOF.md, and none is invented. + +| Record | What it establishes | Relation to this packet | +|---|---|---| +| D. Beliaev, V. Cammarota, I. Wigman, *Two point function for critical points of a random plane wave*, IMRN 2019(9), 2661–2689; arXiv:1704.04943 | Short-range asymptotics of the two-point function of critical points of the random plane wave: the second factorial moment of the count in a small disc of radius `rho` scales as `rho^4`; separate treatment of extrema and saddles. | Unconditioned pair statistics at short range. This packet's object is the law of the count *conditioned on a pinned birth–death pair* at separation `r` and gap `k r^3`, where the conditional second factorial moment is `Theta(r^3)` ([C6], [EDL]); different normalization and question. No cluster law there. | +| J.-M. Azaïs, C. Delmas, *Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices*, arXiv:1911.02300 | Correlation function of critical points of isotropic fields: attraction in dimension `> 2`, neutrality in dimension `2`, repulsion in dimension `1`; the attraction comes from critical points of adjacent indices; exact GOE eigenvalue densities. | Qualitatively consistent with §6.5 and Lemma 3.6 here (the extra near points are saddles of index `d - 1`, adjacent to the pin indices `d` and `d - 1`). Unconditioned; no pinned-pair conditioning; no count law. | +| L. Gass, M. Stecconi, *The number of critical points of a Gaussian field: finiteness of moments*, arXiv:2305.17586 | All moments of the critical-point count are finite as soon as the Taylor polynomial of the relevant order is nondegenerate at every point; general method, not specific to critical points. | Background for moment methods; not the pinned conditional regime, not a small-scale rare-cluster statement. | +| M. Ancona, L. Gass, T. Letendre, M. Stecconi, *Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems*, arXiv:2501.10226 | Cumulant asymptotics, strong law and CLT for nodal volumes and critical-point counts in the large-volume regime for stationary fields with decaying covariance. | Large-volume Gaussian fluctuation regime; no rare-event compound-Poisson or conditional cluster statement. | +| P. Pranav, *Topology and geometry of Gaussian random fields II: on critical points, excursion sets, and persistent homology*, arXiv:2109.08721 | Persistence diagrams of 3D Gaussian fields from simulations; intensity and difference maps; dependence on the power spectrum. | Numerical; no rigorous short-lifetime law for birth–death pairs. | + +Conclusion of the check: no located source states or proves the conditional law of the number of additional critical +points in the height window of a pinned birth–death pair (Theorem N), its support on `{1, 2}`, or the explicit +near coefficients; the closest published relatives are the short-range two-point analyses above, which are +unconditioned. The classical pieces used in §§3.4 and 3.8 (a planar cubic normal form, Bézout, Hadamard's inequality, +Weyl's integration formula) are not claimed as new. Novelty is asserted only relative to this search; a full-text +survey was not performed. diff --git a/frontiers/c6_cluster_law_20260929/RESULTS.json b/frontiers/c6_cluster_law_20260929/RESULTS.json new file mode 100644 index 000000000..381831f0b --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/RESULTS.json @@ -0,0 +1,32 @@ +{ + "groups": [ + "NF", + "RS", + "EX", + "BD", + "EU", + "SH", + "TW", + "LG" + ], + "ledger": { + "cross_term_exponent": "9/2", + "hadamard_row_factors": { + "3": 6, + "4": 8 + }, + "soft_slice_exponent": 3, + "ui_tail": "1/(M-1)" + }, + "mathematical_acceptance": false, + "object": "CL-C6-CLUSTER-LAW-20260929-v1.2", + "passed": true, + "q2_terms": 12, + "schema": 1, + "scientific_effect": "NONE", + "scope": "exact identities, exact configurations and exponent bookkeeping only", + "typed_window": { + "family_size": 201720, + "typed_in_window": 2590 + } +} diff --git a/frontiers/c6_cluster_law_20260929/SOURCE_FILES.json b/frontiers/c6_cluster_law_20260929/SOURCE_FILES.json new file mode 100644 index 000000000..f0b9ede14 --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/SOURCE_FILES.json @@ -0,0 +1,48 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "NONAUTHOR_ACCEPTED", + "stacked_on": [ + "LP, DL, C6, RM, RC, EDL, LM merged on main 7a1cb09 at the SOURCE_MAP identities", + "RCL (Math-#153) consumed for Corollary S only" + ], + "files": [ + { + "path": "PROOF.md", + "bytes": 70131, + "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", + "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5" + }, + { + "path": "README.md", + "bytes": 7560, + "sha256": "e4b35f2b0e919ba57ebfb121c403896a32782d58dd99d7ec6a3d6d9e5107c6ac", + "git_blob": "1166376f7b4d917b1bda3f9438adfd94d8831139" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 6457, + "sha256": "de84ddcc0d3df3079e0e7b0cae9804c2a8ea268d9404871b7dd7c18e03464912", + "git_blob": "ca622d39d995b4ce483fc93cae7a1b8852004ebf" + }, + { + "path": "RESULTS.json", + "bytes": 601, + "sha256": "26f0af06567ea8f33497bf64fbfff07f0e9154017c9a3e280c49100a9247a197", + "git_blob": "381831f0b98f1ee6e0f47555ffb4de7141391fba" + }, + { + "path": "SOURCE_MAP.json", + "bytes": 12566, + "sha256": "0584c3808b633fceb3c308186b07d862eb4916053daf4e7bf1110592ec41f107", + "git_blob": "ef51c582d0b1c874aabb01fce3b35c10631f7fc6" + }, + { + "path": "cluster_law_check.py", + "bytes": 20299, + "sha256": "06dbfaaee703f9649f7a5f24cc5f5dc6a3bac9d9e88fc420348c00f069263e5f", + "git_blob": "91f485f529e09bebcd92f5a8f41d87bf01efc6d6" + } + ], + "review_record": "OpenAI review 5360192822 at f7c3395 (PROOF blob ba492c8e); source-exposed, zero organizational independence; see SOURCE_MAP.json nonauthor_acceptance_record" +} diff --git a/frontiers/c6_cluster_law_20260929/SOURCE_MAP.json b/frontiers/c6_cluster_law_20260929/SOURCE_MAP.json new file mode 100644 index 000000000..ef51c582d --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/SOURCE_MAP.json @@ -0,0 +1,157 @@ +{ + "schema": 1, + "object": "CL-C6-CLUSTER-LAW-20260929-v1.2", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "source_commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "note": "All consumed sources are on main at the source commit; none is unmerged. Nothing consumed is HOLD. Theorem N is conditional on the consumed sources at these exact bytes and on this note's own steps; at d = 2 the regional inputs are reviewed planar rows (section 1.4).", + "inputs": [ + { + "id": "LP", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "use": "Model; section 2 distinct-site jet rank; section 3 contact frame U_r, target v_r -> v_0, covariance sandwich, (3.5); section 4 conditional moments; (5.1)-(5.5) endpoint identities, block determinant identity, Z_r/r^2 -> z_0; (13.3) target-growth conditional C^m moments.", + "author_and_review_state": "OpenAI / ChatGPT (parent); reviewed at main#106 (A1-A4) and reconciled in reviews/d1_chain_reconciliation_20260928" + }, + { + "id": "DL", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 53727, + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "use": "Theorem I_d (1.5)-(1.6) shell bound uniform in A_0 >= 4 and rho <= s_0; regime frames of sections 4-6 as re-expressed in [C6] section 6 (Lemma 5.1 of this note).", + "author_and_review_state": "Anthropic Claude (this session); merged via Math-#141 at e1ca400; xAI Slice A/B identities 5894274124/5894406926; OpenAI continuum reviews 5357858391/5357882570 (source-exposed); xAI planar record 5894512272" + }, + { + "id": "C6", + "path": "frontiers/c6_palm_route_20260929/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 53364, + "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "git_blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5", + "use": "Theorem Q (uniform integrability, section 6.3); Lemma 5.1 marked Kac-Rice with gradient-only kernel; (5.3) and the section 6 regime table with the squared determinant weight and the (1+beta)^(3d) absorption; Lemma 4.1 frame parameter beta and Lemma 4.4.", + "author_and_review_state": "Anthropic Claude (this session); merged via Math-#145 at 820d443; OpenAI reviews 5356233690 (section 4), 5358116559 (sections 5-7, Theorem Q); engineering 5358643154" + }, + { + "id": "RM", + "path": "frontiers/remote_window_20260924/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 18355, + "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", + "use": "Theorem A (2), the Kac-Rice representation (12), the contact kernel Lambda_j (13) with continuity and positivity, the remote nondegeneracy of sections 2-3.", + "author_and_review_state": "OpenAI / ChatGPT; reviewed ACCEPT at main#76 on the fixed-rho interfaces" + }, + { + "id": "RC", + "path": "frontiers/remote_collision_20260928/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 26002, + "sha256": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2", + "git_blob": "7b48a88e2af54e759e89a8c8219573bb450a65ea", + "use": "Corollary D (fixed-remote second factorial moment <= C r^5 |E|) and Corollary E (Q{N_j(E) >= 1} = k r^3 integral_E Lambda_j + O(r^4 |E|)) for E in D_rho, every fixed d.", + "author_and_review_state": "Anthropic Claude (object CL-D5-REMOTE-COLLISION-20260928-v1; same lane as this note, so no nonauthor credit accrues from it here); nonauthor records: OpenAI review 5342280263 on Math-#110 (near-diagonal mechanism, O(r^5) order, Corollaries E and F at stated scopes), xAI review 5342481786 on Math-#110 (finite identities, the (3.1) mark, Lemma 1) frozen in reviews/d5_remote_collision_grok_20260928; provenance corrected in v1.1 (the v1.0 entry wrongly said OpenAI / ChatGPT)" + }, + { + "id": "EDL", + "path": "frontiers/elder_dimension_lift_20260928/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 23971, + "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669", + "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006", + "use": "Section 2 (complete third-order jets, positive definite joint covariance with U_0), section 3 (unit-Jacobian shear chart (A8)-(A10)), (A11)-(A16) (rare set, conditional C^4 control), (A17) (normal form on the sheared soft plane), (A26)-(A28) (rescaled full gradient, stable block), section 9 (block-triangular stability). The lower bound (A4) is not consumed.", + "author_and_review_state": "OpenAI / ChatGPT (object OA-ELDER-DIMENSION-LIFT-20260928-v1); nonauthor review reviews/elder_dimension_lift_claude_20260928 (Anthropic): (A2)-(A4) ACCEPT" + }, + { + "id": "LM", + "path": "frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 13100, + "sha256": "e6272e8709835ca6290fecb7034fa84832dde08c3c4cc990a48b8019e89e7c8d", + "git_blob": "b5c18aac5529c5f975ee5bb056b304debafc6697", + "use": "(L4)-(L8) the planar normal form (L6), its C^2 remainder (L7), the exact cubic (L4) with two extra saddles (positivity witness), section 6 exponent arithmetic r . r^4 / r^2.", + "author_and_review_state": "OpenAI / ChatGPT; on main; consumed by [RCL] section 1 as the prior Bernoulli obstruction" + }, + { + "id": "RCL", + "path": "frontiers/c6_rare_cluster_laws_20260929/PROOF.md", + "branch": "main", + "head": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "bytes": 11811, + "sha256": "c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb", + "git_blob": "2ab625cedfc2e47575c4a7fa4853413418c532da", + "use": "Sections 2, 4, 6, 7 statements (Theorems C, S, identity (6.2), replica limit (7.1)) and the section 6 non-implication examples; consumed only in Corollary S as the statements whose hypotheses this note identifies.", + "author_and_review_state": "OpenAI / ChatGPT (object OA-C6-RARE-CLUSTER-20260929-v1); merged via Math-#153 at 190cf2b; Claude review 5358565704 (sections 2, 5-7), xAI review 5358623521 (sections 3-4)" + } + ], + "planar_sources_behind_DL": [ + { + "id": "PP", + "path": "reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md", + "bytes": 23771, + "sha256": "f972e50bc07674ebce9d971d1a1bd4dba2f8036dde37ecb57d8a44a35650f76a", + "git_blob": "d8acf0bcb9025b1562c03c077c8e0094ae282060", + "use": "cited where [DL] cites them; at d = 2 the regional bounds are these reviewed planar rows" + }, + { + "id": "CP", + "path": "reviews/d5_local_collar_20260928/COLLAR_PROOF.md", + "bytes": 20318, + "sha256": "794babe0fd039c3a93c401c0b8978316331af009fad45f953e63fbe7e1e31aab", + "git_blob": "b5647907a8a589635855b59cc4246b23efce2801", + "use": "cited where [DL] cites them; at d = 2 the regional bounds are these reviewed planar rows" + }, + { + "id": "IW", + "path": "frontiers/intermediate_window_20260928/PROOF.md", + "bytes": 27775, + "sha256": "b3eb9456d7058b5b78149cfca4e072679beda85cd1a0293d209664a542db66cf", + "git_blob": "f53a527ce0204fda271f24730b62b4223a5e43ec", + "use": "cited where [DL] cites them; at d = 2 the regional bounds are these reviewed planar rows" + } + ], + "new_obligations": [ + "Lemma 3.1 normal form in the spectral (rotated) chart with stable slaving and the bound on |W| for critical points in the physical scaled ball", + "Lemma 3.2 resultant factorization (X^2 - 1/4) Q_2 and the at-most-two-extra-critical-points statement off the null exceptional set E_k (exact identity in the finite control RS)", + "Lemma 3.3 exclusion of near critical points for large scaled soft curvature, in the full transverse space on {K_4 <= kappa}, threshold (3.8)", + "Lemma 3.4 pin Hessians, block-determinant expansion (3.9) and the Hadamard bound (3.10); Lemma 3.5 weight limit r^-4 W_r -> (det D)^2 w_0", + "Section 3.2 spectral chart and Weyl density (3.3); section 3.7 reduced weight (3.11)", + "Lemma 4.1 count convergence by C^1 stability under the converging conditional laws", + "Lemma 4.2 large-K_4 tail (4.3) by marked Kac-Rice; Lemma 4.3 domination on {K_4 <= kappa} without D^-1", + "Proposition 4.4 near limit for n in {1, 2}, vanishing for n >= 3, finiteness by Fatou against [DL], A -> oo by dominated convergence", + "Lemma 3.6 sheared cubic (3.12): every extra window critical point on the typed domain is a nondegenerate saddle (det = -3k(B + 4su) < 0) and -s <= 2|B| + (32 k D^2)^(1/3) (3.13); proved here, no #158 input", + "Lemma 5.1 remote conditional covariance floor (5.3a) by whitening the joint frame (U_r, Lambda) against the midpoint five-jet: (5.3b) distinct-site floor from [LP] section 2, (5.3c) frame remainders from [LP] section 3 and [DL] (4.2), (4.5), section 5.2, (5.2)-(5.3); Lemma 5.2 near-far cross term O(r^(9/2)) with the full normalizer once", + "Choice of kappa outside the countable atom set Kap_0 of the K_4-marginal of the limit measure, by Fubini (after Lemma 4.1)", + "Section 6 sandwich, positivity, explicit integrable majorant and continuity (from Lemma 3.6), Corollaries Lambda, S (with uniform integrability from [C6] Theorem Q), X (index d - 1 of the near points)" + ], + "consumed_reviews": [ + "main#76 (remote Theorem A, every d)", + "Math-#106 A1-A4 (LP sections 2-5)", + "Math-#141 reviews 5894274124, 5894406926, 5894512272, 5357858391, 5357882570 (DL)", + "Math-#145 reviews 5356233690, 5358116559 (C6)", + "reviews/elder_dimension_lift_claude_20260928 (EDL, Anthropic nonauthor)", + "Math-#153 reviews 5358565704, 5358623521 (RCL)" + ], + "nonauthor_acceptance": true, + "governing_status_changed": false, + "source_commit_note": "All eleven identities above re-derived from the git objects of main 7a1cb09 on 29 September 2026.", + "revision_note": "v1.1 (29 September 2026) answers OpenAI Codex review 5359552832 on Math-#159; consumed-source identities unchanged. Provenance of [RC] corrected on 29 September 2026 (Claude-authored, not OpenAI), after OpenAI comment 5901054091 on Math-#159. v1.2 (30 September 2026) answers OpenAI nonauthor review 5359895261 of v1.1 (normalizer in Lemmas 4.2/5.2, Lemma 5.1 whitening proof, kappa by Fubini, explicit section 6.2 majorant, section 8 saddle correction); consumed-source identities unchanged. Nonauthor acceptance recorded on 30 September 2026 (review 5360192822); proof bytes frozen.", + "nonauthor_acceptance_record": { + "review": "OpenAI / GPT-6 Astra Pro nonauthor successor analytic review 5360192822 on Math-#159, 30 September 2026 00:44Z, at head f7c33955d005c03d1010835f52f500c4b853a847, PROOF.md blob ba492c8e62e58bfc055fc8254c790e893d346ba5", + "verdict": "ACCEPT Theorem N, Corollaries Lambda and S, and the sequential near point-intensity Corollary X at the stated fixed-d, fixed-torus, compact-positive-gap scope, conditional only on the exact SOURCE_MAP inputs; closes AMEND 5359895261 and the residual-frame follow-up 5901687719; narrow closure 5360171253 of 5359999003 (normalizer displays, Corollary X wording)", + "exposure": "source-exposed (reviewer authored [LP], [LM], [RM], [RCL] and Math-#161); same GitHub account; zero organizational-independence credit; not a merge receipt or register transition", + "nonblocking_clarification": "the [DL] (4.2) jets J are written at the pin-centred origin; expressing them through the midpoint five-jet J_5 is a Taylor translation by +-r/2 whose O_(L^2)(r) term belongs to the remainder R^perp of Lemma 5.1 (ii): J = T_J J_5 + O(r), not literal equality. Recorded here; PROOF.md bytes are frozen at the accepted blob and not edited for this remark.", + "proof_bytes_frozen": "ba492c8e62e58bfc055fc8254c790e893d346ba5 (70131 bytes)" + } +} diff --git a/frontiers/c6_cluster_law_20260929/cluster_law_check.py b/frontiers/c6_cluster_law_20260929/cluster_law_check.py new file mode 100644 index 000000000..91f485f52 --- /dev/null +++ b/frontiers/c6_cluster_law_20260929/cluster_law_check.py @@ -0,0 +1,470 @@ +"""Exact finite controls for CL-C6-CLUSTER-LAW-20260929-v1.2. + +Standard library only; exact rationals and exact polynomial arithmetic over Q. +Verifies identities, exact configurations and exponent bookkeeping only. It does +not verify the Gaussian, regression, dominated-convergence or Kac-Rice steps. + + python -B -S cluster_law_check.py # rc 0, prints RESULTS.json + python -B -S cluster_law_check.py --mutant M # rc 1 for each named mutant +""" +import argparse +import json +import math +import random +from fractions import Fraction as F + +MUTANTS = ('pins-not-critical', 'wrong-pin-hessian', 'drop-sigma-jacobian', + 'three-extra-points', 'cross-term-not-small', 'window-closed', 'index-sign', + 'shear-drop-cubic', 's-bound-constant') + +# ---------------------------------------------------------------- polynomials +V = ('X', 'Z', 's', 'a', 'b', 'c', 'k') +NV = len(V) + + +def var(name, coef=F(1)): + e = [0] * NV + e[V.index(name)] = 1 + return {tuple(e): F(coef)} + + +def const(v): + return {tuple([0] * NV): F(v)} if v != 0 else {} + + +def add(p, q, sg=1): + r = dict(p) + for e, v in q.items(): + r[e] = r.get(e, F(0)) + sg * v + return {e: v for e, v in r.items() if v != 0} + + +def mul(p, q): + r = {} + for e1, u in p.items(): + for e2, v in q.items(): + e = tuple(x + y for x, y in zip(e1, e2)) + r[e] = r.get(e, F(0)) + u * v + return {e: v for e, v in r.items() if v != 0} + + +def smul(c, p): + return {e: F(c) * v for e, v in p.items()} if c != 0 else {} + + +def powr(p, m): + r = const(1) + for _ in range(m): + r = mul(r, p) + return r + + +def diff(p, name): + i = V.index(name) + r = {} + for e, v in p.items(): + if e[i] > 0: + e2 = list(e) + e2[i] -= 1 + r[tuple(e2)] = r.get(tuple(e2), F(0)) + v * e[i] + return {e: v for e, v in r.items() if v != 0} + + +def subs(p, name, q): + i = V.index(name) + r = {} + for e, v in p.items(): + e2 = list(e) + m = e2[i] + e2[i] = 0 + r = add(r, mul({tuple(e2): v}, powr(q, m))) + return r + + +def coeff_in(p, name, j): + i = V.index(name) + r = {} + for e, v in p.items(): + if e[i] == j: + e2 = list(e) + e2[i] = 0 + r[tuple(e2)] = r.get(tuple(e2), F(0)) + v + return {e: v for e, v in r.items() if v != 0} + + +def degree_in(p, name): + i = V.index(name) + return max((e[i] for e in p), default=-1) + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +X, Z, s, a, b, c, k = [var(t) for t in V] +HALF = F(1, 2) + + +def cubic(mutant=None): + """F_0(X,Z) = 2kX^3 - (3k/2)X - k/2 + (s/2)Z^2 + (a/2)(X^2-1/4)Z + (b/2)XZ^2 + (c/6)Z^3.""" + const_term = F(-1, 3) if mutant == 'pins-not-critical' else F(-1, 2) + terms = [smul(2, mul(k, powr(X, 3))), smul(F(-3, 2), mul(k, X)), smul(const_term, k), + smul(HALF, mul(s, powr(Z, 2))), + smul(HALF, mul(a, mul(add(powr(X, 2), const(F(-1, 4))), Z))), + smul(HALF, mul(b, mul(X, powr(Z, 2)))), smul(F(1, 6), mul(c, powr(Z, 3)))] + p = {} + for t in terms: + p = add(p, t) + return p + + +def evaluate(p, **vals): + for name, v in vals.items(): + p = subs(p, name, const(v)) + return p + + +def as_number(p): + require(all(all(x == 0 for x in e) for e in p), 'not a constant: ' + repr(p)) + return sum(p.values(), F(0)) + + +# ------------------------------------------------------------------ checks +def check_normal_form(mutant): + F0 = cubic(mutant) + gx, gz = diff(F0, 'X'), diff(F0, 'Z') + at = lambda p, x: subs(subs(p, 'X', const(x)), 'Z', const(0)) + require(at(F0, -HALF) == {}, 'F_0(M) must be 0') + require(at(F0, HALF) == smul(-1, k), 'F_0(S) must be -k') + for x in (-HALF, HALF): + require(at(gx, x) == {} and at(gz, x) == {}, 'pins must be critical') + Hxx, Hxz, Hzz = diff(gx, 'X'), diff(gx, 'Z'), diff(gz, 'Z') + sign = -1 if mutant == 'wrong-pin-hessian' else 1 + expect = {-HALF: (smul(-6, k), smul(-sign * HALF, a), add(s, smul(-HALF, b))), + HALF: (smul(6, k), smul(sign * HALF, a), add(s, smul(HALF, b)))} + for x, (exx, exz, ezz) in expect.items(): + require(at(Hxx, x) == exx and at(Hxz, x) == exz and at(Hzz, x) == ezz, 'pin Hessian block') + return F0, gx, gz + + +def check_resultant(gx, gz, mutant): + A = [coeff_in(gx, 'Z', j) for j in range(3)] + B = [coeff_in(gz, 'Z', j) for j in range(3)] + t1 = add(mul(A[2], B[0]), mul(A[0], B[2]), -1) + t2 = add(mul(A[2], B[1]), mul(A[1], B[2]), -1) + t3 = add(mul(A[1], B[0]), mul(A[0], B[1]), -1) + res = add(mul(t1, t1), mul(t2, t3), -1) # Sylvester determinant of two quadratics + d = add(powr(X, 2), const(F(-1, 4))) + q, rem = {}, dict(res) + while degree_in(rem, 'X') >= 2: + m = degree_in(rem, 'X') + lead = {e: v for e, v in rem.items() if e[0] == m} + sh = {} + for e, v in lead.items(): + e2 = list(e) + e2[0] -= 2 + sh[tuple(e2)] = v + q = add(q, sh) + rem = add(rem, mul(sh, d), -1) + require(rem == {}, 'resultant not divisible by X^2 - 1/4') + expected_degree = 3 if mutant == 'three-extra-points' else 2 + require(degree_in(q, 'X') == expected_degree, 'quotient degree') + c2 = add(add(add(add(smul(F(1, 4), mul(powr(a, 3), c)), smul(F(-3, 16), mul(powr(a, 2), powr(b, 2)))), + smul(F(-9, 2), mul(mul(a, b), mul(c, k)))), smul(3, mul(powr(b, 3), k))), + smul(9, mul(powr(c, 2), powr(k, 2)))) + c1 = add(add(smul(F(-1, 4), mul(powr(a, 2), b)), smul(-3, mul(mul(a, c), k))), smul(6, mul(powr(b, 2), k))) + inner = add(smul(F(1, 8), mul(a, b)), smul(F(-3, 2), mul(c, k))) + c0 = add(smul(3, mul(mul(powr(s, 2), b), k)), smul(-1, mul(inner, inner))) + Q2 = add(add(mul(c2, powr(X, 2)), mul(mul(s, c1), X)), c0) + require(q == Q2, 'Q_2 coefficients (3.6)') + return Q2 + + +def q2_coeffs(sv, av, bv, cv, kv): + c2 = av ** 3 * cv / 4 - 3 * av ** 2 * bv ** 2 / 16 - F(9, 2) * av * bv * cv * kv + 3 * bv ** 3 * kv + 9 * cv ** 2 * kv ** 2 + c1 = -av ** 2 * bv / 4 - 3 * av * cv * kv + 6 * bv ** 2 * kv + c0 = 3 * sv ** 2 * bv * kv - (av * bv / 8 - 3 * cv * kv / 2) ** 2 + return c2, sv * c1, c0 + + +def weights(sv, av, bv, kv, mutant): + wM = -6 * kv * (sv - bv / 2) - av * av / 4 + wS = av * av / 4 - 6 * kv * (sv + bv / 2) + if mutant == 'index-sign': + wS = -wS + return wM, wS + + +def check_examples(F0, gx, gz, mutant): + kv = F(1) + ev = lambda p, x, z, sv, av, bv, cv: as_number(evaluate(p, X=x, Z=z, s=sv, a=av, b=bv, c=cv, k=kv)) + lo, hi = -kv, F(0) + if mutant == 'window-closed': + in_window = lambda h: lo <= h <= hi + else: + in_window = lambda h: lo < h < hi + # rational configurations: jets, extra points, expected heights, expected N_oo, expected weights + table = [ + ((F(-55, 48), F(5, 2), F(-5, 3), F(-5, 2)), [(F(-2), F(3)), (F(-783, 1616), F(-129, 1616))], + [F(-27, 32), F(-4735, 83566592)], 2, (F(5, 16), F(215, 16))), + ((F(-1, 2), F(3), F(1, 3), F(0)), [(F(-3, 2), F(3)), (F(45, 22), F(-357, 11))], + [F(-1, 2), F(-9947, 121)], 1, (F(7, 4), F(17, 4))), + ((F(-33, 10), F(3), F(-12, 5), F(-3)), [(F(-2), F(5, 2)), (F(-282, 541), F(-1495, 1082))], + [F(-41, 16), F(-3199921, 4682896)], 1, (F(207, 20), F(117, 4))), + ((F(-396), F(-3), F(-204), F(-3)), [(F(-2), F(1, 2)), (F(-263314, 140087), F(-115623, 280174))], + [F(-119, 8), None], 0, (F(7047, 4), F(11961, 4))), + ] + counts = [] + for (sv, av, bv, cv), pts, heights, n_expected, (wM_e, wS_e) in table: + c2, c1s, c0 = q2_coeffs(sv, av, bv, cv, kv) + n_in = 0 + for (x, z), h in zip(pts, heights): + require(ev(gx, x, z, sv, av, bv, cv) == 0 and ev(gz, x, z, sv, av, bv, cv) == 0, 'not a critical point') + require(c2 * x * x + c1s * x + c0 == 0, 'X not a root of Q_2') + hv = ev(F0, x, z, sv, av, bv, cv) + if h is not None: + require(hv == h, 'height') + else: + require(hv < -kv, 'height below window') + n_in += 1 if in_window(hv) else 0 + # the two roots of Q_2 are exactly the two X coordinates (Vieta) + require(c2 != 0 and pts[0][0] + pts[1][0] == -c1s / c2 and pts[0][0] * pts[1][0] == c0 / c2, 'Vieta') + if mutant == 'window-closed': + n_in += sum(1 for h in (F(0), -kv) if in_window(h)) # pins would be counted with a closed window + require(n_in == n_expected, 'N_oo of a configuration') + wM, wS = weights(sv, av, bv, kv, mutant) + require((wM, wS) == (wM_e, wS_e) and wM > 0 and wS > 0, 'pin weights') + counts.append(n_expected) + # the [LM] cubic: s=-3k/2, a=0, b=-2k, c=0; extra points (-3/4, +-sqrt(15/8)) at height -7k/32 + sv, av, bv, cv = F(-3, 2), F(0), F(-2), F(0) + c2, c1s, c0 = q2_coeffs(sv, av, bv, cv, kv) + require(c1s * c1s == 4 * c2 * c0 and -c1s / (2 * c2) == F(-3, 4), '[LM] double root of Q_2 at -3/4') + zz = add(powr(Z, 2), const(F(-15, 8))) # Z^2 - 15/8 + for p, target in ((gx, F(0)), (gz, F(0)), (F0, F(-7, 32))): + red = evaluate(p, X=F(-3, 4), s=sv, a=av, b=bv, c=cv, k=kv) + # reduce modulo Z^2 - 15/8 + while degree_in(red, 'Z') >= 2: + m = degree_in(red, 'Z') + lead = {e: v for e, v in red.items() if e[1] == m} + sh = {} + for e, v in lead.items(): + e2 = list(e) + e2[1] -= 2 + sh[tuple(e2)] = v + red = add(red, mul(sh, zz), -1) + require(red == const(target) or (target == 0 and red == {}), '[LM] configuration on Z^2 = 15/8') + wM, wS = weights(sv, av, bv, kv, mutant) + require((wM, wS) == (F(3), F(15)) and wM * wS == 45, '[LM] weight 45 k^4') + require(in_window(F(-7, 32)), '[LM] heights in the window') + # large soft curvature: no real extra critical point + c2, c1s, c0 = q2_coeffs(F(1000), F(2), F(1), F(-1), kv) + require(c1s * c1s - 4 * c2 * c0 < 0, 'large s: Q_2 has no real root') + wM, wS = weights(F(1000), F(2), F(1), kv, mutant) + require(max(wM, 0) * max(wS, 0) == 0, 'large positive s: index conditions fail') + return sorted(set(counts)) == [0, 1, 2] + + +def det(M): + n = len(M) + if n == 1: + return M[0][0] + tot = F(0) + for j in range(n): + minor = [row[:j] + row[j + 1:] for row in M[1:]] + tot += (-1) ** j * M[0][j] * det(minor) + return tot + + +def lagrange_coeffs(points): + """Exact polynomial coefficients through the given (x, y) points.""" + n = len(points) + coeffs = [F(0)] * n + for i, (xi, yi) in enumerate(points): + basis = [F(1)] + denom = F(1) + for j, (xj, _) in enumerate(points): + if j == i: + continue + basis = [F(0)] + basis + for t in range(len(basis) - 1): + basis[t] -= xj * basis[t + 1] + denom *= (xi - xj) + for t in range(n): + coeffs[t] += yi * basis[t] / denom + return coeffs + + +def check_block_determinant(): + random.seed(20260929) + for dim in (3, 4): + n = dim - 2 + for _ in range(3): + kv, av, sv, bv = [F(random.randint(-9, 9), random.randint(1, 5)) for _ in range(4)] + cross = [[F(random.randint(-5, 5), random.randint(1, 3)) for _ in range(n)] for _ in range(2)] + D = [[F(0)] * n for _ in range(n)] + for i in range(n): + for j in range(i, n): + D[i][j] = D[j][i] = F(random.randint(-5, 5), random.randint(1, 3)) + if det(D) == 0: + D[0][0] += 1 + M2 = [[-6 * kv, -av / 2], [-av / 2, sv - bv / 2]] + pts = [] + for rr in [F(1, p) for p in (3, 5, 7, 11, 13)][:dim + 1]: + H = [[F(0)] * dim for _ in range(dim)] + for i in range(2): + for j in range(2): + H[i][j] = rr * M2[i][j] + for j in range(n): + H[i][2 + j] = H[2 + j][i] = rr * cross[i][j] + for i in range(n): + for j in range(n): + H[2 + i][2 + j] = D[i][j] + pts.append((rr, det(H))) + co = lagrange_coeffs(pts) + require(co[0] == 0 and co[1] == 0, 'block determinant: no r^0, r^1 terms') + require(co[2] == det(D) * det(M2), 'block determinant: r^2 coefficient is det D det M_2') + return True + + +def check_euler(): + xi, zeta, av, bv, cv, kv = X, Z, a, b, c, k # reuse symbols: xi=X, zeta=Z + G3 = add(add(add(smul(2, mul(kv, powr(xi, 3))), smul(HALF, mul(av, mul(powr(xi, 2), zeta)))), + smul(HALF, mul(bv, mul(xi, powr(zeta, 2))))), smul(F(1, 6), mul(cv, powr(zeta, 3)))) + G = add(G3, smul(HALF, powr(zeta, 2))) + euler = add(add(mul(xi, diff(G, 'X')), mul(zeta, diff(G, 'Z'))), add(smul(3, G3), powr(zeta, 2)), -1) + require(euler == {}, 'Euler identity') + ident = add(add(smul(6, G), smul(-1, powr(zeta, 2))), smul(2, add(mul(xi, diff(G, 'X')), mul(zeta, diff(G, 'Z')))), -1) + require(ident == {}, '6G - zeta^2 = 2(xi G_xi + zeta G_zeta), so G = zeta^2/6 at critical points') + require(subs(diff(G, 'X'), 'Z', const(0)) == smul(6, mul(kv, powr(xi, 2))), 'zeta = 0 forces xi = 0') + return True + + +def check_ledger(mutant): + sigma_jac = 0 if mutant == 'drop-sigma-jacobian' else 1 + require(sigma_jac + 4 - 2 == 3, 'exponent: r (soft slice) . r^4 (weight) / r^2 (normalizer) = r^3') + cross = F(3) + (F(3, 2) if mutant != 'cross-term-not-small' else F(0)) + require(cross > 3, 'near-far cross term is o(r^3)') + for A, rho in ((4, F(1, 2)), (100, F(1, 100))): + require(F(1, A * A) + rho * rho <= F(1, 16) + F(1, 4), 'shell error A^-2 + rho^2 decreases') + # Hadamard bound (3.10): per pin determinant, two soft rows, the lambda_2 row and d - 3 stable rows; + # W_r is a product of two such determinants, so the row factors number 2d in all. + hadamard = {} + for d in (3, 4): + require(2 * (2 + 1 + (d - 3)) == 2 * d, 'Hadamard: 2d row factors for two pin determinants') + require(4 + 2 + (2 * d - 6) == 2 * d, 'exponents 4 (soft), 2 (lambda_2), 2d - 6 (stable) of (3.10) sum to 2d') + hadamard[d] = 2 * d + # large-K_4 tail (4.3): kappa^(-p/2) -> 0; checked exactly for even p on kappa = 1, 10, 100 + for half_p in (1, 2, 10): + require(F(1, 100) ** half_p < F(1, 10) ** half_p < F(1) ** half_p, 'kappa^(-p/2) decreases in kappa') + # uniform integrability (§6.3, §6.5): n 1{n > M} <= n(n - 1)/(M - 1) for every integer n > M >= 2 + for M in (2, 5, 10): + for n in range(M + 1, M + 12): + require(F(n) <= F(n * (n - 1), M - 1), 'E[N 1{N > M}] <= E[(N)_2]/(M - 1)') + require(F(1, 100 - 1) < F(1, 10 - 1) < F(1, 2 - 1), 'uniform-integrability tail 1/(M - 1) -> 0') + return {'soft_slice_exponent': 3, 'cross_term_exponent': str(cross), 'hadamard_row_factors': hadamard, + 'ui_tail': '1/(M-1)'} + + +# ------------------------------------------------------- shear and typed window +def check_shear(F0, mutant): + """(3.12) with k cleared: (12k)^3 F_0(X, Z) at 12k X = 12k u - a Z equals + 1728 k^3 C(u) + 864 k^3 s Z^2 + 72 k^2 (12k b - a^2) u Z^2 + 4k (72 k^2 c - 18 k a b + a^3) Z^3. + The variable slot 'X' of the polynomial ring is used for u.""" + u, Z, s, a, b, c, k = (var(n) for n in V) + twelve_k = smul(12, k) + tkX = add(mul(twelve_k, u), smul(-1, mul(a, Z))) # 12k X + # (12k)^3 F_0 written in 12kX: 2k (12kX)^3 - (3k/2)(12k)^2 (12kX) - (k/2)(12k)^3 + # + (12k)^3 [ (s/2) Z^2 + (c/6) Z^3 ] + (a/2)(12k) [ (12kX)^2 - (12k)^2/4 ] Z + (b/2)(12k)^2 (12kX) Z^2 + k2, k3 = powr(twelve_k, 2), powr(twelve_k, 3) + lhs = add(smul(2, mul(k, powr(tkX, 3))), smul(F(-3, 2), mul(mul(k, k2), tkX))) + lhs = add(lhs, smul(F(-1, 2), mul(k, k3))) + lhs = add(lhs, mul(k3, add(smul(F(1, 2), mul(s, powr(Z, 2))), smul(F(1, 6), mul(c, powr(Z, 3)))))) + lhs = add(lhs, smul(F(1, 2), mul(mul(a, twelve_k), mul(add(powr(tkX, 2), smul(F(-1, 4), k2)), Z)))) + lhs = add(lhs, smul(F(1, 2), mul(mul(b, k2), mul(tkX, powr(Z, 2))))) + Cu = add(add(smul(2, mul(k, powr(u, 3))), smul(F(-3, 2), mul(k, u))), smul(F(-1, 2), k)) + twelve_kB = add(mul(twelve_k, b), smul(-1, powr(a, 2))) # 12k B + D144 = add(add(smul(72, mul(powr(k, 2), c)), smul(-18, mul(k, mul(a, b)))), powr(a, 3)) # 144 k^2 D + if mutant == 'shear-drop-cubic': + D144 = add(smul(72, mul(powr(k, 2), c)), smul(-18, mul(k, mul(a, b)))) + rhs = smul(1728, mul(powr(k, 3), Cu)) + rhs = add(rhs, smul(864, mul(powr(k, 3), mul(s, powr(Z, 2))))) + rhs = add(rhs, smul(72, mul(powr(k, 2), mul(twelve_kB, mul(u, powr(Z, 2)))))) + rhs = add(rhs, smul(4, mul(k, mul(D144, powr(Z, 3))))) + require(add(lhs, smul(-1, rhs)) == {}, 'shear identity (3.12) with k cleared') + # the shear is unimodular: u = X + a Z/(12k), Z = Z + require(F(1) * F(1) - F(0) * F(1) == 1, 'unimodular shear') + + +def check_typed_window(mutant): + """Exact rational family of critical points of G(u, Z) = C(u) + (s + B u) Z^2/2 + (D/3) Z^3: + for rational (u, Z != 0, D, k), B = 3k(1 - 4u^2)/Z^2 and s = -B u - D Z make (u, Z) critical. + Checks the determinant identity, the height identity, and Lemma 3.6 on the typed in-window samples.""" + bound = F(1) if mutant == 's-bound-constant' else F(32) + grid = sorted(set(F(n, d) for n in range(-9, 10) for d in (1, 2, 3))) + total = typed = 0 + for k in (F(1), F(1, 2), F(2)): + for u in grid: + for Z in grid: + if Z == 0: + continue + for D in grid: + B = 3 * k * (1 - 4 * u * u) / (Z * Z) + s = -B * u - D * Z + w = s + B * u + gu = 6 * k * u * u - F(3, 2) * k + B * Z * Z / 2 + gZ = Z * (w + D * Z) + require(gu == 0 and gZ == 0, 'family consists of critical points') + G = F(k, 2) * (u - 1) * (2 * u + 1) ** 2 + w * Z * Z / 2 + D * Z ** 3 / 3 + det = 12 * k * u * (w + 2 * D * Z) - (B * Z) ** 2 + require(det == -3 * k * (B + 4 * s * u), 'Hessian determinant -3k(B + 4su) at critical points') + if D != 0: + require(G == F(k, 2) * (u - 1) * (2 * u + 1) ** 2 + w ** 3 / (6 * D * D), 'height C(u) + w^3/(6D^2)') + total += 1 + if s < -abs(B) / 2 and -k < G < 0: + typed += 1 + require(det < 0, 'every typed in-window extra point is a saddle') + excess = max(F(0), -s - 2 * abs(B)) + require(excess ** 3 <= bound * k * D * D, '(-s - 2|B|)_+^3 <= 32 k D^2 (Lemma 3.6)') + require(typed >= 1000, 'enough typed in-window samples') + return {'family_size': total, 'typed_in_window': typed} + + +def run(mutant=None): + groups = [] + F0, gx, gz = check_normal_form(mutant) + groups.append('NF') + Q2 = check_resultant(gx, gz, mutant) + groups.append('RS') + require(check_examples(F0, gx, gz, mutant), 'configurations with 0, 1, 2 extra points') + groups.append('EX') + check_block_determinant() + groups.append('BD') + check_euler() + groups.append('EU') + check_shear(F0, mutant) + groups.append('SH') + tw = check_typed_window(mutant) + groups.append('TW') + ledger = check_ledger(mutant) + groups.append('LG') + return {'schema': 1, 'object': 'CL-C6-CLUSTER-LAW-20260929-v1.2', 'passed': True, 'groups': groups, + 'ledger': ledger, 'q2_terms': len(Q2), 'typed_window': tw, 'scientific_effect': 'NONE', 'mathematical_acceptance': False, + 'scope': 'exact identities, exact configurations and exponent bookkeeping only'} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--mutant', choices=MUTANTS) + args = parser.parse_args() + try: + result = run(args.mutant) + except ValueError as exc: + print(json.dumps({'passed': False, 'error': str(exc)}, sort_keys=True)) + return 1 + print(json.dumps(result, indent=2, sort_keys=True)) + return 0 + + +if __name__ == '__main__': + raise SystemExit(main()) diff --git a/frontiers/c6_marked_fourier_20260929/PROOF.md b/frontiers/c6_marked_fourier_20260929/PROOF.md new file mode 100644 index 000000000..efb702f60 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/PROOF.md @@ -0,0 +1,292 @@ +# Marked Fourier cap: exponential-power control of rare critical-point clusters + +Object: OA-C6-MARKED-FOURIER-20260929-v1.1 (publication successor). +Author: OpenAI / ChatGPT. Date: 29 September 2026. +Disposition: AUTHOR-SIDE CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect NONE. This note changes no source, review, graph, register or prize. + +## 1. Exact model and proposed strengthening + +Use precisely the original variance-one periodized Gaussian field, fixed dimension +`d>=2`, fixed torus side `T`, compact birth and positive-gap marks with `k>=k_->0`, +and original maximum/saddle value-and-gradient pins of sources F and P below. +The original endpoint-conditioned law is Q_r, the typed determinant weight is +W_r, and the full endpoint-only normalizer is Z_r. Write Q_r^W=(W_r/Z_r)Q_r. +N counts additional all-index critical points in (b-kr^3,b), excluding both pins. +No random observation set, changing dimension, vanishing gap floor, or additional +normalizer is introduced. + +Source F provides a Borel functional Psi_F of the FULL field, constructed from +Fourier truncation and successful Rouche radii, which bounds its total critical +count. Source P provides canonical Gaussian witness kernels, the witness-excluded +annulus geometry, the joint-frame floors, and the regional marked Kac--Rice +estimates. This note must extend F under those witness kernels; F's unconditional +tail by itself is not enough. + +**Proposed theorem.