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/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/local-cluster-tightness.yml b/.github/workflows/local-cluster-tightness.yml new file mode 100644 index 000000000..365bad977 --- /dev/null +++ b/.github/workflows/local-cluster-tightness.yml @@ -0,0 +1,40 @@ +name: Local cluster tightness and conditional closure controls + +'on': + pull_request: + paths: + - 'frontiers/local_cluster_tightness_20260929/**' + - '.github/workflows/local-cluster-tightness.yml' + - 'frontiers/c6_palm_route_20260929/PROOF.md' + - 'frontiers/d5_dimension_lift_20260929/PROOF.md' + - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/remote_collision_20260928/PROOF.md' + - 'frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md' + - 'frontiers/c6_rare_cluster_laws_20260929/PROOF.md' + push: + branches: [main] + paths: + - 'frontiers/local_cluster_tightness_20260929/**' + - '.github/workflows/local-cluster-tightness.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-tightness: + runs-on: ubuntu-latest + timeout-minutes: 10 + 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 modes, semantic controls and exact current/historical sources + run: | + set -euo pipefail + python -B -S frontiers/local_cluster_tightness_20260929/verify.py --repo "${GITHUB_WORKSPACE}" --output "${RUNNER_TEMP}/tightness-${GITHUB_RUN_ID}-${GITHUB_RUN_ATTEMPT}" + 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/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/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/local_cluster_tightness_20260929/PROOF.md b/frontiers/local_cluster_tightness_20260929/PROOF.md new file mode 100644 index 000000000..adffa6916 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/PROOF.md @@ -0,0 +1,398 @@ +# Rare-intensity tightness and a conditional planar cluster-law closure + +Object: OA-LOCAL-CLUSTER-TIGHTNESS-20260929-v1. +Author: OpenAI / GPT-6 Astra Pro, 29 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE. Nonauthor analytic review required. +Scientific effect NONE. No prior proof, verdict, graph, register or prize changed. + +## 1. Purpose, law and exact dependency boundary + +This is the bounded companion requested by the author of Math-#158 in #157 +comment 5900564419. It does not duplicate that source's cubic classification. +It supplies rare-event field-norm control, strict-domain tightness and a +local/remote gluing argument. The requested deterministic transverse reduction +was also written by the #158 author in source D before this companion was +published. Section3 gives a self-contained alternate-constant derivation of +that SAME mechanism, not a priority claim or an independent review of D. + +The original variance-one periodized Gaussian field, its fixed torus side T, +fixed dimension, birth b, positive gap k, and pin frame are unchanged. The pins +are M=(-r/2,0), S=(r/2,0), with f(M)=b, f(S)=b-kr^3 and both gradients zero. +Q_r is their canonical Gaussian regression law. W_r is the original typed +maximum/saddle determinant product, Z_r=E_Qr W_r is the FULL normalizer, and +P_r=Q_r^W=(W_r/Z_r)Q_r. N counts additional all-index critical points in +I_r=(b-kr^3,b), excluding M,S, on the entire torus. No other normalizer is used. + +Source P (landed Palm proof) supplies its canonical field-mark Kac--Rice formula, +all fixed conditional C6 moments and the regional density/determinant ledgers. +Source DL supplies the shell mean estimate; RM/RC the fixed-remote singleton and +second-factorial results. Source L supplies the planar ten-jet regression and +pinned cubic expansion. Exact immutable identities are in SOURCE_MAP.json. + +Source C is the NEW, still author-side #158 manuscript at325c5ebd. Its Theorems +C/G/F (classifier, compact-sector limit, finite positive coefficient integrals) +are EXPLICIT PREMISES only of Sections6--8 below. They are not revalidated or +accepted by this companion. Sections2--5 do not consume those new theorems. +Source D is the companion DEEP_TRANSVERSE_EXCLUSION.md at bfcc67dc, recovered +from comment5900634812 during this continuation. Its attribution is mandatory +and byte-bound. No new independent-provider credit is claimed by repeating its +implicit-graph mechanism with different constants. +The marked-Fourier candidate #157 is NOT an input anywhere. Thus review of #157 +is not a prerequisite to