** There are theta>0, C0 such that, uniformly +on the stated fixed compact family, + + E_{Q_r^W}[ N exp(theta N^(2/d)) ] <= C r^3, 02 it is stretched +exponential with exponent 2/d, NOT an ordinary exponential moment. + +With the accepted additional-point lower event (P's source EDL), P(N>=1)>=c r^3. +Consequently both of the following are uniformly bounded: + + E[exp(theta N^(2/d)) | N>=1], + E_sizebiased[exp(theta N^(2/d))], + dP_sizebiased = N dQ_r^W / E_{Q_r^W}N. (MF2) + +These two probability laws are different. The theorem does not claim equality +or independence of their marks. Also + + P(N>=n) <= C r^3 n^(-1) exp(-theta n^(2/d)), n>=1. (MF3) + +This is a new proposed quantitative strengthening of the source's fixed-order +moment conclusions, not another acceptance of the source theorem. + +## 2. Source identities and dependencies + +F: `frontiers/c6_fourier_cutoff_20260929/PROOF.md`, on Math- main +`e1ca400b3414e5a7bec13b91bc4f1e5293df32be`, Git blob +`1d9177a259654df0fb7c558fcb803685e09608d2`, SHA256 +`c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733`. +Consumed: conditioned Fourier-coefficient method, fixed cover, local exponential +coordinate injectivity, isolated-zero Bezout and the successful-frequency cap. +The unconditioned Fourier tail is not assumed to hold under a witness kernel. + +P: `frontiers/c6_palm_route_20260929/PROOF.md`, merged by PR145 at +`820d4435c5f13c12b9c2f3590e859212d2648dc0`, Git blob +`89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5`, SHA256 +`aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b`. +Consumed: Sections 4.1--4.4 definitions and witness slab; Section 5 canonical +kernel / nonnegative Borel field-mark formula; Section 6 joint floors and +regional determinants/density bounds; EDL only for the lower probability used +in MF2. Its mathematical Sections 3--7 are the source-reviewed successor; +the PR was open during the earlier local draft but is now merged. The proof blob +is unchanged from the originally read `78ac1100217279e7b692f0959f86eee34cd1910e`. +The exact landed dependency and its source map are verified by this packet workflow. + +G: the dimension-matched first-moment source was integrated by PR141 at the +above main commit; P binds its proof blob `9d82c707fdb17d3072a8930f26dabedf59e456fc`. +The marked argument below uses P's REGIONAL proof, not merely G's small mean. + +Prior-result credit: the independently reviewed and merged #153 packet +`frontiers/c6_rare_cluster_laws_20260929/PROOF.md`, blob +`2ab625cedfc2e47575c4a7fa4853413418c532da`, already supplies the independent-replica +and polynomially weighted compactness consequences from fixed-order moments. +Section 7 repeats the elementary replica proof for self-containment, not priority. +The new contribution there is exponential-power weighted compactness; MF1 is NOT +inferred from #153's finite-moment hypotheses or its ordinary TV bounds. + +This note is exposed to the earlier OpenAI work and reviews. An actual nonauthor +review of the new conditional Fourier argument and its exponential-mark insertion +is still required. No new theorem is inferred just from the supplied programs. + +## 3. Fourier coefficients under the actual witness regression + +Fix a witness kernel Q'=Q_r(. | Lambda=tau), where Lambda is one of P's regional +frames with invertible covariance Sigma, and let d_tau=tau-E_Qr Lambda. Put + + beta = sqrt(sum_i Var_Qr Lambda_i) ||Sigma^(-1)|| |d_tau|. + +For a real component c_n of the Fourier coefficient, let a_n be its prior +variance scale. Source F and bounded original pin-regression energy give +|E_Qr c_n|<=C sqrt(a_n), and Var_Qr(c_n)<=C a_n. +The EXTRA regression correction obeys + + |Cov_Qr(c_n,Lambda) Sigma^(-1)d_tau| + <= sqrt(Var_Qr c_n) sqrt(sum_i Var_Qr Lambda_i) + ||Sigma^(-1)|| |d_tau| + <= C sqrt(a_n) beta. + +Hence |E_Q'c_n|<=C sqrt(a_n)(1+beta), while Var_Q'c_n<=C a_n by projection. +For every fixed p, ||c_n||_{L^p(Q')}<=C_p sqrt(a_n)(1+beta). Real/imaginary parts +are treated separately; they need not be conditionally independent. + +Let P_m be the gradient of the frequency-m truncation of the FULL realized +field, in its fixed torus lattice. On the same fixed complex tube as F, Gaussian +spectral decay and Minkowski therefore give constants a>0,C independent of the +particular regional kernel such that + + E_Q' R_m^2 <= C(1+beta^2) exp(-4a m^2), + E_Q' L^2 <= C(1+beta^2), (MF4) + +where R_m=sup|grad f-P_m| and L is the fixed-annulus supremum of its derivative. +The frequency truncation is not a new pinned field. Only the original field +is used by N,W,Z, by the kernel, and by the conditional covariance floors. +The conditional MEAN is included in MF4; omitting it would invalidate the result. + +## 4. Witness-excluded Fourier tail + +Retain exactly the Psi_F of F, independent of which witness is being integrated. +Its definition allows all test radii in each interval [eta_j,2eta_j]. At a fixed +witness X, remove the slab of radii within zeta_0 of its local periodic lift's +distance from the centre, as in P. The actual cover has zeta_0<=eta_j/8. Failure +of the ORIGINAL cap test at all radii implies failure on this retained subset. +Thus one does not define or insert a different witness-dependent mark. + +Let Phi=max(1,phi^(-1/2)) with phi the infimum of det Cov_Q'G on the protected, +witness-excluded annuli, G=(Re grad f, Im grad f/t). All following bounds are +vacuous if phi=0; P's regional lemmas supply positive floors where used. +P's small-gradient volume bound reads, for small h, + + E_Q' Vol{ |grad f|<=3h in retained annuli } + <= C Phi h^(2d)(1+log(1/h)). (MF5) + +Take h_m=exp(-a m^2), lambda_m=exp(a m^2/(4d)), and delta_m=h_m/lambda_m. +For all m above a FIXED geometric threshold, delta_m<=eta_j/8 and <=zeta_0/2. +On R_m<=h_m and L<=lambda_m, failure on every retained radius yields separated +open balls in the sublevel set. Their number is at least c eta_j/delta_m. +P's corrected packing gives volume at least c eta_j delta_m^(2d-1). +Mean-volume Markov and MF4 consequently bound the three failure contributions by + + C Phi (1+a m^2) exp[-a(2d+1)m^2/(4d)], + C(1+beta^2) exp(-2a m^2), + C(1+beta^2) exp[-a m^2/(2d)]. (MF6) + +Absorb the fixed polynomial by reducing the rate. With A=1+beta^2+Phi>=1, + + Q'(k_j>m) <= C A exp(-c m^2), + Q'(Psi_F>x) <= C A exp(-c x^(2/d)), x>=0, (MF7) + +after constants absorb finitely many small cutoffs. No monotonicity of success +at the individual frequency m is assumed; 'success by m' is the finite union +used by F. The source's zero-count cap is pathwise, so no new analytic zero-set +or generic-coefficient hypothesis has entered. + +Set alpha=2/d and Y=Psi_F^alpha. From MF7, for 0t)dt + <= 1 + C A u/(c-u) <= C_u A. (MF8) + +The prefactor A is retained. A uniform exponential moment without this prefactor +has NOT been proved for arbitrary witness regressions. + +## 5. The exponential mark uses the same regional absorption + +Choose theta>0 with 2theta=2)>=c r^3, hence EN>=P(N>=1)>=c r^3; G gives +EN<=C r^3. On N>=1, exp(theta N^alpha)<=N exp(theta N^alpha), proving both MF2 +bounds after normalizing by the proper denominator. MF3 follows by Markov with +the increasing function n exp(theta n^alpha). + +For integer p>=2, (N)_p<=N N^(p-1), and elementary calculus gives + + sup_{x>=0} x^(p-1) exp(-theta x^alpha) + = [(p-1)/(e theta alpha)]^((p-1)/alpha). + +Consequently the proposed theorem makes the growth of the earlier unspecified +factorial constants explicit: + + E(N)_p <= C r^3 [(p-1)/(e theta alpha)]^((p-1)/alpha). (MF10) + +It does not give a lower bound of order r^3 for p>=3. A count taking only values +0 and2 is still a counterexample to that unwarranted inference. + +Since MF3 makes the rescaled tail uniformly small, choose a FIXED M sufficiently +large that C M^-1 exp(-theta M^alpha)<=c/2. Then + + P(2<=N<=M)>=c r^3/2. + +Thus the known multiple-point lower event cannot owe all its order-r^3 mass to +clusters of diverging size. This is not a limit law for cluster shape or location. + +## 7. A subsequential independent-replica consequence, not a Poisson field claim + +Fix one parameter sequence r_n down to zero in the admissible family. Define +nu_n(k)=r_n^-3 P(N_{r_n}=k), k>=1. MF1 gives +sum_k k exp(theta k^alpha) nu_n(k)<=C. Thus there is a subsequence converging +coordinatewise to a finite measure nu on the positive integers. For theta'=2}nu(k)>0, using the original +lower event and tightness. Uniqueness of nu is NOT implied by these bounds. + +For independent copies of the field at each r_n, take floor(t/r_n^3) copies and +sum their window counts. Its probability generating function is exactly + + [1+r_n^3 sum_{k>=1}nu_n(k)(z^k-1)]^floor(t/r_n^3), 0<=z<=1, + +which tends along the selected subsequence to + + exp[t sum_{k>=1}nu(k)(z^k-1)]. + +The replica sums are tight because their means are uniformly bounded. Every +subsequential weak limit has the displayed probability generating function, +which uniquely determines a probability measure on the nonnegative integers. +This is a compound-Poisson law with finite activity sum nu(k) and jump-size law +nu/sum nu. In d=2 the generating functions have a common complex neighborhood +of the closed unit disk, because alpha=1; in d>2 no such analytic-disk conclusion +follows from the stretched-exponential estimate alone. + +This conclusion concerns INDEPENDENT REPLICAS on a diverging sample-count scale. +It is not a Poisson or compound-Poisson spatial limit within one field, nor a +unique r->0 count distribution. Alternating rare counts of size2 and3 satisfy +the moment/tail hypotheses but produce different subsequential jump measures. + +## 8. Review boundary and useful falsifiers + +No full-field simulation or finite algebra test proves MF1. Review must check: +extra-conditioned coefficient means; a cap fixed before witness selection; +retained-radius geometry at tiny imaginary parts and periodic lifts; the uniform +positive rate in MF7 versus its nonuniform prefactor A; 2theta2 stretched exponential. +The prefactor under arbitrary witness conditioning is retained, not suppressed. + +The independent-replica argument is already present in merged #153 and is +credited explicitly. New exponential-weight compactness is separate. No unique +cluster law, spatial Poisson theorem, numerical constants or higher-factorial +lower bounds are claimed. Prior handoff bytes stay in their original delivery; +this is a documented successor, not a recovery of missing historical code. + +Run from this directory: + + python -B -S verify.py --output /path/to/new-output + python -B -O -S verify.py --output /path/to/other-new-output + python -B -S -m unittest -v test_sources + python -B -O -S -m unittest -v test_sources + +Each full verifier runs 17 finite mathematical controls and eight intended +semantic variants in BOTH Python modes. The ten additional custody controls use +synthetic local sources; they are not Gaussian checks or full upstream replays. +With a complete historical repository checkout, additionally run: + + python -B -S source_check.py --repo /path/to/repository + +That checks six current and historical proof/reading-rule identities, including +the credited prior result. The hosted workflow executes it with fetch-depth:0. +Passing finite or formal-scalar checks cannot accept the new analytic argument. diff --git a/frontiers/c6_marked_fourier_20260929/RECONNAISSANCE.md b/frontiers/c6_marked_fourier_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..a0ed64c1a --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/RECONNAISSANCE.md @@ -0,0 +1,17 @@ +# Dependency-focused reconnaissance, 29 September 2026 + +Consensus.search was actually attempted for Gaussian critical-point exponential +moments and Palm/Fourier cluster control; it returned the monthly 30-search quota +error with October 1 reset. No search record was invented from that response. + +Primary abstract and publisher metadata inspected via web: +Gass and Stecconi, The number of critical points of a Gaussian field: finiteness +of moments, arXiv:2305.17586; PTRF 190 (2024), 1167-1197, +DOI 10.1007/s00440-024-01273-5. +https://arxiv.org/abs/2305.17586 + +Read scope was abstract/metadata only, not full text. The record establishes the +existing general finite-moment context; it is not imported as a theorem for the +uniform coalescing-pin exponential mark proved here. No claim of exhaustive +novelty search, first priority, or new Gaussian-conditioning principle is made. +The actual technical premises are the pinned project sources in SOURCE_MAP.json. diff --git a/frontiers/c6_marked_fourier_20260929/SOURCE_FILES.json b/frontiers/c6_marked_fourier_20260929/SOURCE_FILES.json new file mode 100644 index 000000000..66607f283 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/SOURCE_FILES.json @@ -0,0 +1,46 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "files": { + "PROOF.md": { + "bytes": 15304, + "sha256": "a5163679400d7d68c1c51538c9ed4ac111015487f79e820057f4ff3e20d09544" + }, + "PUBLICATION_NOTE.md": { + "bytes": 900, + "sha256": "ce8c1215a2310411d4f367b5ba0a1ef241fb4e008458beee2c3c9010a5ebc6d3" + }, + "README.md": { + "bytes": 1815, + "sha256": "8325998b807fbcabb791bcc35a4e7e518772742c586450154fb0eb8d49215c7f" + }, + "RECONNAISSANCE.md": { + "bytes": 959, + "sha256": "20849f4240b2994e5051d494e8266f466c9f6f57370decc16736cbb2a312ab9d" + }, + "SOURCE_MAP.json": { + "bytes": 2650, + "sha256": "371c1259d1f352101fae614f4c40601a8d2ac36036145b276d1d0be913ef4ca0" + }, + "marked.py": { + "bytes": 2950, + "sha256": "0711e16b4e28903d04214c9b87de2e9f5c8a37c26df9142f081e9fe944a66d3f" + }, + "source_check.py": { + "bytes": 2797, + "sha256": "0a2fc856ca362dbc5e60ab7c06a90214934af308e9bd9ef1c2423bb659ec35bb" + }, + "test_marked.py": { + "bytes": 3564, + "sha256": "1834c399d250617e064f2ce0b137fbe54cdfc1ec9922342602db6bbf3af059ad" + }, + "test_sources.py": { + "bytes": 2476, + "sha256": "0b42855a27ff27287d4258d396e90477330633eb3f37a6d9c27d48d7d06befa4" + }, + "verify.py": { + "bytes": 3607, + "sha256": "39a17d9e3d363b957c5874cb23b0c91ee76615da9483bf725552a8a0d2ae5444" + } + } +} diff --git a/frontiers/c6_marked_fourier_20260929/SOURCE_MAP.json b/frontiers/c6_marked_fourier_20260929/SOURCE_MAP.json new file mode 100644 index 000000000..473085282 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/SOURCE_MAP.json @@ -0,0 +1,62 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "sources": [ + { + "id": "F", + "path": "frontiers/c6_fourier_cutoff_20260929/PROOF.md", + "bytes": 18174, + "sha256": "c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733", + "git_blob": "1d9177a259654df0fb7c558fcb803685e09608d2", + "commit": "8ba851fb2addeccb7ff9befb6bdedac456b4cb9b", + "role": "Fourier coefficient method, fixed cover and pathwise cutoff cap; 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not premise for MF1" + } + ], + "local_upstream_verification": false, + "new_analytic_review_required": true +} diff --git a/frontiers/c6_marked_fourier_20260929/marked.py b/frontiers/c6_marked_fourier_20260929/marked.py new file mode 100644 index 000000000..e0d6bcf37 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/marked.py @@ -0,0 +1,72 @@ +"""Exact finite controls for the marked Fourier candidate, not a continuum prover.""" +from fractions import Fraction as F +from math import prod +MUTANT=None + +def alpha(d): + if type(d) is not int or d<2:raise ValueError('fixed dimension d>=2 required') + return F(1) if MUTANT=='exponential-every-d' else F(2,d) + +def admissible_rate(c,theta): + c,theta=F(c),F(theta) + return c>0 and theta>0 and (theta if MUTANT=='forget-holder-two' else 2*theta)var*obsvar:raise ValueError('invalid joint covariance') + mean=base_mean+cov*target/obsvar + return var-cov*cov/obsvar+(0 if MUTANT=='drop-mean' else mean*mean) + +def exponential_moment_bound(c,u,A): + c,u,A=map(F,(c,u,A)) + if not (0=1):raise ValueError('positive rate gap and A>=1 required') + return 1+(1 if MUTANT=='drop-prefactor' else A)*u/(c-u) + +def cutoff_rates(d,a): + alpha(d);a=F(a) + if a<=0:raise ValueError('positive rate required') + return 2*a,a*F(2*d+1,4*d),a/F(2*d) + +def check_law(p,n): + p=list(map(F,p));n=list(n) + if len(p)!=len(n) or not p or any(x<0 for x in p) or sum(p)!=1 or any(type(k) is not int or k<0 for k in n):raise ValueError('finite probability count law required') + return p,n + +def weighted_mark(p,n,w,b): + p,n=check_law(p,n);w=list(map(F,w));b=F(b) + if len(w)!=len(p) or min(w)<0 or b<=1:raise ValueError('nonnegative weights and b>1 required') + Z=sum(x*y for x,y in zip(p,w)) + if Z<=0:raise ValueError('positive normalizer required') + return sum(x*(y*y if MUTANT=='duplicate-weight' else y)*(1 if MUTANT=='drop-count-mark' else k)*b**k for x,y,k in zip(p,w,n))/Z + +def nonempty(p,n): + p,n=check_law(p,n);mass=sum(x for x,k in zip(p,n) if k>0) + if not mass:raise ValueError('nonempty event must have positive mass') + return [x/mass if k else F(0) for x,k in zip(p,n)] + +def sizebiased(p,n): + p,n=check_law(p,n) + if MUTANT=='nonempty-is-palm':return nonempty(p,n) + mean=sum(x*k for x,k in zip(p,n)) + if not mean:raise ValueError('positive count mean required') + return [x*k/mean for x,k in zip(p,n)] + +def tail_bound(M,epsilon,n,b): + M,epsilon,b=map(F,(M,epsilon,b)) + if M<0 or epsilon<0 or type(n) is not int or n<1 or b<=1:raise ValueError('invalid tail parameters') + return M*epsilon/((1 if MUTANT=='drop-tail-count' else n)*b**n) + +def factorial_power(d,p): + a=alpha(d) + if type(p) is not int or p<2:raise ValueError('integer moment p>=2 required') + return F(p-1)/a + +def levy_exponent(p,n,epsilon,z): + p,n=check_law(p,n);epsilon,z=F(epsilon),F(z) + if epsilon<=0 or not 0<=z<=1:raise ValueError('positive epsilon and real pgf domain required') + return sum(x*(z**k-1)/epsilon for x,k in zip(p,n) if k>0) + +def factorial_mean(p,n,q): + p,n=check_law(p,n) + if type(q) is not int or q<1:raise ValueError('positive integer order required') + return sum(x*prod(k-j for j in range(q)) for x,k in zip(p,n) if k>=q) diff --git a/frontiers/c6_marked_fourier_20260929/source_check.py b/frontiers/c6_marked_fourier_20260929/source_check.py new file mode 100644 index 000000000..055036f98 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/source_check.py @@ -0,0 +1,69 @@ +"""Byte-bound upstream evidence, including explicitly credited prior results.""" +from pathlib import Path, PurePosixPath +import argparse +import hashlib +import json +import subprocess + + +def decode(text): + def pairs(items): + result = {} + for key, value in items: + if key in result: + raise ValueError('duplicate JSON key: '+key) + result[key] = value + return result + return json.loads(text, object_pairs_hook=pairs) + + +def identity(data): + return {'bytes': len(data), 'sha256': hashlib.sha256(data).hexdigest(), + 'git_blob': hashlib.sha1(f'blob {len(data)}\0'.encode()+data).hexdigest()} + + +def verify_files(root, entries): + root = Path(root).resolve() + seen = set() + ids = set() + for item in entries: + rel = PurePosixPath(item['path']) + if (not rel.parts or rel.is_absolute() or '..' in rel.parts or + '\\' in item['path'] or str(rel) in seen or item['id'] in ids): + raise ValueError('unsafe or duplicate source path/id') + seen.add(str(rel)); ids.add(item['id']) + path = root.joinpath(*rel.parts) + if any(root.joinpath(*rel.parts[:i]).is_symlink() for i in range(1, len(rel.parts)+1)): + raise ValueError('symlink source component') + if not path.is_file(): + raise ValueError('missing regular source: '+str(rel)) + expected = {key:item[key] for key in ('bytes','sha256','git_blob')} + if identity(path.read_bytes()) != expected: + raise ValueError('source byte identity differs: '+item['id']) + return len(entries) + + +def verify_git(root, entries): + for item in entries: + commit = item['commit'] + if len(commit)!=40 or any(c not in '0123456789abcdef' for c in commit): + raise ValueError('full lowercase commit identity required') + target = commit+':'+item['path'] + run = subprocess.run(['git','cat-file','blob',target],cwd=root,capture_output=True,timeout=30) + if run.returncode or run.stderr: + raise ValueError('historical source unavailable: '+item['id']) + if identity(run.stdout) != {key:item[key] for key in ('bytes','sha256','git_blob')}: + raise ValueError('historical source identity differs: '+item['id']) + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--repo',type=Path,required=True) + args=parser.parse_args() + entries=decode((Path(__file__).resolve().parent/'SOURCE_MAP.json').read_text())['sources'] + count=verify_files(args.repo,entries) + verify_git(args.repo,entries) + print(json.dumps({'upstream_files':count,'current_and_historical_bytes':'PASS', + 'analytic_acceptance':False},sort_keys=True)) + +if __name__=='__main__':main() diff --git a/frontiers/c6_marked_fourier_20260929/test_marked.py b/frontiers/c6_marked_fourier_20260929/test_marked.py new file mode 100644 index 000000000..5beb2f10b --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/test_marked.py @@ -0,0 +1,69 @@ +"""Finite normalization and boundary controls; not Gaussian continuum verification.""" +import unittest +from fractions import Fraction as F +import marked as m + +class MarkedTests(unittest.TestCase): + def test_holder_needs_twice_the_rate(self): + self.assertFalse(m.admissible_rate(F(1),F(3,4))) + self.assertFalse(m.admissible_rate(F(1),F(1,2))) + self.assertTrue(m.admissible_rate(F(1),F(1,4))) + def test_dimension_exponent(self): + self.assertEqual(m.alpha(2),1);self.assertEqual(m.alpha(3),F(2,3));self.assertEqual(m.alpha(4),F(1,2)) + def test_conditional_mean_is_not_dropped(self): + self.assertEqual(m.regression_second(1,F(1,2),1,3,0),3) + self.assertEqual(m.regression_second(1,F(1,2),1,3,1),7) + def test_residual_projection_decreases_variance(self): + for c in (F(-1),F(-1,2),F(0),F(1,2),F(1)): + self.assertLessEqual(m.regression_second(1,c,1,0,0),1) + def test_tail_prefactor_is_retained(self): + self.assertEqual(m.exponential_moment_bound(2,F(1,2),7),F(10,3)) + self.assertGreater(m.exponential_moment_bound(2,F(1,2),7),m.exponential_moment_bound(2,F(1,2),1)) + def test_cutoff_three_positive_rates(self): + for d in range(2,11): + v=m.cutoff_rates(d,F(3,5)) + self.assertTrue(all(x>0 for x in v)) + self.assertEqual(v[1],F(3,5)*F(2*d+1,4*d)) + def test_one_original_weight_and_normalizer(self): + probs=[F(1,2),F(1,3),F(1,6)];counts=[0,1,2];weights=[1,2,3] + self.assertEqual(m.weighted_mark(probs,counts,weights,F(2)),F(16,5)) + def test_count_weight_is_required(self): + self.assertEqual(m.weighted_mark([F(3,4),F(1,4)],[0,3],[1,1],2),6) + def test_nonempty_and_sizebiased_are_distinct(self): + p=[F(7,10),F(2,10),F(1,10)];n=[0,1,3] + self.assertEqual(m.nonempty(p,n),[F(0),F(2,3),F(1,3)]) + self.assertEqual(m.sizebiased(p,n),[F(0),F(2,5),F(3,5)]) + self.assertNotEqual(m.nonempty(p,n),m.sizebiased(p,n)) + def test_cluster_tail_includes_n(self): + self.assertEqual(m.tail_bound(F(2),F(1,100),4,2),F(1,3200)) + def test_exact_finite_exponential_cluster_family(self): + eps=F(1,1024);n=list(range(17));p=[F(0)]+[eps/F(k*4**k) for k in range(1,17)] + p[0]=1-sum(p[1:]);M=m.weighted_mark(p,n,[1]*17,2) + self.assertEqual(M,eps*(1-F(1,2**16)));self.assertLess(M,eps) + for k in range(1,17):self.assertLessEqual(sum(p[k:]),m.tail_bound(1,eps,k,2)) + def test_factorial_constant_growth_exponent(self): + for d in range(2,8): + for p in range(2,8):self.assertEqual(m.factorial_power(d,p),F(d*(p-1),2)) + def test_pgf_first_order_is_exact(self): + eps=F(1,100);probs=[1-3*eps,eps,2*eps];counts=[0,2,3] + for z in (F(0),F(1,4),F(1,2),F(1)): + self.assertEqual(sum(p*z**n for p,n in zip(probs,counts)),1+eps*m.levy_exponent(probs,counts,eps,z)) + def test_replica_independence_is_an_extra_assumption(self): + eps=F(1,10);z=F(1,2) + single=1-eps+eps*z**2 + independent=single**2;identical_copies=1-eps+eps*z**4 + self.assertNotEqual(independent,identical_copies) + def test_no_unique_limit_from_uniform_moment_bounds(self): + eps=F(1,100) + self.assertNotEqual(m.levy_exponent([1-eps,eps],[0,2],eps,F(1,2)),m.levy_exponent([1-eps,eps],[0,3],eps,F(1,2))) + def test_higher_factorial_lower_bound_not_implied(self): + eps=F(1,100) + self.assertEqual(m.factorial_mean([1-eps,eps],[0,2],2),2*eps) + self.assertEqual(m.factorial_mean([1-eps,eps],[0,2],3),0) + def test_bad_inputs(self): + with self.assertRaises(ValueError):m.alpha(1) + with self.assertRaises(ValueError):m.exponential_moment_bound(1,1,2) + with self.assertRaises(ValueError):m.sizebiased([1],[0]) + with self.assertRaises(ValueError):m.regression_second(1,2,1,0,0) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/c6_marked_fourier_20260929/test_sources.py b/frontiers/c6_marked_fourier_20260929/test_sources.py new file mode 100644 index 000000000..eaf284b51 --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/test_sources.py @@ -0,0 +1,46 @@ +"""Synthetic byte-custody controls; not analytic or upstream theorem verification.""" +from pathlib import Path +import hashlib +import tempfile +import unittest +import source_check as m + +class SourceTests(unittest.TestCase): + def setUp(self): + self.tmp = tempfile.TemporaryDirectory() + self.addCleanup(self.tmp.cleanup) + self.root = Path(self.tmp.name) + (self.root/'sources').mkdir() + self.data = b'original source\n' + (self.root/'sources'/'proof.md').write_bytes(self.data) + self.entry = {'id':'F','path':'sources/proof.md','bytes':len(self.data), + 'sha256':hashlib.sha256(self.data).hexdigest(), + 'git_blob':hashlib.sha1(f'blob {len(self.data)}\0'.encode()+self.data).hexdigest()} + def test_original_passes(self): + self.assertEqual(m.verify_files(self.root,[self.entry]),1) + def test_changed_byte_fails(self): + (self.root/'sources'/'proof.md').write_bytes(b'changed source!\n') + with self.assertRaises(ValueError):m.verify_files(self.root,[self.entry]) + def test_missing_file_fails(self): + (self.root/'sources'/'proof.md').unlink() + with self.assertRaises(ValueError):m.verify_files(self.root,[self.entry]) + def test_wrong_length_fails(self): + with self.assertRaises(ValueError):m.verify_files(self.root,[dict(self.entry,bytes=1)]) + def test_wrong_sha_fails(self): + with self.assertRaises(ValueError):m.verify_files(self.root,[dict(self.entry,sha256='0'*64)]) + def test_wrong_blob_fails(self): + with self.assertRaises(ValueError):m.verify_files(self.root,[dict(self.entry,git_blob='0'*40)]) + def test_traversal_fails(self): + with self.assertRaises(ValueError):m.verify_files(self.root,[dict(self.entry,path='../proof.md')]) + def test_leaf_link_fails(self): + (self.root/'original').write_bytes(self.data) + p=self.root/'sources'/'proof.md';p.unlink();p.symlink_to('../original') + with self.assertRaises(ValueError):m.verify_files(self.root,[self.entry]) + def test_parent_link_fails(self): + (self.root/'sources').rename(self.root/'actual') + (self.root/'sources').symlink_to('actual',target_is_directory=True) + with self.assertRaises(ValueError):m.verify_files(self.root,[self.entry]) + def test_duplicate_fails(self): + with self.assertRaises(ValueError):m.verify_files(self.root,[self.entry,self.entry]) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/c6_marked_fourier_20260929/verify.py b/frontiers/c6_marked_fourier_20260929/verify.py new file mode 100644 index 000000000..5e30a8c9f --- /dev/null +++ b/frontiers/c6_marked_fourier_20260929/verify.py @@ -0,0 +1,52 @@ +"""Exact packet controls and named negative variants. No analytic acceptance.""" +from pathlib import Path +import sys,unittest,io,json,argparse,hashlib,subprocess +ROOT=Path(__file__).resolve().parent +MUTANTS={'forget-holder-two':'test_holder_needs_twice_the_rate','exponential-every-d':'test_dimension_exponent','drop-mean':'test_conditional_mean_is_not_dropped','drop-prefactor':'test_tail_prefactor_is_retained','duplicate-weight':'test_one_original_weight_and_normalizer','drop-count-mark':'test_count_weight_is_required','nonempty-is-palm':'test_nonempty_and_sizebiased_are_distinct','drop-tail-count':'test_cluster_tail_includes_n'} +def require(ok,msg): + if not ok:raise ValueError(msg) +def identity(p): + require(p.is_file() and not p.is_symlink(),'regular file required') + b=p.read_bytes();return {'bytes':len(b),'sha256':hashlib.sha256(b).hexdigest()} +def decode(s): + def pairs(xs): + d={} + for k,v in xs: + require(k not in d,'duplicate JSON key');d[k]=v + return d + return json.loads(s,object_pairs_hook=pairs) +def sources(): + identity(ROOT/'SOURCE_FILES.json');m=decode((ROOT/'SOURCE_FILES.json').read_text())['files'] + require(sorted(p.name for p in ROOT.iterdir())==sorted([*m,'SOURCE_FILES.json']),'packet membership differs') + for n,e in m.items(): + require(Path(n).name==n and identity(ROOT/n)==e,'source identity differs: '+n) + return m + +def child(mutant): + import marked,test_marked + marked.MUTANT=mutant + r=unittest.TextTestRunner(stream=io.StringIO()).run(unittest.defaultTestLoader.loadTestsFromModule(test_marked)) + print(json.dumps({'tests':r.testsRun,'passed':r.wasSuccessful(),'failures':sorted(t.id().split('.')[-1] for t,_ in r.failures),'errors':sorted(t.id().split('.')[-1] for t,_ in r.errors)},sort_keys=True)) + return 0 if r.wasSuccessful() else 1 + +def replay(out): + out=out.resolve();require(not out.exists() and not out.is_relative_to(ROOT),'new external output required');before=sources();out.mkdir(parents=True) + cases={} + for name in ['baseline',*MUTANTS]: + pair=[] + for mode,flags in [('normal',[]),('optimized',['-O'])]: + cmd=[sys.executable,'-B',*flags,'-S',str(ROOT/'verify.py'),'--child'] + if name!='baseline':cmd+=['--mutant',name] + p=subprocess.run(cmd,capture_output=True,timeout=30);r=decode(p.stdout.decode()) + (out/(name+'_'+mode+'.stdout')).write_bytes(p.stdout);(out/(name+'_'+mode+'.stderr')).write_bytes(p.stderr) + require(not p.stderr and not r['errors'] and r['tests']==17,'execution error or test-count mismatch') + if name=='baseline':require(p.returncode==0 and r['passed'],'baseline failed') + else:require(p.returncode==1 and not r['passed'] and MUTANTS[name] in r['failures'],'intended semantic failure absent') + pair.append(p.stdout) + require(pair[0]==pair[1],'mode mismatch');cases[name]=decode(pair[0].decode()) + require(before==sources(),'source mutation') + report={'tests_per_mode':17,'semantic_mutants_per_mode':8,'identical_output_pairs':9,'all_passed':True,'sources_unchanged':True,'analytic_acceptance':False,'verification_scope':'local packet; upstream check separate','cases':cases} + (out/'REPORT.json').write_text(json.dumps(report,indent=2,sort_keys=True)+'\n');print(json.dumps({k:v for k,v in report.items() if k!='cases'},sort_keys=True)) +if __name__=='__main__': + p=argparse.ArgumentParser();g=p.add_mutually_exclusive_group(required=True);g.add_argument('--child',action='store_true');g.add_argument('--output',type=Path);p.add_argument('--mutant',choices=MUTANTS);a=p.parse_args() + require(a.child or a.mutant is None,'mutant requires child');raise SystemExit(child(a.mutant) if a.child else replay(a.output)) diff --git a/frontiers/cluster_exponential_weight_20260930/INTERNAL_REVIEWS.md b/frontiers/cluster_exponential_weight_20260930/INTERNAL_REVIEWS.md new file mode 100644 index 000000000..a2f77a2d8 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/INTERNAL_REVIEWS.md @@ -0,0 +1,141 @@ +# Review chronology and current scoped disposition + +The original author-side snapshot below is preserved verbatim as published at +commit `0e70540b70e5e3183df5ed1a6bd70a212b734c9c` on 30 September 2026 UTC. +Its pending-review wording describes that original snapshot. The appended +current disposition records later actual native reviews of the same proof. + +## Original author-side snapshot (historical) + +# Internal challenges, exposure and unresolved review + +Target in both mathematical slices: PROOF.md, 12576 bytes, Git blob +`bf1268b9c18ec05561fd9b14c645ef0b8e530484`, SHA256 +`3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`. +The target was not amended after either mathematical review. + +Root authored the proof. The two OpenAI agents below made no proof edits and +were noncontributors to this implication; both had the root's draft and source +copies as exposure. Same-provider work supplies **zero organizational +independence**. Exact model version was not independently verified. These are +internal challenge results, not external organizational review or acceptance +of either upstream theorem. Public nonauthor review of this packet remains +required; its disposition is AUTHOR_SIDE / CONDITIONAL. + +## Slice A: endpoint tails and normalization (agent exp_weight_tail_review) + +**ACCEPT — §§1–3, conditional on MF1 and SC(3). No amendment required.** + +The declared input interfaces agree: original variance-one periodized field, +endpoint regression, typed determinant weight, full endpoint-only normalizer, +and all-index additional count in the open height window with both pins +excluded. Setting T=L resolves the side-length notation. A fixed admissible +positive gap lies inside a compact family with positive gap floor. No extra +conditioning or size bias enters E1–E2. + +E1 gives sum n w(n)nu_r(n)<=C; retaining n yields the endpoint tail +sum_(n>M)w(n)nu_r(n)<=C/(M+1). Fatou gives the same bound for the limit. +Finite-coordinate convergence then proves EW1 at theta itself. Since w>=1, +a_r->a>0. Centering at zero gives E4; positive conditioning correctly uses +the denominator a_r. For q and theta'=0, exp(-s a_r) sum_j s^j nu_r^{*j}/j! is nonnegative with mass one; +the comparisons therefore use genuine compound-Poisson laws. Independence +supplies the convolution power of the one-copy law. m=0, t=0 and T=0 are +handled. For alpha=1, absolute weighted convergence controls PGFs uniformly +on the closed disk of radius exp(theta), with analyticity in the interior +and continuity at the boundary. For alpha<1, the proposed +n^-3 exp(-2theta n^alpha) contamination obeys the premise but yields +prelimit radius exactly one. No larger disk follows. + +Excluded stronger claims have concrete falsifiers: all replicas equal to +the same N_r defeats the compound-Poisson limit; a coefficient perturbation +1/|log r| defeats an automatic O(r³) replica rate and O(r6) one-copy +limiting-mark error; alpha>1 defeats E11 using two unit atoms. + +Custody caveat at the first read: SOURCES.json was not yet present and the +historical Git objects were unavailable locally. The root later supplied +SOURCES.json and separately verified the source identities through the +public GitHub connector. This is not attributed to the first reviewer as +a historical-tree replay. + +## Custody challenge, repair and separate re-review (agent exp_weight_tail_review) + +Initial verifier SHA256 `45bddfbdd3e4c8b9cf5dd7311f46db870856f28c93c3d6a2bbbf9d4a3ecc78bb`: +**AMEND**. Eight controls passed, but a tree could masquerade as a commit, +and local Git replacement refs could make absent paths pass under an empty +commit's identity. A boolean JSON schema also passed as integer 1. + +Root reproduced all three defects with failing tests before repair. Every +Git read now uses --no-replace-objects; cat-file requires object type commit; +schema requires an actual integer. Repaired verifier SHA256 +`0bc355e6b5bc4459d5395f1730456bca8bb26aa6220f94e657f6dd57f93d7784`: +**PASS — bounded custody re-review**, 11 controls in each Python mode. +The real manifest correctly fails locally when historical objects are absent. +This repair concerns custody code only, not a proof amendment or parent verdict. + +--- + +## Current disposition after actual native reviews (30 September 2026 UTC) + +The unchanged PROOF.md now has **scoped nonauthor ACCEPT, CONDITIONAL** for +Slice A (§§1–3) and Slice B (§§4–5). This supersedes the historical pending-review +status above; it does not rewrite the author-side challenges or make them +independent reviews. The complete actual bodies are retained in +[REVIEW_RECORD.json](REVIEW_RECORD.json): + +- [Anthropic/Claude A+B, review5360109371](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360109371): + actual analytic acceptance of the elementary implications of E1–E2, at fixed + parameters with existential constants, independent copies and no rate. +- [OpenAI/Codex A, review5360144037](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360144037): + §§1–3, E1–E10, conditional on the exact MF1 and SC(3) interfaces; Slice B + outside that individual verdict. +- [OpenAI/Codex B, review5360159070](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360159070): + §§4–5, EW3/EW4, E11–E15 and dimension/independence/rate qualifications, + retaining E1–E2 and Slice A's endpoint consequence as dependencies. + +All three native objects identify original commit +`0e70540b70e5e3183df5ed1a6bd70a212b734c9c`, PROOF.md's 12,576 bytes, +Git blob `bf1268b9c18ec05561fd9b14c645ef0b8e530484`, and SHA256 +`3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`. +Their source exposure is preserved in their complete native bodies, not +replaced by this summary. Claude identifies provider distinction from the +author and reviewer exposure to SC; both Codex reviewers disclose their earlier +source reviews and challenges. All use the same GitHub account and carry +**zero organizational-independence credit**. + +The original MF1/SC(3) statements remain imported hypotheses; these reviews +do not prove their Gaussian, spectral or regional parents, supply rates or +moving-mark uniformity, identify a spatial event, or justify spatial +independence. Scientific effect remains NONE. The later engineering repair +preserves every mathematical source byte. Its implementer previously reviewed +Slice B; that does not self-accept the engineering repair. A separate engineering +review, current-base CI and authorized integration remain distinct gates. + +VALIDATION.json is retained unchanged as the original author execution/history +record. The new packet guard binds its proof hash and REVIEW_RECORD.json's +proof identity to actual PROOF.md and checks MANIFEST.json membership and hashes. +Passing that guard establishes local custody only, not GitHub authenticity or +mathematical acceptance. diff --git a/frontiers/cluster_exponential_weight_20260930/MANIFEST.json b/frontiers/cluster_exponential_weight_20260930/MANIFEST.json new file mode 100644 index 000000000..b854d28a8 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/MANIFEST.json @@ -0,0 +1,85 @@ +{ + "files": [ + { + "path": "INTERNAL_REVIEWS.md", + "bytes": 8160, + "sha256": "3df02b032e97e735a3a52973312d6459aa103e069fde38359e250741e11b0f5d", + "git_blob": "a2f77a2d8e5516e1a851e6db06266edac7489cc3" + }, + { + "path": "PROOF.md", + "bytes": 12576, + "sha256": "3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee", + "git_blob": "bf1268b9c18ec05561fd9b14c645ef0b8e530484" + }, + { + "path": "README.md", + "bytes": 4224, + "sha256": "c4fe795143c4b922bea2f215e73b97ca30cd697f3aca2eabbd58f26ebea8f894", + "git_blob": "31bfd56bce487a695a46b257e95ef57a9af75689" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 1440, + "sha256": "fb0ecd7559e5d451c567896fcc8e5a614ad4f6cb010fdc9699b138ac4c62d847", + "git_blob": "e0b3d9ea52aff5b1abc4b808f96be416289adcb1" + }, + { + "path": "REVIEW_RECORD.json", + "bytes": 42755, + "sha256": "0d9239543a0c23eb476843f6c122884f2b6258c2bf371adad598266b2fa5b56e", + "git_blob": "ee34321999980716e320b8c25eb0f12a7abd3e15" + }, + { + "path": "SOURCES.json", + "bytes": 1650, + "sha256": "478bd0ad93200b0993562d1a7ea3a6f96c7e15620781df469df1e5c127a1b4a1", + "git_blob": "4b3eeb938bca54aa28ce4746028a2f152a94dac6" + }, + { + "path": "VALIDATION.json", + "bytes": 1932, + "sha256": "b1e6b78b10c61820e3ee25db4602348690a2f20dd63348de37f5aa0dec0e1347", + "git_blob": "ab9e91fce0c8b8dba5c69a3538ed6dad826794b2" + }, + { + "path": "finite_checks.py", + "bytes": 7409, + "sha256": "143e3a64f7f07d2d2e0c29bb5dae5cc1ad59b06d50650ff72dc1daf4e1be8ad2", + "git_blob": "86d7f2482acf9326f47b9e39ebe3d56a92fce8f1" + }, + { + "path": "test_finite_checks.py", + "bytes": 11876, + "sha256": "134f7e3074afa8b5fa2e5fbc67fd7867b246434469e00d96575e51a3df477787", + "git_blob": "d350286f0b1291813d6c0dcbb9c3b849e1b0e2f3" + }, + { + "path": "test_verify_packet.py", + "bytes": 9862, + "sha256": "6409627e94f41358d880aacb25f90fc58b693c641c5c9251d54424c4188243e4", + "git_blob": "e597b179f3d116ae79e15c616f3b17b4f13630ea" + }, + { + "path": "test_verify_sources.py", + "bytes": 4863, + "sha256": "ac8a0ae72e3f1982ea722992ae3c9af2684454301a7fc7d407ea3e23e5b3186f", + "git_blob": "0cde3642d6fff16223e94003978d9a00e1de9383" + }, + { + "path": "verify_packet.py", + "bytes": 5441, + "sha256": "a505f52939e10eff494aab7ae81f7dce96f5ea8f78a94a34e49d689fa15aceab", + "git_blob": "70b540e53f9921885c62a2a09724d022fdadcadf" + }, + { + "path": "verify_sources.py", + "bytes": 3410, + "sha256": "0bc355e6b5bc4459d5395f1730456bca8bb26aa6220f94e657f6dd57f93d7784", + "git_blob": "a8ac3636c011337d9d4a12f4cd052d666572afb8" + } + ], + "manifest_self_excluded": true, + "schema": 1, + "scientific_effect": "NONE" +} diff --git a/frontiers/cluster_exponential_weight_20260930/PROOF.md b/frontiers/cluster_exponential_weight_20260930/PROOF.md new file mode 100644 index 000000000..bf1268b9c --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/PROOF.md @@ -0,0 +1,259 @@ +# Endpoint exponential-weight closure and independent-replica error + +Object: OA-CLUSTER-EXP-WEIGHT-20260930-v1. +Author: OpenAI / Codex. Date: 30 September 2026 UTC. +Disposition: AUTHOR_SIDE conditional proof; nonauthor review required. +Scientific effect NONE. No source theorem, review, register, graph, prize or +Boolean is changed. This is a new implication of two exact interfaces, not a +proof or retrospective acceptance of either interface. + +## 1. Exact consumption and the mathematical purpose + +Fix one dimension d>=2, torus side L>0, real birth b, gap k>0 and frame. Use the +same variance-one periodized field, endpoint observations, ORIGINAL typed +determinant weight W_r, full endpoint-only Z_r and law Q_r^W=(W_r/Z_r)Q_r as +the two sources. N_r is their ALL-INDEX additional critical-point count in +(b-k r^3,b), with both endpoint pins excluded. In the marked source T=L; a +fixed admissible mark can be placed inside its compact family. No further +normalizer, height conditioning, or Palm size bias enters this definition. + +The source interface [MF] is MF1 at Math-#157, exact commit +cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b. For alpha=2/d there exist +theta>0, C0 such that + + E_QW[N_r exp(theta N_r^alpha)] <= C r^3. (E1) + +The source interface [SC] is (3) at Math-#162, exact commit +7a44130b41b827577424a73c4feac4cde20b5568. At these fixed parameters, + + nu_r(n)=r^-3 Q_r^W(N_r=n), n>=1, + nu=nu1 delta_1+nu2 delta_2, 0 nu(n) for each n>=1. (E2) + +[SC] proves more than the coordinate convergence consumed here. Its separate +A/B/C review requirements and parent premises remain separate; an open slice +is not discharged by this implication. [MF] has its own Fourier-regression +and regional marked-Kac--Rice premises. SOURCES.json binds both full texts. + +[MF] Section7 previously gives only subsequential weighted compactness from +(E1). [SC] identifies the coefficients in polynomial weights without consuming +(E1). Here their intersection removes the subsequence in exponential weights, +including a boundary exponent that does not require slack. It also gives a +finite error bound for replica sums in that stronger norm. The generic +probability argument below is self-contained and does not use a new Gaussian +regression, Kac--Rice formula, spatial independence, or a coefficient rate. + +## 2. The endpoint norm theorem + +For a signed or complex measure sigma on the nonnegative integers define + + ||sigma||_w = sum_(n>=0) w(n)|sigma(n)|, + w(n)=exp(theta n^alpha), w(0)=1. (E3) + +Measures nu_r and nu have zero mass at0. Put epsilon_r=||nu_r-nu||_w. + +**Theorem EW1.** Under (E1)--(E2), epsilon_r ->0 as r->0. This is convergence +at theta ITSELF, not only at theta'0. (E4) + +The zero mass is centered; r^-3 Q(N_r=0) has no finite limit. + +**Proof.** Equation(E1) is exactly + + sum_(n>=1) n w(n)nu_r(n)<=C. (E5) + +For every integer M>=1, + + sum_(n>M) w(n)nu_r(n)<= C/(M+1). (E6) + +Fatou on finite sums in(E2) gives sum n w(n)nu(n)<=C. Therefore the same +tail estimate holds for nu. On1<=n<=M, coordinate convergence and finiteness +of the sum give convergence in the w-norm. The norm outside that set is at +most2C/(M+1). First let r->0 at fixed M, then M->infinity. This proves EW1. +The retained n factor in(E5), not an extra exponential moment, supplies the +endpoint uniform integrability. Since |a_r-a|<=epsilon_r and the exact law is +delta_0+r^3 H_r, (E4) follows. The full original tilt was used once in(E1) +and(E2); this step introduces no additional field normalizer. QED. + +**Theorem EW2.** For every fixed integer q>=0 and0<=theta'=1) n^q exp(theta' n^alpha)|nu_r(n)-nu(n)| ->0. (E7) + +In particular the positive-conditioned law converges in the ENDPOINT w-norm +to nu/a. The size-biased law converges in every STRICTLY smaller exponential +weight exp(theta'n^alpha) to +(nu1 delta_1+2nu2 delta_2)/(nu1+2nu2). + +**Proof.** Put gamma=theta-theta'>0. On n>M the weighted tail of nu_r is bounded +by + + C sup_(n>M) n^(q-1) exp(-gamma n^alpha), (E8) + +which tends to zero. The same holds for nu by Fatou. Finite-coordinate +convergence proves(E7). For positive conditioning use a_r->a>0 and + + ||nu_r/a_r-nu/a||_w + <= epsilon_r/a_r+|a_r-a| ||nu||_w/(a a_r). + +For size bias let z_r=sum n nu_r(n), z=nu1+2nu2. The q=1 case of(E7) gives +convergence of n nu_r in the smaller exponential norm and z_r->z>0. Divide +by z_r as above. These are distinct denominators and distinct probability +laws. QED. + +## 3. Why endpoint size bias is excluded: an exact escaping-mass example + +The endpoint exponent for EW1 does NOT justify n times that weight. For any +00 take m=m(r)>=3 tending to infinity and + + nu_r=delta_1+delta_2+eta_m delta_m, + eta_m=1/[m exp(theta m^alpha)]. (E9) + +For small r, define the genuine probability law +mu_r=(1-r^3(2+eta_m))delta_0+r^3 nu_r. Its nonempty intensity satisfies(E2) +with nu1=nu2=1 and satisfies(E5) with the constant +exp(theta)+2exp(theta 2^alpha)+1. Every fixed polynomial moment of nu_r +converges. Its endpoint norm error is exactly1/m, as EW1 requires. + +But z_r=3+exp(-theta m^alpha). The size-biased law has w-weighted mass + + w(m) m eta_m/z_r=1/[3+exp(-theta m^alpha)] ->1/3 (E10) + +at m, where the limiting size-biased law has zero mass. It therefore fails +endpoint weighted convergence. This is a counterexample to a tempting stronger +inference, not a counterexample to the Gaussian source law. Extra uniform +integrability at n exp(theta n^alpha) would be needed for that inference. + +## 4. Weighted convolution and an explicit replica bound + +**Lemma EW3 (convolution algebra).** For0=0, +(x+y)^alpha<=x^alpha+y^alpha, so w(x+y)<=w(x)w(y). Thus on measures with finite +w-norm, convolution satisfies + + ||sigma*tau||_w<=||sigma||_w ||tau||_w. (E11) + +This normed space is complete, has identity delta_0, and +exp_*(A)=sum_(j>=0) A^(*j)/j! converges in norm with +||exp_*(A)||_w<=exp(||A||_w). + +**Proof.** For alpha<1, the derivative in y of (x+y)^alpha-y^alpha is +nonpositive for y>0, and its value at y=0 is x^alpha. For alpha=1 equality +holds. Multiplying absolute convolution coefficients by w, summing and using +nonnegative Fubini gives(E11). Completeness is that of weighted l1, equivalently +ordinary l1 after multiplying coordinates by w. The exponential series is +dominated in norm by the scalar exponential series. QED. + +Let delta=r^3, m_r(t)=floor(t/delta), and take m_r(t) deliberately INDEPENDENT +copies of N_r. Write their sum as S_r(t). By(E5), ||H_r||_w<=2C and +||H||_w<=2C. Choose B=2C. Let + + P_t=Law(P1+2P2), P1~Pois(t nu1), P2~Pois(t nu2), + P1 and P2 independent. + +**Theorem EW4.** For every finite T>=0 and all0<=t<=T, + + ||Law(S_r(t))-P_t||_w + <= exp(TB)[delta(TB^2/2+B)+2T epsilon_r]. (E12) + +In particular the replica laws converge uniformly on bounded t intervals in +the endpoint exponential-weight norm. This includes weighted tail control, +not only an ordinary TV or PGF limit. It claims no rate for epsilon_r. + +**Proof.** One-copy law is u_r=delta_0+delta H_r. Independence gives the exact +law u_r^(*m). The convolution exponential + + exp_*(s H_r)=exp(-s a_r) sum_(j>=0) s^j nu_r^(*j)/j! + +is a compound-Poisson probability law. Thus exp_*(t H)=P_t. These are +convolution exponentials of signed generators, not signed approximating +probability laws. The series remainder obeys + + ||exp_*(delta H_r)-u_r||_w + <= delta^2 B^2 exp(delta B)/2. (E13) + +Indeed for every j>=2,1/j!<=1/[2(j-2)!], so the scalar remainder is bounded +by that right side. Also ||u_r||_w<=1+delta B<=exp(delta B). +Telescoping m convolution factors with(E11) gives, for m>=1, + + ||u_r^(*m)-exp_*(m delta H_r)||_w + <= m delta^2 B^2 exp(m delta B)/2 + <= t delta B^2 exp(tB)/2. (E14) + +For m=0 the first difference is zero. For any A,D in this commutative algebra, +norm differentiation of the exponential series and integration in s imply + + exp_*(A)-exp_*(D) + = integral_0^1 exp_*((1-s)D+sA)*(A-D) ds, + ||exp_*(A)-exp_*(D)||_w + <= ||A-D||_w exp(max(||A||_w,||D||_w)). (E15) + +The floor error0<=t-m delta=1)(nu_r(n)-nu(n))(z^n-1)|<=2epsilon_r. + +Each PGF is analytic in its interior and continuous on the boundary. By(E12) +the replica PGFs converge uniformly on that closed disk to +exp[t nu1(z-1)+t nu2(z^2-1)], uniformly0<=t<=T. This does not claim an analytic +neighborhood beyond that disk for each prelimit. + +For d>2, alpha<1. No common disk extending the unit disk follows from(E1) +and(E2), despite the finite-support limit. To see this, let q(n),n>=3, be a +probability proportional to n^-3 exp(-2theta n^alpha), and set +nu_r=delta_1+delta_2+epsilon(r)q with epsilon(r)>0 tending to zero. It satisfies +(E1)--(E2): sum n exp(theta n^alpha)q(n)1, +sum z^n q(n)=infinity, since n log z-2theta n^alpha-3log n ->infinity. +The associated small-r probability laws are constructed exactly as in(E9). +So prelimit PGFs may have radius1. The endpoint convolution norm result still +holds. This generic counterexample does not assert that Gaussian counts +actually have radius1; a larger radius needs another argument. + +## 6. Scope, source exposure, falsifiers and review division + +The elementary theorems EW1--EW4 apply to ANY nonnegative-integer laws satisfying +(E1)--(E2), with delta=r^3; the proofs also work for another positive delta->0. +Their application here is conditional on BOTH full mathematical interfaces +at the listed fixed parameters. This does not upgrade [SC]'s fixed-parameter +claim to mark-uniform convergence, or [MF]'s compact-family statement to +unbounded birth/gap/dimension/volume. Constants theta,C,B are existential; +no extra coefficient digits or numerical enclosure is claimed. + +The author is exposed to both OpenAI source packets and the prior Slice-C +review of [SC]. That source-exposed work is not independent acceptance of +[SC] or [MF]. Internal OpenAI challenge agents have zero organizational +independence. A contributor who changes this proof cannot self-accept it. +Finite checks, custody hashes and CI establish finite/custody facts only. + +Falsifiers of the conditional application: unequal original W/Z/count +interfaces; failure of(E1) at any sufficiently small r; failure of(E2)'s +coefficient limit; replacement of independent copies by coupled copies. +Falsifiers of the NEW implication: an arithmetic/algebra error in the tail +bound(E6), absolute convolution(E11), floor/telescoping ledger(E12)--(E15), +or normalized-law passage. The exact examples(E9)--(E10) preserve the failure +of endpoint SIZE-BIASED convergence; they must not be removed from an honest +stronger-weight statement. The dimension counterexample preserves the +non-implication of a complex neighborhood for alpha<1. + +Proposed nonauthor slices, each independently rejectable: +A: endpoint intensity tails, strict-weight moments, positive/size-biased +normalizations and the escaping-mass falsifier (§§2--3). +B: submultiplicative convolution, uniform replica error and actual-versus-limit +mark rate (§4), plus the complex-disk boundary (§5). +The parent Gaussian/spectral and Fourier arguments are OUTSIDE these slices. +No merge or scientific-status transition is justified solely by this packet. diff --git a/frontiers/cluster_exponential_weight_20260930/README.md b/frontiers/cluster_exponential_weight_20260930/README.md new file mode 100644 index 000000000..31bfd56bc --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/README.md @@ -0,0 +1,71 @@ +# Endpoint exponential-weight closure + +Read [the complete conditional proof](PROOF.md) and [the exact inputs](SOURCES.json). +The packet combines Math-#157 MF1 with Math-#162's fixed-parameter coefficient +limit. Neither parent is accepted or promoted by this combination. + +It proves convergence of the nonempty count intensity in the endpoint norm +sum exp(theta n^(2/d))|nu_r(n)-nu(n)|, removes the subsequence from #157's +weighted consequence, and gives an explicit uniform finite-time replica bound. +Positive conditioning retains that endpoint weight. Size bias requires a +strictly smaller exponent; an exact escaping-mass counterexample preserves +this boundary. The replica construction uses independent copies of the field. + +The earlier polynomial and ordinary-TV consequences remain credited to the +input packets. No spatial independence, new Gaussian theorem, convergence +rate for coefficients, uniformity in moving marks, or numerical enclosure +is introduced. The unchanged proof now has actual scoped nonauthor +**ACCEPT, CONDITIONAL** dispositions for Slice A (§§1–3) and Slice B (§§4–5). +The [source-bound review record](REVIEW_RECORD.json) preserves the actual native +bodies, original commit, exact proof identity and exposure. These reviews +retain MF1 and SC(3) as imported premises; they do not accept the parents or +establish engineering merge readiness. Scientific effect NONE. + +The three recorded reviews are [Claude A+B](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360109371), +[Codex A](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360144037), +and [Codex B](https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360159070), +all on original commit `0e70540b70e5e3183df5ed1a6bd70a212b734c9c` and +proof SHA256 `3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`. +They use the same GitHub account and carry zero organizational-independence +credit. The original author-side [validation history](VALIDATION.json) and +the dated original section of [internal challenges](INTERNAL_REVIEWS.md) +remain historical snapshots; their pending-review language is not the current +disposition. + +## Bounded plan and verification + +1. Bind the two existing interfaces at exact commit/path/blob/SHA256 identities. +2. Prove endpoint tail tightness and retain the failure of endpoint size bias. +3. Prove the weighted convolution and floor/Euler/intensity error ledger. +4. Run independent finite rational controls and deliberate wrong variants. +5. Preserve the completed source-bound reviews of §§1–3 and §§4–5, including + exact original source exposure, in REVIEW_RECORD.json. +6. Publish on an isolated additive branch; require real source-tree verification + and current-head hosted checks before any engineering integration. + +From the repository root, run the commands used by the packet workflow: + +```sh +packet=frontiers/cluster_exponential_weight_20260930 +python -B -S "$packet/verify_packet.py" +python -B -O -S "$packet/verify_packet.py" +python -B -S "$packet/finite_checks.py" +python -B -O -S "$packet/finite_checks.py" +python -B -S -m unittest discover -s "$packet" -p 'test_*.py' -v +python -B -O -S -m unittest discover -s "$packet" -p 'test_*.py' -v +python -B -S "$packet/verify_sources.py" --repo-root . +``` + +The two finite-control outputs must be byte-identical; they establish finite +arithmetic only. The packet guard enforces exact file membership and recorded +byte counts, SHA256 and Git blobs, excluding the manifest itself. It separately +binds VALIDATION.json and REVIEW_RECORD.json to the unchanged PROOF.md. +This checks custody, not remote review authenticity or mathematical acceptance. +VALIDATION.json is a historical record, not finite_checks.py output. Historical +source verification requires both pinned Git objects in a complete checkout; +connector-acquired copies do not substitute for that historical-tree check. + +Publication adds this packet and its isolated workflow only. Existing sources, +root indexes, live integration branches and scientific registers stay under +their own owners. Internal OpenAI challenge reviews carry zero organizational +independence and never self-accept the author's amendments. diff --git a/frontiers/cluster_exponential_weight_20260930/RECONNAISSANCE.md b/frontiers/cluster_exponential_weight_20260930/RECONNAISSANCE.md new file mode 100644 index 000000000..e0b3d9ea5 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/RECONNAISSANCE.md @@ -0,0 +1,27 @@ +# Reconnaissance memo — 30 September 2026 UTC + +Purpose: check the literature context for the elementary compound-Poisson +convolution step, without importing an unverified external theorem or making +novelty claims. + +Consensus was queried for compound-Poisson approximation with weighted total +variation and submultiplicative convolution weights. That lookup could not +complete. No Consensus result is claimed or consumed. + +Two primary author records were retrieved directly afterward: + +- Bero Roos, *Refined total variation bounds in the multivariate and compound + Poisson approximation*, arXiv:1509.04167v2, revised7 March2017: + https://arxiv.org/abs/1509.04167v2 . The abstract concerns approximation of + convolutions by compound-Poisson and signed measures. Only the abstract was + read here; no exact bound or weighted theorem from that paper is consumed. +- A.D. Barbour, Oliver Johnson, Ioannis Kontoyiannis and Mokshay Madiman, + *Compound Poisson Approximation via Information Functionals*, + arXiv:1004.3692v1, submitted21 April2010: + https://arxiv.org/abs/1004.3692v1 . The abstract gives independent-sum + compound-Poisson approximation context. Only the abstract was read; its + information-functional machinery is not used. + +The new implication is proved directly in PROOF.md using a weighted l1 +convolution algebra. This memo gives limited context, not a priority/novelty +audit and not an additional mathematical premise. diff --git a/frontiers/cluster_exponential_weight_20260930/REVIEW_RECORD.json b/frontiers/cluster_exponential_weight_20260930/REVIEW_RECORD.json new file mode 100644 index 000000000..ee3432199 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/REVIEW_RECORD.json @@ -0,0 +1,57 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "recorded_at_utc": "2026-09-30", + "original_review_head": "0e70540b70e5e3183df5ed1a6bd70a212b734c9c", + "proof": { + "path": "PROOF.md", + "bytes": 12576, + "sha256": "3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee", + "git_blob": "bf1268b9c18ec05561fd9b14c645ef0b8e530484" + }, + "source_map": { + "path": "SOURCES.json", + "bytes": 1650, + "sha256": "478bd0ad93200b0993562d1a7ea3a6f96c7e15620781df469df1e5c127a1b4a1", + "git_blob": "4b3eeb938bca54aa28ce4746028a2f152a94dac6" + }, + "current_disposition": "Scoped nonauthor ACCEPT, CONDITIONAL: Slice A and Slice B at the exact original proof bytes. Imported MF1 and SC(3), fixed parameters, independent copies and all stated limits remain hypotheses or restrictions. No parent acceptance, register transition or engineering merge readiness is inferred.", + "identity_policy": "verify_packet.py binds this declared proof identity to the actual proof and verifies this record through MANIFEST.json. It does not authenticate GitHub, recompute analytic review, or accept mathematical conclusions. The native records below were read back through authenticated GitHub access; their exact bodies and original source exposure are retained.", + "exposure_and_credit": [ + "Review5360109371 identifies Anthropic/Claude, provider-distinct from the mathematical author, same GitHub account, zero organizational-independence credit, reviewer-only prior exposure to SC.", + "Review5360144037 identifies OpenAI/Codex agent /root/c17_metadata_repair, same provider as the mathematical author; it discloses prior SC Slice A review and later complementary Claude exposure.", + "Review5360159070 identifies OpenAI/Codex agent /root/c17_pr158_slice_d_audit, same provider as the mathematical author; it discloses prior157 engineering,158D,162 spectral challenge and triage exposure, plus one same-provider internal challenge. It did not consume the complementary Claude analytic review before its own verdict.", + "All three native reviews use the same GitHub account. Provider distinction is not organizational independence. These records add no vote beyond the original scoped reviews.", + "The later C23 engineering repair was implemented by the prior Slice B reviewer, preserving the proof and mathematical sources. That contribution cannot self-accept the engineering repair; a separate nonauthor engineering review and current-base CI remain required." + ], + "historical_records": "VALIDATION.json and the original portion of INTERNAL_REVIEWS.md remain the original author-side snapshot, including pending-review language and historical run counts. This record supplies the later actual scoped dispositions without rewriting that history.", + "native_reviews": [ + { + "id": 5360109371, + "html_url": "https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360109371", + "commit_id": "0e70540b70e5e3183df5ed1a6bd70a212b734c9c", + "submitted_at": "2026-09-30T00:30:29Z", + "state": "COMMENTED", + "body": "## Anthropic/Claude NONAUTHOR analytic review — **Slices A (§§2–3) and B (§§4–5)** — exact head `0e70540b70e5e3183df5ed1a6bd70a212b734c9c`\n\n`frontiers/cluster_exponential_weight_20260930/PROOF.md`, blob `bf1268b9c18ec05561fd9b14c645ef0b8e530484`, SHA256 `3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`, 12576 B. Scientific effect **NONE**. Will not merge. Exposure and provenance as in pickup [5901664947](https://github.com/d6g8k5htny-coder/Math-/pull/164#issuecomment-5901664947): provider-distinct from the author, same GitHub account, zero organizational-independence credit; not the author of [MF] or [SC]; reviewer-only exposure to [SC].\n\n### Interface check (§1) before the slices\n- [MF] at `cbb2f978`, blob `efb702f6…`, SHA256 `a5163679…` (verified from the object store): MF1 is `E_{Q_r^W}[N exp(θ N^{2/d})] ≤ C r³` for the original variance-one periodized field, original value/gradient pins, typed `W_r`, endpoint-only `Z_r`, `Q_r^W = (W_r/Z_r)Q_r`, all-index count in `(b − kr³, b)` excluding both pins, uniformly on a fixed compact family (`T` is the torus side). Fixing one parameter inside the family is legitimate. ✔\n- [SC] at `7a44130`, blob `16c56821…`: (3) at fixed `d, L, b, k, frame` with the same `W_r`, `Z_r`, `N_r` (§1 of that proof) gives, at `q = 0`, the coordinate convergence `r^{−3}Q(N_r = n) → ν1 1{n=1} + ν2 1{n=2}` with `0 < ν1 = a1 + β_far, ν2 = a2 < ∞`. ✔ So (E1)–(E2) are the two sources' statements at one common parameter, with identical `W/Z/count` interfaces; the packet's \"unequal interfaces\" falsifier does not fire.\n\n### VERDICT: **ACCEPT Slice A** (EW1, EW2, the escaping-mass example §3) and **ACCEPT Slice B** (Lemma EW3, EW4 with (E12)–(E15), the actual-versus-limit mark remark, and the §5 dimension boundary) — as elementary implications of (E1)–(E2) for any nonnegative-integer laws with `δ = r³`, at fixed `d, L, b, k, frame`, existential `θ, C, B`, independent copies only, no rate. No defect found. Four non-blocking wording notes.\n\n### Slice A — what I checked\n- **(E5)/(E6).** `r^{−3}E[N_r w(N_r)] = Σ_{n≥1} n w(n) ν_r(n)` (the `n = 0` term vanishes), so (E1) is (E5); for `n > M`, `1 ≤ n/(M+1)` gives (E6). Passing to the limit in the finite sums `Σ_{n≤M} n w(n) ν_r(n) ≤ C` and letting `M → ∞` gives `Σ n w(n) ν(n) ≤ C`, hence the same tail for `ν`. ✔\n- **EW1.** `ε_r ≤ Σ_{n≤M} w(n)|ν_r(n) − ν(n)| + 2C/(M+1)`; `r → 0` then `M → ∞`. Convergence at `θ` itself, as claimed: the retained factor `n` is what supplies the uniform integrability. **(E4):** `Law(N_r) = (1 − r³a_r)δ_0 + r³ν_r`, so `r^{−3}(Law − δ_0) = H_r`; `‖H_r − H‖_w = |a_r − a| + Σ_{n≥1} w|ν_r − ν| ≤ 2ε_r` since `w ≥ 1` gives `|a_r − a| ≤ ε_r`. ✔ The zero mass is centred, not scaled. ✔\n- **EW2.** With `γ = θ − θ' > 0`, `Σ_{n>M} n^q e^{θ'n^α}ν_r(n) ≤ C sup_{n>M} n^{q−1}e^{−γn^α} → 0` for every fixed `q` (E8); same for `ν`; finite coordinates converge. Positive conditioning: `ν_r/a_r` is `Law(N_r | N_r ≥ 1)`, `a_r → a > 0`, and the displayed two-term bound is the triangle inequality with `‖ν‖_w ≤ C`. Size bias: `nν_r(n)/z_r` is `N_r dQ_r^W/EN_r` on `{n ≥ 1}`; `q = 1` in (E7) gives the strictly-smaller-weight convergence and `z_r → ν1 + 2ν2`; limit `(ν1δ_1 + 2ν2δ_2)/(ν1 + 2ν2)`. ✔ The two denominators are correctly kept distinct.\n- **§3 example.** `Σ n w(n)ν_r(n) = e^θ + 2e^{θ2^α} + 1` (the `m` term is exactly 1) ✔; `ν_r(n) → ν(n)` coordinatewise as `m(r) → ∞` ✔; polynomial moments `m^{q−1}e^{−θm^α} → 0` ✔; `‖ν_r − ν‖_w = w(m)η_m = 1/m` ✔; `z_r = 3 + e^{−θm^α}` ✔; size-biased `w`-mass at `m` equals `1/(3 + e^{−θm^α}) → 1/3` (E10) while the limit size-biased law has no mass there ✔; `μ_r` is a probability law once `r³(2 + η_m) ≤ 1` ✔. This is a correct counterexample to endpoint size-biased convergence under (E1)–(E2) alone.\n\n### Slice B — what I checked\n- **Lemma EW3.** `(x+y)^α ≤ x^α + y^α` on `x, y ≥ 0`, `0 < α ≤ 1` (the derivative argument is right: `α[(x+y)^{α−1} − y^{α−1}] ≤ 0`, value `x^α` at `y = 0`), hence `w(x+y) ≤ w(x)w(y)` with `w(0) = 1`; (E11) by nonnegative Fubini; `(ℓ¹_w, *)` is a unital commutative Banach algebra (`‖δ_0‖_w = 1`), so `exp_*` converges with `‖exp_*(A)‖_w ≤ e^{‖A‖_w}`. ✔\n- **`B = 2C`.** `‖H_r‖_w = a_r + Σ_{n≥1} wν_r ≤ 2Σ_{n≥1} wν_r ≤ 2Σ nwν_r ≤ 2C`; same for `H`. ✔\n- **Compound Poisson identification.** `exp_*(sH_r) = e^{−sa_r}Σ_j s^jν_r^{*j}/j!` (commutativity, `exp_*(−sa_rδ_0) = e^{−sa_r}δ_0`) is the compound-Poisson law with intensity `sν_r`; `exp_*(tH) = Law(P1 + 2P2)` ✔.\n- **(E13).** `‖Σ_{j≥2}δ^jH_r^{*j}/j!‖ ≤ Σ_{j≥2}δ^jB^j/j! ≤ (δ²B²/2)e^{δB}` by `j! ≥ 2(j−2)!` ✔; `‖u_r‖_w ≤ 1 + δB ≤ e^{δB}` ✔.\n- **(E14).** Telescoping `u^{*m} − E^{*m} = Σ_{i 2`: `q(n) ∝ n^{−3}e^{−2θn^α}` has `Σ n e^{θn^α}q(n) < ∞` yet `Σ z^n q(n) = ∞` for every `z > 1`, so the prelimit PGF radius is 1; the laws are built as in (E9). Correct, and correctly limited to \"no common disk follows\", not a claim about the Gaussian counts. ✔\n\n### Custody (finite/custody facts only, never proof)\nWorktree at `0e70540`: `test_finite_checks` + `test_verify_sources` → **23/23** in `-B -S` and `-B -O -S`; `finite_checks.py` output byte-identical across the two modes (`12` checks, `8` positive, `4` intended negatives all rejected); `verify_sources.py --repo-root` reproduces both pinned identities (MF `efb702f6…`/`a5163679…`, SC `16c56821…`/`e971b2cb…`); tree clean. Hosted checks on `0e70540` read SUCCESS (`exact-exponential`, downstream, formal, gates). `VALIDATION.json` is a record file, not the checker's output, so no byte comparison to it is claimed.\n\n### Non-blocking wording notes\n1. §2, EW1 proof: \"Fatou on finite sums\" — the step is a limit of finite sums followed by `M → ∞` (monotone convergence of the partial sums); naming it so avoids a reader looking for a lim inf of a series.\n2. (E6) and (E8) use `ν_r ≥ 0` and `w ≥ 1`; one clause \"(`w ≥ 1`, so `|a_r − a| ≤ ε_r`)\" before (E4) would make the factor 2 self-evident.\n3. §3: say once that `μ_r` is a probability law for `r³(2 + η_m) ≤ 1`, i.e. for small `r` (the text says \"for small r\" in the definition but not why).\n4. §5, `d = 2`: the closed disk `|z| ≤ e^θ` is where the *intensity* PGFs converge by (E4); the replica PGF statement uses (E12) — already so, but one cross-reference to (E12) in the sentence would help.\n\n### Not established by this review\nNothing beyond Slices A and B at the scope above: no revalidation of [MF] or [SC] or of their parents; no mark-uniformity, no rate for `ε_r`, no within-field Poisson process, no spatial independence, no numerical constant; no register transition, no merge, and no second independent vote (same account; provider-distinct from the author and from the Codex pickup, but Claude sibling sessions share my provider).\n\n---\n_Generated by [Claude Code](https://claude.ai/code)_", + "github_login": "d6g8k5htny-coder" + }, + { + "id": 5360144037, + "html_url": "https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360144037", + "commit_id": "0e70540b70e5e3183df5ed1a6bd70a212b734c9c", + "submitted_at": "2026-09-30T00:36:55Z", + "state": "COMMENTED", + "body": "# C22 — nonauthor analytic review of PR164 Slice A\n\n**Verdict: ACCEPT Slice A, §§1–3, equations (E1)–(E10), as an elementary implication conditional on the exact MF1 and SC(3) interfaces. No blocking defect found.** The two imported statements are checked for matching objects and scope; their Gaussian, Fourier, spectral and regional proofs are not reaccepted by this review. Slice B, §§4–5, is outside this verdict.\n\nThe reviewed object is `OA-CLUSTER-EXP-WEIGHT-20260930-v1`, Math PR164, exact head `0e70540b70e5e3183df5ed1a6bd70a212b734c9c`, path `frontiers/cluster_exponential_weight_20260930/PROOF.md`: **12,576 bytes**, SHA256 `3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`, Git blob `bf1268b9c18ec05561fd9b14c645ef0b8e530484`. I read the complete actual proof, then reconstructed §§1–3 and the consumed source interfaces. Later sections were read for boundaries, not reviewed as Slice B.\n\nScientific effect **NONE**. This review does not authorize a register transition, a source amendment, a whole-PR acceptance or a merge. Organizational-independence credit is **zero**.\n\n## Authorship, exposure and coordination\n\nI am the OpenAI/Codex review agent `/root/c17_metadata_repair`, operating in the GPT-6 agent environment. I did not author or modify PR164's proof, its implementation, or either consumed mathematical source. My earlier work includes an unpublished C17 metadata repair that was stopped after a competing author's successor, triage exposure, the actual PR162 Slice A nonauthor review, and the C21 delivery builder. Reviewing SC Slice A is prior exposure; its verdict is not transferred into acceptance of SC(3) or this new implication. This is fresh review of PR164's new conditional probability argument, with disclosed source exposure and the same provider as its author.\n\nThe first independent current timeline read showed only the author's request5901532627 and our actual pickup5901632715. Claim `cb43a8b2-57e1-4e51-8cab-21c9efa7fe6a` was read back in Work Events296, with expiry2026-09-30T02:26:20Z. I verified the exact row in root's saved actual readback before creating review evidence. Source and evidence writes are confined to this review directory; no source file or remote object was changed.\n\nAfter my analytic reconstruction and exact control runs were complete, the final coordination read exposed complementary Claude pickup5901664947 and actual review5360109371 at the same original head. That review's body was returned in the live timeline and therefore constitutes disclosed later exposure. It is not the basis for my reconstruction, tests or verdict, supplies no additional organizational independence, and is not counted as my work. Its remarks do not change the calculations recorded below. The timeline also identifies the separate C23 engineering pickup5901702618; that is not a competing analytic claim and is not integration evidence for this report. The shared GitHub account does not identify an independent organization. I did not use author-side internal challenges as review substitutes.\n\n## Exact parent interfaces and the common probability law\n\nThe target `SOURCES.json` contains two immutable proof inputs. Both full texts were read. Existing source bytes were reused only after byte count, SHA256 and computed Git-blob verification; no whole-Drive or repository recrawl occurred.\n\n| Input | Exact source | Identity | Consumption here |\n| --- | --- | --- | --- |\n| MF | `cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b`, `frontiers/c6_marked_fourier_20260929/PROOF.md` | 15,304 bytes; SHA256 `a5163679400d7d68c1c51538c9ed4ac111015487f79e820057f4ff3e20d09544`; blob `efb702f601eb0c0a24dad6d125061158158867b9` | MF1 at one fixed admissible parameter; its Fourier-regression and regional marked-Kac–Rice premises remain imported. Author: OpenAI / ChatGPT. |\n| SC | `7a44130b41b827577424a73c4feac4cde20b5568`, `frontiers/spectral_cluster_closure_20260929/PROOF.md` | 25,006 bytes; SHA256 `e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb`; blob `16c56821b52fd76b0be791622b9c3809eafde75a` | Coordinate consequence of (3), together with finite positive ν1,ν2, at fixed d,L,b,k,frame. Its other slices and imported premises remain separate. Author: OpenAI / GPT-6 Astra Pro. |\n\nMF §1 fixes the original variance-one periodized field, original maximum/saddle value-and-gradient endpoint pins, endpoint-conditioned Q_r, typed determinant product W_r, full endpoint-only Z_r, and Q_r^W=(W_r/Z_r)Q_r. Its N counts additional **all-index** critical points in the open height window `(b-k r³,b)`, excluding both pins. SC §1 explicitly defines the same observations at M=(-r/2,0), S=(r/2,0), the same typed product `F_d(H_M)F_(d-1)(H_S)`, same tilt, and same count. MF's torus-side notation T agrees with SC's L. There is no restriction to a single extra-point index, elder-selected points, or branch-adjacent points in the consumed count.\n\nFor any one admissible fixed d≥2, L>0, real b, k>0 and frame, the fixed mark can be placed inside the declared compact family with a positive gap floor. Restricting MF's compact-family assertion to that point supplies θ>0, C<∞ and r_* without making them explicit, uniform in unbounded b/k, or uniform in dimension/volume. SC(3) is explicitly fixed-parameter. Its q=0 consequence implies every coordinate limit in E2, with ν1=a1+β_far and ν2=a2 strictly positive and finite. None of these premises gives a convergence rate.\n\nConsequently the common law can consistently satisfy E1 and E2 without a change of normalization. All later divisions by a_r or z_r define new count laws only where the proof explicitly asks for them. They are not inserted into MF1, SC(3), W_r or Z_r. As a check on the distinction, the probability law with masses μ(1)=μ(2)=δ and μ(0)=1-2δ has full-law `E[N2^N]=10δ`, while conditioning on N≥1 gives expectation5. The latter cannot replace an O(δ) input bound. The retained original full tilt is essential.\n\n## EW1: retained-index endpoint tail and centered zero\n\nSet α=2/d, so 0<α≤1, and w(n)=exp(θn^α), w(0)=1. The measures ν_r and ν are nonnegative and have zero mass at0. Multiplying E1 by r^-3 gives the exact identity\n\n`sum_{n≥1} n w(n)ν_r(n) = r^-3 E_QW[N_r w(N_r)] ≤ C`.\n\nThere is no zero-atom contribution because of the factor n. For integer n>M, n≥M+1, hence\n\n`sum_{n>M} w(n)ν_r(n) ≤ (M+1)^-1 sum_{n>M} n w(n)ν_r(n) ≤ C/(M+1)`.\n\nThe denominator M+1 is sharp under this sole moment bound: a measure concentrated at M+1 with mass `C/[(M+1)w(M+1)]` attains equality. The stronger denominator M+2 is false. This also identifies precisely what would be lost if the retained n were dropped. A bound only on `sum wν_r` allows a mass `1/w(m)` to move to m→∞ while keeping endpoint weighted mass1; coordinate convergence alone would not eliminate it.\n\nFor each finite K, coordinate convergence gives `sum_{n≤K} n w(n)ν(n)≤C`. Increasing K proves the same full bound for ν by monotone convergence of nonnegative partial sums. This is the elementary content of the text's “Fatou on finite sums.” No dominated convergence of the whole infinite series is being assumed. In particular ν has the same C/(M+1) tail, and `||ν||_w≤C`.\n\nWriting ε_r=||ν_r−ν||_w, split the norm at a fixed M. The finite part tends to0; the rest is at most2C/(M+1). Thus `limsup_{r→0} ε_r≤2C/(M+1)` for every M, and sending M→∞ proves ε_r→0. The θ in the norm is exactly the θ from E1. No larger exponential moment or smaller-exponent slack was used. The order of limits is fixed M first, then M→∞.\n\nLet a_r=sumν_r and a=sumν=ν1+ν2>0. Since w≥1, `|a_r−a|≤ε_r`. The exact complete probability law is\n\n`μ_r=Law_QW(N_r)=(1−r³a_r)δ_0+r³ν_r=δ_0+r³H_r`, where `H_r=ν_r−a_rδ_0`.\n\nSupports at0 and at positive integers are disjoint, so there is an exact norm identity, stronger than the displayed inequality:\n\n`||r^-3(μ_r−δ_0)−H||_w = ε_r+|a_r−a| ≤ 2ε_r`, with `H=ν−aδ_0`.\n\nThis verifies E4. The uncentered scaled zero atom is `r^-3μ_r(0)=r^-3−a_r→+∞`; it cannot be silently included in a finite nonempty intensity measure. H and H_r have total signed mass0. The actual tests exercise this identity on genuine probability laws, including signed coefficient differences and the zero-error case.\n\n## EW2: strict exponent gap and the two normalized laws\n\nFix an integer q≥0 and 0≤θ'<θ, and set γ=θ−θ'>0. On n≥1,\n\n`n^q exp(θ'n^α) = n w(n) · n^(q−1) exp(−γn^α)`.\n\nTherefore the tail in E8 is bounded by `C sup_{n>M} n^(q−1)exp(−γn^α)`. The ratio tends to0 because `(q−1)log n−γn^α→−∞` for every fixed q and α>0. For q=0 the n^-1 factor is defined on the positive support and presents no singular zero-coordinate issue. The same bound holds for ν from the just-established moment bound. Finite-coordinate convergence then proves E7. This argument covers θ'=0, which supplies all polynomial moments and in particular first-moment convergence. It supplies no rate for ε_r or E7: coordinate convergence on the finite portion has no specified rate.\n\nFor conditioning on positivity, `Q(N_r≥1)=r³a_r`, so the conditional count distribution is ν_r/a_r. Because a_r→a>0, its denominator is bounded away from0 for all sufficiently small r. The triangle decomposition\n\n`ν_r/a_r−ν/a = (ν_r−ν)/a_r + ν(1/a_r−1/a)`\n\ngives the displayed bound `ε_r/a_r+|a_r−a|||ν||_w/(aa_r)`, which tends to0 at the **endpoint θ**. Its limit is `(ν1δ_1+ν2δ_2)/(ν1+ν2)`.\n\nFor size bias, the original first moment is `E_QW N_r = r³z_r`, where z_r=sum nν_r(n). The count law is `nν_r(n)/z_r`. E7 with q=1 and any strictly smaller θ' gives convergence of nν_r in that smaller weighted norm. Since its weights are at least1, it also gives z_r→z=ν1+2ν2>0. The same triangle argument with denominator z_r yields the asserted strict-weight convergence. Its limit is `(ν1δ_1+2ν2δ_2)/(ν1+2ν2)`.\n\nThe denominators a_r and z_r are distinct. With ν1,ν2>0 the two limiting distributions are distinct as well. The proof does not infer endpoint size-biased convergence from mere boundedness of its endpoint moment, and MF2's boundedness is not such a convergence statement. No additional field normalizer is introduced by these explicitly named count-law operations.\n\nA concrete no-rate family reinforces the qualifier: take θ=log2, α=1, r→0 and ν_r=(1+r)δ_1+δ_2. It satisfies the same type of uniform E5 bound and the required positive finite coordinate limit, but ε_r=2r and ε_r/r³=2/r²→∞. A sufficiently small-r probability law is again formed with zero mass1−r³sumν_r. Consequently coefficient convergence cannot be assigned an automatic O(r³) rate. This is an implication counterexample, not a statement about the source field's actual rate.\n\n## E9–E10: the genuine escaping-mass counterexample\n\nFix any 0<α≤1, θ>0 and any integer m=m(r)≥3 tending to∞. Let `t_m=exp(−θm^α)` and `η_m=t_m/m`. Then ν_r=δ_1+δ_2+η_mδ_m has total mass2+η_m, and η_m≤1/3. Hence, for all sufficiently small r, `r³(2+η_m)≤1` and the displayed μ_r is a genuine nonnegative probability law with positive nonempty intensity exactly ν_r.\n\nFor every fixed coordinate n≥3, the moving atom eventually leaves n; coordinates1 and2 stay1. Thus E2 holds with ν1=ν2=1. The full retained-index exponential budget is exactly\n\n`sum n w(n)ν_r(n) = exp(θ)+2exp(θ2^α)+m exp(θm^α)η_m = exp(θ)+2exp(θ2^α)+1`.\n\nIt is independent of r. Thus the actual μ_r satisfies E1 with that fixed constant. For every fixed polynomial moment q, the moving atom contributes `m^qη_m=m^(q−1)exp(−θm^α)→0`. The endpoint intensity error is exactly `w(m)η_m=1/m→0`, in accordance with EW1.\n\nBut z_r=3+t_m, and the size-biased law assigns m mass `mη_m/z_r=t_m/(3+t_m)`. Weighting it by w(m)=1/t_m gives exactly `1/(3+t_m)→1/3`. The limiting size-biased distribution is supported on1,2 and has zero mass at m. Absolute weighted norm cannot cancel that moving positive contribution. Indeed, the entire endpoint norm error is\n\n`1/(3+t_m) + [w(1)+2w(2)] t_m/[3(3+t_m)] → 1/3`.\n\nThis directly verifies the failure of endpoint weighted convergence, not merely failure of one proposed proof. The ordinary and strict-weight size-biased limits remain consistent with EW2 because the escaping weighted contribution then contains the decaying factor exp[−(θ−θ')m^α]. No contradiction with the endpoint positive-conditioned law occurs: its escaped weighted mass is `1/[m(2+η_m)]→0`.\n\nThe example is fully within the abstract assumptions E1–E2, so it refutes the stronger implication from those assumptions alone. It is expressly not asserted to arise from the Gaussian field. An additional uniform-integrability condition on n w(n)ν_r, or a genuinely stronger moment premise sufficient for it, would be needed to infer endpoint size-biased convergence.\n\n## Actual finite evidence and falsification controls\n\nThe independent script `exact_controls.py` is **8,655 bytes**, SHA256 `f8f0b8af48ca8cfa697cdc91076a96d4cfa9f761b006a839077a1c79681c966d`. It does not import or execute the author's verifier or finite controls. It uses exact Fraction arithmetic, explicit exceptions rather than removable assertions, and transparent rational interval bounds. The analytic derivation above proves the general statements; these finite calculations detect concrete algebraic/scope failures and do not prove Gaussian inputs or infinite-limit theorems by testing.\n\nThere were **20 actual executions**, ten with `-B -S` and ten with `-B -S -O`. In each mode the baseline exited0, all eight deliberate mutants exited1, and the unknown-mutant check exited2. Every corresponding stdout is byte-identical between modes. Original command vectors, return codes, stdout/stderr files and hashes are preserved in `controls/`. `execution-summary.json` is11,186 bytes, SHA256 `8b8faf298e8bfe7acf032ef7a3b3053f06373405f518fc4eaa2cc17be9b97d1f`. The deterministic baseline `RESULTS.json` is580 bytes, SHA256 `6e591f2400d7482695328aa664a53f90e0a88147ed8a9a65ee44c0c6cdb153a8`.\n\nThe positive checks comprise 1,458 retained-index tail inequalities,16 sharp M+1 atoms and M+2 falsifiers,12 exact centered-law identities,12 distinct normalized-law cases,62 genuine escaping probability laws,217 exact strict-exponent-gap identities,33 higher-dimensional rational interval cases, and6 no-automatic-rate cases. For d=2 the weight is exactly2^n. For d=4,6,8, choosing m=j^(d/2) makes m^(2/d)=j and w(m)=2^j exact; the sole irrational small-support weight w(2) lies in[2,4]. E5 is affine in that value, giving the rigorous rational budget interval[7,11]. This interval control is not a numerical coefficient claim.\n\nThe eight deliberately rejected alterations are:\n\n1. Dropping the retained n while retaining the endpoint tail conclusion.\n2. Replacing the sharp denominator M+1 by M+2.\n3. Scaling the raw zero atom without centering it.\n4. Using the size-biased denominator for positive conditioning.\n5. Using the positive-conditioning denominator for size bias.\n6. Inferring endpoint size-biased convergence from E1–E2.\n7. Assigning an automatic r³ coefficient-convergence rate.\n8. Substituting a nonempty-conditioned normalization for the original full tilt in MF1.\n\nThese controls supplement the explicit counterexamples and exact equalities in this report. No full master verification was run or is claimed for this read-only analytic review.\n\n## Retained obligations\n\nAcceptance is confined to the conditional implication in §§1–3 at the original immutable head. MF1 and SC(3) and all their imported hypotheses retain their separate review obligations and scopes. This report supplies no new global field estimate, coefficient value or rate, mark-uniform result, varying-dimension/volume assertion, spatial independence, elder pairing, event-identification result, or RN/JETMOD/prize closure. Independent-copy consequences belong to separately reviewed Slice B. A source amendment would require an affected fresh review; hash equality, CI and this scoped verdict do not authorize whole-PR integration by themselves.\n", + "github_login": "d6g8k5htny-coder" + }, + { + "id": 5360159070, + "html_url": "https://github.com/d6g8k5htny-coder/Math-/pull/164#pullrequestreview-5360159070", + "commit_id": "0e70540b70e5e3183df5ed1a6bd70a212b734c9c", + "submitted_at": "2026-09-30T00:38:56Z", + "state": "COMMENTED", + "body": "# PR164 Slice B — actual nonauthor analytic review\n\nDisposition: **ACCEPT, CONDITIONAL**, limited to PROOF.md §§4–5 (EW3, EW4,\nE11–E15, the actual-versus-limit mark-rate qualification, and the dimension\nboundary for complex PGFs). No actionable defect found in that scope.\n\nThis is an actual reconstruction by the OpenAI/Codex agent\n`/root/c17_pr158_slice_d_audit`, completed 30 September 2026 UTC. It is neither\nauthor self-acceptance nor an organizationally independent vote. The parent\nagent owns any native GitHub publication and disposition. This directory makes\nno remote change and cannot itself merge or change scientific status.\n\n## Exact object and authority\n\nRepository: `d6g8k5htny-coder/Math-`, PR164, exact head\n`0e70540b70e5e3183df5ed1a6bd70a212b734c9c`.\n\nReviewed full source:\n`frontiers/cluster_exponential_weight_20260930/PROOF.md`.\n\n- Bytes: `12576`.\n- Git blob: `bf1268b9c18ec05561fd9b14c645ef0b8e530484`.\n- SHA256: `3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee`.\n- Source-map bytes: `1650`; SHA256\n `478bd0ad93200b0993562d1a7ea3a6f96c7e15620781df469df1e5c127a1b4a1`.\n\nFresh PR metadata was read back during this review: same head, draft/open,\nbase `aa288385af9392c509233c25a0f4012c9ba023e5`. This review is of immutable\nmathematical bytes; those metadata do not establish current-base merge readiness.\n\nRoot R17 claim `cb43a8b2-57e1-4e51-8cab-21c9efa7fe6a` was read back in Work\nEvents row296, with expiry `2026-09-30T02:26:20Z`; pickup comment5901632715.\nThe exact claim/readback envelope is preserved as `claim-and-entry.json`.\nRoot authorized material output only in this Slice-B directory. The later\nClaude complementary pickup reported by root was coordination information,\nnot a premise or source of this verdict; I did not consume its analytic review.\n\n## Provider, authorship, and exposure\n\nActual provider/model identity available here: OpenAI/Codex. No unverifiable\nmodel version or external provider is imputed. I did not author PR164, either\nmathematical source, or the prior #157 mathematical proof. Prior exposure\nincludes #157's engineering workflow repair, #158 Slice D analytic review,\n#162's spectral-Jacobian challenge, and coordination triage. This review read\nthe full PR164 proof, its exact source map, and the consumed parent interfaces.\nI did not read PR164's author internal-review packet or import its test program.\n\nA separate internal read-only challenge agent,\n`/root/c17_pr158_slice_d_audit/c22_pgf_challenge`, directly fetched and read the\nsame full proof and challenged only §5 and the §4 independence/no-rate\nqualifications. It inherited our context and used the same provider. Its report\nis summarized separately in `CHALLENGE.md`; it supplies **zero additional\norganizational independence and no separate vote**. My derivations of the full\nSlice B ledger and the persisted test implementation remain this review's\nown work. Slice A is another agent's assignment and is not accepted here.\n\n## Pinned hypotheses, retained without re-proving them\n\nThe exact parent source bytes were hash-verified and preserved:\n\n| Interface | Immutable source | Bytes | SHA256 |\n|---|---|---:|---|\n| MF1: weighted first moment | commit `cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b`, `frontiers/c6_marked_fourier_20260929/PROOF.md`, blob `efb702f601eb0c0a24dad6d125061158158867b9` | 15304 | `a5163679400d7d68c1c51538c9ed4ac111015487f79e820057f4ff3e20d09544` |\n| SC(3): coordinate limit | commit `7a44130b41b827577424a73c4feac4cde20b5568`, `frontiers/spectral_cluster_closure_20260929/PROOF.md`, blob `16c56821b52fd76b0be791622b9c3809eafde75a` | 25006 | `e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb` |\n\nMF1 uses the variance-one periodized field, endpoint regression, original typed\nweight W, full endpoint-only Z, and the all-index additional count in the open\nwindow `(b-k*r^3,b)`, with both pins excluded. SC §1 names the same objects and\npins. MF's torus notation T corresponds to the fixed torus side L here; its\ncompact-family guarantee may be evaluated at this one fixed admissible mark.\nSC(3) supplies positive finite nu1,nu2 and coordinate convergence at these\nfixed parameters. None of the matching read or byte verification re-proves\nthe Gaussian, Fourier, regional Kac–Rice, spectral, or global-assembly parents.\n\nFor this slice I retain E1/E2 and Slice A's endpoint conclusion\n`epsilon_r=||nu_r-nu||_w -> 0` as hypotheses/dependencies. In particular, my\nverdict does not independently accept §§2–3, source-status changes, unresolved\nparent premises, or mark-uniformity. E5 is the exact re-expression of E1:\n`sum n*w(n)*nu_r(n) <= C`. All statements below apply for sufficiently small r\nwhere that interface holds, with one fixed dimension, torus, birth, gap, frame,\ntheta and C. No source of an epsilon rate or additional normalizer is supplied.\n\n## Analytic reconstruction\n\n### EW3 and the algebra required by EW4\n\nFor `00, the derivative of\n`(x+y)^alpha-y^alpha` is nonpositive. Its limit at y=0 is x^alpha, proving\nsubadditivity; the x=0 and alpha=1 cases are immediate. Consequently\n`w(n+m)<=w(n)*w(m)` on nonnegative integers. For signed or complex measures,\nthe triangle inequality followed by nonnegative Fubini bounds the convolution\nnorm by the product of norms. This simultaneously justifies absolute\nconvergence and rearrangement. The coordinate map `sigma(n) -> w(n)*sigma(n)`\nis an isometry to ordinary l1, giving completeness; delta0 is a norm-one unit.\nNo positivity of signed generators is needed for E11.\n\nThe exponential series converges absolutely in this unital commutative Banach\nalgebra, with norm bounded by the scalar exponential. The use of a commutative\nalgebra matters for the simple E15 derivative identity; it holds here because\naddition of the nonnegative integer indices commutes. There is no missing\nnoncommutative correction.\n\n### The actual one-copy law and B\n\nBy definition of nu_r on the positive indices, the one-copy law is exactly\n`u_r=delta0+delta*H_r`, with `delta=r^3` and\n`H_r=nu_r-a_r*delta0`. The coefficient at zero is centered. For small r this\nis an actual probability measure, already supplied by the original law, not\nan arbitrarily signed approximation.\n\nSince n>=1 and w(n)>=1,\n`||nu_r||_w<=C`, `a_r<=C`, and hence `||H_r||_w<=2C=B`.\nThe limit has the same bounds by finite-sum Fatou under coordinate convergence.\nAlso `|a_r-a|<=epsilon_r`, so `||H_r-H||_w<=2*epsilon_r`; the zero-coordinate\nerror is essential even when not sharp. For s>=0, splitting the commuting\ngenerator gives\n`exp_*(s*H_r)=exp(-s*a_r)*sum_j s^j*nu_r^(*j)/j!`, which is a compound-Poisson\nprobability law. For `H=nu1*delta1+nu2*delta2-(nu1+nu2)*delta0`, it is exactly\nthe law of the stated independent Poisson counts with jump sizes one and two.\n\n### E13, E14, E15, and the complete error ledger\n\nFor E13, the jth remainder norm is at most `(delta*B)^j/j!`. For j>=2,\n`1/j!<=1/[2*(j-2)!]`, giving the displayed\n`delta^2*B^2*exp(delta*B)/2`. The power delta^2 and coefficient1/2 are correct.\n\nLet `v=exp_*(delta*H_r)`. For m>=1, the exact telescoping difference of m\nconvolution factors has m terms. Each remaining product contains m-1 factors,\neach with norm at most `exp(delta*B)`: for u use `1+delta*B<=exp(delta*B)`;\nfor v use the exponential norm bound. Thus each error term contributes at\nmost `delta^2*B^2*exp(m*delta*B)/2`, and their sum has the factor m in E14.\nThere is no extra uncompensated `exp(delta*B)` after counting all factors.\nFor m=0 both powers are the unit and that difference is exactly zero.\n\nFor E15, set `G(s)=D+s*(A-D)`. Norm-convergent differentiation, justified\nuniformly on the compact s interval by the exponential majorant, gives\n`d exp_*(G(s))/ds=exp_*(G(s))*(A-D)`. Its Bochner integral is the desired\ndifference. Convexity of the norm gives\n`||G(s)||<=max(||A||,||D||)`, proving the stated Lipschitz bound. This\nargument is valid for both signed-real and complex sequences.\n\nThe triangle decomposition is precisely\n\n1. `u_r^(*m)` to `exp_*(m*delta*H_r)` (Euler term);\n2. `exp_*(m*delta*H_r)` to `exp_*(t*H_r)` (floor term);\n3. `exp_*(t*H_r)` to `exp_*(t*H)` (intensity replacement).\n\nWith `m=floor(t/delta)`, `m*delta<=t` and `0<=t-m*delta2, alpha<1. The normalization constant for\n`q(n) proportional to n^-3*exp(-2*theta*n^alpha), n>=3`, is finite and\nstrictly positive. Its required weighted first moment is the finite series\n`constant*sum n^-2*exp(-theta*n^alpha)`. A positive function epsilon(r)->0\nis bounded for sufficiently small r, so the perturbed intensities have one\nuniform E1 constant and converge at each fixed index as in E2. Choosing r\nsmall makes `1-r^3*(2+epsilon(r))` nonnegative, giving genuine probability\nlaws. For every real z>1, the logarithm of the unnormalized PGF term is\n`n*log(z)-2*theta*n^alpha-3*log(n)`, which tends to positive infinity, since\nalpha<1. Thus the terms do not even tend to zero. Their radius is exactly\none because the laws still have finite total mass. The conclusion is a\nfailure of implication for the abstract interfaces; it says nothing negative\nabout the actual Gaussian count's analytic radius.