checking the field-norm lemma requested for #158. + +Let K4=1+||f||C4; any fixed full derivative norm may be used. On a fixed torus, +K4 is bounded by a fixed multiple of source P's K6. Norms in an orthonormal pin +frame are uniformly equivalent, but no rotational invariance of the torus law +is invoked. Constants depend on the fixed model/mark family, and on R or moment +orders where explicitly displayed. No numerical Gaussian constants are provided. +The final coefficient limits fix b,k,T and frame, as in source C. + +## 2. Lemma N: field-norm moments at the rare intensity scale + +For every fixed integer p>=1, in every fixed d>=2, + + E_Pr[N K4^p] <= C_p r^3. (N1) + +Consequently, for M>=1, + + r^-3 E_Pr[N 1{K4>M}] <= C_p M^-p, + r^-3 P_r(N_D>=1, K4>M) <= C_p M^-p, (N2) + +for any deterministic subregion D and its count N_D<=N. The second bound also +holds for any event implying N>=1. These are RARE-INTENSITY bounds, not a claim +that division of an unconditional derivative tail by r^3 stays small. + +Proof. Apply P Lemma5.1 first with the bounded mark W_r min(K4^p,A), and pass +monotonically to the limit A->infinity. Its explicitly fixed Gaussian regression +kernels avoid any evaluation of an arbitrary a.e. version at zero gradient. +In each region, the source proof bounds W_r |det H_X| by a deterministic +geometric factor times K6^(3d). Multiplication by K4^p changes only the moment +order. P Lemma4.4 gives + + E_Q'[K6^(3d+p)] <= C_p(1+beta)^(3d+p). (N3) + +The same conclusion for a noninteger p follows by increasing to an integer, +but it is enough to state (N1) for integers. Here Q' is the ACTUAL witness law, +including the height only in the regions that use it. The original W occurs +once inside the integrand, and the original Z once outside. + +For completeness the additional losses and their absorptions are: + +- Pin balls R1: multiply by (1+chi)^p. In the ordinary cone the penalty + exp(-c chi^2) absorbs this fixed polynomial without changing |q|^(2-d). + In the tiny axial cone chi<=1/r, so exp(-c/r^2) absorbs the extra r^-p. +- Collar R2a: beta<=C r^-10; exp(-c r^-2/3) absorbs r^-10p. +- Collar R2b: beta<=C |v|^-4; exp(-c/|v|^2) absorbs |v|^-4p. +- Collar R2c: beta is bounded, so the moment factor is a constant. +- Shell R3a: beta<=C s^-10; exp(-c s^-1/4) absorbs s^-10p. Use the + endpoint-short-column/Hadamard bound throughout the WHOLE strip, not only + at v=0. That determinant bound is already explicit in P's repaired source. +- Shell R3b: beta<=C |v|^-6; exp(-c/|v|^2) absorbs |v|^-6p. Away from zero + the same factor is uniformly bounded. The shell and height powers do not change. +- Remote R4: beta is bounded for fixed exclusion and window targets. + +For each fixed p, sup_{t>0} t^a exp(-c t^b) is finite whenever a,b,c>0, so +these absorptions cost C_p only. In the all-height inner regions their bounds +dominate the window count. The shell sum is still summable and its height +length remains kr^3. Sum exactly P/DL's tiling. This proves (N1); Markov in the +valid count-weighted direction proves (N2). No count exponential moment is used. + +## 3. Lemma T: an extra nearby point forces a soft transverse entry + +This is a self-contained version of the DETERMINISTIC planar mechanism already +written in source D, with alternative conservative constants and an absolute- +value conclusion that does not require endpoint types. It is not claimed as a +new mechanism distinct from D. Fix R>=1. Let f be C4 on a neighborhood +of the closed rectangle |x|,|z|<=Rr, with the six exact pins above. Suppose each +coordinate partial derivative through order4 is bounded by K>=1 there. Put + + A_R = 64(R+1)^2, a0=f_zz(0,0). + +If + + r <= k/[2(R+1)K], |a0| >= A_R r K^2/k, (T1) + +then the rectangle contains no critical points other than M,S. In particular, +any additional critical point in B(0,Rr), of ANY height or type, implies + + |f_zz(0)| <= A_R r K^2/k (T2) + +when the first inequality of (T1) holds. For a torus use a local lift with the +rectangle embedded, e.g. r=12k, so K>=k automatically. For +|x|<=Rr and |z|<=r/8, the C4 bound gives + + |f_xxx(x,z)-12k| <= (R+1)rK. (T4) + +Let phi(x)=f_z(x,0). Its values at +-h vanish, and |phi''|<=K. The ordinary +interpolation remainder (also valid outside the two nodes in this rectangle) +and Rolle's theorem give + + |phi(x)| <= (K/2)|x^2-h^2| <= (R+1)^2 K