\n\n## Actual finite and interval controls\n\n`verify_slice_b.py` is a new independent implementation using only Python's\nstandard library. It imports neither the author verifier nor any parent test.\nI executed it once successfully; `execution.json` captures the actual command,\nPython version, start/end time, exit code, and exact stdout/stderr hashes.\n`verification.stdout` and `.stderr` are the captured original byte streams.\nThe source identities are checked again during that execution.\n\n- 81 finite signed-rational convolution pairs satisfy E11 and commutativity.\n- 567 exact finite telescoping identities include the zero-factor case.\n- 567 homogeneous coefficient/integral identities reconstruct E15 term by\n term, using exact integration of powers of the interpolation variable.\n- 41 scalar Taylor coefficient inequalities test E13's remainder bound.\n- 25 rational square comparisons test alpha=1/2 subadditivity, including zero.\n- 60 actual replica-law comparisons use d=2, theta=log2, w(n)=2^n, rational\n one/two-jump intensities and times. Of these, 24 have m=0 and 45 have a\n nonzero actual/limiting intensity difference. The entire weighted norm\n against the infinite compound-Poisson law is enclosed: rational Taylor\n bounds enclose the exponential, exact recurrences enclose every retained\n coefficient, and a positive geometric bound covers the omitted weighted\n Poisson jump-count tail. Comparing this norm upper bound with a lower\n enclosure of E12's right side verifies the actual inequality for each case.\n\nEight negative controls were rejected as intended: extending alpha above\none; deleting the floor error; deleting intensity replacement; claiming a\ndelta^3 Euler remainder; replacing independent copies by fully coupled ones;\ndeducing an automatic delta^2 actual-to-limit one-copy rate; removing zero\ncentering; and inferring a disk beyond one from a stretched-exponential bound.\nIn particular, for `delta=k^-2` and positive intensities\n`nu_r=(1+1/k)*delta1+delta2`, the exact one-copy weighted error is `3/k^3`;\nits ratio to delta^2 is `3*k`, recorded as12,24,48,96,192. For fully coupled\ncopies with delta=1/100 and both limiting intensities one, the zero-mass\ndiscrepancy at t=1 is rigorously greater than4/5.\n\nThese finite controls supplement the preceding analytic derivation. Their\npassing does not prove an infinite theorem, validate either parent, or add a\nsecond acceptance. No master or author workflow was rerun by this reviewer.\n\n## Remaining gates\n\nNo amendment is requested for this exact Slice B. It remains conditional on\nE1/E2, the fixed-parameter identification, and Slice A's endpoint norm\nconvergence. The proof's existential constants, missing epsilon rate,\nindependent-copy scope, original W/Z normalization, distinction between\npositive conditioning and size bias, all-index count, and unaccepted parent\nobligations are retained. This verdict grants no all-angle/all-radius result,\nspatial Poisson process, new Gaussian moment proof, event-identification\ntheorem, organizational independence, prize closure, or merge readiness.\n\nExact mathematical changes require a new affected review. Source-preserving\nintegration, current-base CI, native-review custody, R17 release, and delivery\nremain with root under their own applicable authority.\n", + "github_login": "d6g8k5htny-coder" + } + ] +} diff --git a/frontiers/cluster_exponential_weight_20260930/SOURCES.json b/frontiers/cluster_exponential_weight_20260930/SOURCES.json new file mode 100644 index 000000000..4b3eeb938 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/SOURCES.json @@ -0,0 +1,30 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "policy": "Direct mathematical interfaces only. Verifying immutable bytes does not accept either parent theorem or this new implication.", + "sources": [ + { + "id": "MF", + "commit": "cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b", + "path": "frontiers/c6_marked_fourier_20260929/PROOF.md", + "bytes": 15304, + "git_blob": "efb702f601eb0c0a24dad6d125061158158867b9", + "sha256": "a5163679400d7d68c1c51538c9ed4ac111015487f79e820057f4ff3e20d09544", + "author": "OpenAI / ChatGPT", + "use": "MF1 only at a fixed parameter inside the declared compact family, alpha=2/d, ORIGINAL W/Z and ALL-INDEX count. Parent proof not revalidated here.", + "source_url": "https://github.com/d6g8k5htny-coder/Math-/blob/cbb2f978f9b3e8e2eaca94f3b359b18d7c144f1b/frontiers/c6_marked_fourier_20260929/PROOF.md" + }, + { + "id": "SC", + "commit": "7a44130b41b827577424a73c4feac4cde20b5568", + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "author": "OpenAI / GPT-6 Astra Pro", + "use": "Coordinate consequence of (3), finite positive nu1,nu2 at FIXED d,L,b,k,frame. Pending parent slices and imported premises remain conditional; no source status promotion.", + "source_url": "https://github.com/d6g8k5htny-coder/Math-/blob/7a44130b41b827577424a73c4feac4cde20b5568/frontiers/spectral_cluster_closure_20260929/PROOF.md" + } + ] +} diff --git a/frontiers/cluster_exponential_weight_20260930/VALIDATION.json b/frontiers/cluster_exponential_weight_20260930/VALIDATION.json new file mode 100644 index 000000000..ab9e91fce --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/VALIDATION.json @@ -0,0 +1,28 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "scope": "finite rational algebra, immutable source custody and conditional prose", + "proof_sha256": "3f32fdd744350a15359451bbbee6c6bcc71af47e4f57f6110d7afb4978e941ee", + "author_disposition": "AUTHOR_SIDE / CONDITIONAL; external scoped nonauthor review required", + "internal_reviews": "INTERNAL_REVIEWS.md; zero organizational independence", + "local_runs": { + "finite_controls": {"tests": 12, "positive": 8, "intended_negative": 4, "normal_exit": 0, "optimized_exit": 0, "stdout_byte_identical": true}, + "signed_convolution_pairs": 729, + "custody_controls": {"tests": 11, "normal_exit": 0, "optimized_exit": 0}, + "packet_discovery": {"tests": 23, "normal_exit": 0, "optimized_exit": 0} + }, + "preserved_failures": [ + "Initial finite-control implementation omitted zero centering; a mass check failed before correction.", + "Four checker-bypass mutation runs each caused one named negative control to fail and exit 1.", + "Initial repository-root discovery found zero tests and exited 5; targeted packet discovery is the supported command.", + "Initial source verifier accepted tree-as-commit, replacement-ref substitution and boolean schema; all three added controls failed before the minimal repairs. Separate re-review passed." + ], + "source_exposure": { + "connector_commit_path_blob_size_sha256_checks": true, + "current_MF_successor_proof_equals_pinned_text": true, + "local_complete_historical_git_objects_available": false, + "local_full_repository_suite_run": false, + "hosted_actual_checkout_source_verification": "REQUIRED, not yet claimed" + }, + "excluded_inferences": ["Gaussian proof from tests", "parent theorem acceptance", "spatial independence", "coefficient convergence rate", "larger-disk analyticity for d>2", "endpoint size-bias convergence", "register transition", "external organizational review", "engineering merge"] +} diff --git a/frontiers/cluster_exponential_weight_20260930/finite_checks.py b/frontiers/cluster_exponential_weight_20260930/finite_checks.py new file mode 100644 index 000000000..86d7f2482 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/finite_checks.py @@ -0,0 +1,218 @@ +"""Deterministic exact finite algebra controls for the conditional EW packet. + +All arithmetic is rational at alpha=1 and theta=log(2), so w(n)=2**n. +These controls validate selected finite identities and inequalities only. +They do not prove MF1, SC, a coefficient rate, or any field independence. +No source-cache content is read or exported by this module. + +Run ``python finite_checks.py`` (also with ``-O``) for one small JSON result. +The only intended failures are the four named variants in the test module. +""" + +from fractions import Fraction +from io import StringIO +import json +from math import factorial +import sys +import unittest + + +class ControlFailure(ValueError): + """An explicit finite invariant failed; never relies on Python assert.""" + + +def require(condition, label): + if not condition: + raise ControlFailure(label) + + +def rational(value): + require(isinstance(value, (int, Fraction)), "exact rational required") + return Fraction(value) + + +def canonical(measure): + result = {} + for n, coefficient in measure.items(): + require(isinstance(n, int) and n >= 0, "nonnegative integer atom required") + coefficient = rational(coefficient) + if coefficient: + result[n] = coefficient + return result + + +def add(left, right): + result = canonical(left) + for n, q in canonical(right).items(): + result[n] = result.get(n, Fraction(0)) + q + return canonical(result) + + +def scale(measure, scalar): + scalar = rational(scalar) + return canonical({n: scalar * q for n, q in canonical(measure).items()}) + + +def subtract(left, right): + return add(left, scale(right, -1)) + + +def weighted_norm(measure): + return sum((2**n * abs(q) for n, q in canonical(measure).items()), Fraction(0)) + + +def positive_intensity(intensity): + intensity = canonical(intensity) + require(all(n >= 1 and q >= 0 for n, q in intensity.items()), "positive intensity required") + return intensity + + +def center(intensity): + intensity = positive_intensity(intensity) + return add(intensity, {0: -sum(intensity.values(), Fraction(0))}) + + +def probability_from_intensity(intensity, delta): + delta = rational(delta) + require(delta > 0, "delta must be positive") + law = add({0: Fraction(1)}, scale(center(intensity), delta)) + require(all(q >= 0 for q in law.values()), "one-copy law must be nonnegative") + check_equal(sum(law.values(), Fraction(0)), Fraction(1), "one-copy probability mass") + return law + + +def escaping_intensity(m): + require(isinstance(m, int) and m >= 3, "escaping atom m must be at least 3") + return {1: Fraction(1), 2: Fraction(1), m: Fraction(1, m * 2**m)} + + +def positive_conditioned(intensity): + intensity = positive_intensity(intensity) + normalizer = sum(intensity.values(), Fraction(0)) + require(normalizer > 0, "positive-conditioned normalizer") + return scale(intensity, 1 / normalizer) + + +def size_biased(intensity): + intensity = positive_intensity(intensity) + biased = {n: n * q for n, q in intensity.items()} + normalizer = sum(biased.values(), Fraction(0)) + require(normalizer > 0, "size-biased normalizer") + return scale(biased, 1 / normalizer) + + +def convolve(left, right): + result = {} + for n, q in canonical(left).items(): + for k, p in canonical(right).items(): + result[n + k] = result.get(n + k, Fraction(0)) + q * p + return canonical(result) + + +def convolution_power(measure, exponent): + require(isinstance(exponent, int) and exponent >= 0, "nonnegative replica count required") + result, base = {0: Fraction(1)}, canonical(measure) + while exponent: + if exponent % 2: + result = convolve(result, base) + exponent //= 2 + if exponent: + base = convolve(base, base) + return result + + +def replica_law(intensity, delta, t): + delta, t = rational(delta), rational(t) + require(delta > 0 and t >= 0, "replica time and delta") + # This operation encodes deliberately independent replicas only. + return convolution_power(probability_from_intensity(intensity, delta), t // delta) + + +def exp_polynomial(generator, t, order=8): + """Signed convolution Taylor polynomial, not an infinite CP law. + + Its mass is exactly 1 when the generator has total mass zero, by + cancellation. For an uncentered positive generator even this mass + statement fails. The remainder must be added to every comparison + with exp_*(t*generator). + """ + t = rational(t) + require(t >= 0 and isinstance(order, int) and order >= 0, "Taylor time and order") + argument = scale(generator, t) + total = term = {0: Fraction(1)} + for j in range(1, order + 1): + term = scale(convolve(term, argument), Fraction(1, j)) + total = add(total, term) + return total + + +def scalar_exp_prefix(x, order=8): + x = rational(x) + require(x >= 0 and isinstance(order, int) and order >= 0, "scalar Taylor argument and order") + return sum((x**j / factorial(j) for j in range(order + 1)), Fraction(0)) + + +def scalar_tail_bound(x, order=8): + """Rigorous rational upper bound for sum_(j>order) x**j/j!. + + The first omitted term is x**(order+1)/(order+1)!. Each subsequent + ratio is at most x/(order+2), so a geometric series bounds the entire + infinite scalar tail when that ratio is below 1. Submultiplicativity + then bounds the omitted convolution tail at x=||t*generator||_w. + Fixtures keep x<=1/4 and use the fixed order 8 for replica comparisons; + no floating-point exponential or adaptive digit refinement is used. + """ + x = rational(x) + require(x >= 0 and isinstance(order, int) and order >= 0, "scalar remainder argument and order") + ratio = x / (order + 2) + require(ratio < 1, "geometric scalar remainder requires ratio below 1") + return x**(order + 1) / factorial(order + 1) / (1 - ratio) + + +def check_zero_mass(measure, label): + require(sum(canonical(measure).values(), Fraction(0)) == 0, label) + + +def check_upper(actual, upper, label): + require(actual <= upper, label) + + +def check_equal(actual, expected, label): + require(actual == expected, label) + + +def run_controls(): + import test_finite_checks + + suite = unittest.defaultTestLoader.loadTestsFromModule(test_finite_checks) + + def flatten(tests): + for test in tests: + if isinstance(test, unittest.TestSuite): + yield from flatten(test) + else: + yield test + + tests = list(flatten(suite)) + negative_ids = {test.id() for test in tests if getattr(test, "NEGATIVE_CONTROL", False)} + diagnostics = StringIO() + result = unittest.TextTestRunner(stream=diagnostics, verbosity=0).run(suite) + failed_ids = {test.id() for test, _ in result.failures + result.errors} + summary = { + "all_passed": result.wasSuccessful(), + "checks": result.testsRun, + "failed": len(failed_ids), + "positive_controls": len(tests) - len(negative_ids), + "intended_negative_controls": len(negative_ids), + "intended_negative_controls_rejected": len(negative_ids - failed_ids), + "scientific_effect": "NONE", + "scope": "finite_algebra_only", + } + print(json.dumps(summary, sort_keys=True, separators=(",", ":"))) + if not result.wasSuccessful(): + sys.stderr.write(diagnostics.getvalue()) + return 0 if result.wasSuccessful() else 1 + + +if __name__ == "__main__": + sys.exit(run_controls()) diff --git a/frontiers/cluster_exponential_weight_20260930/test_finite_checks.py b/frontiers/cluster_exponential_weight_20260930/test_finite_checks.py new file mode 100644 index 000000000..d350286f0 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/test_finite_checks.py @@ -0,0 +1,233 @@ +"""Exact finite controls only: no MF1/SC validation or field simulation.""" + +from fractions import Fraction as F +from itertools import product +from math import factorial +import unittest + +import finite_checks as fc + + +def coefficient_product(left, right): + """Independent oracle: evaluate PGF products, then Newton-interpolate. + + This never forms pairs of measure atoms. Exact product evaluations at + degree+1 distinct integers determine the polynomial coefficients. + """ + degree = max(left, default=0) + max(right, default=0) + values = [ + sum(q * x**n for n, q in left.items()) + * sum(q * x**n for n, q in right.items()) + for x in range(degree + 1) + ] + coefficients = [F(0)] * (degree + 1) + basis = [F(1)] # binomial(x, 0) + for j in range(degree + 1): + for n, q in enumerate(basis): + coefficients[n] += values[0] * q + values = [b - a for a, b in zip(values, values[1:])] + next_basis = [F(0)] * (len(basis) + 1) + for n, q in enumerate(basis): + next_basis[n] -= F(j, j + 1) * q + next_basis[n + 1] += F(1, j + 1) * q + basis = next_basis + return {n: q for n, q in enumerate(coefficients) if q} + + +class PositiveFiniteControls(unittest.TestCase): + def test_original_probability_and_zero_centering(self): + """An omitted negative atom at zero must fail this control.""" + for m in (3, 4, 8, 12): + eta = F(1, m * 2**m) + nu = {1: F(1), 2: F(1), m: eta} + self.assertEqual(fc.escaping_intensity(m), nu) + centered = fc.center(nu) + self.assertEqual(sum(centered.values()), 0) + delta = F(1, 128 * m) + law = fc.probability_from_intensity(nu, delta) + self.assertEqual(sum(law.values()), 1) + self.assertGreater(law[0], 0) + self.assertTrue(all(q >= 0 for q in law.values())) + self.assertEqual(law[m], delta * eta) + scaled = {n: q / delta for n, q in law.items()} + scaled[0] -= 1 / delta + self.assertEqual(scaled, centered) + self.assertEqual(centered[0], -(2 + eta)) + + def test_endpoint_errors_and_distinct_normalizers(self): + """Omitting n in size bias, or equating its denominator with a, fails.""" + limit = {1: F(1), 2: F(1)} + for m in (3, 4, 8, 12): + nu = fc.escaping_intensity(m) + eta = F(1, m * 2**m) + a, z = sum(nu.values()), sum(n * q for n, q in nu.items()) + self.assertEqual(a, 2 + eta) + self.assertEqual(z, 3 + F(1, 2**m)) + self.assertNotEqual(a, z) + self.assertEqual(fc.weighted_norm(fc.subtract(nu, limit)), F(1, m)) + centered_error = fc.weighted_norm( + fc.subtract(fc.center(nu), fc.center(limit)) + ) + self.assertEqual(centered_error, F(1, m) + eta) + self.assertLessEqual(centered_error, F(2, m)) + ordinary, biased = fc.positive_conditioned(nu), fc.size_biased(nu) + self.assertEqual(sum(ordinary.values()), 1) + self.assertEqual(sum(biased.values()), 1) + self.assertEqual(ordinary[1], 1 / a) + self.assertEqual(biased[1], 1 / z) + self.assertEqual(biased[2], 2 / z) + self.assertEqual(2**m * ordinary[m], 1 / (m * a)) + self.assertEqual(2**m * biased[m], 1 / z) + self.assertGreaterEqual(2**m * biased[m], F(1, 4)) + + def test_retained_n_factor_certifies_endpoint_tail(self): + """Replacing the first weighted moment by the zeroth breaks E6.""" + for m in (3, 4, 8, 12): + nu = fc.escaping_intensity(m) + first_moment = sum(n * 2**n * q for n, q in nu.items()) + self.assertEqual(first_moment, 11) + for cutoff in (1, 2, m - 1, m, m + 1): + tail = sum(2**n * q for n, q in nu.items() if n > cutoff) + fc.check_upper(tail, F(11, cutoff + 1), "E6 endpoint tail") + + def test_signed_convolution_against_coefficient_product(self): + """Lost signs, missed coefficients and a wrong weight fail the oracle.""" + measures = [ + {n: q for n, q in enumerate(entries) if q} + for entries in product((F(-2, 3), F(0), F(1, 2)), repeat=3) + ] + for left, right in product(measures, repeat=2): + convolution = fc.convolve(left, right) + self.assertEqual(convolution, coefficient_product(left, right)) + fc.check_upper( + fc.weighted_norm(convolution), + fc.weighted_norm(left) * fc.weighted_norm(right), + "E11 signed convolution", + ) + cancellation = fc.convolve({0: F(1), 1: F(-1)}, {0: F(1), 1: F(1)}) + self.assertEqual(cancellation, {0: F(1), 2: F(-1)}) + self.assertEqual(fc.weighted_norm(cancellation), 5) + + def test_taylor_polynomial_remainder_and_mass_are_explicit(self): + """Dropping the tail or treating every Taylor polynomial as a law fails.""" + generator = {0: F(-1), 1: F(1)} + polynomial = fc.exp_polynomial(generator, F(1, 16), order=2) + self.assertEqual(polynomial, {0: F(481, 512), 1: F(15, 256), 2: F(1, 512)}) + self.assertEqual(sum(polynomial.values()), 1) # centered cancellation + positive_polynomial = fc.exp_polynomial({1: F(1), 2: F(1)}, F(1, 128), 8) + positive_mass = sum(positive_polynomial.values()) + expected_mass = sum(F(1, 64)**j / factorial(j) for j in range(9)) + self.assertEqual(positive_mass, expected_mass) + self.assertNotEqual(positive_mass, 1) + self.assertGreater(F(1, 64)**9 / factorial(9), 0) # omitted CP-series mass + for x in (F(0), F(1, 16), F(1, 4)): + for order in (0, 2, 8): + remainder = fc.scalar_tail_bound(x, order) + tail_prefix = sum(x**j / factorial(j) for j in range(order + 1, order + 10)) + self.assertGreaterEqual(remainder, tail_prefix) + self.assertEqual(fc.scalar_exp_prefix(x, order), sum(x**j / factorial(j) for j in range(order + 1))) + + def test_one_copy_euler_error_with_infinite_tail_allowance(self): + """E13 comparison adds a rigorous remainder instead of truncating truth.""" + bound = F(22) # B=2C, C=11 in these escaping-mass fixtures. + for m in (3, 4, 8, 12): + nu, delta = fc.escaping_intensity(m), F(1, 128 * m) + generator = fc.center(nu) + x = delta * fc.weighted_norm(generator) + self.assertLessEqual(x, F(1, 4)) + law = fc.probability_from_intensity(nu, delta) + polynomial = fc.exp_polynomial(generator, delta, 8) + certified_error = fc.weighted_norm(fc.subtract(law, polynomial)) + fc.scalar_tail_bound(x, 8) + # A lower scalar enclosure makes this a stronger finite check. + e13_lower = delta**2 * bound**2 * fc.scalar_exp_prefix(delta * bound, 8) / 2 + fc.check_upper(certified_error, e13_lower, "E13 with Taylor remainder") + + def test_actual_replica_convolution_and_floor_ledger(self): + """Wrong replica multiplication or omitting m=floor(t/delta) fails.""" + bound = F(22) + for escape_m in (3, 4, 8, 12): + nu = fc.escaping_intensity(escape_m) + delta = F(1, 128 * escape_m) + generator = fc.center(nu) + for t in (F(0), delta / 2, 2 * delta, 7 * delta / 3): + copies = t // delta + law = fc.replica_law(nu, delta, t) + one_copy = fc.probability_from_intensity(nu, delta) + self.assertEqual(law, fc.convolution_power(one_copy, copies)) + self.assertEqual(sum(law.values()), 1) + self.assertGreaterEqual(t - copies * delta, 0) + self.assertLess(t - copies * delta, delta) + s = copies * delta + polynomial = fc.exp_polynomial(generator, s, 8) + x = s * fc.weighted_norm(generator) + self.assertLessEqual(x, F(1, 4)) + certified_error = fc.weighted_norm(fc.subtract(law, polynomial)) + fc.scalar_tail_bound(x, 8) + e14_lower = copies * delta**2 * bound**2 * fc.scalar_exp_prefix(s * bound, 8) / 2 + fc.check_upper(certified_error, e14_lower, "E14 with Taylor remainder") + bernoulli = {0: F(7, 8), 1: F(1, 8)} + self.assertEqual(fc.convolution_power(bernoulli, 2), {0: F(49, 64), 1: F(7, 32), 2: F(1, 64)}) + self.assertEqual(fc.convolution_power(bernoulli, 0), {0: F(1)}) + + def test_e12_actual_laws_against_limit_exponential_with_remainder(self): + """Endpoint mark error, floor and infinite-series tail all enter E12.""" + limit = {1: F(1), 2: F(1)} + generator = fc.center(limit) + bound, horizon = F(22), F(1, 100) + self.assertLessEqual(horizon * bound, F(1, 4)) + exponential_lower = fc.scalar_exp_prefix(horizon * bound, 8) + for m in (3, 4, 8, 12): + nu, delta = fc.escaping_intensity(m), F(1, 128 * m) + epsilon = fc.weighted_norm(fc.subtract(nu, limit)) + self.assertEqual(epsilon, F(1, m)) + for t in (F(0), delta / 2, 2 * delta, 7 * delta / 3): + self.assertLessEqual(t, horizon) + actual = fc.replica_law(nu, delta, t) + polynomial = fc.exp_polynomial(generator, t, 8) + x = t * fc.weighted_norm(generator) + certified_error = fc.weighted_norm(fc.subtract(actual, polynomial)) + fc.scalar_tail_bound(x, 8) + e12_lower = exponential_lower * ( + delta * (horizon * bound**2 / 2 + bound) + 2 * horizon * epsilon + ) + fc.check_upper(certified_error, e12_lower, "E12 with Taylor remainder") + + +class IntendedNegativeControls(unittest.TestCase): + NEGATIVE_CONTROL = True + + def test_reject_omitted_zero_centering(self): + """Deliberately wrong generator nu must be rejected as noncentered.""" + wrong_generator = fc.escaping_intensity(8) + with self.assertRaisesRegex(fc.ControlFailure, "zero-centering"): + fc.check_zero_mass(wrong_generator, "omitted-zero-centering") + + def test_reject_omitted_n_factor(self): + """A bounded zeroth weighted moment cannot justify the endpoint tail.""" + m = 16 + wrong_intensity = {1: F(1), 2: F(1), m: F(1, 2**m)} + wrong_constant = fc.weighted_norm(wrong_intensity) + self.assertEqual(wrong_constant, 7) + self.assertEqual(sum(n * 2**n * q for n, q in wrong_intensity.items()), 26) + with self.assertRaisesRegex(fc.ControlFailure, "omitted-n-factor"): + fc.check_upper(F(1), wrong_constant / m, "omitted-n-factor") + + def test_reject_endpoint_size_bias_convergence_inference(self): + """The false extension of the intensity's 2/m bound is rejected.""" + limit_biased = {1: F(1, 3), 2: F(2, 3)} + for m in (16, 32, 64): + biased = fc.size_biased(fc.escaping_intensity(m)) + error = fc.weighted_norm(fc.subtract(biased, limit_biased)) + self.assertGreaterEqual(error, F(1, 4)) + with self.assertRaisesRegex(fc.ControlFailure, "endpoint-size-bias"): + fc.check_upper(error, F(2, m), "endpoint-size-bias") + + def test_reject_coupled_replicas_as_independent(self): + """Perfectly coupled Bernoulli copies omit the positive middle atom.""" + coupled_sum = {0: F(7, 8), 2: F(1, 8)} + independent_sum = {0: F(49, 64), 1: F(7, 32), 2: F(1, 64)} + self.assertEqual(fc.weighted_norm(fc.subtract(coupled_sum, independent_sum)), F(63, 64)) + with self.assertRaisesRegex(fc.ControlFailure, "coupled-replicas"): + fc.check_equal(coupled_sum, independent_sum, "coupled-replicas") + + +if __name__ == "__main__": + unittest.main() diff --git a/frontiers/cluster_exponential_weight_20260930/test_verify_packet.py b/frontiers/cluster_exponential_weight_20260930/test_verify_packet.py new file mode 100644 index 000000000..e597b179f --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/test_verify_packet.py @@ -0,0 +1,249 @@ +"""Fail-closed packet custody regressions; no mathematical claim is tested.""" +import hashlib +import json +from pathlib import Path +import subprocess +import sys +import tempfile +import unittest + + +CHECKER = Path(__file__).with_name('verify_packet.py') + + +class PacketIdentityTests(unittest.TestCase): + def setUp(self): + self.temp = tempfile.TemporaryDirectory() + self.root = Path(self.temp.name) / 'packet' + self.root.mkdir() + (self.root / 'PROOF.md').write_bytes(b'Exact synthetic proof.\n') + (self.root / 'SOURCES.json').write_bytes(b'{"synthetic":true}\n') + (self.root / 'README.md').write_bytes(b'Original custody fixture.\n') + self.validation = {'schema': 1, 'scientific_effect': 'NONE', + 'proof_sha256': hashlib.sha256(b'Exact synthetic proof.\n').hexdigest(), + 'preserved_failures': ['Historical fixture, not finite output.']} + self.write_validation() + proof = (self.root / 'PROOF.md').read_bytes() + self.review = {'schema': 1, 'scientific_effect': 'NONE', 'proof': { + 'path': 'PROOF.md', 'bytes': len(proof), + 'sha256': hashlib.sha256(proof).hexdigest(), + 'git_blob': hashlib.sha1(b'blob '+str(len(proof)).encode()+b'\0'+proof).hexdigest()}} + (self.root / 'REVIEW_RECORD.json').write_text(json.dumps(self.review) + '\n') + self.resign() + + def tearDown(self): + self.temp.cleanup() + + def write_validation(self): + (self.root / 'VALIDATION.json').write_text(json.dumps(self.validation) + '\n') + + def resign(self): + self.manifest = {'schema': 1, 'scientific_effect': 'NONE', + 'manifest_self_excluded': True, 'files': []} + for p in sorted(self.root.iterdir()): + if p.name != 'MANIFEST.json': + b = p.read_bytes() + self.manifest['files'].append({'path': p.name, 'bytes': len(b), + 'sha256': hashlib.sha256(b).hexdigest(), + 'git_blob': hashlib.sha1(b'blob '+str(len(b)).encode()+b'\0'+b).hexdigest()}) + self.write_manifest() + + def write_manifest(self): + (self.root / 'MANIFEST.json').write_text(json.dumps(self.manifest) + '\n') + + def check(self, accepted=False): + flags = ['-B', '-S'] + (['-O'] if sys.flags.optimize else []) + p = subprocess.run([sys.executable, *flags, str(CHECKER), '--packet-root', str(self.root)], + capture_output=True) + if accepted: + self.assertEqual(p.returncode, 0, p.stderr.decode()) + result = json.loads(p.stdout) + self.assertEqual(result['packet_files_verified'], 5) + self.assertTrue(result['validation_proof_identity_verified']) + self.assertTrue(result['review_proof_identity_verified']) + self.assertFalse(result['mathematical_acceptance']) + else: + self.assertEqual(p.returncode, 1, 'bad packet accepted or wrong failure: '+p.stdout.decode()+p.stderr.decode()) + self.assertEqual(p.stdout, b'') + self.assertIn(b'packet identity failed:', p.stderr) + + def test_valid_packet_cli_and_historical_validation(self): + self.check(accepted=True) + + def test_stale_file_bytes_rejected(self): + (self.root / 'README.md').write_bytes(b'Changed but unrecorded.\n') + self.check() + + def test_each_recorded_identity_is_enforced(self): + original = json.loads(json.dumps(self.manifest)) + for key, wrong in [('bytes', 1), ('sha256', '0'*64), ('git_blob', '0'*40)]: + with self.subTest(key=key): + self.manifest = json.loads(json.dumps(original)) + self.manifest['files'][0][key] = wrong + self.write_manifest() + self.check() + + def test_unlisted_file_rejected(self): + (self.root / 'EXTRA.txt').write_bytes(b'Unlisted.\n') + self.check() + + def test_missing_declared_file_rejected(self): + (self.root / 'README.md').unlink() + self.check() + + def test_duplicate_manifest_path_rejected(self): + self.manifest['files'].append(dict(self.manifest['files'][0])) + self.write_manifest() + self.check() + + def test_empty_manifest_rejected(self): + self.manifest['files'] = [] + self.write_manifest() + self.check() + + def test_essential_proof_cannot_be_removed_and_resigned(self): + (self.root / 'PROOF.md').unlink() + self.resign() + self.check() + + def test_essential_validation_cannot_be_removed_and_resigned(self): + (self.root / 'VALIDATION.json').unlink() + self.resign() + self.check() + + def test_manifest_self_exclusion_must_be_true_boolean(self): + for wrong in [False, 1, 'true', None]: + with self.subTest(value=wrong): + self.manifest['manifest_self_excluded'] = wrong + self.write_manifest() + self.check() + + def test_boolean_schema_rejected(self): + self.manifest['schema'] = True + self.write_manifest() + self.check() + + def test_invalid_size_types_rejected(self): + for wrong in [True, -1, 1.0, '1', None]: + with self.subTest(value=wrong): + self.manifest['files'][0]['bytes'] = wrong + self.write_manifest() + self.check() + + def test_unsafe_or_nonflat_paths_rejected(self): + for wrong in ['', '.', '..', '../PROOF.md', '/PROOF.md', './PROOF.md', + 'a//b', 'a/b', 'a\\b', 'C:proof', 'bad\0name', 7]: + with self.subTest(path=wrong): + self.manifest['files'][0]['path'] = wrong + self.write_manifest() + self.check() + + def test_duplicate_manifest_json_key_rejected(self): + text = (self.root / 'MANIFEST.json').read_text() + (self.root / 'MANIFEST.json').write_text(text.replace('"schema": 1', '"schema": 1, "schema": 1')) + self.check() + + def test_duplicate_record_key_rejected(self): + text = (self.root / 'MANIFEST.json').read_text() + (self.root / 'MANIFEST.json').write_text(text.replace('"bytes": ', '"bytes": 0, "bytes": ', 1)) + self.check() + + def test_manifest_symlink_rejected(self): + manifest = self.root / 'MANIFEST.json' + outside = self.root.parent / 'outside.json' + manifest.replace(outside) + manifest.symlink_to(outside) + self.check() + + def test_payload_symlink_rejected_even_when_bytes_match(self): + path = self.root / 'README.md' + outside = self.root.parent / 'outside.md' + path.replace(outside) + path.symlink_to(outside) + self.check() + + def test_unlisted_directory_rejected(self): + (self.root / 'extra').mkdir() + self.check() + + def test_listed_directory_rejected(self): + p = self.root / 'README.md' + p.unlink() + p.mkdir() + self.check() + + def test_symlink_packet_root_rejected(self): + original = self.root + self.root = original.parent / 'linked' + self.root.symlink_to(original, target_is_directory=True) + self.check() + + def test_false_validation_proof_rejected_after_manifest_resign(self): + self.validation['proof_sha256'] = '0'*64 + self.write_validation() + self.resign() + self.check() + + def test_boolean_validation_schema_rejected_after_manifest_resign(self): + self.validation['schema'] = True + self.write_validation() + self.resign() + self.check() + + def test_missing_validation_digest_rejected_after_manifest_resign(self): + del self.validation['proof_sha256'] + self.write_validation() + self.resign() + self.check() + + def test_duplicate_validation_key_rejected_after_manifest_resign(self): + p = self.root / 'VALIDATION.json' + text = p.read_text().replace('"schema": 1', '"schema": 1, "schema": 1') + p.write_text(text) + self.resign() + self.check() + + def test_nonfinite_json_rejected_after_manifest_resign(self): + self.validation['unexpected_number'] = float('nan') + self.write_validation() + self.resign() + self.check() + + def test_validation_history_not_compared_to_finite_output(self): + self.validation['local_runs'] = {'old_count': 23, 'old_missing_objects': True} + self.write_validation() + self.resign() + self.check(accepted=True) + + def test_overflowed_json_number_rejected_after_manifest_resign(self): + p = self.root / 'VALIDATION.json' + text = p.read_text().replace('"schema": 1', '"unbounded": 1e999, "schema": 1') + p.write_text(text) + self.resign() + self.check() + + def test_review_cannot_transfer_after_proof_validation_manifest_resign(self): + (self.root / 'PROOF.md').write_bytes(b'New proof with stale old review.\n') + self.validation['proof_sha256'] = hashlib.sha256((self.root / 'PROOF.md').read_bytes()).hexdigest() + self.write_validation() + self.resign() + self.check() + + def test_each_review_proof_identity_is_enforced_after_resign(self): + original = json.loads(json.dumps(self.review)) + for key, wrong in [('bytes', True), ('git_blob', '0'*40), ('sha256', '0'*64), ('path', 'README.md')]: + with self.subTest(key=key): + self.review = json.loads(json.dumps(original)) + self.review['proof'][key] = wrong + (self.root / 'REVIEW_RECORD.json').write_text(json.dumps(self.review) + '\n') + self.resign() + self.check() + + def test_review_record_cannot_be_removed_and_manifest_resigned(self): + (self.root / 'REVIEW_RECORD.json').unlink() + self.resign() + self.check() + + +if __name__ == '__main__': + unittest.main() diff --git a/frontiers/cluster_exponential_weight_20260930/test_verify_sources.py b/frontiers/cluster_exponential_weight_20260930/test_verify_sources.py new file mode 100644 index 000000000..0cde3642d --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/test_verify_sources.py @@ -0,0 +1,112 @@ +"""Hostile custody controls in synthetic Git trees; no Gaussian claim is tested.""" +import copy +import hashlib +import json +from pathlib import Path +import subprocess +import tempfile +import unittest + +from verify_sources import no_duplicates, verify + + +class SourceCustodyTests(unittest.TestCase): + def setUp(self): + self.temp = tempfile.TemporaryDirectory() + self.root = Path(self.temp.name) + subprocess.run(["git", "init", "-q", str(self.root)], check=True) + entries, sources = [], [] + for label in ("MF", "SC"): + data = ("Synthetic " + label + " custody fixture\n").encode() + blob = self.git("hash-object", "-w", "--stdin", data=data).decode().strip() + name = "parent/" + label + ".md" + entries.append("100644 blob " + blob + "\t" + name + "\n") + sources.append({"id": label, "path": name, "git_blob": blob, + "sha256": hashlib.sha256(data).hexdigest(), "bytes": len(data)}) + sub = self.git("mktree", data="".join(e.replace("parent/", "") for e in entries).encode()).decode().strip() + tree = self.git("mktree", data=("040000 tree " + sub + "\tparent\n").encode()).decode().strip() + commit = self.git("-c", "user.name=Fixture", "-c", "user.email=fixture@example.invalid", + "commit-tree", tree, data=b"Synthetic fixture\n").decode().strip() + for source in sources: + source["commit"] = commit + self.manifest = {"schema": 1, "scientific_effect": "NONE", "sources": sources} + + def tearDown(self): + self.temp.cleanup() + + def git(self, *args, data=None): + return subprocess.run(["git", "-C", str(self.root), *args], input=data, + check=True, stdout=subprocess.PIPE, stderr=subprocess.PIPE).stdout + + def test_valid_historical_trees(self): + self.assertEqual(len(verify(self.root, self.manifest)["historical_sources_verified"]), 2) + + def test_nested_caller_has_full_tree_semantics(self): + nested = self.root / "frontiers" / "new-packet" + nested.mkdir(parents=True) + self.assertEqual(verify(nested, self.manifest), verify(self.root, self.manifest)) + + def test_wrong_path_even_with_correct_blob(self): + bad = copy.deepcopy(self.manifest) + bad["sources"][0]["path"] = "MF.md" + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_wrong_digest_or_size(self): + for key, wrong in (("sha256", "0" * 64), ("bytes", 1)): + bad = copy.deepcopy(self.manifest) + bad["sources"][0][key] = wrong + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_wrong_blob_in_real_commit(self): + bad = copy.deepcopy(self.manifest) + bad["sources"][0]["git_blob"] = bad["sources"][1]["git_blob"] + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_absent_commit_fails(self): + bad = copy.deepcopy(self.manifest) + bad["sources"][0]["commit"] = "0" * 40 + with self.assertRaises(subprocess.CalledProcessError): + verify(self.root, bad) + + def test_tree_must_not_pass_as_commit(self): + bad = copy.deepcopy(self.manifest) + tree = self.git("rev-parse", bad["sources"][0]["commit"] + "^{tree}").decode().strip() + for source in bad["sources"]: + source["commit"] = tree + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_local_replace_cannot_substitute_historical_commit(self): + empty_tree = self.git("mktree", data=b"").decode().strip() + empty_commit = self.git("-c", "user.name=Fixture", "-c", "user.email=fixture@example.invalid", + "commit-tree", empty_tree, data=b"No source paths\n").decode().strip() + self.git("replace", empty_commit, self.manifest["sources"][0]["commit"]) + bad = copy.deepcopy(self.manifest) + for source in bad["sources"]: + source["commit"] = empty_commit + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_boolean_schema_rejected(self): + bad = copy.deepcopy(self.manifest) + bad["schema"] = True + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_unsafe_path_fails(self): + for name in ("../parent/MF.md", "/parent/MF.md", "parent//MF.md"): + bad = copy.deepcopy(self.manifest) + bad["sources"][0]["path"] = name + with self.assertRaises(ValueError): + verify(self.root, bad) + + def test_duplicate_json_keys_fail(self): + with self.assertRaises(ValueError): + json.loads('{"schema":1,"schema":2}', object_pairs_hook=no_duplicates) + + +if __name__ == "__main__": + unittest.main() diff --git a/frontiers/cluster_exponential_weight_20260930/verify_packet.py b/frontiers/cluster_exponential_weight_20260930/verify_packet.py new file mode 100644 index 000000000..70b540e53 --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/verify_packet.py @@ -0,0 +1,135 @@ +"""Check local packet custody; this neither authenticates reviews nor proves mathematics.""" +import argparse +import hashlib +import json +import math +from pathlib import Path +import re +import stat +import sys + + +ESSENTIAL = {'PROOF.md', 'SOURCES.json', 'VALIDATION.json', 'REVIEW_RECORD.json'} + + +def unique_object(pairs): + result = {} + for key, value in pairs: + if key in result: + raise ValueError('duplicate JSON key: '+key) + result[key] = value + return result + + +def reject_constant(value): + raise ValueError('nonfinite JSON constant: '+value) + + +def finite_float(value): + result = float(value) + if not math.isfinite(result): + raise ValueError('nonfinite JSON number: '+value) + return result + + +def read_regular(path): + if not stat.S_ISREG(path.lstat().st_mode): + raise ValueError('not a regular, nonsymlink file: '+str(path)) + return path.read_bytes() + + +def read_json(data, label): + result = json.loads(data, object_pairs_hook=unique_object, parse_constant=reject_constant, + parse_float=finite_float) + if not isinstance(result, dict): + raise ValueError(label+' must be a JSON object') + if type(result.get('schema')) is not int or result['schema'] != 1: + raise ValueError(label+' must have integer schema 1') + if result.get('scientific_effect') != 'NONE': + raise ValueError(label+' scientific_effect must be NONE') + return result + + +def safe_name(value): + if (not isinstance(value, str) or not value or value in {'.', '..'} + or any(c in value for c in '/\\:\0')): + raise ValueError('unsafe or nonflat packet path: '+repr(value)) + return value + + +def check_identity(record, data, label): + if not isinstance(record, dict): + raise ValueError(label+' identity must be an object') + size = record.get('bytes') + if type(size) is not int or size < 0 or size != len(data): + raise ValueError(label+' byte count mismatch or invalid type') + actual = {'sha256': hashlib.sha256(data).hexdigest(), + 'git_blob': hashlib.sha1(b'blob '+str(len(data)).encode('ascii')+b'\0'+data).hexdigest()} + for key, length in [('sha256', 64), ('git_blob', 40)]: + value = record.get(key) + if not isinstance(value, str) or re.fullmatch('[0-9a-f]{'+str(length)+'}', value) is None: + raise ValueError(label+' invalid '+key) + if value != actual[key]: + raise ValueError(label+' '+key+' mismatch') + + +def verify(packet_root): + root = Path(packet_root) + if not stat.S_ISDIR(root.lstat().st_mode): + raise ValueError('packet root must be a nonsymlink directory') + manifest = read_json(read_regular(root / 'MANIFEST.json'), 'MANIFEST.json') + if manifest.get('manifest_self_excluded') is not True: + raise ValueError('MANIFEST.json must explicitly exclude itself') + records = manifest.get('files') + if not isinstance(records, list) or not records: + raise ValueError('MANIFEST.json files must be a nonempty list') + indexed = {} + for record in records: + if not isinstance(record, dict): + raise ValueError('manifest file entry must be an object') + name = safe_name(record.get('path')) + if name == 'MANIFEST.json' or name in indexed: + raise ValueError('self or duplicate manifest path: '+name) + indexed[name] = record + if not ESSENTIAL <= indexed.keys(): + raise ValueError('manifest lacks essential packet files') + observed = {p.name for p in root.iterdir()} + expected = set(indexed) | {'MANIFEST.json'} + if observed != expected: + raise ValueError('packet membership mismatch; missing='+repr(sorted(expected-observed)) + +'; undeclared='+repr(sorted(observed-expected))) + payloads = {} + for name, record in indexed.items(): + data = read_regular(root / name) + check_identity(record, data, name) + payloads[name] = data + proof = payloads['PROOF.md'] + validation = read_json(payloads['VALIDATION.json'], 'VALIDATION.json') + # VALIDATION is a retained author execution/history record, not finite_checks output. + if validation.get('proof_sha256') != hashlib.sha256(proof).hexdigest(): + raise ValueError('VALIDATION.json proof_sha256 does not bind current PROOF.md') + reviews = read_json(payloads['REVIEW_RECORD.json'], 'REVIEW_RECORD.json') + proof_record = reviews.get('proof') + if not isinstance(proof_record, dict) or proof_record.get('path') != 'PROOF.md': + raise ValueError('REVIEW_RECORD.json must bind PROOF.md') + check_identity(proof_record, proof, 'REVIEW_RECORD.json proof') + return {'schema': 1, 'scientific_effect': 'NONE', 'packet_files_verified': len(indexed), + 'validation_proof_identity_verified': True, 'review_proof_identity_verified': True, + 'remote_review_authenticity_checked': False, 'mathematical_acceptance': False} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--packet-root', type=Path, default=Path(__file__).absolute().parent) + args = parser.parse_args() + try: + result = verify(args.packet_root) + except (ValueError, OSError) as error: + print('packet identity failed: '+str(error), file=sys.stderr) + return 1 + print(json.dumps(result, indent=2, sort_keys=True)) + return 0 + + +if __name__ == '__main__': + sys.exit(main()) diff --git a/frontiers/cluster_exponential_weight_20260930/verify_sources.py b/frontiers/cluster_exponential_weight_20260930/verify_sources.py new file mode 100644 index 000000000..a8ac3636c --- /dev/null +++ b/frontiers/cluster_exponential_weight_20260930/verify_sources.py @@ -0,0 +1,72 @@ +#!/usr/bin/env python3 +"""Verify immutable upstream Git trees; byte custody is not mathematical acceptance.""" +import argparse +import hashlib +import json +from pathlib import Path, PurePosixPath +import re +import subprocess + + +def no_duplicates(pairs): + result = {} + for key, value in pairs: + if key in result: + raise ValueError("duplicate JSON key: " + key) + result[key] = value + return result + + +def git(repo, *args): + return subprocess.run(["git", "--no-replace-objects", "-C", str(repo), *args], check=True, + stdout=subprocess.PIPE, stderr=subprocess.PIPE).stdout + + +def verify(repo, manifest): + if type(manifest.get("schema")) is not int or manifest["schema"] != 1 or manifest.get("scientific_effect") != "NONE": + raise ValueError("unexpected source manifest scope") + sources = manifest.get("sources") + if not isinstance(sources, list) or [s.get("id") for s in sources] != ["MF", "SC"]: + raise ValueError("exactly the MF and SC interfaces are required") + verified = [] + for source in sources: + commit, blob, digest = (source.get(k, "") for k in ("commit", "git_blob", "sha256")) + if not (re.fullmatch("[0-9a-f]{40}", commit) + and re.fullmatch("[0-9a-f]{40}", blob) + and re.fullmatch("[0-9a-f]{64}", digest)): + raise ValueError("invalid immutable identity") + name = source.get("path", "") + path = PurePosixPath(name) + if not name or path.is_absolute() or ".." in path.parts or str(path) != name: + raise ValueError("invalid repository-relative source path") + if type(source.get("bytes")) is not int or source["bytes"] < 1: + raise ValueError("invalid source size") + if git(repo, "cat-file", "-t", commit).strip() != b"commit": + raise ValueError(source["id"] + ": expected an actual historical commit") + # --full-tree is essential: callers may run from a nested packet directory. + listing = git(repo, "ls-tree", "--full-tree", "-z", commit, "--", name) + expected = ("100644 blob " + blob + "\t" + name + "\0").encode() + if listing != expected: + raise ValueError(source["id"] + ": historical commit/path/blob mismatch") + data = git(repo, "cat-file", "blob", blob) + if len(data) != source["bytes"] or hashlib.sha256(data).hexdigest() != digest: + raise ValueError(source["id"] + ": source bytes mismatch") + if hashlib.sha1(b"blob " + str(len(data)).encode() + b"\0" + data).hexdigest() != blob: + raise ValueError(source["id"] + ": Git blob content mismatch") + verified.append({"id": source["id"], "commit": commit, "path": name, + "git_blob": blob, "sha256": digest, "bytes": len(data)}) + return {"scientific_effect": "NONE", "historical_sources_verified": verified} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--repo-root", required=True) + args = parser.parse_args() + manifest = json.loads(Path(__file__).with_name("SOURCES.json").read_text(), + object_pairs_hook=no_duplicates, + parse_constant=lambda value: (_ for _ in ()).throw(ValueError(value))) + print(json.dumps(verify(Path(args.repo_root).resolve(), manifest), sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/frontiers/downstream_gate_20260925/GRAPH.json b/frontiers/downstream_gate_20260925/GRAPH.json index fa6ba930f..220b56867 100644 --- a/frontiers/downstream_gate_20260925/GRAPH.json +++ b/frontiers/downstream_gate_20260925/GRAPH.json @@ -332,26 +332,58 @@ "math.rn-region.mesoscopic-scaled-annulus": { "layer": "D5", "kind": "region", - "classification": "OPEN_ACTIVE", + "classification": "PROVED_REVIEWED", "controlling": false, "fingerprint": "x=r y with A<=|y|<=B away from pins", - "notes": "Owned by Math- PR7 reduction; exact J0 ledger still open" + "notes": "Reviewed only at D5-d with R >= B (covers the reviewed fixed annulus); see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-d with R >= B (covers the reviewed fixed annulus)", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.punctured-pin-proof", + "math.d5-component.punctured-pin-review-111", + "math.d5-component.punctured-pin-continuum-review", + "math.d5-component.punctured-pin-continuum-check", + "math.d5-component.collar-proof", + "math.d5-component.collar-review", + "math.d5-component.annulus-proof", + "math.d5-component.annulus-review" + ] }, "math.rn-region.pin-collision": { "layer": "D5", "kind": "region", - "classification": "OPEN_ACTIVE", + "classification": "PROVED_REVIEWED", "controlling": false, "fingerprint": "near +-e1/2 scaled pin sites", - "notes": "Complement of fixed-remote and fixed scaled annulus" + "notes": "Reviewed only at D5-a, D5-b, D5-d; see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-a, D5-b, D5-d", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.punctured-pin-proof", + "math.d5-component.punctured-pin-review-111", + "math.d5-component.punctured-pin-continuum-review", + "math.d5-component.punctured-pin-continuum-check", + "math.d5-component.collar-proof", + "math.d5-component.collar-review" + ] }, "math.rn-region.intermediate-r-to-rho": { "layer": "D5", "kind": "region", - "classification": "OPEN_ACTIVE", + "classification": "PROVED_REVIEWED", "controlling": false, "fingerprint": "r<<|x|<2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-e (I3/I4, window I_r)", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.intermediate-window-proof", + "math.d5-component.i5-review-xai", + "math.d5-component.i5-review-claude" + ] }, "math.rn-region.witness-collision": { "layer": "D5", @@ -359,7 +391,13 @@ "classification": "OPEN_ACTIVE", "controlling": false, "fingerprint": "eta->0 mutual witness separation", - "notes": "Fixed-eta factorial moments do not cover shrinking separation" + "notes": "Off-pin fixed-separation second-moment evidence is reviewed and sharp global order is reviewed and merged through Math-#145, but the regional shrinking pin/witness-collision mechanism and its source-bound graph fold remain open.", + "scope": "catalog C6", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "OPEN", + "reading_rule": [ + "math.d5-component.offpin-second-moment-review" + ] }, "math.rn-mesoscopic-reduction": { "layer": "D5", @@ -457,6 +495,215 @@ "fingerprint": "regional-bypass-fixed-annulus-window-v1", "scope": "fixed d=2 scaled annulus; compact positive gaps; between-pin height window", "notes": "For this scope only, the reviewed analytic route bypasses CH-LIFT / Piece-2 / OBL-H5-JETMOD. Historical predicates remain OPEN/ABSENT in their original replay scopes." + }, + "math.d5-pin-neighborhood-first-moment": { + "layer": "D5", + "kind": "candidate", + "classification": "PROVED_REVIEWED", + "controlling": false, + "scope": "D5-a..D5-f of RECONCILIATION.md section 2; existential constants; fixed T; compact marks with k >= k_- > 0; d = 2", + "source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md" + }, + "math.d5-component.punctured-pin-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "f972e50bc07674ebce9d971d1a1bd4dba2f8036dde37ecb57d8a44a35650f76a", + "source": "reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md", + "review_basis": [ + { + "review": "#111", + "provider": "xAI", + "verdict": "ACCEPT algebra; continuum HOLD", + "depth": "algebraic" + }, + { + "review": "d5_punctured_pin_continuum_claude_20260929", + "provider": "Anthropic", + "verdict": "ACCEPT existential", + "depth": "full" + } + ] + }, + "math.d5-component.punctured-pin-review-111": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "668d74825f156c929177bc1a3481cf421de4dd7874e7b25b2aece6c10aa7dc0e", + "source": "reviews/d5_punctured_pin_nonauthor_20260928/REVIEW.md" + }, + "math.d5-component.punctured-pin-continuum-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "Anthropic", + "fingerprint": "124c472cc064ba24021cb4bb8da1002836a4ce26c8df506228f5c20609f6575a", + "source": "reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md" + }, + "math.d5-component.punctured-pin-continuum-check": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_checker", + "author_provider": "Anthropic", + "fingerprint": "57bc87d11271b47ebccec6ea73abf823fa90e3a8c3c7d7cd2165bdc6f7e4175c", + "source": "reviews/d5_punctured_pin_continuum_claude_20260929/pp_continuum_check.py" + }, + "math.d5-component.collar-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "794babe0fd039c3a93c401c0b8978316331af009fad45f953e63fbe7e1e31aab", + "source": "reviews/d5_local_collar_20260928/COLLAR_PROOF.md", + "review_basis": [ + { + "review": "d5_collar_count_20260928", + "provider": "not named in record", + "verdict": "ACCEPT existential", + "depth": "full" + }, + { + "review": "Math-#105 comment 5872932723 (external)", + "provider": "Anthropic", + "verdict": "CONFIRMED", + "depth": "full" + } + ] + }, + "math.d5-component.collar-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "not named in record", + "fingerprint": "8c8c30cc3b19601a7421341d5da8cb6c982fbc1fcc5ae4791acafa11cc81ee4e", + "source": "reviews/d5_collar_count_20260928/REVIEW.md" + }, + "math.d5-component.annulus-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05", + "source": "frontiers/rn_annulus_bridge_20260925/PROOF.md", + "review_basis": [ + { + "review": "pr28_annulus_bridge_nonauthor_20260925", + "provider": "xAI", + "verdict": "ACCEPT R1-R7", + "depth": "full" + } + ] + }, + "math.d5-component.annulus-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "69ea7a479a9e8a5150cbe07b373772ab22413a1d81194d5a6a78adf6ada5f03f", + "source": "reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md" + }, + "math.d5-component.intermediate-window-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "b3eb9456d7058b5b78149cfca4e072679beda85cd1a0293d209664a542db66cf", + "source": "frontiers/intermediate_window_20260928/PROOF.md", + "review_basis": [ + { + "review": "d5_i5_planar_f_20260928", + "provider": "xAI", + "verdict": "ACCEPT existential", + "depth": "full" + }, + { + "review": "d5_intermediate_window_claude_20260928", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full" + } + ] + }, + "math.d5-component.i5-review-xai": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "527d66e72b079ef7c854987f7e9d44f2252b81aa0e8d9e7ef3b45deb02388659", + "source": "reviews/d5_i5_planar_f_20260928/REVIEW.md" + }, + "math.d5-component.i5-review-claude": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "Anthropic", + "fingerprint": "fb3207802bf2d92ad138bc338a36f39bcd3a94440669b9aba9558e566f767976", + "source": "reviews/d5_intermediate_window_claude_20260928/REVIEW.md" + }, + "math.d5-component.remote-window-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "source": "frontiers/remote_window_20260924/PROOF.md", + "review_basis": [ + { + "review": "STATUS D4 (main#76, external)", + "provider": "nonauthor", + "verdict": "ACCEPT", + "depth": "full" + } + ] + }, + "math.d5-component.offpin-second-moment-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision", + "component_role": "review_record", + "author_provider": "nonauthor", + "fingerprint": "43beb4ef7da5bf71e15164d5c4bc1862db387d83f906aebdf9831625b1ca6369", + "source": "reviews/d5_offpin_second_moment_20260928/REVIEW.md" } }, "edges": [ @@ -609,6 +856,186 @@ "to": "hist.OBL-H5-JETMOD", "required": false, "relation": "regional_bypass_only" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.annulus-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.annulus-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.intermediate-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.i5-review-xai", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.i5-review-claude", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.remote-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.annulus-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.annulus-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.intermediate-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.i5-review-xai", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.i5-review-claude", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.d5-component.offpin-second-moment-review", + "required": true, + "relation": "requires_evidence" } ] } diff --git a/frontiers/downstream_gate_20260925/RESULTS.json b/frontiers/downstream_gate_20260925/RESULTS.json index 65b2bdb3f..657594914 100644 --- a/frontiers/downstream_gate_20260925/RESULTS.json +++ b/frontiers/downstream_gate_20260925/RESULTS.json @@ -6,7 +6,7 @@ "hist.rnu_env.py" ], "illegal_controlling_count": 0, - "open_or_author_side_count": 20 + "open_or_author_side_count": 17 }, "d0_ci_unblock": { "ci_error": "PROBLEM packet: unexpected files ['DOWNSTREAM_CROSSWALK_20260925.md']", @@ -41,30 +41,32 @@ { "classification": "OPEN_ACTIVE", "coverage_source": null, + "fingerprint": "eta->0 mutual witness separation", + "id": "math.rn-region.witness-collision", + "notes": "Off-pin fixed-separation second-moment evidence is reviewed and sharp global order is reviewed and merged through Math-#145, but the regional shrinking pin/witness-collision mechanism and its source-bound graph fold remain open." + } + ], + "proved_reviewed_regions": [ + { + "classification": "PROVED_REVIEWED", + "coverage_source": null, "fingerprint": "r<<|x|<2, numerical, elder-selection, JETMOD, or witness-collision closure." }, { - "classification": "OPEN_ACTIVE", + "classification": "PROVED_REVIEWED", "coverage_source": null, "fingerprint": "x=r y with A<=|y|<=B away from pins", "id": "math.rn-region.mesoscopic-scaled-annulus", - "notes": "Owned by Math- PR7 reduction; exact J0 ledger still open" + "notes": "Reviewed only at D5-d with R >= B (covers the reviewed fixed annulus); see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure." }, { - "classification": "OPEN_ACTIVE", + "classification": "PROVED_REVIEWED", "coverage_source": null, "fingerprint": "near +-e1/2 scaled pin sites", "id": "math.rn-region.pin-collision", - "notes": "Complement of fixed-remote and fixed scaled annulus" - }, - { - "classification": "OPEN_ACTIVE", - "coverage_source": null, - "fingerprint": "eta->0 mutual witness separation", - "id": "math.rn-region.witness-collision", - "notes": "Fixed-eta factorial moments do not cover shrinking separation" + "notes": "Reviewed only at D5-a, D5-b, D5-d; see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure." } ] }, diff --git a/frontiers/downstream_gate_20260925/SOURCE_FILES.json b/frontiers/downstream_gate_20260925/SOURCE_FILES.json index 19363814c..1fa1e59e4 100644 --- a/frontiers/downstream_gate_20260925/SOURCE_FILES.json +++ b/frontiers/downstream_gate_20260925/SOURCE_FILES.json @@ -5,16 +5,16 @@ "sha256": "2423fa7d3519a5ba06739cb43cc72e26db70020350338a1fa508e5adfb109643" }, "GRAPH.json": { - "bytes": 22830, - "sha256": "53057a110d64e8bf135a19b8e7681a594c3c3fda18b1133136ba1b0ebba7ce20" + "bytes": 38753, + "sha256": "8822e9618678321a342d69cd0b8ae6552de1b5d578c331de5072b2892ee9dd09" }, "README.md": { "bytes": 4025, "sha256": "13fc1accebebd6017f34b0e5473bf6338f2604948c25a002051b5f0620fc327a" }, "RESULTS.json": { - "bytes": 6535, - "sha256": "a513d80f0e54e229955cd2f63e1119e029ee13105c5531359cda1cb411801545" + "bytes": 7095, + "sha256": "0b95624e08e3eff7ca386a68534960b89ef63b10312ad5120159c0e29961264a" }, "SCOPE.md": { "bytes": 2218, @@ -29,8 +29,8 @@ "sha256": "393d407c2e35d2b6f3230f5a60a9aaa7691172544d3b59f72cc880c81be20701" }, "hard_gate.py": { - "bytes": 26557, - "sha256": "1954723d143a3d68cbf3834b2c18e4198fd89ba49145f012bdc425c80050d052" + "bytes": 26735, + "sha256": "a78f3e25f3b0cfe113e618a4c31a7a25d7f22af638c46dec1ecba221fa333ac8" }, "patches/d0_packet_allowlist.patch": { "bytes": 3096, @@ -41,8 +41,8 @@ "sha256": "276e69c2b29e012c386db615ed64e24badf3726a9f122fd9d77776a9c4f00711" }, "test_hard_gate.py": { - "bytes": 22691, - "sha256": "0f28ed996a50a1676b91a17198cf090086976e9eaf6a61aa9652f03acc427cf0" + "bytes": 23161, + "sha256": "c444e6c347da680aff5bef7f43900e45d52a0cc45081fb7b4def45266d97aac1" }, "test_transition_contract.py": { "bytes": 12679, diff --git a/frontiers/downstream_gate_20260925/hard_gate.py b/frontiers/downstream_gate_20260925/hard_gate.py index 52ec00305..3a334f11e 100644 --- a/frontiers/downstream_gate_20260925/hard_gate.py +++ b/frontiers/downstream_gate_20260925/hard_gate.py @@ -486,6 +486,7 @@ def d4_region_complement(graph: dict[str, Any]) -> dict[str, Any]: """Explicit D4/D5 region map with coverage attributed to its exact source.""" fixed_remote = [] other_scoped = [] + proved_reviewed = [] open_regions = [] for nid, node in sorted(graph['nodes'].items()): if node.get('kind') != 'region': @@ -502,11 +503,14 @@ def d4_region_complement(graph: dict[str, Any]) -> dict[str, Any]: fixed_remote.append(entry) else: other_scoped.append(entry) + elif node['classification'] == 'PROVED_REVIEWED': + proved_reviewed.append(entry) else: open_regions.append(entry) return { 'covered_by_fixed_remote_candidate': fixed_remote, 'covered_by_other_scoped_candidates': other_scoped, + 'proved_reviewed_regions': proved_reviewed, 'open_complement': open_regions, 'no_event_to_expectation_reversal': True, 'legacy_24jet_discharged': False, @@ -657,4 +661,4 @@ def main() -> None: if __name__ == '__main__': - main() \ No newline at end of file + main() diff --git a/frontiers/downstream_gate_20260925/test_hard_gate.py b/frontiers/downstream_gate_20260925/test_hard_gate.py index 1c8c9f63e..0c8263e96 100644 --- a/frontiers/downstream_gate_20260925/test_hard_gate.py +++ b/frontiers/downstream_gate_20260925/test_hard_gate.py @@ -226,6 +226,7 @@ def test_d4_region_complement(self): regions = m.d4_region_complement(self.graph) covered_ids = {r['id'] for r in regions['covered_by_fixed_remote_candidate']} scoped_ids = {r['id'] for r in regions['covered_by_other_scoped_candidates']} + reviewed_ids = {r['id'] for r in regions['proved_reviewed_regions']} open_ids = {r['id'] for r in regions['open_complement']} self.assertEqual(covered_ids, {'math.rn-region.fixed-remote'}) self.assertEqual(scoped_ids, {'math.rn-region.fixed-annulus-window'}) @@ -233,10 +234,18 @@ def test_d4_region_complement(self): regions['covered_by_other_scoped_candidates'][0]['coverage_source'], 'math.rn-fixed-annulus-window', ) - self.assertIn('math.rn-region.mesoscopic-scaled-annulus', open_ids) - self.assertIn('math.rn-region.pin-collision', open_ids) - self.assertIn('math.rn-region.intermediate-r-to-rho', open_ids) + self.assertEqual(reviewed_ids, { + 'math.rn-region.mesoscopic-scaled-annulus', + 'math.rn-region.pin-collision', + 'math.rn-region.intermediate-r-to-rho', + }) + self.assertTrue(reviewed_ids.isdisjoint(open_ids)) self.assertIn('math.rn-region.witness-collision', open_ids) + witness = next(r for r in regions['open_complement'] + if r['id'] == 'math.rn-region.witness-collision') + self.assertIn('sharp global order is reviewed and merged through Math-#145', + witness['notes']) + self.assertNotIn('sharp full-window order remain open', witness['notes']) self.assertFalse(regions['legacy_24jet_discharged']) self.assertTrue(regions['no_event_to_expectation_reversal']) @@ -445,4 +454,4 @@ def test_duplicate_edge_fails_closed(self): if __name__ == '__main__': - unittest.main() \ No newline at end of file + unittest.main() diff --git a/frontiers/planar_cubic_cluster_20260929/DEEP_TRANSVERSE_EXCLUSION.md b/frontiers/planar_cubic_cluster_20260929/DEEP_TRANSVERSE_EXCLUSION.md new file mode 100644 index 000000000..309a38cab --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/DEEP_TRANSVERSE_EXCLUSION.md @@ -0,0 +1,95 @@ +# Addendum: an extra nearby critical point forces a soft transverse jet + +Object: OA-PLANAR-CUBIC-CLUSTER-20260929-D1. +Author: OpenAI / GPT-6 Astra Pro, 29 September 2026. +Disposition: author-side deterministic proof candidate; nonauthor review OPEN. +Scientific effect NONE. This addendum leaves PROOF.md unchanged. It discharges the deterministic part of the follow-up proposed in Math-#157 comment5900564419; the weighted Gaussian derivative-tail estimate in that comment is NOT established here. + +## Statement + +Fix R>=1, k>0 and K>0. Let f be C^4 on a neighborhood of the closed square [-Rr,Rr]^2, with every coordinate derivative of orders one through four bounded in absolute value by K. Retain exactly the two pins + + M=(-r/2,0), S=(r/2,0), + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Assume r>0 and + + r K <= k/(4R), + A=f_zz(0,0), L_R=8RK+56RK^2/k. (D1) + +**Lemma D.** If A < -L_R r, the square contains no critical point other than M,S. Consequently, if H_f(M) is negative definite and ANY additional critical point lies in that square, then + + -L_R <= A/r < K/2, + |f_xxz(0)|, |f_xzz(0)|, |f_zzz(0)| <= K. (D2) + +The conclusion concerns all additional critical points, not only window points. No global Morse assumption or Gaussian probability estimate is used. + +## Proof + +Suppose lambda=-A>L_R r. The derivative bound gives, throughout the square, + + f_zz(x,z) <= -lambda+2RK r <= -lambda/2. (D3) + +For g(x)=f_z(x,0), the two gradient pins give g(-r/2)=g(r/2)=0 and |g''|<=K. The two-node interpolation remainder, also for x outside the pin interval but within [-Rr,Rr], yields + + |g(x)| <= (K/2)|x-r/2||x+r/2| <= KR^2 r^2. (D4) + +This follows directly by subtracting the quadratic with the same value at x and applying Rolle's theorem twice, with repeated nodes handled by continuity. Rolle applied once to the two pin zeros also supplies xi in (-r/2,r/2) with g'(xi)=0, so + + |f_xz(x,0)| <= K(R+1/2)r <= (3/2)RK r. (D5) + +Put rho=4R^2 K r^2/lambda. Since lambda>8RK r, rho0 and f_z(x,rho)<0 for every x in [-Rr,Rr]. Strict decrease in z therefore gives a unique zero z=zeta(x), with |zeta(x)|=11k>0. Hence psi' is strictly convex on [-Rr,Rr]. It already has zeros at -r/2 and r/2. A strictly convex function cannot have three distinct zeros: the middle would lie strictly below the chord joining the outer two. Thus psi' has only those two zeros, and the graph contains only M,S as critical points. This proves the first assertion. + +For the consequence, the existence of an additional critical point implies A>=-L_R r by contraposition. Negative definiteness at M gives f_zz(M)<0, while |f_zz(0)-f_zz(M)|<=Kr/2, so A/r0} r^-3 E[N_{r,R}; Theta_r outside rectangle] + <= C_p(1+K)^(-p). (D12) + +The pointwise containment is applied only when N_{r,R}>0; otherwise the counted integrand is zero. The global count N_r bounds the local count. This explains exactly how the proposed derivative-weighted insertion would supply a missing rare-intensity tail, without dividing an unconditioned Gaussian tail by r^3. + +(D11) is an OPEN input, not a consequence asserted from the new count-exponential candidate #157. Further spatial and mixed local/remote estimates remain separate. Neither (D10) nor the conditional implication (D12) establishes uniqueness of the global cluster law. + +## Finite checks + +The companion test_deep_exclusion.py checks the constants of (D3)-(D9) over an exact rational grid, the graph sign bracket, and (D7) against explicitly reducible polynomial potentials. These are finite checks of the displayed algebra, not a formal proof of the implicit-function or convexity argument. Review requested: all of Lemma D and the conditional-only interpretation of (D11)-(D12). diff --git a/frontiers/planar_cubic_cluster_20260929/MANIFEST.json b/frontiers/planar_cubic_cluster_20260929/MANIFEST.json new file mode 100644 index 000000000..111e4a191 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/MANIFEST.json @@ -0,0 +1,73 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "NONAUTHOR_REVIEW_OPEN", + "files": [ + { + "path": "DEEP_TRANSVERSE_EXCLUSION.md", + "bytes": 6174, + "sha256": "95c03e59262cb9607a21a296a6837932839dd47ec8e15e67bdfa46782adbcd5a", + "git_blob": "309a38cab16ef67dd909351f40a48e1d83ba2c28" + }, + { + "path": "PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758" + }, + { + "path": "README.md", + "bytes": 3053, + "sha256": "785c30cf471273d9a4191572cd70b2c93b0d4c6daba4bc130ccab3b4e5a42189", + "git_blob": "07e319b4e0be13cae4d4e8853398e678c549ed69" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 2047, + "sha256": "1ac5692687f97bcc874372bfc59d5b9c581346adf0671e6084913e9d4bdbbd74", + "git_blob": "474974ba0997d6a6ed476d5d2f61121a8e0d654b" + }, + { + "path": "RESULTS.json", + "bytes": 294, + "sha256": "6b029e4e066386ae6f54d52b21b702cb3a7e9e7d6ad9f8855c8ed5d09160fb5c", + "git_blob": "b7aa090e792afbb2e5316b37c241a826bbea6e33" + }, + { + "path": "SOURCES.json", + "bytes": 1137, + "sha256": "382a7c773f9cb321ad48a2f57e4ab6d55899d914303266b94be051a263d11774", + "git_blob": "a62d49679f296589f9686248fe5630d6aa7e8e40" + }, + { + "path": "cubic.py", + "bytes": 6213, + "sha256": "d8567e99fee9ac2a6204d44367e91ab38c5a188c28a887337a11323fe6bd7932", + "git_blob": "570e369da96a3bd08f188a2c28a540ffae5e6d08" + }, + { + "path": "test_cubic.py", + "bytes": 5593, + "sha256": "eda7d69dbdfccd3fd7d4b17f692ed55a18629730ef0451b203de65d107438299", + "git_blob": "081c698af198f6803ef79cfa0f930087027c4b6b" + }, + { + "path": "test_deep_exclusion.py", + "bytes": 2182, + "sha256": "8cdc2622a808f453109a53ec287ca74c7885c3d0d1eafbf12aad1dd539764fbe", + "git_blob": "e5b89fd9881b22ec18078245d0ea4fe38e747690" + }, + { + "path": "test_verify.py", + "bytes": 2271, + "sha256": "d96f277e7d6fe2d19c9f13653b16a02c09065afa099f64c56a69e52f404bbbbf", + "git_blob": "441b2e38409e69f9d625eeac6eadf4828b1aad35" + }, + { + "path": "verify.py", + "bytes": 3400, + "sha256": "402be79b490983f1f4d53f0e0981655e4a9d0cefecdbada7ad00d8a484d7ff79", + "git_blob": "0fa5833c0f000dba7cc939835514dabc94301c72" + } + ] +} diff --git a/frontiers/planar_cubic_cluster_20260929/PROOF.md b/frontiers/planar_cubic_cluster_20260929/PROOF.md new file mode 100644 index 000000000..bb446d08d --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/PROOF.md @@ -0,0 +1,270 @@ +# Exact planar cubic cluster classification and a Gaussian soft-layer coefficient + +Object: OA-PLANAR-CUBIC-CLUSTER-20260929-v1. +Author: OpenAI / GPT-6 Astra Pro, foreground continuation session, 29 September 2026. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW OPEN. +Scientific effect: NONE. No parent proof, governing status, Boolean, prize or review disposition is changed. Same GitHub account as other lanes; no organizational-independence credit. + +## 1. Scope and exact inputs + +This is a planar result, d=2. Fix the exact variance-one periodized Gaussian field on the torus of side L>0, one orthonormal frame, a birth b and a gap mark k>0. Coordinates are (x,z). The pins are + + M=(-r/2,0), S=(r/2,0), + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r is the original Gaussian regression law and 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. In particular Z_r is the FULL normalizer, not one restricted to the event below. All results are for fixed b,k,L and frame. No result uniform as these parameters escape compacts is asserted. + +The exact sources at commit 7a1cb09a9d58d179393e6d29146252f714b9c9bf are in SOURCES.json: + +- [L] frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md, equations (L6)-(L13): pinned cubic, its C^2 Taylor remainder, the full ten-dimensional planar jet rank and extra-conditioned C^4 moments. +- [P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md, Sections 1-5: exact covariance and law, contact frame, positive Fourier spectrum, smooth Gaussian regression and full normalizer (5.4). + +Only these displayed interfaces are consumed; their author-time headers are not being used as a current acceptance register. This note neither revalidates those inputs nor needs the global C6 factorial upper theorem. The results below are new consumers of the pinned Taylor and Gaussian interfaces, not a claimed novelty audit of polynomial critical-point theory. + +## 2. Complete deterministic classifier + +For real s,a,beta,c and k>0, take exactly [L, (L6)]: + + P(X,Z)=2kX^3-3kX/2-k/2+(s/2)Z^2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3. (C1) + +It has the exact values 0,-k and zero gradients at (-1/2,0),(1/2,0). Set + + u=X+aZ/(12k), + B=beta-a^2/(12k), + D=(c-a beta/(4k)+a^3/(72k^2))/2. (C2) + +This determinant-one shear fixes both pins and gives + + P=2ku^3-3ku/2-k/2+(s/2)Z^2+(B/2)uZ^2+(D/3)Z^3. (C3) + +Both Hessian inertia and determinant are preserved under the shear. At the pins the determinants are 3k(B-2s), 3k(B+2s), respectively; the axial entries are -6k,6k. Thus the first pin is a strict maximum and the second a saddle exactly when + + s < -|B|/2. (C4) + +Write D_k for this open parameter set in theta=(s,a,beta,c), using (C2). On it put + + T=-(s-B)^2(B+2s)/(12k) >= 0, + w(theta)=9k^2(4s^2-B^2)>0. (C5) + +Here w is the positive endpoint determinant product. Let n(theta) count ALL additional critical points of this cubic, anywhere in R^2, with height strictly between -k and 0. + +**Theorem C (exact classifier).** Every such critical point is a nondegenerate saddle. There are at most two, and + + if s>B: n=2 when D^2=T; + if s<=B: n=1 when D^2>T, and n=0 when D^2<=T. (C6) + +These formulas include B=0, D=0, s=B, conic tangencies, and degree-drop cases. Equality is NOT rounded into the open height window. In particular the old witness (s,a,beta,c)=(-3k/2,0,-2k,0) has n=2 and w=45k^4. + +### Proof: conic, line, height and inertia + +From (C3), stationary points obey + + Q=6k(u^2-1/4)+(B/2)Z^2=0, + Z(s+Bu+DZ)=0. (C7) + +For Z=0 these are precisely the two pins. Every extra point is therefore on the conic Q=0 and the line s+Bu+DZ=0, with Z!=0. At such a point direct substitution gives + + P=-k(u+1/2)+(s/6)Z^2, + det Hess P=-3k(B+4su). (C8) + +For example the Hessian on the line is [[12ku,BZ],[BZ,DZ]], whose determinant reduces using Q=0 to the second identity. The height identity follows also by Euler's identity for the cubic, quadratic and linear homogeneous parts. + +First take B!=0. On Q=0, + + Z^2=-12k(u^2-1/4)/B, + P/k=-(u+1/2)[1+(2s/B)(u-1/2)]. (C9) + +If B>0, extra real points have -1/2-k is equivalent to + + -1/21/2. The right branch u>1/2 lies below -k. On the left branch the height is negative, and the open window is equivalent to + + u_w0: for B<0 use u<-1/2 and B-2s>0; for B>0, at the largest allowed u=u_w it equals -B-2s>0, and it increases as u decreases because s<0. Equation (C8) therefore proves that every in-window point is a nondegenerate saddle. In particular a conic tangency can never occur inside this window. + +On the positive-Z branch of the window conic write + + Z(u)=sqrt(-12k(u^2-1/4)/B), + g(u)=(-s-Bu)/Z(u). + +The line condition is D=g(u) on that branch and -D=g(u) on the negative-Z branch. Differentiation, retaining the conic relation, gives + + g'(u)=det Hess P / (B Z(u)^3). (C12) + +For B>0, g decreases strictly from +infinity at u=-1/2 to sqrt(T) at u=u_w. Thus there is one extra window point precisely when |D|>sqrt(T). Since sg(u_w)} + 1{-D>g(u_w)}. + +If s>B, the starting value is negative: both branches contribute below the squared threshold, and only one contributes at or above it. If s<=B, the starting value is nonnegative: precisely one branch contributes strictly above the squared threshold, none at or below. This proves (C6) for B!=0. The endpoint identity g(u_w)^2=T follows from + + Z(u_w)^2=-3k(B+2s)/s^2, + -s-Bu_w=(B-s)(B+2s)/(2s). + +For B=0, Q forces u=+/-1/2. If D=0, s<0 forces Z=0, so there are no extras. Otherwise Z=-s/D and the two extra heights are s^3/(6D^2) and -k+s^3/(6D^2). Only the first can be in the window, precisely when D^2>-s^3/(6k)=T; its determinant is 6ks<0. This proves all B=0 cases. + +No root at infinity has been silently counted. For B,D!=0, eliminating Z gives + + (12kD^2+B^3)u^2+2B^2s u+Bs^2-3kD^2=0. (C13) + +If its leading coefficient vanishes, its linear coefficient is nonzero because B!=0 and s<0. It has one finite root, included by the monotone-branch argument. If D=0 and B!=0, the line fixes u=-s/B and Q supplies at most two values of Z. The B=D=0 case was already separated. This also proves the claimed upper bound on the number of extra points. QED. + +## 3. Window stability without an unnecessary global Morse hypothesis + +The transition set is + + Sigma_k={12kD^2+(s-B)^2(B+2s)=0} inside D_k. (C14) + +It is Lebesgue-null in (s,a,beta,c): after clearing fixed positive k denominators it is a nonzero polynomial (its c^2 coefficient is nonzero). No claim is made that the entire cubic is Morse outside Sigma_k: critical points below the window can be degenerate, and that causes no problem here. + +**Lemma S.** For theta in D_k outside Sigma_k, the window count is stable under sufficiently small C^2 perturbations on a sufficiently large fixed disk, provided both pins, their zero gradients and their values 0,-k are retained exactly. On a compact subset of D_k disjoint from Sigma_k, the disk and perturbation size can be chosen uniformly. + +**Proof.** Equations (C10)-(C11), and the B=0 case, give |u|<=3/2 for all critical points with heights in [-k,0]. From (C8), P>=-k also gives + + Z^2 <= 12k/|s|. (C15) + +A compact subset of D_k has |s| bounded below and a bounded. The inverse shear therefore bounds all these points in one disk. Choose a strictly larger disk. The only critical points at the boundary heights are the pins, unless theta is in Sigma_k. Theorem C makes all critical points inside the window nondegenerate; the two pin Hessians are nondegenerate by (C4). + +Choose disjoint small neighborhoods of those finitely many roots. In each, the implicit-function theorem gives one and only one nearby zero for a small C^2 perturbation. The pins stay the same by hypothesis. On the remaining compact set where the height is in a small enlargement of [-k,0], the gradient has a positive minimum; hence no new in-window zero is possible. Points of sufficiently different height cannot enter the window under a small C^0 perturbation. This argument uses no assumption on critical points at lower heights. A finite-cover argument gives uniformity over a compact parameter set avoiding Sigma_k. QED. + +For any fixed compact C subset of D_k, choose R_C large enough using (C15) and the shear. For each fixed R>=R_C, Lemma S applies pointwise for almost every theta in C. It is not necessary to assume C avoids Sigma_k for integration below, since that set is null. + +## 4. An explicit Gaussian law in every compact soft-jet sector + +Return to the exact field. At its midpoint define the physical jet + + J=(A,a,beta,c)=(f_zz,f_xxz,f_xzz,f_zzz), + Theta_r=(f_zz/r, f_xxz, f_xzz, f_zzz). (G1) + +The field remains conditioned on the original six endpoint observations. Let h_r(A,a,beta,c) be the density of J under Q_r. The nonsingular contact limit is + + U_0=(f,f_x,f_xx,f_xxx,f_z,f_xz), + U_0=(b,0,0,12k,0,0), (G2) + +and h_0 is the corresponding joint density of J under this contact law Q_0. [L] and [P] give full rank for (U_0,J), hence h_0 is a strictly positive Gaussian density. Their regression argument gives h_r->h_0 locally uniformly. + +Let A_0=f_zz under Q_0. The FULL normalizer limit from [P, (5.4)] is + + Z_r/r^2 -> z_0=36k^2 E_Q0[A_0^2 1{A_0<0}]>0. (G3) + +Fix any compact Borel C subset of D_k. For R>=R_C let N_{r,R} count additional critical points in the physical disk of radius Rr, excluding M,S, with heights strictly in (b-kr^3,b). + +**Theorem G (compact-sector limit).** As r decreases to zero, the finite measures + + mu_{r,C}(E,j)=r^-3 Q_r^W{Theta_r in E cap C, N_{r,R}=j} + +converge in full total variation on C times the nonnegative integers to + + d mu_C(theta,j) + = 1_C(theta) 1{j=n(theta)} + [w(theta)/z_0] h_0(0,a,beta,c) ds da d beta dc. (G4) + +The limit is supported on j=0,1,2. In particular, for j in {0,1,2}, + + r^-3 Q_r^W{Theta_r in C, N_{r,R}=j} -> Lambda_j(C), + Lambda_j(C)=z_0^-1 integral_C w h_0(0,a,beta,c) 1{n=j} dtheta. (G5) + +The same scaled probability for N_{r,R}>=3 tends to zero within C. If C has positive four-dimensional Lebesgue measure, its limiting total mass is positive, and the law of (Theta_r,N_{r,R}) conditional on Theta_r in C under Q_r^W converges in probability total variation to mu_C/mu_C(C times N_0). + +### Proof: retain the rare density before any remainder estimate + +Append J to the nonsingular endpoint frame U_r. Its covariance Gamma_r tends to a positive-definite ten-dimensional covariance at r=0: these are exactly all planar jet entries through degree three. Targets (v_r,rs,a,beta,c), theta in C, are bounded. For one unconditioned smooth Gaussian field F, the regression coupling + + f_r^theta = F + Cov(F,V_r) Gamma_r^-1(t_r(theta)-V_r(F)), + V_r=(U_r,J), t_r=(v_r,rs,a,beta,c), (G6) + +has exactly the required conditional law. The coefficient functions converge in C^4 and Gamma_r^-1 stays bounded. The averaged contact functionals are bounded by a fixed multiple of ||F||_{C^3}; [P]'s Fourier summability therefore yields, uniformly for small r and theta in C, + + ||f_r^theta||_{C^4} <= C_C(1+||F||_{C^4}) (G7) + +in this coupling, with all fixed moments finite. Internal Gaussian correlations are retained; no independence of a jet and its remainder is assumed. + +The pinned Taylor argument [L, (L7)] extends from radius 3 to any fixed radius R simply by changing its Taylor-remainder constant: the endpoint identities are unchanged, and powers of R enter only that constant. Thus + + ||(f_r^theta(rX,rZ)-b)/r^3 - P_theta||_{C^2(B_R)} + <= C_R r ||f_r^theta||_{C^4} -> 0 (G8) + +pathwise. For almost every theta in C, Lemma S gives N_{r,R}=n(theta) eventually. The endpoint Hessians divided by r converge to those of P_theta. Consequently + + W_r(f_r^theta)/r^4 -> w(theta), + W_r(f_r^theta)/r^4 <= C_C(1+||F||_{C^4})^4. (G9) + +This is the r^4 soft-sector determinant scale, not the typical r^2 weight scale used in the full normalizer. + +The change A=rs in the PHYSICAL four-dimensional jet density has Jacobian r. Disintegration therefore gives the density of mu_{r,C} at (theta,j) as + + 1_C(theta) [r^2/Z_r] h_r(rs,a,beta,c) + E[ (W_r(f_r^theta)/r^4) 1{N_{r,R}=j} ]. (G10) + +Indeed r^-3 times r from the jet density times r^4 from W_r divided by Z_r equals r^2/Z_r. This is the ledger r*r^4/r^2=r^3 before the r^-3 rescaling. + +For a fixed good theta, the l1 difference over j between the conditional vector inside (G10) and w(theta)delta_{n(theta)} tends to zero. It is bounded by the expectation of |W_r/r^4-w| plus twice the expectation of (W_r/r^4)1{N_{r,R}!=n}; (G7)-(G9) dominate both. On C the Gaussian density h_r is uniformly bounded, and r^2/Z_r is bounded by (G3). Dominated convergence in theta and F proves full l1 convergence of the joint densities, hence (G4). Normalization follows by positive limiting total mass. These limits can equivalently be proved along every sequence r_i->0, avoiding any common-null-set assertion for uncountably many conditional laws. QED. + +## 5. Finite nonempty coefficient integrals and actual global lower bounds + +Define, only for j=1,2, + + alpha_j=z_0^-1 integral_{D_k} w(theta) h_0(0,a,beta,c) + 1{n(theta)=j} dtheta. (G11) + +**Theorem F.** Both alpha_1 and alpha_2 are finite and strictly positive. They are explicit integrals, not numerical enclosures. With N_r the GLOBAL window count, + + liminf r^-3 Q_r^W{N_r>=2} >= alpha_2, + liminf r^-3 E_QrW[N_r(N_r-1)] >= 2 alpha_2, + liminf r^-3 E_QrW[N_r] >= alpha_1+2 alpha_2. (G12) + +**Proof of finiteness.