r^2/2, + |phi'(x)| <= (R+1/2)Kr. (T5) + +The second inequality uses a zero of phi' between the two nodes. From (T1), +|a0|>=64(R+1)^2 rK, whereas |f_zz(x,z)-a0|<=2RrK. Thus f_zz has one sign +throughout the rectangle and |f_zz|>=|a0|/2. The changes of f_z between z=0 +and z=+-Rr dominate (T5) strictly. For each |x|<=Rr there is exactly one root +z=h_r(x) of f_z=0 in the rectangle, and + + |h_r(x)| <= (R+1)^2 K r^2/|a0| <= r/8. (T6) + +The implicit function theorem makes h_r C3. Any critical point in the rectangle +lies on this graph; in particular h_r(+-r/2)=0 by uniqueness and the exact pins. +Let q=f_xz/f_zz evaluated on the graph. Equations(T5)-(T6) yield + + |f_xz(x,h_r(x))| <= (R+1)rK, + |q| <= 2(R+1)rK/|a0| <= k/[32(R+1)K] <= 1. (T7) + +Set g(x)=f_x(x,h_r(x)). Differentiating the Schur complement, NOT discarding +its mixed terms, gives + + g'=f_xx-f_xz^2/f_zz, + g''=f_xxx-3q f_xxz+3q^2 f_xzz-q^3 f_zzz. (T8) + +There is no fifth derivative, spectral frame or invariant-manifold theorem. +Together with r<=k/[2(R+1)K], (T4) and (T7) show + + g'' >=12k-k/2-7K|q| + >=12k-k/2-7k/[32(R+1)] > 10k >0. (T9) + +Thus g is strictly convex. It already vanishes at -r/2 and r/2, and a strictly +convex real function cannot have three distinct zeros. These are its only +zeros, proving the lemma. No maximum/saddle type assumption was used. + +The K^2/k dependence is essential to this argument. Merely keeping f_zz away +from zero by a multiple of rK does not make the mixed correction small relative +to k uniformly over the allowed derivative sizes. + +## 4. Tightness in the strict typed soft-jet domain + +Return to d=2 and fix R. Put + + Theta_r=(s,a,beta,c)=(f_zz(0)/r,f_xxz(0),f_xzz(0),f_zzz(0)), + B=beta-a^2/(12k), + D_k={s<-|B|/2}. + +Let A_{r,R} count additional window points in B(0,Rr). Define the finite +NONEMPTY measure + + eta_r^R(E)=r^-3 P_r(Theta_r in E, A_{r,R}>=1). (J1) + +For every epsilon>0 there is a compact C contained STRICTLY in D_k such that + + limsup_(r->0) eta_r^R(R^4 minus C) <= epsilon. (J2) + +This statement does NOT include the enormous N=0 mass outside the rare layer. + +Proof. By (N2), choose M so the scaled probability of A_{r,R}>=1 and K4>M is +at most epsilon/3. On K4<=M and sufficiently small r (depending on fixed M,R), +Lemma T gives |s|<=A_R M^2/k and |a|,|beta|,|c|<=C M. This is a compact box. + +Source L's planar ten-jet rank implies a uniform upper bound for the density +of (f_zz,f_xxz,f_xzz,f_zzz) under Q_r: appending these four variables to U_r +converges to all independent planar jets of order<=3. The physical density +change f_zz=rs has Jacobian r. On the above compact box and K4<=M, endpoint +Taylor identities show every endpoint Hessian entry is O_M(r), hence W_r<=C_Mr^4. +Because Z_r>=c r^2, for every Borel E in that box, + + r^-3 P_r(Theta_r in E,K4<=M) <= C_M Leb_4(E). (J3) + +No independence between the remainder and the jets is asserted. + +The limiting endpoint Hessians divided by r have determinants +-6k(s-B/2) at M and 6k(s+B/2) at S, with uniform O_M(r) errors on this box. +Under P_r the maximum/saddle types hold wherever W_r>0. Thus + + s-B/2 <= C_M r, s+B/2 <= C_M r. (J4) + +These inequalities put Theta_r in a shrinking neighborhood of the closed domain +D_k. Its boundary s=-|B|/2 has four-dimensional Lebesgue measure zero: for each +(a,beta,c) it is one value of s. A vertical layer of thickness 2delta in a +bounded box has volume at most 2delta times the other three side lengths. +Use (J3), first choosing delta small and then r small. The mass not satisfying +s<=-|B|/2-delta is arbitrarily small. Intersect this strict inequality with the +fixed box to get C, proving (J2). This also handles the absolute-value corner at +B=0; no differentiable boundary chart is needed. + +## 5. Local and fixed-remote events do not mix at leading order + +Fix R and delta>0. Let C_{r,delta} count points at dist(X,0)>=delta. Then + + P_r(A_{r,R}>=1, C_{r,delta}>=1)=o(r^3). (M1) + +Proof. On K4<=M, (T2) forces |f_zz(0)|<=S_{R,M}r. Upper-bound the joint event +by the marked REMOTE count with these two field restrictions. Disintegrate in +the physical scalar t=f_zz(0), the remote gradient, and its height h in I_r. +The joint frame (U_r,f_zz(0),grad f(X),f(X)) has a uniform covariance floor for +|X|>=delta, through r=0: f_zz is a new midpoint monomial and X is a distinct +remote site. Positive