** Put x=-s>0 and M=|B|. On the nonempty sector, (C6) gives either s>B, which forces xT. If x>=2M, then |s-B|>=x/2 and -B-2s>=x, so T>=x^3/(48k). Therefore every nonempty sector satisfies + + 0=1} ds <=12k^2 H^3 + <=384k^2 |B|^3+2304k^3 D^2. (G13) + +Both |B|^3 and D^2 are bounded by a fixed-degree polynomial in (a,beta,c), at fixed k. The slice h_0(0,a,beta,c) has Gaussian decay and hence all polynomial moments. This proves finiteness. Positivity follows from open neighborhoods of theta=(-3k/2,0,-2k,0), where n=2, and theta=(-k,0,0,2k), where n=1. The weight and the Gaussian density are strictly positive there. + +Take an increasing compact exhaustion C_m of D_k. Theorem G identifies the limits for each fixed C_m and a disk large enough for that compact. Two local points imply at least two global points, and the bounded statistic 1{N_{r,R}=1}+2 1{N_{r,R}=2} is at most N_r. Apply these inequalities, then (G5), and finally monotone convergence as m increases. This proves (G12) without any assertion that the compact exhaustion captures ALL scaled global intensity. QED. + +For evaluating (G11), the s integral is one-dimensional polynomial integration over sectors cut by the cubic T(s)=D^2 and the typed endpoint boundary. Its antiderivative is + + integral 9k^2(4s^2-B^2) ds =12k^2 s^3-9k^2 B^2 s. (G14) + +For checks: at D=0 and B<0, the n=2 slice integrates over B=3 transfer, RN/24-jet/JETMOD closure or formal Lean proof is made. + +Nonauthor review requested: +A. (C2)-(C14): exact shear, endpoint signs, height-window classification, saddle types, boundary and degree-drop cases. +B. Lemma S and (G6)-(G10): compact stability, full-field conditional coupling, the rare Jacobian, domination and joint total-variation limit. +C. (G11)-(G16): finite positive integrals, justified lower-bound exhaustion, birth factorization and the global non-claims. + +The standard-library checker compares (C6) with a separate exact conic-root calculation at 600 rational points; tests include exact boundaries, linear degeneration, shear identities and the r^3 ledger. Those finite controls do not prove the Gaussian or compactness steps. They are not substitutes for the three analytic review slices. diff --git a/frontiers/planar_cubic_cluster_20260929/README.md b/frontiers/planar_cubic_cluster_20260929/README.md new file mode 100644 index 000000000..07e319b4e --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/README.md @@ -0,0 +1,31 @@ +# Planar cubic cluster law: exact classification and a compact Gaussian sector + +OpenAI / GPT-6 Astra Pro. Author-side conditional proof candidate; nonauthor review OPEN. Scientific effect NONE. No governing status or parent source is changed. No self-merge or organizational-independence claim. + +The packet adds an exact zero/one/two classification for the full typed pinned cubic, a Gaussian r^3 coefficient and joint total-variation limit in every fixed compact soft-jet sector, finite positive nonempty coefficient integrals, and birth-height factorization of the limiting sector law. It does NOT identify the full global C6 limiting measure: uniform scaled tails outside compact jets and spatial regions remain unproved. + +Start with PROOF.md. The deterministic classifier is (C6); the conditional Gaussian law is (G4); explicit global LOWER bounds are (G12); birth factorization is (G15). In particular neither the local support {0,1,2} nor limiting sector birth independence is asserted globally. + +## Reproduction + +Python standard library only, with Git needed for full source verification: + + python -B -S verify.py + +Run in a checkout containing the exact source commit in SOURCES.json. This checks every declared source identity, the complete flat packet inventory, tests in normal and optimized Python, deterministic output, eight rejected semantic mutants and unknown-label rejection. Hosted full-repository and Lean checks remain separate; this packet is not Lean-formalized. + + python -B -S verify.py --local-only + +The second command explicitly reports source_pins_checked=false and does not authenticate the project source Git objects. It is appropriate for the standalone download only. Source preservation and full hosted checks are not waived. + +The independent finite oracle counts exact conic roots, including double roots, linear degree drops and open endpoints, without floating point or calls to the classifier's threshold formula. The code does not evaluate the Gaussian coefficients numerically and does not prove the analytic limit. + +## Review requested + +A: shear, endpoint types, classifier, strict window boundary and saddle signatures. +B: compact C^2 stability, conditional full-field coupling, density Jacobian and joint total-variation limit. +C: integrability, positive coefficients and justified lower-bound exhaustion, parity factorization and global non-claims. + +Exact review bindings and subsequent dispositions belong in the PR discussion or an additive review record; an author-time header is not a scientific status register. + +DEEP_TRANSVERSE_EXCLUSION.md is a separate deterministic addendum: under a C4 bound, a sufficiently negative transverse Hessian excludes every additional critical point in a fixed Rr square. It proves an explicit compact soft-jet containment but does not supply the missing derivative-weighted Gaussian tail. Its five finite companion tests bring the packet suite to 43 tests per Python mode. This addendum has its own nonauthor review obligation; PROOF.md and cubic.py remain unchanged. diff --git a/frontiers/planar_cubic_cluster_20260929/RECONNAISSANCE.md b/frontiers/planar_cubic_cluster_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..474974ba0 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/RECONNAISSANCE.md @@ -0,0 +1,23 @@ +# Reconnaissance — 29 September 2026 + +This is a narrow external context check, not an exhaustive novelty or priority audit. No external result is imported as an unverified step of the new proof. + +Consensus.search was actually called with: + + Gaussian random fields short persistence critical points cubic normal form local critical point multiplicity saddles domain:math + +It returned the exhausted 30-search monthly quota (reset October 1), not research results. + +Web queries actually executed: + +- site.arxiv.org Feldbrugge 1908.01619 topology Gaussian random fields persistence +- site.arxiv.org 2305.17586 critical points Gaussian field finiteness moments +- site.arxiv.org 2304.07424 Kac Rice formula + +Primary sources inspected: + +1. Job Feldbrugge, Matti van Engelen, Rien van de Weygaert, Pratyush Pranav and Gert Vegter, Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs towards Betti Numbers and Persistence Diagrams, arXiv:1908.01619 (2019). The arXiv search-result abstract was read; direct opening of https://arxiv.org/abs/1908.01619 failed with UnexpectedStatusCode. It concerns two-dimensional random-field homology and critical-point-based descriptions. No claim about absence of our specific cubic classifier from its full text is made. + +2. Louis Gass and Michele Stecconi, The number of critical points of a Gaussian field: finiteness of moments, arXiv:2305.17586v2 (19 July 2023). https://arxiv.org/abs/2305.17586 opened successfully; metadata and abstract were read. This supplies context for critical-count moment finiteness, not the rare pinned r^3 asymptotic or the new compact-sector formula. The full paper was not audited in this pass. + +The supplied proofs use the exact in-project pinned Taylor identity, contact Gaussian regression and full normalizer, plus elementary algebra, the implicit-function theorem and dominated convergence. The new finite classifier and probability-limit argument are written out for nonauthor review. Neither search metadata nor a checker is substituted for that review. diff --git a/frontiers/planar_cubic_cluster_20260929/RESULTS.json b/frontiers/planar_cubic_cluster_20260929/RESULTS.json new file mode 100644 index 000000000..b7aa090e7 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/RESULTS.json @@ -0,0 +1,16 @@ +{ + "dimension": 2, + "groups": [ + "shear", + "types", + "strict-window", + "weight", + "rare-ledger", + "conic-oracle" + ], + "mathematical_acceptance": false, + "passed": true, + "rational_classifier_cases": 600, + "scientific_effect": "NONE", + "scope": "finite algebra only" +} diff --git a/frontiers/planar_cubic_cluster_20260929/SOURCES.json b/frontiers/planar_cubic_cluster_20260929/SOURCES.json new file mode 100644 index 000000000..a62d49679 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/SOURCES.json @@ -0,0 +1,25 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "sources": [ + { + "id": "L", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md", + "bytes": 13100, + "sha256": "e6272e8709835ca6290fecb7034fa84832dde08c3c4cc990a48b8019e89e7c8d", + "git_blob": "b5c18aac5529c5f975ee5bb056b304debafc6697", + "use": "L6-L13: exact planar pinned cubic, C2 Taylor remainder, ten-jet rank, extra-conditioned C4 moments. Source acceptance is not revalidated." + }, + { + "id": "P", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "use": "Sections 1-5: exact covariance/law, contact frame, smooth regression, positive Fourier spectrum, full normalizer 5.4. No elder-selection or global factorial theorem consumed." + } + ] +} diff --git a/frontiers/planar_cubic_cluster_20260929/cubic.py b/frontiers/planar_cubic_cluster_20260929/cubic.py new file mode 100644 index 000000000..570e369da --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/cubic.py @@ -0,0 +1,197 @@ +"""Exact planar pinned-cubic classifier. Standard library; rational inputs only. + +This checks finite algebra. It does not certify a Gaussian limit theorem. +""" +import argparse +import json +from fractions import Fraction as F + +MUTANTS = ('omit-shear', 'omit-cubic-shear', 'close-window', 'swap-weight-sign', + 'claim-three', 'drop-jet-jacobian', 'restrict-normalizer', 'lose-linear-case') + + +def rational(x): + if type(x) not in (int, F): + raise ValueError('exact int or Fraction required') + return F(x) + + +def positive_k(k): + k = rational(k) + if k <= 0: + raise ValueError('k must be positive') + return k + + +def canonical(k, a, beta, c): + k = positive_k(k) + a, beta, c = map(rational, (a, beta, c)) + return beta-a*a/(12*k), (c-a*beta/(4*k)+a**3/(72*k*k))/2 + + +def typed(k, s, B): + positive_k(k) + s, B = map(rational, (s, B)) + return s < -abs(B)/2 + + +def threshold(k, s, B): + k = positive_k(k) + s, B = map(rational, (s, B)) + if not typed(k, s, B): + raise ValueError('strict maximum/saddle endpoint types required') + return -(s-B)**2*(B+2*s)/(12*k) + + +def window_count(k, s, B, D): + s, B, D = map(rational, (s, B, D)) + T = threshold(k, s, B) + if s > B: + return 2 if D*D < T else 1 + return int(D*D > T) + + +def weight(k, s, B): + k = positive_k(k) + s, B = map(rational, (s, B)) + if not typed(k, s, B): + raise ValueError('strict typed support required') + return 9*k*k*(4*s*s-B*B) + + +def integrated_weight(k, B, lo, hi): + """Polynomial integral only; does not assert the interval is typed.""" + k = positive_k(k) + B, lo, hi = map(rational, (B, lo, hi)) + if lo > hi: + raise ValueError('ordered integration endpoints required') + return 9*k*k*(F(4,3)*(hi**3-lo**3)-B*B*(hi-lo)) + + +def rare_exponent(): + # Four-dimensional physical jet Jacobian r, two determinants r^4, full Z~r^2. + return 1+4-2 + + +def sign(x): + return (x > 0)-(x < 0) + + +def surd_sign(a, b, d): + """Exact sign of a+b*sqrt(d), d>=0, without floats.""" + a, b, d = map(rational, (a, b, d)) + if d < 0: + raise ValueError('nonnegative radicand required') + if b == 0 or d == 0: + return sign(a) + if a == 0 or sign(a) == sign(b): + return sign(b) + comparison = sign(a*a-b*b*d) + return 0 if comparison == 0 else (sign(a) if comparison > 0 else sign(b)) + + +def roots_in_open_interval(A, b, c, lo, hi): + """Independent algebraic root counter, including linear and double roots.""" + A, b, c, lo, hi = map(rational, (A, b, c, lo, hi)) + if lo >= hi: + raise ValueError('nonempty open interval required') + if A == 0: + if b == 0: + if c == 0: + raise ValueError('identically zero polynomial') + return 0 + return int(lo < -c/b < hi) + disc = b*b-4*A*c + if disc < 0: + return 0 + choices = (0,) if disc == 0 else (-1, 1) + total = 0 + for eps in choices: + left = surd_sign(-b-2*A*lo, eps, disc)*sign(A) + right = surd_sign(-b-2*A*hi, eps, disc)*sign(A) + total += int(left > 0 and right < 0) + return total + + +def conic_oracle(k, s, B, D): + """Independent count from the stationary conic/line and height intervals. + + Never calls window_count or threshold. Handles B=0 and D=0 separately. + """ + k = positive_k(k) + s, B, D = map(rational, (s, B, D)) + if not typed(k, s, B): + raise ValueError('strict typed support required') + if B == 0: + if D == 0: + return 0 + h = s**3/(6*D*D) + return int(-k < h < 0) + int(-k < h-k < 0) + if D == 0: + u = -s/B + z2 = -12*k*(u*u-F(1,4))/B + if z2 <= 0: + return 0 + h = -k*(u+F(1,2))+s*z2/6 + return 2*int(-k < h < 0) + end = -F(1,2)-B/(2*s) + lo, hi = sorted((-F(1,2), end)) + return roots_in_open_interval(12*k*D*D+B**3, 2*B*B*s, + B*s*s-3*k*D*D, lo, hi) + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def run_checks(mutant=None): + B, D = canonical(1, 6, 1, F(-3,2)) + if mutant == 'omit-shear': B = F(1) + if mutant == 'omit-cubic-shear': D = F(-3,2) + require((B,D) == (F(-2),F(0)), 'exact shear coefficients') + count = window_count(1,F(-3,2),B,D) + if mutant == 'claim-three': count = 3 + require(count == 2, 'two-saddle source example') + boundary = window_count(F(5,12),-2,-1,1) + if mutant == 'close-window': boundary = 1 + require(boundary == 0, 'strict lower height boundary') + w = weight(1,F(-3,2),-2) + if mutant == 'swap-weight-sign': w = -w + require(w == 45, 'typed determinant product') + exponent = rare_exponent() + if mutant == 'drop-jet-jacobian': exponent = 4-2 + if mutant == 'restrict-normalizer': exponent = 1+4-5 + require(exponent == 3, 'physical jet and full normalizer ledger') + linear = conic_oracle(F(1,12),-1,-1,1) + if mutant == 'lose-linear-case': linear = 0 + require(linear == 1, 'degree-drop conic case') + tested = 0 + for k in (F(1,12),F(1),F(3,2)): + for s in (F(-3),F(-2),F(-3,2),F(-1),F(-1,3)): + for B in map(F,(-4,-3,-2,-1,0,1,2,3,4)): + if not typed(k,s,B): continue + for D in (F(-3),F(-1),F(-1,3),F(0),F(1,10),F(1,3),F(1),F(3)): + require(window_count(k,s,B,D) == conic_oracle(k,s,B,D), + 'classifier disagrees with exact conic roots') + tested += 1 + return {'passed': True, 'rational_classifier_cases': tested, + 'scientific_effect': 'NONE', 'mathematical_acceptance': False, + 'dimension': 2, 'scope': 'finite algebra only', + 'groups': ['shear','types','strict-window','weight','rare-ledger','conic-oracle']} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--mutant', choices=MUTANTS) + args = parser.parse_args() + try: + result = run_checks(args.mutant) + except ValueError as exc: + print(json.dumps({'passed':False, 'error':str(exc)}, sort_keys=True)) + return 1 + print(json.dumps(result, sort_keys=True, indent=2)) + return 0 + + +if __name__ == '__main__': raise SystemExit(main()) diff --git a/frontiers/planar_cubic_cluster_20260929/test_cubic.py b/frontiers/planar_cubic_cluster_20260929/test_cubic.py new file mode 100644 index 000000000..081c698af --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/test_cubic.py @@ -0,0 +1,112 @@ +import unittest +from fractions import Fraction as F +import cubic + +class CubicTests(unittest.TestCase): + def test_known_two_saddle_model(self): + self.assertEqual(cubic.window_count(1,F(-3,2),-2,0), 2) + def test_single_saddle_sector(self): + self.assertEqual(cubic.window_count(1,-1,0,1), 1) + def test_zero_saddle_sector(self): + self.assertEqual(cubic.window_count(1,-1,1,0), 0) + def test_shear(self): + self.assertEqual(cubic.canonical(1,6,1,F(-3,2)), (F(-2),F(0))) + def test_two_sector_boundary_is_one(self): + self.assertEqual(cubic.window_count(F(5,48),F(-3,2),-2,1), 1) + def test_zero_sector_boundary_is_zero(self): + self.assertEqual(cubic.window_count(F(5,12),-2,-1,1), 0) + def test_scalar_boundary(self): + self.assertEqual(cubic.window_count(F(1,6),-1,0,1), 0) + def test_typed_weight(self): + self.assertEqual(cubic.weight(1,F(-3,2),-2),45) + def test_slice_integrals(self): + self.assertEqual(cubic.integrated_weight(1,-2,-2,-1),48) + def test_exponent_ledger(self): + self.assertEqual(cubic.rare_exponent(),3) + + +class ExtendedTests(unittest.TestCase): + def test_sign_reflection(self): + for D in (F(0),F(1,10),F(1),F(3)): + self.assertEqual(cubic.window_count(1,F(-3,2),-2,D), + cubic.window_count(1,F(-3,2),-2,-D)) + def test_typed_boundary_rejected(self): + with self.assertRaises(ValueError): cubic.window_count(1,-1,-2,0) + def test_float_rejected(self): + with self.assertRaises(ValueError): cubic.window_count(1.0,-1,0,1) + def test_bool_rejected(self): + with self.assertRaises(ValueError): cubic.canonical(True,0,0,0) + def test_nonpositive_k(self): + with self.assertRaises(ValueError): cubic.canonical(0,0,0,0) + def test_linear_conic(self): + self.assertEqual(cubic.conic_oracle(F(1,12),-1,-1,1),1) + def test_tangent_below_window(self): + self.assertEqual(cubic.conic_oracle(F(1,4),-1,1,1),0) + def test_s_equals_B(self): + self.assertEqual(cubic.window_count(1,-1,-1,0),0) + self.assertEqual(cubic.window_count(1,-1,-1,1),1) + def test_B_D_zero(self): + self.assertEqual(cubic.conic_oracle(1,-1,0,0),0) + def test_surd_exact_equality(self): + self.assertEqual(cubic.surd_sign(-2,1,4),0) + def test_surd_both_signs(self): + self.assertEqual(cubic.surd_sign(-1,1,2),1) + self.assertEqual(cubic.surd_sign(1,-1,2),-1) + def test_open_root_endpoints(self): + self.assertEqual(cubic.roots_in_open_interval(1,0,-1,-1,1),0) + self.assertEqual(cubic.roots_in_open_interval(1,0,-1,-2,2),2) + def test_double_root(self): + self.assertEqual(cubic.roots_in_open_interval(1,-2,1,0,2),1) + def test_linear_root(self): + self.assertEqual(cubic.roots_in_open_interval(0,2,-1,0,1),1) + def test_no_real_roots(self): + self.assertEqual(cubic.roots_in_open_interval(1,0,1,-3,3),0) + def test_zero_polynomial_rejected(self): + with self.assertRaises(ValueError): cubic.roots_in_open_interval(0,0,0,-1,1) + def test_closed_form_D_zero_integral(self): + for k in (F(1,2),F(1),F(3)): + for B in (F(-1),F(-2),F(-5)): + self.assertEqual(cubic.integrated_weight(k,B,B,B/2),-6*k*k*B**3) + def test_scalar_sector_integral(self): + # k=1/6, D=1 gives lower s=-1 and integral 72*k^3*D^2=1/3. + self.assertEqual(cubic.integrated_weight(F(1,6),0,-1,0),F(1,3)) + def test_conic_discriminant_grid(self): + self.assertEqual(cubic.run_checks()['rational_classifier_cases'],600) + def test_shear_polynomial_identity(self): + for k in (F(1,2),F(1),F(2)): + for a in (F(-3),F(0),F(6)): + beta,c,s=F(2,3),F(-5,2),F(-7,3) + B,D=cubic.canonical(k,a,beta,c) + for u,z in ((F(0),F(1)),(F(2,3),F(-3,2)),(F(-1,2),F(0))): + X=u-a*z/(12*k) + original=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 + canonical=2*k*u**3-F(3,2)*k*u-k/2+s*z*z/2+B*u*z*z/2+D*z**3/3 + self.assertEqual(original,canonical) + def test_stationary_height_and_det(self): + # Construct exact critical points rather than taking a floating square root. + for u in (F(-3,4),F(-1),F(1,4),F(3,4)): + k,z=F(1),F(2) + B=-12*k*(u*u-F(1,4))/(z*z) + s=-2-abs(B) + D=(-s-B*u)/z + h=2*k*u**3-F(3,2)*k*u-k/2+s*z*z/2+B*u*z*z/2+D*z**3/3 + det=12*k*u*(s+B*u+2*D*z)-(B*z)**2 + self.assertEqual(h,-k*(u+F(1,2))+s*z*z/6) + self.assertEqual(det,-3*k*(B+4*s*u)) + def test_cli_modes_and_mutants(self): + import subprocess,sys + from pathlib import Path + script=str(Path(cubic.__file__).resolve()) + reference=None + for flags in (['-B','-S'],['-B','-O','-S']): + run=subprocess.run([sys.executable,*flags,script],capture_output=True,timeout=30) + self.assertEqual(run.returncode,0,run.stderr) + if reference is None: reference=run.stdout + self.assertEqual(run.stdout,reference) + for name in cubic.MUTANTS: + run=subprocess.run([sys.executable,*flags,script,'--mutant',name],capture_output=True,timeout=30) + self.assertEqual(run.returncode,1,(name,run.stdout,run.stderr)) + run=subprocess.run([sys.executable,*flags,script,'--mutant','unknown'],capture_output=True,timeout=30) + self.assertEqual(run.returncode,2) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/planar_cubic_cluster_20260929/test_deep_exclusion.py b/frontiers/planar_cubic_cluster_20260929/test_deep_exclusion.py new file mode 100644 index 000000000..e5b89fd98 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/test_deep_exclusion.py @@ -0,0 +1,51 @@ +"""Finite algebra checks for the separate deterministic exclusion addendum.""" +import unittest +from fractions import Fraction as F + + +class DeepExclusionTests(unittest.TestCase): + def test_rational_constant_grid(self): + for R in map(F, (1, 2, 5)): + for K in map(F, (1, 3, 12)): + for k in (F(1, 10), F(1), F(3)): + r=k/(8*R*K) + lam=2*(8*R*K+56*R*K*K/k)*r + rho=4*R*R*K*r*r/lam + t=4*R*K*r/lam + self.assertLess(2*R*K*r,lam/2) + self.assertLess(rho,R*r/2) + self.assertLess(t,F(1,2)) + self.assertLess(28*R*K*K*r/lam,k/2) + self.assertLessEqual(2*R*K*r,k/2) + self.assertGreater(12*k-2*R*K*r-7*K*t,11*k) + def test_vertical_sign_bracket(self): + # At +/-rho the guaranteed vertical variation is twice the axis bound. + for R,K,r,lam in ((F(2),F(3),F(1,100),F(2)), + (F(1),F(1),F(1,1000),F(1))): + gmax=K*R*R*r*r; rho=4*R*R*K*r*r/lam + self.assertEqual(lam*rho/2,2*gmax) + self.assertGreater(lam*rho/2-gmax,0) + def test_reduced_third_polynomial_identity(self): + # f=g(x)+h(x)z-lambda*z^2/2, zeta=h/lambda, + # g'''=12k, h=1+2x+3x^2+4x^3. Compute both formulas independently. + for x in (F(-2),F(-1,2),F(0),F(3,4)): + for lam in (F(1),F(3),F(7)): + k=F(2); h=1+2*x+3*x*x+4*x**3 + hp=2+6*x+12*x*x; hpp=6+24*x; hppp=F(24) + direct=12*k+(3*hp*hpp+h*hppp)/lam + fxxx=12*k+hppp*h/lam + t=-hp/lam + reduced=fxxx-3*hpp*t + self.assertEqual(direct,reduced) + def test_mixed_term_is_load_bearing(self): + k,lam,h,hp,hpp,hppp=map(F,(1,2,1,2,6,24)) + direct=12*k+(3*hp*hpp+h*hppp)/lam + omitted=12*k+hppp*h/lam + self.assertNotEqual(direct,omitted) + def test_convexity_margin(self): + for k in (F(1,100),F(1),F(100)): + self.assertEqual(12*k-k/2-k/2,11*k) + self.assertGreater(11*k,0) + + +if __name__=='__main__': unittest.main() diff --git a/frontiers/planar_cubic_cluster_20260929/test_verify.py b/frontiers/planar_cubic_cluster_20260929/test_verify.py new file mode 100644 index 000000000..441b2e384 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/test_verify.py @@ -0,0 +1,44 @@ +"""Verifier regression tests using a real isolated Git fixture, no network.""" +import copy +import hashlib +import json +from pathlib import Path +import subprocess +import tempfile +import unittest +import verify as v + +class VerifyTests(unittest.TestCase): + def setUp(self): + t=tempfile.TemporaryDirectory(); self.addCleanup(t.cleanup) + self.root=Path(t.name); self.data=b'exact local test fixture\n' + (self.root/'source.md').write_bytes(self.data) + self.git('init','-q'); self.git('add','source.md') + self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid','commit','-qm','fixture') + self.entry={'id':'source','path':'source.md','commit':self.git('rev-parse','HEAD').decode().strip(), + 'bytes':len(self.data),'sha256':hashlib.sha256(self.data).hexdigest(), + 'git_blob':self.git('rev-parse','HEAD:source.md').decode().strip()} + def git(self,*args): + return subprocess.run(['git',*args],cwd=self.root,check=True,capture_output=True,timeout=15).stdout + def check(self,entry): + (self.root/'SOURCES.json').write_text(json.dumps({'sources':[entry]})) + v.verify_sources(self.root) + def test_valid_source(self): self.check(self.entry) + def test_wrong_identity_fields(self): + for field,value in [('bytes',0),('sha256','0'*64),('git_blob','0'*40)]: + entry=copy.deepcopy(self.entry); entry[field]=value + with self.subTest(field=field), self.assertRaises(ValueError): self.check(entry) + def test_missing_source(self): + entry=copy.deepcopy(self.entry); entry['path']='missing.md' + with self.assertRaises(subprocess.CalledProcessError): self.check(entry) + def test_traversal_rejected(self): + entry=copy.deepcopy(self.entry); entry['path']='../source.md' + with self.assertRaises(ValueError): self.check(entry) + def test_wrong_commit(self): + entry=copy.deepcopy(self.entry); entry['commit']='0'*40 + with self.assertRaises(subprocess.CalledProcessError): self.check(entry) + def test_duplicate_json(self): + p=self.root/'duplicate.json'; p.write_text('{"a":1,"a":2}') + with self.assertRaises(ValueError): v.read_json(p) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/planar_cubic_cluster_20260929/verify.py b/frontiers/planar_cubic_cluster_20260929/verify.py new file mode 100644 index 000000000..0fa5833c0 --- /dev/null +++ b/frontiers/planar_cubic_cluster_20260929/verify.py @@ -0,0 +1,79 @@ +"""Verify packet identities and exact controls; default includes project Git sources.""" +import argparse +import hashlib +import json +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + + +def read_json(path): + def unique(pairs): + ans = {} + for k,v in pairs: + if k in ans: raise ValueError('duplicate JSON key: '+k) + ans[k] = v + return ans + return json.loads(Path(path).read_text(encoding='utf-8'), object_pairs_hook=unique) + + +def identity(data, entry): + blob = hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if (len(data)!=entry['bytes'] or hashlib.sha256(data).hexdigest()!=entry['sha256'] + or blob!=entry['git_blob']): + raise ValueError('identity mismatch: '+entry['path']) + + +def verify_sources(root): + entries = read_json(root/'SOURCES.json')['sources'] + ids = [e['id'] for e in entries] + if not entries or len(ids)!=len(set(ids)): raise ValueError('invalid source inventory') + for e in entries: + commit,path=e['commit'],e['path'] + p=PurePosixPath(path) + if not re.fullmatch('[0-9a-f]{40}',commit) or not path or p.is_absolute() or '..' in p.parts: + raise ValueError('unsafe source identity') + data=subprocess.run(['git','show',commit+':'+path],cwd=root,check=True, + capture_output=True,timeout=60).stdout + identity(data,e) + + +def verify_inventory(root): + entries=read_json(root/'MANIFEST.json')['files'] + names=[e['path'] for e in entries] + 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('packet inventory mismatch') + for e in entries: + path=root/e['path'] + if PurePosixPath(e['path']).name!=e['path'] or path.is_symlink() or not path.is_file(): + raise ValueError('flat regular file required') + identity(path.read_bytes(),e) + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--local-only',action='store_true') + args=parser.parse_args() + root=Path(__file__).resolve().parent + verify_inventory(root) + if not args.local_only: verify_sources(root) + expected=(root/'RESULTS.json').read_bytes() + mutants=['omit-shear','omit-cubic-shear','close-window','swap-weight-sign', + 'claim-three','drop-jet-jacobian','restrict-normalizer','lose-linear-case'] + for flags in (['-B','-S'],['-B','-O','-S']): + subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'],cwd=root, + check=True,timeout=120) + command=[sys.executable,*flags,str(root/'cubic.py')] + result=subprocess.run(command,capture_output=True,check=True,timeout=30) + if result.stdout!=expected: raise ValueError('deterministic output mismatch') + for name in mutants+['unknown']: + result=subprocess.run(command+['--mutant',name],capture_output=True,timeout=30) + if result.returncode!=(2 if name=='unknown' else 1): + raise ValueError('negative control not rejected: '+name) + print(json.dumps({'passed':True,'source_pins_checked':not args.local_only, + 'scientific_effect':'NONE','mathematical_acceptance':False},sort_keys=True)) + + +if __name__=='__main__': main() diff --git a/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md b/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md new file mode 100644 index 000000000..23dadc6a3 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md @@ -0,0 +1,29 @@ +# Source-bound audit and explicit replacement + +Scientific effect NONE. The old mathematical bytes are preserved. This document does not rewrite another author's proof or change a global disposition. + +## Audited object + +Math-#159, author Anthropic Claude, head f0d11805854b1938a2f89235ec590a37b04a7cb9, PROOF.md blob1ce3769e76c540ab03b42f2deb6f2d1124459a81. Full byte identity is AUDITED-159 in SOURCES.json and is checked by the verifier although its results are NOT consumed. + +OpenAI source-exposed review5359646638 gave AMEND REQUIRED. Exposure correction5901054091: RC is Claude-authored, not OpenAI-authored as inadvertently grouped in the initial exposure sentence. The relevant OpenAI author-side exposure is LM/LP/RM/RCL. The mathematical findings remain unchanged; no independent revalidation of the parents is claimed. + +## Findings and replacement steps + +1. Proposition4.4 includes a finite limit for r^-3 P(N_near=0). Since P(N_near=0)=1-O(r^3), that quantity diverges. PROOF.md excludes count0 from the intensity measure and states explicitly that its mass must be centered before scaling. + +2. Lemma4.3 uses r/|det D|^2 <= r/eta^2 on |det D|0 makes |raw J|>=c|tau|. The replacement retains compact orthogonal O, all tensor coordinates, and the actual non-isotropic Gaussian density under the Haar integral. + +4. Fixed-radius observables depend on the old q-dependent ellipse, so q cannot first be integrated out and then used outside the integral. Exact-count indicators are not monotone in radius (0 then1 then2 is possible). The new formula retains every angular variable and uses actual soft-plane spatial coordinates. Dominated convergence, not monotone convergence of exact-count indicators, takes the radius limit. + +5. The prior near-witness conditioning route requires uniform rank estimates through coalescence. The replacement bounds the far first moment after conditioning on a fixed-origin COMPLETE physical jet, first on a compact spectral box. It proves O(r^(9/2)) there and only o(r^3) after exhaustion. It does not falsely transfer the compact-box rate to the whole space. + +## Collaboration record + +main#95 pickup5900871219 and PR159 comments5900894321/5900933928 presented the planar absorption repair and its orthogonal all-d extension. The new packet is a separate author-side candidate requiring nonauthor review, not an ACCEPT of PR159. Claude's lane can compare or consume this replacement only with its new review status respected. The new proof uses #158's accepted deterministic classifier, not its still-separate Gaussian-sector disposition. + +## Test limits + +The finite tests catch the reversed cutoff, zero mass, missing spectral r, missing Vandermonde, wrong hard power, dropped far branch, dropped mixed derivative and wrong normalizer. They do not certify dominated convergence, the implicit-function theorem, Gaussian rank, or the local-to-global passage. Those obligations are explicitly written in PROOF.md. diff --git a/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md b/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md new file mode 100644 index 000000000..3f63edb54 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md @@ -0,0 +1,161 @@ +# Addendum: the global derivative-weighted rare-count bound + +Object: OA-SPECTRAL-CLUSTER-CLOSURE-20260929-GD1. +Author: OpenAI / GPT-6 Astra Pro, 29 September 2026. +Disposition: author-side conditional derivation; nonauthor review REQUIRED. +Scientific effect NONE. The main PROOF.md is unchanged and does NOT depend on +this addendum. This closes a separately proposed sufficient route if its source +interfaces and the new insertion argument below are accepted. + +## 1. Exact statement and inputs + +Use the ORIGINAL count N_r, Q_r, W_r, full Z_r, and Q_r^W of PROOF.md Section1. +Fix d>=2 and L. Birth and positive-gap marks may range over fixed compact sets, +with k bounded away from zero, and the axial frame over O(d). For each fixed +p>=0 there are C_p,r_p>0 such that + + E_(Q_r^W)[ N_r (1+||f||_C4)^p ] <= C_p r^3, 0=1 satisfies E_Q' K^u <= C_u(1+beta)^u for fixed u. +- Lemma5.1: Kac-Rice for a nonnegative Borel full-field mark, with canonical + Gaussian kernels and monotone exhaustion. It is stated under Q_r. +- Section5.2 and Section6: pathwise geometric bounds on W_r F_j(H_X), density + bounds and the regional beta/penalty table; the normalized integration ledgers + of the pin balls, collar, intermediate shells and fixed remote region. + +All those inputs are retained at their source scope, not revalidated by this +addendum. SOURCES.json already binds the complete [C6] and its parent [D5]/[P] +identities. No use is made of #157, #159, or the new spectral proof's conclusions. + +## 2. One full normalizer, one derivative mark + +Let Xi=K^p. It is nonnegative and Borel on the smooth field space. Applying the +source's formula with mark W_r Xi, and then dividing by the ORIGINAL normalizer, + + E_QW[N_r K^p] + = Z_r^-1 sum_j integral_X integral_Ir p_(grad f(X),f(X)|pins)(0,h) + E_Q'[W_r F_j(H_X) K^p] dh dX. (GD2) + +The two conditioned pins are removed by compact exhaustion. In the all-height +pin-ball/collar regions one may integrate the gradient-only kernel instead; +that gives an upper bound on the window count. A height window of length kr^3 +is retained in the shell and remote regions. This never becomes an all-height +global O(r^3) assertion. + +If a regional source bound is W_r F_j(H_X)<=G_r(X,h) K^(3d), then directly + + E_Q'[W_r F_j(H_X) K^p] + <= G_r E_Q'[K^(3d+p)] + <= C_p G_r (1+beta)^(3d+p). (GD3) + +The source's unmarked estimate is the same expression with exponent3d. Thus the +ONLY new cost is a factor (1+beta)^p; G_r, the spatial/height Jacobians and Z_r +remain unchanged. Where the source has already used Z_r>=z_*r^2 to write a +normalized pathwise bound, use that version exactly once. No additional Z_r +appears and none is removed. This direct moment bound does not use a +Cauchy-Schwarz factor with a missing square root. + +If p is not an integer, bound K^p by K^ceil(p); it suffices to prove the integer +cases. The full conditional C6 moment is available for every such fixed order. + +## 3. Regional absorption with explicit new losses + +For x>=1, M>=0 and a,c>0, + + x^M exp(-c x^a) <= C_(M,a,c) exp(-(c/2)x^a), + C_(M,a,c)=max{1,[2M/(a c e)]^(M/a)} (GD4) + +with the M=0 expression interpreted as1. Maximize M log x-(c/2)x^a to obtain +this inequality. Thus a fixed polynomial cost is absorbed by half the Gaussian +penalty; the remaining half is still available for the original positive +regional integration. No power of r or shell radius has to be weakened. + +Here is the complete additional-cost ledger from [C6] Section6. All labels and +beta bounds are those of that pinned source, not new nondegeneracy assumptions. + +| Region | Source beta bound | Extra K^p cost | Penalty retained for absorption | Resulting normalized integral | +|---|---|---|---|---| +| R1-I, punctured pin ball | C(1+chi) | C_p(1+chi)^p | exp(-c chi^2) | C_p r^3 | +| R1-II, pin axial region | C/r, since chi<=1/r | C_p r^-p | exp(-c/(2r^2)) | C_p r^3 | +| R2a, collar axial strip | C r^-10 | C_p r^-10p | exp(-c r^-2/3) | C_p r^3 | +| R2b, collar small nonzero v | C norm(v)^-4 | C_p norm(v)^-4p | exp(-c/norm(v)^2) | C_p r^3 | +| R2c, collar away from axis | C | C_p | no new penalty needed | C_p r^3 | +| R3a, shell axial strip | C s^-10 | C_p s^-10p | exp(-c s^-1/4) | C_p r^3 s^2 | +| R3b, shell off axis | C norm(v)^-6 | C_p norm(v)^-6p | exp(-c/norm(v)^2), or bounded away from axis | C_p r^3[(r/s)^2+s^2] | +| R4, fixed remote region | C | C_p | no new penalty needed | C_p r^3 | + +Details of the integrations: + +R1-I: the source's factor r^(3-d)|q|^(2-d) is multiplied by a polynomial in +chi and exp(-c chi^2). The extra (1+chi)^p changes only the finite supremum of +that product. The same omega_d integrability and r^d spatial Jacobian give r^3. +R1-II: chi<=1/r, so use (GD4) with x=1/r, M=p, a=2. The remaining exponential +absorbs the original negative radius powers exactly as in [C6](6.3). The +S-centered pin ball is identical; the full normalizer is unchanged. + +R2a: use x=1/r, M=10p, a=2/3, then the original source's density/weight radius +ledger with c replaced by c/2. R2b: use x=1/|v|, M=4p, a=2. The factor +r^(3-d) remains, and integration over the scaled collar supplies r^d. R2c has +bounded beta and hence no new singular factor. + +R3a: the source's AXIS-SAFE endpoint-short-column bound is valid on the entire +strip, not only at v=0. It gives W_r F_j(H_X)/Z_r<=C K^(3d). The extra moment +cost s^-10p is absorbed using x=1/s, a=1/4. The normalized intensity can still +be bounded by a constant. The source's spatial strip volume is at most C s^d, +which is at most C s^2 for d>=2 and s<=1; the retained height window gives r^3. +Thus this contribution is C_p r^3 s^2. + +R3b: the unchanged geometric and density part leaves +s^(-(d+2))(r+s^2)^2 times an inverse-v polynomial and exp(-c/|v|^2). Absorb the +new |v|^-6p using (GD4). Multiplication by shell volume s^d and height length +kr^3 gives C_p r^3[(r/s)^2+2r+s^2]. Since 2r<=r^2/s^2+s^2, this is at most a +constant times r^3[(r/s)^2+s^2]. The same estimate applies where |v| is bounded +away from zero, by bounded beta rather than a singular exponential inequality. + +R4: the complete witness value/gradient law at fixed distance has bounded beta. +The endpoint short columns supply r^2 in W_r, canceled by the one full Z_r~r^2; +the window supplies kr^3. Any fixed added conditional derivative moment remains +bounded. The spatial region has fixed finite volume. + +Finally sum the dyadic shells s=2^j*4r up to the fixed small cutoff s0: + + sum_j (r/s_j)^2 <= C, sum_j s_j^2 <= C s0^2. + +Together with the two pin balls, collar and fixed remote region, (GD2)-(GD3) +give (GD1). The proof is uniform on the same fixed compact marks/frames because +all conditional-moment and penalty constants in the pinned source are uniform +there, and p is fixed before those constants are chosen. QED, conditional on +the declared regional source interfaces. + +## 4. Consequences and exact relationship to the main proof + +For T>=1, + + r^-3 E_QW[N_r; 1+||f||_C4>T] <= C_p T^-p. (GD5) + +This is an actual rare-intensity derivative tail: it is not obtained by dividing +an unconditioned derivative-tail probability by r^3. It also gives bounded fixed +polynomial derivative moments under the count-size-biased law N_r dQ_r^W/EN_r, +using the already positive order-r^3 lower bound on EN_r. + +If #158's compact-sector and deterministic-exclusion arguments are consumed at +their eventual accepted scopes, (GD5) supplies the sufficient probability input +D11 proposed in that packet. The main spectral proof in this packet instead +uses its own residual absorption and does not need (GD1). Neither route is +circular: this addendum uses only the earlier [C6] regional proof, and the main +proof never imports this addendum. + +No exponential derivative moment, convergence rate of the cluster coefficients, +numerical constant, or change in a parent source's acceptance status follows. +Nonauthor review focus: one Z_r^-1 in (GD2); the exact conditional K moment in +(GD3); all eight table rows, especially R1-II and the full R3a strip; and the +unchanged height-window factor in the intermediate/remote regions. Finite +companion probes verify powers and the elementary absorption inequality only. diff --git a/frontiers/spectral_cluster_closure_20260929/MANIFEST.json b/frontiers/spectral_cluster_closure_20260929/MANIFEST.json new file mode 100644 index 000000000..9d6c6eefd --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/MANIFEST.json @@ -0,0 +1,97 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "review_status": "CORE_AND_GD_NONAUTHOR_ACCEPTED_AT_SCOPED_CONDITIONAL_LEVEL", + "files": [ + { + "path": "AUDIT_AND_REPAIR.md", + "bytes": 3553, + "sha256": "755a2405e7e7925d23adc8b1c7a3abce1fd73bad5fba80e7546618d19a6736ac", + "git_blob": "23dadc6a314492bdf98bfcb84997b541e9656618" + }, + { + "path": "GLOBAL_DERIVATIVE_WEIGHTED.md", + "bytes": 8813, + "sha256": "a445607be26cc2863682c404d702798a984fa2b8994ed0dc5b57fb9c4da153eb", + "git_blob": "3f63edb54ea5422cdc328cc54fb093c8a5d84be1" + }, + { + "path": "PLAN.md", + "bytes": 1851, + "sha256": "531d71ad7009195d25fd42d8ed55593267eb4c8e476e79f106cb3d2512d191ee", + "git_blob": "12f03d8f5082a2c5e7edfbec040804a69a1de3fb" + }, + { + "path": "PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a" + }, + { + "path": "README.md", + "bytes": 5084, + "sha256": "b1e2a0722945a6189d8338c2e5b1f1a74b3b5646b8dc10fe29168c165508342c", + "git_blob": "895dded8aa8e405ad8e9640d3ffebc6449a108d0" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 1458, + "sha256": "c3af589197c895d2521721c41ac7866d0945ec3c9de1c853d348bdc1a0ce42a1", + "git_blob": "aec1f588dcafa4f1605f4fd53feef04cd350bae5" + }, + { + "path": "RESULTS.json", + "bytes": 338, + "sha256": "f7ebe3d9a6767d967a8b0139de02b3d18e9081f36c9f85b847c6f3f01d6ce529", + "git_blob": "7c564dacf616f1b6099c93e05daf564e5c36ec8f" + }, + { + "path": "REVIEW_RECORD.md", + "bytes": 5629, + "sha256": "6031d36697789097185a8e65bfbdf80f447e229689eac5a706ee8b0558db64a5", + "git_blob": "9bcae183fecdf0ffe4d612a209438a4fb1fa0edd" + }, + { + "path": "SOURCES.json", + "bytes": 5677, + "sha256": "c1e84ef7ccf749b7eb19c713d2b0fe4f4547c03ce56ef18c9858010f99929d9e", + "git_blob": "8daf24b7ea01e889102170dcd056634b4f67e911" + }, + { + "path": "VALIDATION.md", + "bytes": 3082, + "sha256": "602ab8a58b2866b63de06923f0fb259b813b031a1e7c0cb6fc995a269be726f5", + "git_blob": "bf00357c5c7bfb6a019c1edc85e37e865a73d804" + }, + { + "path": "check.py", + "bytes": 5877, + "sha256": "d24e07af792fc0538f977489e7af24a7815413487577f94f1a74cb229c8dec94", + "git_blob": "e747e1c81779e665cdb614562579206237720c76" + }, + { + "path": "test_check.py", + "bytes": 6510, + "sha256": "b094624c46756ab771465ce1eeb73c315e3f36e66e61ff8f116d0e07d88e1f46", + "git_blob": "33fdbff05bbae19f9cb9700ba0d7b1ecea5e05c4" + }, + { + "path": "test_derivative_weighted.py", + "bytes": 2922, + "sha256": "39c11a235a6186189d694794078fa9b29b26f1b530b7b79c423c8fff63b5211b", + "git_blob": "5389a7400feebae73dbaeeb96cd0f1940c3bab66" + }, + { + "path": "test_verify.py", + "bytes": 4439, + "sha256": "16bccfbe377e87ecfbfba6db895fe508dfa1f2eab9d9be807a3d310750b6d0bf", + "git_blob": "d813c34e90a77c9eb15248299380dc8184892cfa" + }, + { + "path": "verify.py", + "bytes": 4729, + "sha256": "d3d499b7b4dc1e7ba7414bcd452d7302e001719fccd6abaceff62a748bb4c3f5", + "git_blob": "db4260655b433c393c62b7abe411de2f416b382c" + } + ] +} diff --git a/frontiers/spectral_cluster_closure_20260929/PLAN.md b/frontiers/spectral_cluster_closure_20260929/PLAN.md new file mode 100644 index 000000000..12f03d8f5 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/PLAN.md @@ -0,0 +1,30 @@ +# Spectral cluster closure implementation plan + +Goal: publish an additive, test-backed conditional replacement for the failed +unbounded-shear domination, proving the fixed-d global count law without assuming +a global derivative-weighted estimate. Preserve all source proofs. + +Architecture: PROOF.md contains vector exclusion, orthogonal spectral coordinates, +residual-target absorption and the nonempty-measure limit. check.py implements +finite exact identities only; test_check.py compares separate formulations. +verify.py and test_verify.py authenticate every declared source and inventory. +SOURCES.json binds consumed/credited sources; MANIFEST.json binds packet files. +The workflow checks the full source objects and leaves existing gates unchanged. + +Tech stack: Python standard library, exact Fraction arithmetic, Git, Markdown. + +Tasks: +1. Audit #159 at its exact head and give reproducible blockers; keep a repair log. +2. Write the full replacement proof and enumerate the remaining non-claims. +3. Write finite tests before implementation, observe intended RED, implement and + replay normal/-O, and reject semantic mutations and unknown labels. +4. Test custody with real temporary Git repositories, including wrong hash, + missing source, duplicate identity, unsafe path and symlink rejection. +5. Generate manifests and exact results. Authenticate remote tree blobs; publish + a new branch/PR, never overwrite peer proof bodies. +6. Request distinct-provider analytic review of three scoped slices. Do not infer + acceptance or a merge from passing tests. Check exact-current-head hosted gates. + +Review focus: near-zero hard spectra; the exceptional absorption branch; rotation +versus isotropy; uncentered zero counts; compact-only cross-term rates; actual +versus limiting compound-Poisson parameters. Finite controls do not prove analysis. diff --git a/frontiers/spectral_cluster_closure_20260929/PROOF.md b/frontiers/spectral_cluster_closure_20260929/PROOF.md new file mode 100644 index 000000000..16c56821b --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/PROOF.md @@ -0,0 +1,491 @@ +# Orthogonal spectral closure of the rare window-count law + +Object: OA-SPECTRAL-CLUSTER-CLOSURE-20260929-v1. +Author: OpenAI / GPT-6 Astra Pro, 29 September 2026. +Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. +Scientific effect NONE: no existing proof, acceptance record, global register, +Boolean or prize is changed. Shared GitHub identity is not organizational +independence. All new analytic steps below require their own review. + +## 1. Model, inputs, and the claimed closure + +Fix d>=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/frontiers/spectral_cluster_closure_20260929/README.md b/frontiers/spectral_cluster_closure_20260929/README.md new file mode 100644 index 000000000..895dded8a --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/README.md @@ -0,0 +1,41 @@ +# Orthogonal spectral closure of the window-count law + +OpenAI / GPT-6 Astra Pro. Core conditional proof has completed scoped nonauthor A/B/C acceptance. The separate global derivative-weighted addendum has its own Claude nonauthor acceptance5360082678. All acceptance is at the exact stated conditional scope. Scientific effect NONE. No self-merge, parent-proof modification, or global status/Boolean change. Shared GitHub identity is not organizational independence. + +The new argument proposes the unique limiting nonempty count measure in every FIXED dimension d>=2: nu1 delta1 + nu2 delta2, with explicit positive finite coefficients. Near coefficients are an orthogonal Haar/spectral/Gaussian jet integral; the far contribution is the reviewed remote singleton kernel. The new proof does not consume the audited v1.0 global argument of #159 or the proposed marked Fourier theorem of #157. + +The main new steps are the vector transverse-exclusion lemma; the absorption split into a regular soft layer and a separately controlled exceptional raw-field tail; a polynomial majorant with NO inverse hard eigenvalues; weighted typed-boundary and multiple-soft control; pointwise stable-direction elimination; compact-origin-jet near/far decoupling; and a radius sandwich. Existing C6 moments provide polynomial uniform integrability after, not instead of, this new limit argument. + +Consequences, conditional on the new theorem: complete polynomially weighted count convergence; higher factorial moments o(r^3) for q>=3; identified conditional and Palm laws; independent-replica limit Pois(t nu1)+2 Pois(t nu2); exact limiting ratio 2 between mean-matched and optimized ordinary-Poisson error. No convergence rate or numerical coefficient is supplied. Replacing the actual conditional law in the canonical compound approximant by its limit does not preserve the old O(r^6) rate without new rate information. + +## Read and replay + +Read PROOF.md and AUDIT_AND_REPAIR.md. PLAN.md records the task split. SOURCES.json includes all consumed, credited, and audited identities, including the historical #159 proof; no recorded source is silently skipped. + +In a full checkout containing all pinned commits: + + python -B -S frontiers/spectral_cluster_closure_20260929/verify.py + +In the standalone directory: + + python -B -S verify.py --local-only + +The latter explicitly reports source_pins_checked=false. Local Git fixtures test verifier behavior, not the truth of the actual project sources. The hosted workflow performs the actual source-object checks and full packet replay; existing downstream/formal gates remain separate. + +Current local suite: 55 tests in normal Python and 55 with -O; deterministic result equality; eight semantic mutants rejected with exit1 in each mode and unknown labels with exit2. Controls cover exact algebra only, not Gaussian convergence. The initial12-test stub failed all12 intended assertions. An expanded positive fixture mistakenly violated the absorption premise; that fixture was corrected, not the valid implementation. Source verifier had its own fail-first fixture suite. See VALIDATION.md for the exact scope. + +## Actual scoped core reviews + +A: Claude5359968447 accepts vector exclusion, absorption and the exact spectral/Haar measure. +B: Claude5359845136 accepts the majorant, typed/hard-boundary control, normal form and coefficient integrals. +C: OpenAI/Codex5359879627 accepts the remote conditioning, mixed-event exhaustion, radius sandwich and consequences; this is same-provider nonauthor review, not independent-provider review. + +All bind the unchanged core PROOF.md. The separate OpenAI/Codex A check5360029463 is corroboration, not an additional independent vote. Parent-input acceptance and organizational independence are not renewed by these reviews. A code pass is not their substitute. No Lean formalization of the new proof is claimed. + +The initial hosted nested-directory source-path defect is repaired with git ls-tree --full-tree and a real-Git regression. See VALIDATION.md; a predecessor failure is not concealed by local test success. + +## Separate global derivative-weighted addendum + +GLOBAL_DERIVATIVE_WEIGHTED.md derives E[N_r(1+||f||C4)^p]<=C_p r^3 directly from the earlier C6 regional Kac-Rice proof, in fact with C6 norm. It retains the one full normalizer, adds the conditional-moment cost (1+beta)^p, and writes all eight regional absorptions. It is not a consequence assumed by the core spectral theorem. Nine additional finite ledger tests are included. The addendum has separate Slice D acceptance5360082678; it was not inferred from the earlier core reviews. + +REVIEW_RECORD.md records the completed core A/B/C and separate D coverage, exact byte bindings and provider exposure. GD1 is not an input to the core theorem. The unchanged manuscripts retain their author-time candidate headers to preserve reviewed bytes; this record supplies the later dispositions. Integration still requires successful current-head checks and a nonauthor integrator. The author will not self-merge or change a global scientific register. diff --git a/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md b/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..aec1f588d --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md @@ -0,0 +1,10 @@ +# Narrow primary-source reconnaissance, September29 2026 + +Consensus.search was actually called with “Gaussian random fields critical point rare clusters Palm spectral eigenvalue boundary layer Kac Rice domain:math”. It returned the exhausted monthly30-search quota and October1 reset. No Consensus result is claimed. + +Primary arXiv metadata/abstract pages were opened through web: + +- https://arxiv.org/abs/2304.07424v3 — Diego Armentano, Jean-Marc Azais and Jose Rafael Leon, On a general Kac-Rice formula for the measure of a level set. The repository cites v3 (December5 2023); the page reports a later v4 dated April22 2025. This pass did not call v3 latest or audit changes between versions. The parent Kac-Rice interface is retained at its recorded scope. +- https://arxiv.org/abs/2305.17586 — Louis Gass and Michele Stecconi, The number of critical points of a Gaussian field: finiteness of moments, v2 July19 2023. The abstract concerns finite critical-count moments under jet nondegeneracy. It does not substitute for the rare tilted O(r^3) bounds or the identified cluster law proposed here. + +Access depth: metadata and abstracts only, not full-text theorem audits. The orthogonal spectral Jacobian is derived in the manuscript and its normalization is fixed by an explicit Gaussian test integral. No external unverified theorem is imported as a replacement for a missing project premise, and no exhaustive novelty or priority claim is made. diff --git a/frontiers/spectral_cluster_closure_20260929/RESULTS.json b/frontiers/spectral_cluster_closure_20260929/RESULTS.json new file mode 100644 index 000000000..7c564dacf --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/RESULTS.json @@ -0,0 +1,16 @@ +{ + "absorption_cases": 141, + "groups": [ + "spectral-jacobian", + "hard-weight", + "absorption", + "cutoff-direction", + "nonempty-measure", + "vector-third-derivative", + "full-normalizer" + ], + "mathematical_acceptance": false, + "passed": true, + "scientific_effect": "NONE", + "scope": "finite exact controls only" +} diff --git a/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md b/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md new file mode 100644 index 000000000..9bcae183f --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md @@ -0,0 +1,97 @@ +# Completed scoped reviews — spectral core and derivative-weighted addendum + +Scientific effect NONE. This is a factual record of actual nonauthor reviews, +not a self-issued acceptance, governing-register transition, or organizationally +independent endorsement. All lanes share one GitHub account. Source reviews do +not revalidate every parent result or extend beyond the stated parameter scope. + +## Exact immutable mathematical objects + +Core PROOF.md:25006 bytes; SHA256 +e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb; +Git blob16c56821b52fd76b0be791622b9c3809eafde75a, unchanged since +41814598d37de8446dd538ce3cf2d57982a4077f. + +GLOBAL_DERIVATIVE_WEIGHTED.md:8813 bytes; SHA256 +a445607be26cc2863682c404d702798a984fa2b8994ed0dc5b57fb9c4da153eb; +Git blob3f63edb54ea5422cdc328cc54fb093c8a5d84be1, unchanged since +3892aeed64601350472566c657d954e84ffdd78d. It is NOT an input to the core proof. +The manuscripts retain their historical author-time headers; the later actual +review dispositions are recorded below without changing their mathematical bytes. + +## Actual core coverage + +A — Anthropic Claude5359968447, submitted2026-09-30T00:07:07Z at7a44130: +ACCEPT Sections2-3, complete-jet coupling, ordered spectral/Haar coordinates, +vector exclusion, both absorption branches and the original-Q exception bound. +The reviewer declares no authorship of the consumed sources or competing159. +Different provider from the author; no organizational-independence credit. + +B — Anthropic Claude5359845136, submitted2026-09-29T23:46:11Z at41814598: +ACCEPT Sections4-6, majorant, typed/hard-boundary control, normal form and +coefficient integrals, conditional on A and the pinned classifier. That reviewer +authored consumed C6/D5/RC and competing159; this is source-exposed scrutiny, +not a renewal of those inputs or organizational independence. + +C — OpenAI/Codex5359879627, submitted2026-09-29T23:51:40Z at7a44130: +ACCEPT Sections7-8, compact-origin-jet far conditioning, full mixed o(r^3), radius +sandwich and probability consequences, conditional on A/B and exact inputs. +Different author session but SAME PROVIDER. Independent-provider and +organizational credit are zero. Its read-only challenges are not extra votes. +Comment5901273120 clarifies arbitrary fixed real b, not b>0. + +Additional OpenAI/Codex5360029463 corroborates A at the unchanged bytes. It has +the same-provider limitation and is not an additional independent vote. + +Combined core scope: fixed d>=2,L>0,k>0,b in R and frame, with exact imported +Gaussian/regional interfaces. No numerical coefficient or convergence rate, +growing-volume or vanishing-mark uniformity, or distinct-pin-pair independence. + +## Separate D acceptance + +Anthropic Claude5360082678, submitted2026-09-30T00:26:33Z at3892aeed: +ACCEPT GD1-GD5, exactly one full Z^-1, conditional K^(3d+p) moment insertion, +all eight regional absorptions with height window retained, dyadic integration, +and the global rare-intensity derivative tail. This reviewer declares no +authorship of C6 or the other pinned sources; different provider from the author, +but the same shared GitHub account and zero organizational independence. + +Scope: every fixed d>=2,L, compact birth/positive-gap marks bounded away from +k=0, O(d) frames, and each fixed p, with existential constants. The verdict is +conditional exactly on earlier C6 Lemma4.4/5.1 and regional interfaces, not a +revalidation of them. It does not accept an exponential derivative moment or rate. +The earlier core reviews were never used as substitutes for this D review. + +Nonblocking D reading clarifications adopted in author comment5901687461: +choose K=max(source K,1), which preserves its moment bound; R3a is r^3 s^d +<=r^3 s^2; the unmarked comparison is the underlying DL/RM bound before C6's +extra Psi mark; R4 uses Lemma4.4 with RM's bounded beta. GD4's displayed constant +is a convenient valid upper bound, not a claim of the smallest possible constant. +Core clarifications: the soft direction is largest for A/smallest for -A; +||f_xy||<=K(R+1/2)r+K|zeta|<=2RKr; norms are invariant or uniformly equivalent +coordinate norms, never an assumption of field isotropy; N_R is the Euclidean +Rr-ball count inside the exclusion cylinder. + +## Verification and integration boundary + +Initial target36646437664 failed because git ls-tree inherited the nested cwd. +A real-Git fail-first fixture reproduced it; --full-tree fixed it at7a44130. +Codex P1 comment4139440699 is answered4139467347 and RESOLVED. All identity, +path, symlink and negative controls remain; no source check was bypassed. + +Target/full runs36646903471/36646903884 at7a44130,36649121320/36649121574 at3892aeed, +and36650263358/36650263498 atc433a592 all succeeded. These observations are not +automatically rebound to later metadata-only commits. Final-head evidence belongs +in the PR handoff after a fresh readback. + +Local packet:55 tests per normal/optimized mode,141 exact absorption probes, +eight semantic mutants rejected per mode and unknown-label rejection. Eleven +actual consumed/credited/audited Git-source objects are checked by hosted replay. +The local container did not obtain a full upstream checkout; its explicit +local-only mode never reports actual project source pins checked. Existing Lean +package success does not formalize the new analytic proofs. + +Both manuscripts and all executables/source pins are unchanged by this record. +All four requested analytic review slices are complete at their stated scope. +A nonauthor integrator must still read the final diff, review threads and +current-head/current-base checks. No self-merge or global-status fold is implied. diff --git a/frontiers/spectral_cluster_closure_20260929/SOURCES.json b/frontiers/spectral_cluster_closure_20260929/SOURCES.json new file mode 100644 index 000000000..8daf24b7e --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/SOURCES.json @@ -0,0 +1,129 @@ +{ + "schema": 1, + "repository": "d6g8k5htny-coder/Math-", + "scientific_effect": "NONE", + "source_policy": "Every entry, including credit and audited historical bytes, is mandatory for full source verification. 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Expanded stub checks followed. +- Initial implementation replay exposed one erroneous positive fixture: x=2,U=1,r=1/10,a=1 violates the premise x<=aU^2+ar^2x^2 since2>1.04. Corrected the test to x=1, without weakening the premise or modifying the valid implementation. +- Mathematical suite including CLI checks:32 tests pass in normal and optimized modes. +- Source-custody placeholder:13 tests produced17 expected assertion failures including subtests; valid fixture passed. Implemented complete Git-object/mode and byte identity checks. +- Full local suite:45 tests pass in normal and optimized modes. The deterministic checker output matches RESULTS.json. Eight named false variants exit1 in each mode; unknown labels exit2. All packet identities are generated from exact local bytes. + +Custody fixtures use real isolated Git repositories. They cover wrong size/SHA256/blob, missing path/commit, unsafe paths, duplicate source IDs, empty inventories, duplicate JSON, Git leaf and parent symlinks, and packet symlinks. Source verification authenticates the pinned tree/blob rather than following a worktree link. + +The container could not obtain a complete remote checkout, so actual project Git-object verification is not claimed local. Default hosted replay is configured to fetch and verify EVERY declared source, including historical audit and prior-credit entries. Hosted success and full repository/formal checks must be read at the actual current PR head before any integration. No Lean formalization of the new analytic proof is claimed. + +## Hosted nested-directory repair + +Initial hosted run36646437664 failed at source verification, job109670396383, because `git ls-tree` interpreted the pinned repository-root path relative to the nested packet cwd. Both source fetches and inventory authentication had succeeded; the error was not a missing upstream proof. A new real-Git nested-packet regression failed before repair (None!=1). Adding `--full-tree` makes root-relative source paths independent of cwd. The mathematical PROOF.md and all premise identities remain unchanged. Latest local suite:46 tests per mode. The original failing hosted run is retained; the successor still needs its own hosted result. + +## Addendum and review-metadata completion + +The separate GD1 addendum adds nine exact finite ledger tests. Fresh complete +local replay passes55 tests in normal and55 in optimized Python, with identical +RESULTS and all eight semantic mutations rejected per mode. These test the +normalizer/radius/penalty ledgers, not conditional Gaussian integration. +At addendum head3892aeed64601350472566c657d954e84ffdd78d, hosted target36649121320 +and full36649121574 both succeeded. Later metadata-only commits must be checked +at their own exact PR test merge. Core PROOF.md, GD1, all source pins and every +executable remain unchanged by this final review-metadata update. diff --git a/frontiers/spectral_cluster_closure_20260929/check.py b/frontiers/spectral_cluster_closure_20260929/check.py new file mode 100644 index 000000000..e747e1c81 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/check.py @@ -0,0 +1,155 @@ +"""Exact finite controls for orthogonal spectral cluster closure. + +Standard library only. These controls do not prove Gaussian convergence. +""" +import argparse +import json +from fractions import Fraction as F +from math import prod + +MUTANTS=('drop-spectral-r','drop-soft-vandermonde','wrong-hard-power', + 'drop-far-branch','reverse-cutoff','include-zero', + 'drop-mixed-term','swap-full-normalizer') + + +def rat(x): + if type(x) not in (int,F): raise ValueError('exact int/Fraction required') + return F(x) + + +def absorption(x,u,r,a): + x,u,r,a=map(rat,(x,u,r,a)) + if x<0 or u<1 or r<=0 or a<=0: raise ValueError('invalid absorption arguments') + if x>a*u*u+a*r*r*x*x: raise ValueError('absorption premise does not hold') + if x<=2*a*u*u: return 'near' + if x<1/(2*a*r*r): raise ValueError('absorption conclusion failed') + return 'far' + + +def vandermonde(values): + values=list(map(rat,values)) + if any(b<=a for a,b in zip(values,values[1:])): + raise ValueError('strictly ordered spectrum required') + return prod((values[j]-values[i] for i in range(len(values)) + for j in range(i+1,len(values))),start=F(1)) + + +def spectral_factor(r,s,hard): + r,s=map(rat,(r,s)); hard=list(map(rat,hard)) + if r<=0: raise ValueError('positive radius required') + return r*vandermonde([-r*s]+hard) + + +def limit_hard_factor(hard): + hard=list(map(rat,hard)) + if any(x<=0 for x in hard): raise ValueError('positive hard spectrum required') + return prod((x**3 for x in hard),start=F(1))*vandermonde(hard) + + +def typed_weight(k,s,a,beta): + k,s,a,beta=map(rat,(k,s,a,beta)) + if k<=0: raise ValueError('positive gap required') + return max(F(0),-6*k*(s-beta/2)-a*a/4)*max(F(0),a*a/4-6*k*(s+beta/2)) + + +def nonempty_intensity(law,epsilon): + epsilon=rat(epsilon) + if epsilon<=0 or not isinstance(law,dict) or not law: + raise ValueError('positive scale and PMF required') + out={} + for n,p in law.items(): + if type(n) is not int or n<0: raise ValueError('invalid count') + p=rat(p) + if p<0: raise ValueError('negative probability') + if n>0 and p>0: out[n]=p/epsilon + if sum(map(rat,law.values()))!=1: raise ValueError('PMF must sum to one') + return out + + +def reduced_third(xxx,xxy_v,xyy_vv,yyy_vvv): + xxx,xxy_v,xyy_vv,yyy_vvv=map(rat,(xxx,xxy_v,xyy_vv,yyy_vvv)) + return xxx+3*xxy_v+3*xyy_vv+yyy_vvv + + +def majorant_degree(d): + if type(d) is not int or d<2: raise ValueError('integer d>=2 required') + return 4*d+(d-1)*(d-2) + + +def invalid_cutoff_bound(r,determinant,eta): + r,determinant,eta=map(rat,(r,determinant,eta)) + if not 0=1/(2*a*r*r), + 'absorption dichotomy grid') + cases+=1 + return {'passed':True,'scientific_effect':'NONE','mathematical_acceptance':False, + 'scope':'finite exact controls only','absorption_cases':cases, + 'groups':['spectral-jacobian','hard-weight','absorption','cutoff-direction', + 'nonempty-measure','vector-third-derivative','full-normalizer']} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--mutant',choices=MUTANTS) + args=parser.parse_args() + try: result=run_checks(args.mutant) + except ValueError as exc: + print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True)); return 1 + print(json.dumps(result,indent=2,sort_keys=True)); return 0 + + +if __name__=='__main__': raise SystemExit(main()) diff --git a/frontiers/spectral_cluster_closure_20260929/test_check.py b/frontiers/spectral_cluster_closure_20260929/test_check.py new file mode 100644 index 000000000..33fdbff05 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/test_check.py @@ -0,0 +1,136 @@ +import unittest +from fractions import Fraction as F +import check as c + +class InitialTests(unittest.TestCase): + def test_absorption_near(self): + self.assertEqual(c.absorption(1,1,F(1,10),1),'near') + def test_absorption_far(self): + self.assertEqual(c.absorption(200,1,F(1,10),1),'far') + def test_absorption_invalid(self): + with self.assertRaises(ValueError): c.absorption(5,1,F(1,10),1) + def test_vandermonde(self): + self.assertEqual(c.vandermonde([1,3,5]),16) + def test_spectral_factor(self): + self.assertEqual(c.spectral_factor(F(1,10),-1,[2,4]),F(741,500)) + def test_empty_hard_factor(self): + self.assertEqual(c.spectral_factor(F(1,10),-1,[]),F(1,10)) + def test_limit_hard_factor(self): + self.assertEqual(c.limit_hard_factor([2,4]),1024) + def test_weight(self): + self.assertEqual(c.typed_weight(1,F(-3,2),0,-2),45) + def test_zero_filtered(self): + self.assertEqual(c.nonempty_intensity({0:F(7,8),1:F(1,16),2:F(1,16)},F(1,8)),{1:F(1,2),2:F(1,2)}) + def test_reduced_derivative(self): + self.assertEqual(c.reduced_third(1,2,3,4),20) + def test_majorant_degree(self): + self.assertEqual(c.majorant_degree(3),14) + def test_bad_cutoff(self): + self.assertFalse(c.invalid_cutoff_bound(F(1,10),F(1,100),F(1,10))) + + +class AlgebraTests(unittest.TestCase): + def test_absorption_grid(self): + count=0 + for a in (F(1),F(3)): + for u in (F(1),F(2),F(5)): + for r in (F(1,100),F(1,10),F(1,2)): + for x in map(F,(0,1,2,5,10,20,100,200,1000,20000)): + if x<=a*u*u+a*r*r*x*x: + branch=c.absorption(x,u,r,a) + self.assertTrue(x<=2*a*u*u if branch=='near' else x>=1/(2*a*r*r)) + count+=1 + self.assertGreater(count,80) + def test_float_rejected(self): + with self.assertRaises(ValueError): c.absorption(1.0,1,F(1,10),1) + def test_bad_dimension(self): + with self.assertRaises(ValueError): c.majorant_degree(1) + def test_bool_dimension(self): + with self.assertRaises(ValueError): c.majorant_degree(True) + def test_unordered_spectrum(self): + with self.assertRaises(ValueError): c.spectral_factor(F(1,10),-30,[1,2]) + def test_hard_zero_rejected(self): + with self.assertRaises(ValueError): c.limit_hard_factor([0,1]) + def test_weight_boundary_zero(self): + self.assertEqual(c.typed_weight(1,-1,0,-2),0) + def test_weight_shear(self): + self.assertEqual(c.typed_weight(1,F(-3,2),6,1),45) + def test_bad_probability(self): + with self.assertRaises(ValueError): c.nonempty_intensity({0:F(1,2)},F(1,10)) + def test_zero_mass_diverges(self): + for j in (10,100,1000): + e=F(1,j) + self.assertEqual((1-e)/e,j-1) + self.assertNotIn(0,c.nonempty_intensity({0:1-e,2:e},e)) + def test_full_mixed_third_identity(self): + for g in (F(1),F(12),F(24)): + for vd in (F(2),F(5)): + for tensor in (F(3),F(7)): + terms=(g-3*vd-tensor,vd+tensor,-tensor,tensor) + self.assertEqual(c.reduced_third(*terms),g) + for index in (1,2,3): + changed=list(terms); changed[index]=F(0) + self.assertNotEqual(c.reduced_third(*changed),g) + def test_determinant(self): + self.assertEqual(c.det([[2,1],[1,3]]),5) + self.assertEqual(c.det([]),1) + def test_weyl_local_jacobian_2(self): + for co,si in ((F(1),F(0)),(F(3,5),F(4,5)),(F(5,13),F(12,13))): + for l1,l2 in ((F(-2),F(3)),(F(1,3),F(7))): + gap=l2-l1 + jac=[[co*co,si*si,2*co*si*gap], + [co*si,-co*si,-(co*co-si*si)*gap], + [si*si,co*co,-2*co*si*gap]] + self.assertEqual(abs(c.det(jac)),gap) + def test_two_soft_column_determinant(self): + # Exact d=3 Schur expansion, keeping the r^3 mixed correction. + for r in (F(1,100),F(1,10),F(1),F(2)): + matrix=[[-6*r,-r,2*r],[-r,-2*r,3*r],[2*r,3*r,-5]] + expected=-55*r*r+50*r**3 + self.assertEqual(c.det(matrix),expected) + def test_orthogonal_norms(self): + for co,si in ((F(3,5),F(4,5)),(F(5,13),F(12,13))): + l1,l2=F(-2),F(7) + a=co*co*l1+si*si*l2; b=co*si*(l1-l2); d=si*si*l1+co*co*l2 + self.assertEqual(a*a+2*b*b+d*d,l1*l1+l2*l2) + def test_unbounded_shear_counterexample(self): + # D=-1,v=t,A00=-t^2, raw T_www=1, transformed tau_zzz=t^3. + for t in (10,100,1000): + raw_norm_upper=F(t*t+2*t+2) + self.assertGreater(F(t**3)/raw_norm_upper,F(t,2)) + def test_deep_constants(self): + for R in (F(1),F(3)): + for K in (F(1),F(12)): + for k in (F(1,10),F(1),F(3)): + r=k/(4*R*K); lam=2*(8*R*K+56*R*K*K/k)*r + rho=4*R*R*K*r*r/lam; slope=4*R*K*r/lam + self.assertLess(rho,R*r/2) + self.assertLess(slope,F(1,2)) + self.assertLess(28*R*K*K*r/lam,k/2) + self.assertEqual(2*R*K*r,k/2) + def test_near_far_scaling(self): + self.assertEqual(F(3)+F(3,2),F(9,2)) + def test_higher_factorials_vanish_in_limit(self): + from math import prod + for q in range(3,9): + for n in (1,2): self.assertEqual(prod(n-j for j in range(q)),0) + +class ExecutionTests(unittest.TestCase): + def test_both_modes_and_semantic_controls(self): + import subprocess,sys + from pathlib import Path + command=str(Path(c.__file__).resolve()) + reference=None + for flags in (['-B','-S'],['-B','-O','-S']): + run=subprocess.run([sys.executable,*flags,command],capture_output=True,timeout=30) + self.assertEqual(run.returncode,0,run.stderr) + if reference is None: reference=run.stdout + self.assertEqual(run.stdout,reference) + for label in c.MUTANTS: + run=subprocess.run([sys.executable,*flags,command,'--mutant',label],capture_output=True,timeout=30) + self.assertEqual(run.returncode,1,(label,run.stdout,run.stderr)) + self.assertEqual(run.stderr,b'') + run=subprocess.run([sys.executable,*flags,command,'--mutant','unknown'],capture_output=True,timeout=30) + self.assertEqual(run.returncode,2) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py b/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py new file mode 100644 index 000000000..5389a7400 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py @@ -0,0 +1,73 @@ +"""Exact finite ledgers for GD1. They do not prove conditional Gaussian bounds.""" +from fractions import Fraction as F +from math import ceil, factorial +import unittest + + +class DerivativeWeightTests(unittest.TestCase): + def test_one_full_normalizer(self): + # Remote normalized moment: endpoint short columns, normalizer, height. + self.assertEqual(2-2+3,3) + self.assertNotEqual(2+3,3) + self.assertNotEqual(2-2-2+3,3) + + def test_pin_and_collar_spatial_powers(self): + for d in range(2,10): + self.assertEqual((3-d)+d,3) + + def test_exponential_absorption_budgets(self): + # These overestimate by retaining the older source's cap costs as well. + for d in range(2,9): + for p in range(10): + for alpha,cost in ((F(2),F(6+3*d+p)), + (F(2,3),F(15+39*d+10*p)), + (F(1,4),F(40*d+5+10*p))): + n=ceil(cost/alpha) + self.assertGreaterEqual(alpha*n,cost) + self.assertLess(alpha*(n-1),cost) + + def test_inverse_angle_budgets(self): + for d in range(2,9): + for p in range(10): + for cost in (16*d+10+4*p,22*d+15+6*p): + n=ceil(F(cost,2)) + self.assertGreaterEqual(2*n,cost) + self.assertLess(2*(n-1),cost) + + def test_positive_exponential_series_term(self): + # exp(y)>=y^n/n!: retain the n-th term of a strictly positive polynomial. + for y in (F(1,2),F(1),F(3),F(10)): + for n in (1,2,5,9): + partial=sum((y**j/F(factorial(j)) for j in range(n+1)),F(0)) + self.assertGreater(partial,y**n/F(factorial(n))) + + def test_shell_cross_term(self): + for r in (F(1,100),F(1,20),F(1,10)): + for s in (4*r,F(1,2),F(1)): + if r>s/4: continue + value=(r+s*s)**2/(s*s) + self.assertEqual(value,r*r/(s*s)+2*r+s*s) + self.assertLessEqual(value,2*(r*r/(s*s)+s*s)) + + def test_full_strip_volume(self): + for d in range(2,10): + for s in (F(1,100),F(1,3),F(1,2),F(1)): + self.assertLessEqual(s**d,s*s) + + def test_dyadic_sums(self): + for levels in range(1,13): + r=F(1,2**(levels+3)) + scales=[4*r*2**j for j in range(levels)] + self.assertLess(sum((r/s)**2 for s in scales),F(1,12)) + self.assertLess(sum(s*s for s in scales),F(4,3)*scales[-1]**2) + + def test_count_weighted_tail_is_pointwise(self): + for count in (0,1,2,9): + for norm in (F(1),F(3),F(10)): + for threshold in (F(1),F(2),F(5)): + for p in range(5): + left=count if norm>threshold else 0 + self.assertLessEqual(left,count*norm**p/threshold**p) + + +if __name__=='__main__': unittest.main() diff --git a/frontiers/spectral_cluster_closure_20260929/test_verify.py b/frontiers/spectral_cluster_closure_20260929/test_verify.py new file mode 100644 index 000000000..d813c34e9 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/test_verify.py @@ -0,0 +1,80 @@ +"""Source-custody tests against a real isolated Git repository, not mocked Git.""" +import copy +import hashlib +import json +from pathlib import Path +import subprocess +import tempfile +import unittest +import verify as v + +class CustodyTests(unittest.TestCase): + def setUp(self): + temp=tempfile.TemporaryDirectory(); self.addCleanup(temp.cleanup) + self.root=Path(temp.name); self.data=b'proof fixture\n' + (self.root/'nested').mkdir(); (self.root/'nested'/'source.md').write_bytes(self.data) + (self.root/'leaf.md').symlink_to('nested/source.md') + (self.root/'alias').symlink_to('nested',target_is_directory=True) + self.git('init','-q'); self.git('add','nested/source.md','leaf.md','alias') + self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid','commit','-qm','fixture') + self.entry={'id':'S','commit':self.git('rev-parse','HEAD').decode().strip(), + 'path':'nested/source.md','bytes':len(self.data), + 'sha256':hashlib.sha256(self.data).hexdigest(), + 'git_blob':self.git('rev-parse','HEAD:nested/source.md').decode().strip()} + def git(self,*args): + return subprocess.run(['git',*args],cwd=self.root,check=True,capture_output=True,timeout=15).stdout + def run_pins(self,entries): + p=self.root/'SOURCES.json'; p.write_text(json.dumps({'sources':entries})) + return v.verify_sources(self.root) + def test_valid_source(self): + self.assertEqual(self.run_pins([self.entry]),1) + def test_valid_source_from_nested_packet(self): + packet=self.root/'frontiers'/'packet'; packet.mkdir(parents=True) + (packet/'SOURCES.json').write_text(json.dumps({'sources':[self.entry]})) + try: + result=v.verify_sources(packet) + except (ValueError,subprocess.CalledProcessError): + result=None + self.assertEqual(result,1,'repo-root source identity must work from nested packet') + def test_wrong_size(self): + e=copy.deepcopy(self.entry); e['bytes']=0 + with self.assertRaises(ValueError): self.run_pins([e]) + def test_wrong_sha(self): + e=copy.deepcopy(self.entry); e['sha256']='0'*64 + with self.assertRaises(ValueError): self.run_pins([e]) + def test_wrong_blob(self): + e=copy.deepcopy(self.entry); e['git_blob']='0'*40 + with self.assertRaises(ValueError): self.run_pins([e]) + def test_missing_path(self): + e=copy.deepcopy(self.entry); e['path']='missing.md' + with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e]) + def test_bad_commit(self): + e=copy.deepcopy(self.entry); e['commit']='0'*40 + with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e]) + def test_unsafe_paths(self): + for path in ('../source.md','/source.md','',':source.md','nested/../source.md','nested//source.md'): + e=copy.deepcopy(self.entry); e['path']=path + with self.subTest(path=path),self.assertRaises(ValueError): self.run_pins([e]) + def test_duplicate_source_ids(self): + with self.assertRaises(ValueError): self.run_pins([self.entry,self.entry]) + def test_missing_inventory(self): + with self.assertRaises(ValueError): self.run_pins([]) + def test_git_leaf_symlink(self): + e=copy.deepcopy(self.entry); e['path']='leaf.md' + e['git_blob']=self.git('rev-parse','HEAD:leaf.md').decode().strip() + data=b'nested/source.md'; e['bytes']=len(data); e['sha256']=hashlib.sha256(data).hexdigest() + with self.assertRaises(ValueError): self.run_pins([e]) + def test_git_parent_symlink(self): + e=copy.deepcopy(self.entry); e['path']='alias/source.md' + with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e]) + def test_duplicate_json(self): + p=self.root/'dup.json'; p.write_text('{"a":1,"a":2}') + with self.assertRaises(ValueError): v.read_json(p) + def test_packet_leaf_symlink(self): + packet=self.root/'packet'; packet.mkdir() + (packet/'file.md').symlink_to('../nested/source.md') + e={k:value for k,value in self.entry.items() if k in ('bytes','sha256','git_blob')}; e['path']='file.md' + (packet/'MANIFEST.json').write_text(json.dumps({'files':[e]})) + with self.assertRaises(ValueError): v.verify_inventory(packet) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/spectral_cluster_closure_20260929/verify.py b/frontiers/spectral_cluster_closure_20260929/verify.py new file mode 100644 index 000000000..db4260655 --- /dev/null +++ b/frontiers/spectral_cluster_closure_20260929/verify.py @@ -0,0 +1,102 @@ +"""Fail-closed spectral packet replay. --local-only excludes project Git objects.""" +import argparse +import hashlib +import json +from pathlib import Path, PurePosixPath +import re +import subprocess +import sys + + +def read_json(path): + def unique(pairs): + result={} + for key,value in pairs: + if key in result: raise ValueError('duplicate JSON key: '+key) + result[key]=value + return result + return json.loads(Path(path).read_text(encoding='utf-8'),object_pairs_hook=unique) + + +def safe_path(path): + if not isinstance(path,str) or not path or '\\' in path or ':' in path or '\x00' in path: + raise ValueError('unsafe source path') + p=PurePosixPath(path) + if p.is_absolute() or '..' in p.parts or '.' in path.split('/') or str(p)!=path: + raise ValueError('unsafe source path') + return path + + +def identity(data,entry): + blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() + if (type(entry['bytes']) is not int or len(data)!=entry['bytes'] + or hashlib.sha256(data).hexdigest()!=entry['sha256'] or blob!=entry['git_blob']): + raise ValueError('identity mismatch: '+entry['path']) + + +def verify_sources(root): + entries=read_json(root/'SOURCES.json')['sources'] + ids=[e['id'] for e in entries] + if not entries or len(ids)!=len(set(ids)): + raise ValueError('missing or duplicate source inventory') + for entry in entries: + commit,path=entry['commit'],safe_path(entry['path']) + if not re.fullmatch('[0-9a-f]{40}',commit): raise ValueError('invalid source commit') + # The pinned tree, not a followed worktree symlink, is the authority. + out=subprocess.run(['git','ls-tree','--full-tree','-z',commit,'--',path],cwd=root, + check=True,capture_output=True,timeout=60).stdout + records=[x for x in out.split(b'\0') if x] + if len(records)!=1: raise ValueError('missing or ambiguous pinned path: '+path) + header,actual=records[0].split(b'\t',1) + mode,kind,blob=header.decode().split() + if actual.decode()!=path or mode!='100644' or kind!='blob': + raise ValueError('pinned source must be a regular nonexecutable blob: '+path) + if blob!=entry['git_blob']: raise ValueError('tree blob mismatch: '+path) + data=subprocess.run(['git','cat-file','blob',blob],cwd=root,check=True, + capture_output=True,timeout=60).stdout + identity(data,entry) + return len(entries) + + +def verify_inventory(root): + if (root/'MANIFEST.json').is_symlink(): raise ValueError('manifest symlink') + entries=read_json(root/'MANIFEST.json')['files'] + names=[e['path'] for e in entries] + 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('packet inventory mismatch') + for entry in entries: + name=safe_path(entry['path']); path=root/name + if PurePosixPath(name).name!=name or path.is_symlink() or not path.is_file(): + raise ValueError('flat regular packet file required') + identity(path.read_bytes(),entry) + return len(entries) + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--local-only',action='store_true') + args=parser.parse_args(); root=Path(__file__).resolve().parent + verify_inventory(root) + source_count=0 if args.local_only else verify_sources(root) + expected=(root/'RESULTS.json').read_bytes() + mutants=['drop-spectral-r','drop-soft-vandermonde','wrong-hard-power', + 'drop-far-branch','reverse-cutoff','include-zero', + 'drop-mixed-term','swap-full-normalizer'] + for flags in (['-B','-S'],['-B','-O','-S']): + subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'],cwd=root, + check=True,timeout=120) + cmd=[sys.executable,*flags,str(root/'check.py')] + out=subprocess.run(cmd,check=True,capture_output=True,timeout=30) + if out.stdout!=expected or out.stderr: raise ValueError('deterministic replay mismatch') + for mutant in mutants+['unknown']: + out=subprocess.run(cmd+['--mutant',mutant],capture_output=True,timeout=30) + if out.returncode!=(2 if mutant=='unknown' else 1): + raise ValueError('negative control not rejected: '+mutant) + verify_inventory(root) + print(json.dumps({'passed':True,'source_pins_checked':not args.local_only, + 'source_count':source_count,'scientific_effect':'NONE', + 'mathematical_acceptance':False},sort_keys=True)) + + +if __name__=='__main__': main() diff --git a/imports/gp_data_114_embedded_20260930/GP-DATA-114-v1.0_receipt.json b/imports/gp_data_114_embedded_20260930/GP-DATA-114-v1.0_receipt.json new file mode 100644 index 000000000..77c511eaa --- /dev/null +++ b/imports/gp_data_114_embedded_20260930/GP-DATA-114-v1.0_receipt.json @@ -0,0 +1,197 @@ +{ + "box_id": "P01-UNIFORMQ-D4-BOX-001", + "explicit_exclusions": [ + "NO Gaussian or Palm mass lower bound in this source", + "NO exact-field transfer", + "NO exact full-field Palm or adjacency theorem", + "NO P0.1/P0.2/Theorem B promotion" + ], + "negative_controls": [ + { + "id": "NC1_ZERO_WIDTH", + "observed": "FULL_DIMENSIONAL_BOX_FAIL", + "pass": true, + "required": "FULL_DIMENSIONAL_BOX_FAIL" + }, + { + "id": "NC2_TYPING_DESTROYED", + "observed": [ + "saddle_det_negative_margin", + "maximum_A_negative_margin", + "maximum_det_positive_margin", + "cone_axis_margin", + "central_drift_margin", + "capture_gershgorin_margin_1" + ], + "pass": true, + "required_any_failed_gate": [ + "saddle_det_negative_margin", + "maximum_det_positive_margin" + ] + }, + { + "id": "NC3_Q_BOX_CROSSES_ZERO", + "observed": "Q_NEGATIVITY_FAIL", + "pass": true, + "required": "Q_NEGATIVITY_FAIL" + }, + { + "id": "NC4_CONE_TOO_NARROW", + "observed": [ + "cone_slope_margin", + "cone_lower_inward_margin" + ], + "pass": true, + "required_failed_gate": "cone_slope_margin" + }, + { + "id": "NC5_CAPTURE_CHART_DESTROYED", + "observed": [ + "cone_slope_margin", + "cone_upper_inward_margin", + "cone_lower_inward_margin", + "handoff_margin", + "strip_top_inward_margin", + "strip_bottom_inward_margin", + "central_drift_margin", + "chart_x_margin", + "chart_y_margin", + "capture_gershgorin_margin_1" + ], + "pass": true, + "required_any_failed_gate": [ + "chart_x_margin", + "capture_gershgorin_margin_1" + ] + } + ], + "overall": "PASS_UNIFORM_R_FIXED_Q_FULL_DIMENSIONAL_DEGREE4_INTERVAL_BOX", + "primary": { + "box": { + "a": [ + 0.0024000000000000002, + 0.0026 + ], + "c04": [ + -0.01, + 0.01 + ], + "c13": [ + -0.01, + 0.01 + ], + "c22": [ + -0.01, + 0.01 + ], + "c31": [ + -0.01, + 0.01 + ], + "c40": [ + -0.01, + 0.01 + ], + "q": [ + -1.02, + -0.98 + ], + "w": [ + -0.001, + 0.001 + ], + "z": [ + 0.99, + 1.01 + ] + }, + "derived_intervals": { + "W": [ + 0.016970562748477133, + 0.018974031961839034 + ], + "rho": [ + 0.20285955732969546, + 0.20356830685437158 + ], + "tau_upper": 6.513136004191066e-05 + }, + "failed_gates": [], + "failure_class": null, + "fixed": { + "kappa": 0.01, + "mu_interval": [ + 0.017677669529663684, + 0.01767766952966369 + ], + "q_interval": [ + -1.02, + -0.98 + ], + "r_interval": [ + 0.0, + 0.05 + ], + "s_upper_worst": -19.599999999999998, + "sigma": 0.1 + }, + "lower_bounds": { + "axial_gap_margin": 20.599395833333325, + "capture_gershgorin_margin_1": 0.589931481037461, + "capture_gershgorin_margin_2": 19.391268980780914, + "central_drift_margin": 0.08996598847018418, + "chart_x_margin": 0.04643169314562845, + "chart_y_margin": 0.7964316931456284, + "cone_axis_margin": 0.8999287018324998, + "cone_lower_inward_margin": 0.20363763551749148, + "cone_slope_margin": 0.009869737279916179, + "cone_upper_inward_margin": 0.20600236151749143, + "handoff_margin": 0.007485281374238566, + "maximum_A_negative_margin": 0.9999583333333333, + "maximum_D_negative_margin": 19.599437499999993, + "maximum_det_positive_margin": 19.59861905670138, + "partial_y_contraction_margin": 19.58039234290867, + "saddle_det_negative_margin": 19.598622198506934, + "strip_bottom_inward_margin": 0.33241853805878996, + "strip_top_inward_margin": 0.3325271710449011, + "threshold_slack": 19.562591224677412 + }, + "monotonicity_ledger": { + "affected_gates": [ + "threshold", + "endpoint typing", + "axial gap", + "cone faces", + "strip faces", + "partial-y contraction", + "capture h22" + ], + "domain": "0 float: + return float(x.a) + + +def upper(x) -> float: + return float(x.b) + + +def absmax(x) -> float: + return max(abs(lower(x)), abs(upper(x))) + + +CENTER = { + 'a': 0.0025, + 'w': 0.0, + 'z': 1.0, + 'q': -1.0, + 'c40': 0.0, + 'c31': 0.0, + 'c22': 0.0, + 'c13': 0.0, + 'c04': 0.0, +} + +RADII = { + 'a': 0.0001, + 'w': 0.001, + 'z': 0.01, + 'q': 0.02, + 'c40': 0.01, + 'c31': 0.01, + 'c22': 0.01, + 'c13': 0.01, + 'c04': 0.01, +} + +R = I('0', '0.05') +R_MAX = 0.05 +SIGMA = 0.1 +KAPPA = 0.01 +MU = 1 / (40 * iv.sqrt(I(2))) + + +def flow_x(X, Y, p): + r, a, w = R, p['a'], p['w'] + c40, c31, c22, c13 = p['c40'], p['c31'], p['c22'], p['c13'] + return ( + X**2 - I('0.25') + a*X*Y + w*Y**2/2 + + r*( + c40*(X**3-X/4)/6 + + c31*(3*X**2-I('0.25'))*Y/6 + + c22*X*Y**2/2 + + c13*Y**3/6 + ) + ) + + +def flow_y(X, Y, p): + r, a, w, z, s = R, p['a'], p['w'], p['z'], p['_s'] + c31, c22, c13, c04 = p['c31'], p['c22'], p['c13'], p['c04'] + return ( + a*(X**2-I('0.25'))/2 + s*Y + w*X*Y + z*Y**2/2 + + r*( + c31*(X**3-X/4)/6 + + c22*X**2*Y/2 + + c13*X*Y**2/2 + + c04*Y**3/6 + ) + ) + + +def hessian(X, Y, p): + r, a, w, z, s = R, p['a'], p['w'], p['z'], p['_s'] + c40, c31, c22, c13, c04 = p['c40'], p['c31'], p['c22'], p['c13'], p['c04'] + j11 = ( + 2*X + a*Y + + r*(c40*(3*X**2-I('0.25'))/6 + c31*X*Y + c22*Y**2/2) + ) + j12 = ( + a*X + w*Y + + r*( + c31*(3*X**2-I('0.25'))/6 + + c22*X*Y + + c13*Y**2/2 + ) + ) + j22 = ( + s + w*X + z*Y + + r*(c22*X**2/2 + c13*X*Y + c04*Y**2/2) + ) + return j11, j12, j22 + + +def cone_axis(xi, t, p): + # Continuous quotient xi_dot/xi after X=1/2-xi, Y=t*xi. + r, a, w = R, p['a'], p['w'] + c40, c31, c22, c13 = p['c40'], p['c31'], p['c22'], p['c13'] + return ( + 12*a*t*xi - 6*a*t + - 2*c13*r*t**3*xi**2 + + 6*c22*r*t**2*xi**2 - 3*c22*r*t**2*xi + - 6*c31*r*t*xi**2 + 6*c31*r*t*xi - c31*r*t + + 2*c40*r*xi**2 - 3*c40*r*xi + c40*r + - 6*t**2*w*xi - 12*xi + 12 + )/12 + + +def cone_y_quotient(xi, t, p): + # Continuous quotient y_dot/xi after X=1/2-xi, Y=t*xi. + r, a, w, z, s = R, p['a'], p['w'], p['z'], p['_s'] + c31, c22, c13, c04 = p['c31'], p['c22'], p['c13'], p['c04'] + return ( + 12*a*xi - 12*a + + 4*c04*r*t**3*xi**2 + - 12*c13*r*t**2*xi**2 + 6*c13*r*t**2*xi + + 12*c22*r*t*xi**2 - 12*c22*r*t*xi + 3*c22*r*t + - 4*c31*r*xi**2 + 6*c31*r*xi - 2*c31*r + + 24*s*t + 12*t**2*xi*z - 24*t*w*xi + 12*t*w + )/24 + + +def make_box(radii=None): + radii = RADII if radii is None else radii + return {k: I(CENTER[k]-radii[k], CENTER[k]+radii[k]) for k in CENTER} + + +def certify(radii=None, kappa=KAPPA): + radii = RADII if radii is None else radii + if any(radii[k] <= 0 for k in CENTER): + return {'pass': False, 'failure_class': 'FULL_DIMENSIONAL_BOX_FAIL', 'failed_gates': ['positive_width_all_coordinates']} + + p = make_box(radii) + q_hi = CENTER['q'] + radii['q'] + if not (q_hi < 0): + return {'pass': False, 'failure_class': 'Q_NEGATIVITY_FAIL', 'failed_gates': ['q_strictly_negative']} + s_worst = q_hi / R_MAX + p['_s'] = I(s_worst) + kap = I(kappa) + xi = I(0, SIGMA) + t = I(-kappa, kappa) + + F = abs(p['a'])/8 + R*abs(p['c31'])/48 + A = (abs(p['z']) + R*abs(p['c13'])/2)*F + B = R*abs(p['c04'])*F**2/2 + C = abs(p['w'])/2 + MU + R*abs(p['c22'])/8 + A/MU + B/MU**2 + W = F/MU + + AS = 1 + R*p['c40']/12 + BS = p['a']/2 + R*p['c31']/12 + DS = p['_s'] + p['w']/2 + R*p['c22']/8 + AM = -1 + R*p['c40']/12 + BM = -p['a']/2 + R*p['c31']/12 + DM = p['_s'] - p['w']/2 + R*p['c22']/8 + detS = AS*DS - BS**2 + detM = AM*DM - BM**2 + axial = AS-DS + tau_upper = float('inf') if lower(axial) <= 0 else absmax(BS)/lower(axial) + + axis = cone_axis(xi, t, p) + upper_cone = kap*cone_axis(xi, kap, p) - cone_y_quotient(xi, kap, p) + lower_cone = kap*cone_axis(xi, -kap, p) + cone_y_quotient(xi, -kap, p) + + Xc = I(-0.5+SIGMA, 0.5-SIGMA) + top = -flow_y(Xc, W, p) + bottom = flow_y(Xc, -W, p) + Ystrip = I(-upper(W), upper(W)) + drift = -flow_x(Xc, Ystrip, p) + _, _, py = hessian(Xc, Ystrip, p) + + rho = 2*iv.sqrt(I(SIGMA**2)+W**2) + rho_upper = upper(rho) + Xcap = I(-0.5-rho_upper, -0.5+rho_upper) + Ycap = I(-rho_upper, rho_upper) + h11, h12, h22 = hessian(Xcap, Ycap, p) + offdiag = absmax(h12) + + bounds = { + 'threshold_slack': lower(-p['_s']-C), + 'saddle_det_negative_margin': -upper(detS), + 'maximum_A_negative_margin': -upper(AM), + 'maximum_D_negative_margin': -upper(DM), + 'maximum_det_positive_margin': lower(detM), + 'axial_gap_margin': lower(axial), + 'cone_slope_margin': kappa-2*tau_upper, + 'cone_axis_margin': lower(axis), + 'cone_upper_inward_margin': lower(upper_cone), + 'cone_lower_inward_margin': lower(lower_cone), + 'handoff_margin': lower(W)/2-kappa*SIGMA, + 'strip_top_inward_margin': lower(top), + 'strip_bottom_inward_margin': lower(bottom), + 'central_drift_margin': lower(drift), + 'partial_y_contraction_margin': -upper(py), + 'chart_x_margin': 0.75-(0.5+rho_upper), + 'chart_y_margin': 1-rho_upper, + 'capture_gershgorin_margin_1': lower(-h11)-offdiag, + 'capture_gershgorin_margin_2': lower(-h22)-offdiag, + } + failed = [k for k,v in bounds.items() if not (v > 0)] + return { + 'pass': not failed, + 'failure_class': None if not failed else 'INTERVAL_GATE_FAIL', + 'failed_gates': failed, + 'box': {k: [CENTER[k]-radii[k], CENTER[k]+radii[k]] for k in CENTER}, + 'radii': radii, + 'fixed': {'r_interval': [0.0, R_MAX], 'q_interval': [CENTER['q']-radii['q'], CENTER['q']+radii['q']], 's_upper_worst': s_worst, 'sigma': SIGMA, 'kappa': kappa, 'mu_interval': [lower(MU), upper(MU)]}, + 'derived_intervals': {'W': [lower(W), upper(W)], 'rho': [lower(rho), upper(rho)], 'tau_upper': tau_upper}, + 'lower_bounds': bounds, + 'monotonicity_ledger': { + 'domain': '0 spec.source.bytes or inflater.unconsumed_tail: + raise ValueError("payload size exceeds declared limit") + if not inflater.eof: + raise ValueError("incomplete gzip stream") + if inflater.unused_data: + raise ValueError("trailing gzip bytes or concatenated member") + check_identity(raw, spec.source, "payload") + return raw + + +def declared(section, key): + values = re.findall(r"^" + re.escape(key) + r": ([^\r\n]+)$", section, re.MULTILINE) + if len(values) != 1: + raise ValueError(f"declared {key} count") + return values[0] + + +def decode_wrapper(raw, wrapper_identity=None): + check_identity(raw, wrapper_identity or WRAPPER_IDENTITY, "wrapper") + try: + text = raw.decode("utf-8-sig").replace("\r\n", "\n") + except UnicodeDecodeError as exc: + raise ValueError("wrapper is not UTF-8") from exc + result = {} + for spec in SPECS: + begin = f"BEGIN_{spec.label}_GZIP_BASE64" + end = f"END_{spec.label}_GZIP_BASE64" + if text.count(begin) != 1 or text.count(end) != 1: + raise ValueError("marker count must be exactly one pair per payload") + before, rest = text.split(begin) + encoded, after = rest.split(end) + if before.rfind("\n" + spec.label + "\n") == -1: + raise ValueError("missing payload declaration heading") + section = before[before.rfind("\n" + spec.label + "\n"):] + fields = (("filename", spec.source.name), ("bytes", str(spec.source.bytes)), + ("sha256", spec.source.sha256), ("gzip bytes", str(spec.gzip.bytes)), + ("gzip sha256", spec.gzip.sha256)) + for key, expected in fields: + if declared(section, key) != expected: + raise ValueError(f"declared {key} mismatch") + compact = re.sub(r"[\r\n\t ]", "", encoded) + try: + compressed = base64.b64decode(compact, validate=True) + except (binascii.Error, ValueError) as exc: + raise ValueError("invalid base64") from exc + result[spec.source.name] = decode_member(compressed, spec) + return result + + +def verify_existing(directory, decoded): + directory = Path(directory) + if directory.is_symlink(): + raise ValueError("directory symlink") + for name, raw in decoded.items(): + safe_basename(name) + path = directory / name + if path.is_symlink(): + raise ValueError("payload symlink") + if not path.is_file() or path.read_bytes() != raw: + raise ValueError(f"existing payload mismatch: {name}") + + +def report(decoded): + return {"scientific_effect": "NONE", "archived_code_executed": False, + "complete_single_gzip_member_per_payload": True, + "payloads": [{"path": name, "bytes": len(raw), + "sha256": hashlib.sha256(raw).hexdigest(), "git_blob": git_blob(raw)} + for name, raw in decoded.items()]} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--wrapper-input", required=True, help="Exact UTF-8 wrapper file, or '-' for stdin bytes") + output = parser.add_mutually_exclusive_group(required=True) + output.add_argument("--verify-existing", type=Path, help="Verify the original two payload files") + output.add_argument("--write-new", type=Path, help="Create a NEW directory only after both payloads verify") + args = parser.parse_args() + try: + raw = sys.stdin.buffer.read(65537) if args.wrapper_input == "-" else Path(args.wrapper_input).read_bytes() + decoded = decode_wrapper(raw) + if args.verify_existing: + verify_existing(args.verify_existing, decoded) + else: + if args.write_new.is_symlink() or args.write_new.exists(): + raise ValueError("output directory must not already exist") + args.write_new.mkdir() + for name, payload in decoded.items(): + (args.write_new / name).write_bytes(payload) + print(json.dumps(report(decoded), sort_keys=True)) + return 0 + except (ValueError, OSError) as exc: + print(json.dumps({"passed": False, "error": str(exc)}, sort_keys=True), file=sys.stderr) + return 1 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/imports/gp_data_114_embedded_20260930/test_inert_decoder.py b/imports/gp_data_114_embedded_20260930/test_inert_decoder.py new file mode 100644 index 000000000..b12294300 --- /dev/null +++ b/imports/gp_data_114_embedded_20260930/test_inert_decoder.py @@ -0,0 +1,136 @@ +"""Real malformed-input controls; never import either recovered payload.""" +import argparse +import dataclasses +import gzip +import hashlib +import json +from pathlib import Path +import sys +import tempfile +import unittest + +import inert_decoder as decoder + +WRAPPER = b"" +HERE = Path(__file__).resolve().parent + + +def identity(name, raw): + return decoder.Identity(name, len(raw), hashlib.sha256(raw).hexdigest()) + + +def fixture(raw=b"nonexecutable fixture\n"): + compressed = gzip.compress(raw, mtime=0) + spec = decoder.PayloadSpec("SOURCE", identity("fixture.py", raw), + identity("fixture.gz", compressed)) + return raw, compressed, spec + + +class DecoderControls(unittest.TestCase): + def test_complete_single_member_decodes_exactly(self): + raw, compressed, spec = fixture() + self.assertEqual(decoder.decode_member(compressed, spec), raw) + + def test_truncation_rejects_even_when_compressed_identity_matches(self): + _, compressed, spec = fixture() + broken = compressed[:-1] + spec = dataclasses.replace(spec, gzip=identity("fixture.gz", broken)) + with self.assertRaisesRegex(ValueError, "incomplete gzip"): + decoder.decode_member(broken, spec) + + def test_trailing_byte_rejects_even_when_compressed_identity_matches(self): + _, compressed, spec = fixture() + broken = compressed + b"x" + spec = dataclasses.replace(spec, gzip=identity("fixture.gz", broken)) + with self.assertRaisesRegex(ValueError, "trailing gzip"): + decoder.decode_member(broken, spec) + + def test_concatenated_member_rejects(self): + _, compressed, spec = fixture() + broken = compressed + gzip.compress(b"second member", mtime=0) + spec = dataclasses.replace(spec, gzip=identity("fixture.gz", broken)) + with self.assertRaisesRegex(ValueError, "trailing gzip"): + decoder.decode_member(broken, spec) + + def test_invalid_crc_rejects(self): + _, compressed, spec = fixture() + broken = compressed[:-8] + bytes([compressed[-8] ^ 1]) + compressed[-7:] + spec = dataclasses.replace(spec, gzip=identity("fixture.gz", broken)) + with self.assertRaisesRegex(ValueError, "invalid gzip"): + decoder.decode_member(broken, spec) + + def test_payload_size_mismatch_rejects(self): + raw, compressed, spec = fixture() + spec = dataclasses.replace(spec, source=dataclasses.replace(spec.source, bytes=len(raw)+1)) + with self.assertRaisesRegex(ValueError, "payload size"): + decoder.decode_member(compressed, spec) + + def test_payload_hash_mismatch_rejects(self): + _, compressed, spec = fixture() + spec = dataclasses.replace(spec, source=dataclasses.replace(spec.source, sha256="0"*64)) + with self.assertRaisesRegex(ValueError, "payload sha256"): + decoder.decode_member(compressed, spec) + + def test_unsafe_filename_rejects(self): + _, compressed, spec = fixture() + spec = dataclasses.replace(spec, source=dataclasses.replace(spec.source, name="../fixture.py")) + with self.assertRaisesRegex(ValueError, "unsafe filename"): + decoder.decode_member(compressed, spec) + + def test_exact_public_wrapper_reconstructs_original_basename_pair(self): + decoded = decoder.decode_wrapper(WRAPPER) + self.assertEqual(set(decoded), {"GP-DATA-114-v1.0_uniform_q_degree4_interval.py", + "GP-DATA-114-v1.0_receipt.json"}) + self.assertEqual(len(decoded["GP-DATA-114-v1.0_uniform_q_degree4_interval.py"]), 9787) + self.assertEqual(len(decoded["GP-DATA-114-v1.0_receipt.json"]), 5047) + + def test_altered_wrapper_identity_rejects(self): + with self.assertRaisesRegex(ValueError, "wrapper sha256"): + decoder.decode_wrapper(WRAPPER.replace(b"ACTIVE WRAPPER", b"XCTIVE WRAPPER", 1)) + + def test_duplicate_marker_rejects_after_rebinding_wrapper_fixture(self): + broken = WRAPPER.replace(b"BEGIN_SOURCE_GZIP_BASE64", b"BEGIN_SOURCE_GZIP_BASE64\r\nBEGIN_SOURCE_GZIP_BASE64", 1) + with self.assertRaisesRegex(ValueError, "marker count"): + decoder.decode_wrapper(broken, identity("wrapper.txt", broken)) + + def test_declared_compressed_size_rejects_after_rebinding_wrapper_fixture(self): + broken = WRAPPER.replace(b"gzip bytes: 3416", b"gzip bytes: 3417", 1) + with self.assertRaisesRegex(ValueError, "declared gzip bytes"): + decoder.decode_wrapper(broken, identity("wrapper.txt", broken)) + + def test_invalid_base64_rejects_after_rebinding_wrapper_fixture(self): + broken = WRAPPER.replace(b"BEGIN_SOURCE_GZIP_BASE64\r\nH", b"BEGIN_SOURCE_GZIP_BASE64\r\n!", 1) + with self.assertRaisesRegex(ValueError, "invalid base64"): + decoder.decode_wrapper(broken, identity("wrapper.txt", broken)) + + def test_changed_existing_payload_rejects(self): + decoded = decoder.decode_wrapper(WRAPPER) + with tempfile.TemporaryDirectory(prefix="controls_", dir=HERE) as directory: + for name, raw in decoded.items(): + Path(directory, name).write_bytes(raw) + Path(directory, "GP-DATA-114-v1.0_receipt.json").write_bytes(b"altered") + with self.assertRaisesRegex(ValueError, "existing payload"): + decoder.verify_existing(Path(directory), decoded) + + def test_existing_payload_symlink_rejects(self): + decoded = decoder.decode_wrapper(WRAPPER) + self.assertIn("GP-DATA-114-v1.0_receipt.json", decoded) + with tempfile.TemporaryDirectory(prefix="controls_", dir=HERE) as directory: + for name, raw in decoded.items(): + Path(directory, name).write_bytes(raw) + leaf = Path(directory, "GP-DATA-114-v1.0_receipt.json") + leaf.unlink() + leaf.symlink_to("GP-DATA-114-v1.0_uniform_q_degree4_interval.py") + with self.assertRaisesRegex(ValueError, "symlink"): + decoder.verify_existing(Path(directory), decoded) + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument("--wrapper-input", required=True, help="Exact UTF-8 wrapper file, or '-' for stdin bytes") + args = parser.parse_args() + WRAPPER = sys.stdin.buffer.read() if args.wrapper_input == "-" else Path(args.wrapper_input).read_bytes() + result = unittest.TextTestRunner(verbosity=2).run(unittest.defaultTestLoader.loadTestsFromTestCase(DecoderControls)) + print(json.dumps({"tests_run": result.testsRun, "failures": len(result.failures), + "errors": len(result.errors), "passed": result.wasSuccessful()}, sort_keys=True)) + sys.exit(0 if result.wasSuccessful() else 1) diff --git a/imports/gp_data_114_embedded_20260930/test_verify_packet.py b/imports/gp_data_114_embedded_20260930/test_verify_packet.py new file mode 100644 index 000000000..45b2c48d4 --- /dev/null +++ b/imports/gp_data_114_embedded_20260930/test_verify_packet.py @@ -0,0 +1,188 @@ +"""Packet-boundary controls; archived Python is copied and read as bytes only.""" +import argparse +import hashlib +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile +import unittest + +import verify_packet as verifier + +HERE = Path(__file__).resolve().parent +ORIGINAL_FILES = ( + 'GP-DATA-114-v1.0_receipt.json', + 'GP-DATA-114-v1.0_uniform_q_degree4_interval.py', + 'MANIFEST.json', 'README.md', 'SOURCES.json', 'VALIDATION.json', + 'inert_decoder.py', 'test_inert_decoder.py', +) +WRAPPER_RAW = None + + +class PacketControls(unittest.TestCase): + def setUp(self): + self.temporary = tempfile.TemporaryDirectory(prefix='gpdata114_packet_') + self.addCleanup(self.temporary.cleanup) + self.directory = Path(self.temporary.name) / 'packet' + self.directory.mkdir() + for name in ORIGINAL_FILES: + shutil.copyfile(HERE / name, self.directory / name) + self.resign_manifest() + self.wrapper = WRAPPER_RAW if WRAPPER_RAW is not None else Path('/tmp/gpdata114-public-wrapper.txt').read_bytes() + + def resign_manifest(self): + manifest = json.loads((self.directory / 'MANIFEST.json').read_text()) + entries = [] + for name in ORIGINAL_FILES: + if name == 'MANIFEST.json': + continue + data = (self.directory / name).read_bytes() + framed = b'blob ' + str(len(data)).encode() + b'\0' + data + entries.append({'path': name, 'bytes': len(data), + 'sha256': hashlib.sha256(data).hexdigest(), + 'git_blob': hashlib.sha1(framed).hexdigest()}) + manifest['files'] = entries + (self.directory / 'MANIFEST.json').write_text(json.dumps(manifest, sort_keys=True)) + + def test_reject_undeclared_packet_file(self): + (self.directory / 'undeclared.py').write_bytes(b'inert unexpected bytes\n') + with self.assertRaisesRegex(ValueError, 'file set'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_false_sources_identity_after_manifest_resigning(self): + original = (self.directory / 'SOURCES.json').read_text() + wrong_values = {'label': 'OTHER', 'path': 'wrong.py', 'bytes': 9788, + 'sha256': '0' * 64, 'git_blob': '0' * 40, + 'gzip_bytes': 3417, 'gzip_sha256': '0' * 64} + for field, value in wrong_values.items(): + with self.subTest(field=field): + sources = json.loads(original) + sources['payloads'][0][field] = value + (self.directory / 'SOURCES.json').write_text(json.dumps(sources, sort_keys=True)) + self.resign_manifest() + with self.assertRaisesRegex(ValueError, 'SOURCES payload'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_known_eight_file_fixture_passes(self): + report = verifier.verify_packet(self.directory, self.wrapper) + self.assertTrue(report['passed']) + self.assertEqual(report['packet_files'], 8) + self.assertEqual(report['manifest_entries'], 7) + self.assertEqual(report['sources_payload_bindings'], 2) + self.assertEqual(report['scientific_effect'], 'NONE') + self.assertFalse(report['archived_code_executed']) + + def test_reject_duplicate_manifest_path(self): + manifest = json.loads((self.directory / 'MANIFEST.json').read_text()) + manifest['files'].append(dict(manifest['files'][0])) + (self.directory / 'MANIFEST.json').write_text(json.dumps(manifest)) + with self.assertRaisesRegex(ValueError, 'duplicate manifest path'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_missing_declared_file(self): + (self.directory / 'README.md').unlink() + with self.assertRaisesRegex(ValueError, 'file set'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_unsafe_manifest_paths_before_reading(self): + original = (self.directory / 'MANIFEST.json').read_text() + for path in ('../README.md', 'nested/file', '/absolute', '\\absolute', '.', '..', '', 'bad\0name', 'MANIFEST.json'): + with self.subTest(path=path): + manifest = json.loads(original) + manifest['files'][0]['path'] = path + (self.directory / 'MANIFEST.json').write_text(json.dumps(manifest)) + with self.assertRaisesRegex(ValueError, 'unsafe manifest path'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_file_and_directory_symlinks(self): + leaf = self.directory / 'README.md' + leaf.unlink() + leaf.symlink_to('SOURCES.json') + with self.assertRaisesRegex(ValueError, 'symlink'): + verifier.verify_packet(self.directory, self.wrapper) + link = self.directory.parent / 'linked-packet' + link.symlink_to(self.directory, target_is_directory=True) + with self.assertRaisesRegex(ValueError, 'symlink'): + verifier.verify_packet(link, self.wrapper) + + def test_reject_nonfile_directory_entry(self): + leaf = self.directory / 'README.md' + leaf.unlink() + leaf.mkdir() + with self.assertRaisesRegex(ValueError, 'regular file'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_changed_data_and_each_manifest_identity_field(self): + readme = self.directory / 'README.md' + original_bytes = readme.read_bytes() + readme.write_bytes(original_bytes + b'changed') + with self.assertRaisesRegex(ValueError, 'packet file identity'): + verifier.verify_packet(self.directory, self.wrapper) + readme.write_bytes(original_bytes) + original_manifest = (self.directory / 'MANIFEST.json').read_text() + for field, value in (('bytes', -1), ('sha256', '0' * 64), ('git_blob', '0' * 40)): + with self.subTest(field=field): + manifest = json.loads(original_manifest) + entry = next(item for item in manifest['files'] if item['path'] == 'README.md') + entry[field] = value + (self.directory / 'MANIFEST.json').write_text(json.dumps(manifest)) + with self.assertRaisesRegex(ValueError, 'packet file identity'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_duplicate_and_missing_sources_payloads(self): + original = (self.directory / 'SOURCES.json').read_text() + for variant in ('duplicate', 'missing'): + with self.subTest(variant=variant): + sources = json.loads(original) + payloads = sources['payloads'] + sources['payloads'] = [payloads[0], payloads[0]] if variant == 'duplicate' else payloads[:1] + (self.directory / 'SOURCES.json').write_text(json.dumps(sources)) + self.resign_manifest() + with self.assertRaisesRegex(ValueError, 'SOURCES payload'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_reject_jointly_resigned_archived_bytes_and_sources(self): + name = 'GP-DATA-114-v1.0_uniform_q_degree4_interval.py' + payload = self.directory / name + changed = payload.read_bytes() + b'\n' + payload.write_bytes(changed) + sources = json.loads((self.directory / 'SOURCES.json').read_text()) + source = sources['payloads'][0] + source['bytes'] = len(changed) + source['sha256'] = hashlib.sha256(changed).hexdigest() + source['git_blob'] = hashlib.sha1(b'blob ' + str(len(changed)).encode() + b'\0' + changed).hexdigest() + (self.directory / 'SOURCES.json').write_text(json.dumps(sources)) + self.resign_manifest() + with self.assertRaisesRegex(ValueError, 'SOURCES payload'): + verifier.verify_packet(self.directory, self.wrapper) + + def test_cli_failure_exits_one_and_matches_in_both_modes(self): + (self.directory / 'undeclared.txt').write_bytes(b'extra') + wrapper_path = self.directory.parent / 'wrapper.txt' + wrapper_path.write_bytes(self.wrapper) + outputs = [] + for flags in (('-B', '-S'), ('-B', '-O', '-S')): + command = [sys.executable, *flags, str(HERE / 'verify_packet.py'), + '--packet', str(self.directory), '--wrapper-input', str(wrapper_path)] + result = subprocess.run(command, capture_output=True, text=True) + self.assertEqual(result.returncode, 1) + summary = json.loads(result.stdout) + self.assertFalse(summary['passed']) + self.assertIn('file set', summary['error']) + self.assertEqual(summary['scientific_effect'], 'NONE') + outputs.append(result.stdout) + self.assertEqual(*outputs) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--wrapper-input', default='/tmp/gpdata114-public-wrapper.txt') + args = parser.parse_args() + WRAPPER_RAW = Path(args.wrapper_input).read_bytes() + result = unittest.TextTestRunner(verbosity=2).run(unittest.defaultTestLoader.loadTestsFromTestCase(PacketControls)) + print(json.dumps({'tests_run': result.testsRun, 'failures': len(result.failures), + 'errors': len(result.errors), 'passed': result.wasSuccessful(), + 'scientific_effect': 'NONE', 'archived_code_executed': False}, sort_keys=True)) + sys.exit(0 if result.wasSuccessful() else 1) diff --git a/imports/gp_data_114_embedded_20260930/verify_packet.py b/imports/gp_data_114_embedded_20260930/verify_packet.py new file mode 100644 index 000000000..3b933da10 --- /dev/null +++ b/imports/gp_data_114_embedded_20260930/verify_packet.py @@ -0,0 +1,157 @@ +"""Verify exact packet custody; never execute, import or compile archived code. + +Only the new inert_decoder module is imported. Both recovered payloads, +including the archived .py file, are handled solely as bytes. A re-signed +manifest cannot override the decoder's independent original payload pins. +This verifier supplies custody checks, not mathematical or status acceptance. +""" +import argparse +import hashlib +import json +from pathlib import Path +import sys + +import inert_decoder as decoder + +MANIFEST = 'MANIFEST.json' +DEFAULT_WRAPPER = '/tmp/gpdata114-public-wrapper.txt' + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def unique_object(pairs): + result = {} + for key, value in pairs: + require(key not in result, 'duplicate JSON key: ' + key) + result[key] = value + return result + + +def load_record(raw, context): + value = json.loads(raw, object_pairs_hook=unique_object) + require(isinstance(value, dict), context + ' must be a JSON object') + return value + + +def safe_manifest_path(name): + require(isinstance(name, str) and name and name not in {'.', '..', MANIFEST} + and not any(c in name for c in '/\\\0'), 'unsafe manifest path') + + +def identity(raw): + framed = b'blob ' + str(len(raw)).encode('ascii') + b'\0' + raw + return {'bytes': len(raw), 'sha256': hashlib.sha256(raw).hexdigest(), + 'git_blob': hashlib.sha1(framed).hexdigest()} + + +def exact_field(record, field, expected, context): + actual = record.get(field) + require(type(actual) is type(expected) and actual == expected, + context + ': ' + field) + + +def verify_packet(directory, wrapper_raw): + directory = Path(directory) + require(not directory.is_symlink(), 'packet directory symlink') + require(directory.is_dir(), 'packet must be a directory') + observed = {} + for path in directory.iterdir(): + require(not path.is_symlink(), 'packet file symlink: ' + path.name) + require(path.is_file(), 'packet entry must be a regular file: ' + path.name) + observed[path.name] = path + require(MANIFEST in observed, 'packet file set missing MANIFEST.json') + manifest = load_record(observed[MANIFEST].read_bytes(), 'manifest') + require(manifest.get('manifest_self_excluded') is True, 'manifest self-exclusion required') + require(isinstance(manifest.get('files'), list), 'manifest files must be a list') + entries = {} + for entry in manifest['files']: + require(isinstance(entry, dict), 'manifest entry must be an object') + name = entry.get('path') + safe_manifest_path(name) + require(name not in entries, 'duplicate manifest path: ' + name) + entries[name] = entry + expected_names = set(entries) | {MANIFEST} + actual_names = set(observed) + require(actual_names == expected_names, + 'packet file set mismatch: extra=' + ','.join(sorted(actual_names - expected_names)) + + ' missing=' + ','.join(sorted(expected_names - actual_names))) + + stored = {} + for name, entry in entries.items(): + data = observed[name].read_bytes() + for field, expected in identity(data).items(): + exact_field(entry, field, expected, 'packet file identity mismatch: ' + name) + stored[name] = data + + require('SOURCES.json' in stored, 'SOURCES record missing from manifest') + sources = load_record(stored['SOURCES.json'], 'SOURCES') + payloads = sources.get('payloads') + require(isinstance(payloads, list) and len(payloads) == len(decoder.SPECS), + 'SOURCES payload list must contain exactly the canonical pair') + source_entries = {} + for payload in payloads: + require(isinstance(payload, dict), 'SOURCES payload must be an object') + label = payload.get('label') + require(isinstance(label, str) and label not in source_entries, + 'SOURCES payload duplicate or invalid label') + source_entries[label] = payload + require(set(source_entries) == {spec.label for spec in decoder.SPECS}, + 'SOURCES payload labels differ from canonical pair') + + require(isinstance(wrapper_raw, bytes), 'wrapper input must be exact bytes') + decoded = decoder.decode_wrapper(wrapper_raw) + wrapper = sources.get('wrapper') + require(isinstance(wrapper, dict), 'SOURCES wrapper must be an object') + wrapper_pin = decoder.WRAPPER_IDENTITY + for field, expected in (('bytes', wrapper_pin.bytes), ('sha256', wrapper_pin.sha256), + ('git_blob', wrapper_pin.blob)): + exact_field(wrapper, field, expected, 'SOURCES wrapper identity mismatch') + + for spec in decoder.SPECS: + payload = source_entries[spec.label] + expected_fields = { + 'label': spec.label, 'path': spec.source.name, 'bytes': spec.source.bytes, + 'sha256': spec.source.sha256, 'git_blob': spec.source.blob, + 'gzip_bytes': spec.gzip.bytes, 'gzip_sha256': spec.gzip.sha256, + } + for field, expected in expected_fields.items(): + exact_field(payload, field, expected, 'SOURCES payload identity mismatch: ' + spec.label) + require(spec.source.name in entries, 'payload missing from manifest: ' + spec.label) + # decode_wrapper already binds each compressed member, its declarations, + # and decoded bytes to SPECS. Bind the resulting raw identity again to + # both independently pinned SPECS and the two packet metadata records. + for field, actual in identity(decoded[spec.source.name]).items(): + exact_field(payload, field, actual, 'SOURCES payload decoded identity mismatch: ' + spec.label) + exact_field(entries[spec.source.name], field, actual, 'manifest decoded payload identity mismatch: ' + spec.label) + decoder.verify_existing(directory, decoded) + report = decoder.report(decoded) + report.update({'passed': True, 'packet_files': len(observed), + 'manifest_entries': len(entries), 'sources_payload_bindings': len(source_entries), + 'archived_code_imported': False, 'archived_code_compiled': False}) + return report + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--packet', type=Path, default=Path(__file__).resolve().parent) + parser.add_argument('--wrapper-input', type=Path, default=Path(DEFAULT_WRAPPER)) + args = parser.parse_args() + try: + with args.wrapper_input.open('rb') as stream: + wrapper_raw = stream.read(65537) + report = verify_packet(args.packet, wrapper_raw) + exit_code = 0 + except (OSError, ValueError) as exc: + report = {'passed': False, 'error': str(exc), 'scientific_effect': 'NONE', + 'archived_code_executed': False, 'archived_code_imported': False, + 'archived_code_compiled': False} + exit_code = 1 + print(json.dumps(report, sort_keys=True)) + return exit_code + + +if __name__ == '__main__': + sys.exit(main()) diff --git a/reviews/candidates_pending_20260928/CANDIDATES.md b/reviews/candidates_pending_20260928/CANDIDATES.md index e3fc1a310..02455f056 100644 --- a/reviews/candidates_pending_20260928/CANDIDATES.md +++ b/reviews/candidates_pending_20260928/CANDIDATES.md @@ -1,8 +1,11 @@ -# Theorem candidates pending review and test +# Historical theorem-candidate intake with current review crosswalk Scientific effect: **NONE**. Publication of this list is not acceptance. -Algebra already checked is noted. Continuum or missing public tests are the pending obligation. +This file began as the 28 September 2026 intake list. The source descriptions below +are preserved, while each section now states the source-bound disposition known on +29 September 2026. A later review does not recover the absent original `algebra.py`, +and tests or integration do not substitute for the cited analytic review. ## C1. Planar elder-failure lower bound (Math-#116) @@ -17,19 +20,33 @@ for small r, hence with the parent upper bound `1-p_r = Theta(r^3)` and Finite identities checked: cubic `G_k` pins correctly; path nodes evaluate to `0`, `-7k/32`, `-7k/32`, `+17k/64`; extra saddles at `X=-3/4`, `Z^2=15/8`, height `-7k/32`. -Pending: 10-jet covariance floor; `Q_r` box mass `>= c r`; `W_r >= c r^4` on that box; comparison with `Z_r <= C r^2`; public runnable suite (`algebra.py` is not in the PR). +Current disposition: Anthropic's source-bound +[`reviews/d1_elder_lower_claude_20260928/REVIEW.md`](../d1_elder_lower_claude_20260928/REVIEW.md) +accepts the ten-jet Schur floor and box mass (L9--L11), conditional `C^4` +control (L12--L13), determinant/normalizer ledger (L5/L14), and Theorem E1. +The review packet was integrated by Math-#124. The original author's +`algebra.py` remains absent, so its historical execution claim is still not +publicly reproducible; the review's distinct checker does not recover it. ## C2. Local window multiplicity (Math-#116) Source: `frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md` at `0507e3a1dbd3dede84cbeb805947c70aa16ebc36`. -Candidate: `P(N_global >= 2) = Theta(r^3)` in the planar height window. -If established with the stated first-moment upper bound, this would rule out global -conditional uniqueness. Its lower half would also obstruct a global `O(r^5)` -factorial upper bound; see C6. These are conditional consequences, not an -acceptance of this candidate. - -Pending: the jet-box probability lower bound together with the planar first-moment upper bound (I5), as a matched Theta statement, plus public tests. +Original candidate: `P(N_global >= 2) = Theta(r^3)` in the planar height window. +The same global order conclusion is now available by the separate reviewed Palm +route in `frontiers/c6_palm_route_20260929/PROOF.md`; that theorem does not +validate this candidate's local jet-box mechanism. The separate C1/C2 review cited +below does validate that mechanism with its stated first-moment upper bound, giving +an independent local explanation of the failure of global conditional uniqueness. +That acceptance is not transferred from the Palm theorem. + +Current disposition: the same source-bound +[`reviews/d1_elder_lower_claude_20260928/REVIEW.md`](../d1_elder_lower_claude_20260928/REVIEW.md) +accepts Theorem L, including the jet-box lower event and the I5 first-moment +upper composition. This reviewed local mechanism remains distinct from the open +regional `math.rn-region.witness-collision` graph predicate. The absent original +`algebra.py` remains a publication/execution limitation, not an unreviewed +Gaussian step. ## C3. Remote ordered-pair kernel and beta-gap law (Math-#116) @@ -51,7 +68,10 @@ little-o is for fixed `E`, not arbitrary oscillatory `E_r`. Finite identities checked: Beta(2/3, 2) has mean `1/4` and variance `9/176`. -Pending: the contact Gaussian coefficient, extra-conditioned weight limit, overlap `(k-s^3|t|)_+`, L1 translation, and public tests. +Current disposition: Anthropic's source-bound +[`reviews/remote_pair_law_claude_20260928/REVIEW.md`](../remote_pair_law_claude_20260928/REVIEW.md) +records the analytic verdict integrated by Math-#129. Its exact fixed-remote, +fixed-Borel scope remains controlling; it is not a shrinking-region statement. ## C4. Distance-moment transition (Math-#116) @@ -59,7 +79,11 @@ Source: `frontiers/window_multiplicity_laws_20260928/REMOTE_DISTANCE_MOMENTS.md` Candidate: for each fixed moment order `p>=0`, ordered-pair distance moments `M_p` scale as `r^{5+p} A_p` for `0<=p<1`, `r^6(A_1+B_1)` at `p=1`, and `r^6 B_p` for `p>1`. -Pending: radial envelope `C(r^{12} delta^{-8}+r^6)`, equal-height contact kernel, loss of uniform integrability at moment order 1, public tests. +Current disposition: the amended source-bound +[`reviews/remote_distance_moments_claude_20260929/REVIEW.md`](../remote_distance_moments_claude_20260929/REVIEW.md) +records the accepted moment transition integrated with Math-#130. The finite-r +microscopic and limiting moment ranges remain those stated in the integration +receipt; no pair-event-conditioned law is inferred. ## C5. Remote singleton height mark (Math-#116) @@ -67,48 +91,79 @@ Source: `frontiers/window_multiplicity_laws_20260928/HEIGHT_MARKS.md` at `0507e3 Candidate: conditional on a remote occurrence, height in the window is asymptotically uniform and independent of the joint location/index mark, at `O(r)` TV. -Pending: source-bound use of the pre-height-integrated remote kernel. Math-#115 -was integrated at `4b2aa45b2f995fc27c16ec37cccb081821eda74c` with a scoped -nonauthor review, but its proof explicitly omits the height law. This new height -candidate does not inherit acceptance from that merge. - -## C6. Torus-wide second factorial moment: open optimal order - -The global `O(r^5)` extension is not an unqualified theorem target. There is a -**CONDITIONAL OBSTRUCTION** from C2: pointwise for every nonnegative integer `N`, - - N(N-1) >= 2*1{N>=2}. - -Thus a proved lower bound `P(N>=2)>=c r^3` would force -`E[N(N-1)]>=2c r^3`, incompatible with a uniform `O(r^5)` upper bound as `r->0`. -This is not an accepted disproof until C2's analytic lower mechanism is validated. -No global factorial upper bound or optimal order is supplied here. +Current disposition: Anthropic's source-bound +[`reviews/remote_height_marks_claude_20260929/REVIEW.md`](../remote_height_marks_claude_20260929/REVIEW.md) +records the accepted singleton height-mark statement integrated with Math-#130. +The acceptance comes from that review, not from Math-#115; it retains the fixed +remote and deterministic varying positive-volume-set scope recorded in +[`docs/integration/2026-09-29-reviewed-window-and-inverse.md`](../../docs/integration/2026-09-29-reviewed-window-and-inverse.md). + +## C6. Torus-wide factorial moments: reviewed global order; regional mechanism open + +The reviewed planar Fourier-cutoff theorem now supplies, for fixed `T`, compact +positive-gap marks and existential constants, + + E_{Q_r^W}[N(N-1)] <= C r^3 log(1/r). + +Full proof: `frontiers/c6_fourier_cutoff_20260929/PROOF.md` at Math main +`4d062e2a976abc8800a1ae63a49a943526b6bc0d`. The source-bound reading note is +`reviews/c6_fourier_completion_20260929/READING_NOTE.md`; Anthropic nonauthor +review 5356335148 ACCEPTs the stated planar composition. The complete planar +D5/I5 reading rule is load-bearing. Tests and the merge do not substitute for +that analytic review. + +The later sharp route is a separate reviewed/merged packet, not part of this +planar graph transition. `frontiers/d5_dimension_lift_20260929/PROOF.md` +(Math-#141, merge `e1ca400`) supplies the fixed-`d` first-moment inputs, and +`frontiers/c6_palm_route_20260929/PROOF.md` (Math-#145, merge `820d443`) gives +`E[(N)_q]<=C_q r^3` for fixed `q>=2` and +`E[N(N-1)]=Theta(r^3)` in every fixed `d>=2`, at the exact scopes and review +exposures stated in those packets. The conditional consequences in +`frontiers/c6_rare_cluster_laws_20260929/PROOF.md` (Math-#153, merge +`190cf2b`) are likewise separate: they do not identify a unique limiting +cluster law and do not move a graph node. + +These separate reviewed/merged packets resolve the global sharp-order and +higher-dimensional first-moment statements at their recorded scopes. They do +not close the regional shrinking-witness mechanism: the planar graph node +`math.rn-region.witness-collision` remains `OPEN_ACTIVE`. This catalog line +therefore distinguishes a proved global result from an unresolved regional +explanation rather than treating the global result as open. The fixed-remote `O(r^5)` result remains separate: both witnesses stay a fixed -distance from the pins. Source: `frontiers/remote_collision_20260928/PROOF.md` -at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78`, Corollary D / P_eta. - -Open task: determine a valid full-window factorial upper bound with the -pin-neighborhood collisions included, compatible with the eventual disposition -of C2. Cauchy-Schwarz on the first moment (I5) does not supply such a bound. +distance from the pins. Source: `frontiers/remote_collision_20260928/PROOF.md`. -## C7. Unrestricted selection-difference rate +Open task: prove or refute the shrinking-separation witness-collision mechanism +encoded by the open graph node, without inferring it from the global Palm +bound. Numerical constants and a unique limiting cluster law also remain +outside the reviewed packets cited above. +## C7. Unrestricted selection-difference rate (historical proposal — REFUTED) -Candidate: Theorem C's unrestricted densities satisfy +Historical candidate: Theorem C's unrestricted densities satisfy 0 <= nu_cand^{all}(ell) - nu_eld^{all}(ell) <= C ell^{2/3}. -Written support: the compact-mark case is the parent difference estimate under +The compact-mark case is the parent difference estimate under Theorem A in `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` -at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78`. #116 C1 would make the compact exponent sharp. +at `5a5b97d2c518dd5c04c89d17029ec96e721d3b78`; it is not contradicted by +the unrestricted counterexample. -Pending: a rate in the unbounded-mark dominated-convergence argument of parent §13; parent text presently does not claim this unrestricted remainder. +Current disposition: **REFUTED**. Theorem U in +[`frontiers/unrestricted_selection_difference_20260929/PROOF.md`](../../frontiers/unrestricted_selection_difference_20260929/PROOF.md) +proves that the unrestricted difference is bounded below by a positive constant, +so its ratio to `ell^alpha` diverges for every `alpha>0`. The source-bound +[`reviews/c7_nonvanishing_openai_20260929/REVIEW.md`](../c7_nonvanishing_openai_20260929/REVIEW.md) +accepts that conclusion at fixed `d,L,r_0`; the packet landed via Math-#133 at +`5bd924969bb261241885cd224d5dfb31acf64509`. This does not assert an +individual expansion for `nu_eld` or transfer the compact-mark rate to the +unrestricted densities. -## C8. Numerical constants +## C8. Numerical constants (still open as of 2026-09-29) Candidates: explicit finite `C`, `r_*`, `z_*`, `c_{B,K}`, `c_{d,L}` on a declared compact-mark / torus band. -Pending: a certified covariance-enclosure / finite-band program. No numerical value is recorded here. +Current disposition: **OPEN**. A certified covariance-enclosure / finite-band +program is still missing. No numerical value is recorded here. ## Not candidates diff --git a/reviews/d5_graph_fold_20260929/README.md b/reviews/d5_graph_fold_20260929/README.md new file mode 100644 index 000000000..b90db1f0e --- /dev/null +++ b/reviews/d5_graph_fold_20260929/README.md @@ -0,0 +1,21 @@ +# D5 graph/proof-index fold — 2026-09-29 + +This additive integration executes the already reviewed, declarative D5 reconciliation against the live downstream graph and repairs the public proof index/catalog. It does not alter proof bodies, `STATUS.md`, `LANDING_CLAIMS.json`, prizes or scientific Booleans. + +Scope: +- planar `d=2`, fixed `T`, compact marks with `k>=k_->0`, existential constants; +- D5-a–D5-d all-height first moments and D5-e–D5-f window-only first moments; +- reviewed planar C6 upper bound `E N(N-1)<=C r^3 log(1/r)`. + +Separate reviewed/merged packets are not folded by this planar transition: the fixed-dimensional D5 first moments in `frontiers/d5_dimension_lift_20260929/` (Math-#141), the sharp Palm route in `frontiers/c6_palm_route_20260929/` (Math-#145), and the conditional rare-cluster consequences in `frontiers/c6_rare_cluster_laws_20260929/` (Math-#153). + +Explicitly open in this graph fold: the regional shrinking witness-collision mechanism and its `OPEN_ACTIVE` node. Numerical constants and RN/24-jet/JETMOD remain outside this fold; no elder-selection or other register transition is inferred. + +Run: + +```sh +python -B -S reviews/d5_graph_fold_20260929/verify.py +python -B -O -S reviews/d5_graph_fold_20260929/verify.py +``` + +The verifier checks exact proposed nodes/edges and negative controls for a missing node, stale fingerprint, missing edge and accidental witness closure. Passing software checks are not mathematical acceptance. diff --git a/reviews/d5_graph_fold_20260929/verify.py b/reviews/d5_graph_fold_20260929/verify.py new file mode 100644 index 000000000..989033732 --- /dev/null +++ b/reviews/d5_graph_fold_20260929/verify.py @@ -0,0 +1,114 @@ +#!/usr/bin/env python3 +"""Fail-closed checks for the D5 graph/proof-index/C6 catalog fold.""" +import copy +import json +import pathlib +import sys +import unittest + +ROOT=pathlib.Path(__file__).resolve().parents[2] +GRAPH=ROOT/"frontiers/downstream_gate_20260925/GRAPH.json" +PROP=ROOT/"reviews/d5_reconciliation_20260929/PROPOSED_TRANSITIONS.json" +INDEX=ROOT/"PROOF_INDEX.md" +CAND=ROOT/"reviews/candidates_pending_20260928/CANDIDATES.md" + +def load(path): + def pairs(items): + out={} + for k,v in items: + if k in out: raise ValueError("duplicate JSON key: "+k) + out[k]=v + return out + return json.loads(path.read_text(encoding="utf-8"),object_pairs_hook=pairs, + parse_constant=lambda x: (_ for _ in ()).throw(ValueError("nonfinite JSON: "+x))) + +def realized(graph,proposal): + nodes=graph["nodes"]; edges=graph["edges"] + if len({(e["from"],e["to"],e.get("relation")) for e in edges})!=len(edges): return False + for n in proposal["proposed_graph_nodes"]: + actual=nodes.get(n["id"]) + if actual is None: return False + for k,v in n.items(): + if k!="id" and actual.get(k)!=v: return False + have={(e["from"],e["to"],e.get("relation"),e.get("required")) for e in edges} + for e in proposal["proposed_graph_edges"]: + if (e["from"],e["to"],e.get("relation"),True) not in have: return False + for tr in proposal["graph"]: + if nodes.get(tr["node"],{}).get("classification")!=tr["proposed"]: return False + return nodes["math.rn-region.witness-collision"]["classification"]=="OPEN_ACTIVE" + +class FoldTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.g=load(GRAPH); cls.p=load(PROP) + cls.index=INDEX.read_text(encoding="utf-8") + cls.cand=CAND.read_text(encoding="utf-8") + def test_proposal_realized(self): self.assertTrue(realized(self.g,self.p)) + def test_missing_node_rejected(self): + g=copy.deepcopy(self.g);g["nodes"].pop("math.d5-component.collar-proof") + self.assertFalse(realized(g,self.p)) + def test_stale_fingerprint_rejected(self): + g=copy.deepcopy(self.g);g["nodes"]["math.d5-component.punctured-pin-proof"]["fingerprint"]="0"*64 + self.assertFalse(realized(g,self.p)) + def test_missing_edge_rejected(self): + g=copy.deepcopy(self.g);g["edges"]=[e for e in g["edges"] if not (e["from"]=="math.d5-pin-neighborhood-first-moment" and e["to"]=="math.d5-component.collar-proof")] + self.assertFalse(realized(g,self.p)) + def test_witness_closure_rejected(self): + g=copy.deepcopy(self.g);g["nodes"]["math.rn-region.witness-collision"]["classification"]="PROVED_REVIEWED" + self.assertFalse(realized(g,self.p)) + def test_index_scope_and_open_boundary(self): + self.assertIn("D5 planar pin/collar/intermediate first moment",self.index) + self.assertIn("C6 planar Fourier upper bound",self.index) + self.assertIn("C6 sharp Palm route",self.index) + self.assertIn("NO COMPLETE REGIONAL PROOF YET",self.index) + self.assertIn("`math.rn-region.witness-collision` remains `OPEN_ACTIVE`",self.index) + self.assertNotIn("the claimed summed pin-neighborhood bound remains AMEND",self.index) + def test_catalog_records_only_reviewed_bound(self): + sec=self.cand.split("## C6.",1)[1].split("\n## C7.",1)[0] + self.assertIn("r^3 log(1/r)",sec) + self.assertIn("E[N(N-1)]=Theta(r^3)",sec) + self.assertRegex(sec,r"They do\s+not close the regional shrinking-witness mechanism") + self.assertIn("`math.rn-region.witness-collision` remains `OPEN_ACTIVE`",sec) + self.assertNotIn("No global factorial upper bound",sec) + def test_later_packets_are_separate_from_open_regional_mechanism(self): + for source in ( + "frontiers/d5_dimension_lift_20260929/PROOF.md", + "frontiers/c6_palm_route_20260929/PROOF.md", + "frontiers/c6_rare_cluster_laws_20260929/PROOF.md", + ): + self.assertIn(source,self.index) + sec=self.cand.split("## C6.",1)[1].split("\n## C7.",1)[0] + self.assertIn("separate reviewed/merged packets",sec) + self.assertRegex(sec,r"They do\s+not close the regional shrinking-witness mechanism") + heading=self.cand.split("## C6.",1)[1].splitlines()[0] + self.assertNotIn("sharp order open",heading) + def test_catalog_current_review_crosswalk(self): + c1=self.cand.split("## C1.",1)[1].split("\n## C2.",1)[0] + c2=self.cand.split("## C2.",1)[1].split("\n## C3.",1)[0] + self.assertIn("reviews/d1_elder_lower_claude_20260928/REVIEW.md",c1) + self.assertIn("Theorem E1",c1) + self.assertNotIn("Pending:",c1) + self.assertIn("reviews/d1_elder_lower_claude_20260928/REVIEW.md",c2) + self.assertIn("Theorem L",c2) + self.assertNotIn("Pending for this candidate",c2) + def test_graph_note_separates_global_order_from_regional_gap(self): + note=self.g["nodes"]["math.rn-region.witness-collision"]["notes"] + self.assertIn("sharp global order is reviewed and merged through Math-#145",note) + self.assertIn("regional shrinking pin/witness-collision mechanism",note) + self.assertNotIn("sharp full-window order remain open",note) + def test_catalog_c7_historical_proposal_is_refuted(self): + c7=self.cand.split("## C7.",1)[1].split("\n## C8.",1)[0] + self.assertIn("REFUTED",c7) + self.assertIn("frontiers/unrestricted_selection_difference_20260929/PROOF.md",c7) + self.assertIn("reviews/c7_nonvanishing_openai_20260929/REVIEW.md",c7) + self.assertIn("Math-#133",c7) + self.assertIn("5bd924969bb261241885cd224d5dfb31acf64509",c7) + self.assertNotIn("Math-#149",c7) + self.assertNotIn("Current disposition: **OPEN**",c7) + def test_declarative_source_preserved(self): + self.assertIs(self.p["declarative"],True) + self.assertIs(self.p["executed"],False) + +if __name__=="__main__": + result=unittest.TextTestRunner(verbosity=2).run(unittest.defaultTestLoader.loadTestsFromTestCase(FoldTests)) + raise SystemExit(0 if result.wasSuccessful() else 1)