Fourier spectrum and compactness give the density bound. +On K4<=M and |t|<=S_{R,M}r, W_r<=C_{R,M}r^4 and |det H_X|<=C_M. Hence the +unnormalized integral is bounded by a constant times + + r^4 * r * r^3 = r^8, + +where the factors are the endpoint weight, width of the PHYSICAL f_zz band, +and remote height window. Division by the one full Z_r>=c r^2 gives O(r^6). +All constants here may depend on fixed R,M,delta. Outside K4<=M, the joint event +implies N>=1, so (N2) bounds its probability by C_p M^-p r^3. Therefore + + limsup_(r->0) r^-3 P_r(joint) <= C_p M^-p. + +Now let M->infinity. No quantitative rate in (M1) is claimed after that limit, +and no mixed-event independence assumption is used. + +## 6. Conditional closure theorem for source C's coefficient law + +From this point ASSUME the exact Theorems C/G/F of source C (#158), at their +fixed planar parameter scope. Let n(theta) in {0,1,2}, w(theta), h_0 and z_0 be +its classifier, soft determinant weight, contact four-jet density and FULL +normalizer. Define the finite positive coefficients already supplied there: + + alpha_j = integral_(D_k) (w/z_0)h_0(0,a,beta,c)1{n=j} dtheta, + j=1,2. (G1) + +Source RM defines the full contact singleton kernel Lambda_j(X) with the +endpoint weight INSIDE its conditional expectation. Set + + mu_delta = k integral_(dist(X,0)>=delta) sum_j Lambda_j(X) dX, + mu = lim_(delta->0) mu_delta. (G2) + +The factor k is required: the physical height window has length kr^3. +By RM/RC, for each FIXED delta>0, + + r^-3 Law_+(C_{r,delta}) -> mu_delta delta_1 in l1, (G3) + +where Law_+(Y)(n)=P(Y=n), n>=1. Indeed its mean is mu_delta r^3+O_delta(r^4), +and E(C_{r,delta})_2=O_delta(r^5). Its distance to (E C)delta_1 is at most twice +that second factorial moment. Also 0=1, + nu_r -> nu in l1(n^p), for every fixed integer p>=0, + nu(1)=alpha_1+mu, nu(2)=alpha_2, nu(n>=3)=0. (G4) + +This is a conditional consumer theorem until source C and the new lemmas here +have their own complete analytic reviews. It is NOT an unconditional acceptance +of #158, and it must not be used to change a scientific register. + +### Proof: the spatial and jet limits in their correct order + +DL's shell bound, summed over dyadic shells, gives for R>=4 and 0=R, its limsup after division by +r^3 is at most C R^-2. Constants in this inequality are independent of R,R',delta +within the stated geometric range, since the source shell constants are uniform; +only the small-r requirement depends on fixed R'. Circle boundaries have zero +expected count by spatial Kac--Rice and can be assigned once. + +We first establish the large-R local approximation + + limsup_(r->0) ||r^-3 Law_+(A_{r,R}) + -(alpha_1 delta_1+alpha_2 delta_2)||_1 + <= C R^-2. (G6) + +Fix R and epsilon. By(J2) choose compact C_0 strictly in D_k carrying all but +epsilon of the limsup NONEMPTY local intensity. By source C's integrability of +the nonempty coefficient, enlarge to a compact C containing C_0 such that the +coefficient mass alpha_1+alpha_2 outside C is below epsilon. Choose a fixed +R'>=max(R,R_C) as required by source C's compact-sector theorem. If A_{r,R'} +differs from A_{r,R}, there is a point in the shorter annulus, so its probability +is at most C r^3 R^-2+o(r^3), by(G5). + +Consequently the scaled nonempty law of A_{r,R'} outside C has limsup mass at +most epsilon+C R^-2. On C, source C Theorem G gives full total-variation +convergence, hence l1 convergence of the count marginal restricted to positive +counts, to alpha_1(C)delta_1+alpha_2(C)delta_2. Comparing A_{r,R} with A_{r,R'} +costs at most twice their probability of inequality. Comparing alpha_j(C) with +alpha_j costs at most epsilon. Let epsilon->0. This proves(G6). No interchange +of R_C with an unbounded jet set occurred; R' was fixed only AFTER C was chosen. +In particular we did not assume source C's compact theorem for arbitrary R. + +Now write N=A_{r,R}+B_{r,R,delta}+C_{r,delta} with disjoint regions. For any +nonnegative integer A,B,C, + + ||Law_+(A+B+C)-Law_+(A)-Law_+(C)||_1 + <=3 P(B>=1)+3 P(A>=1,C>=1). (G7) + +This follows by comparing the corresponding atomic signed measures for each +outcome; their l1 norm is at most3, and it vanishes when B=0 and at most one of +A,C is positive. Thus(G3),(G5),(M1),(G6) give + + limsup_(r->0) ||nu_r-[(alpha_1+mu)delta_1+alpha_2 delta_2]||_1 + <= C(R^-2+delta^2)+|mu_delta-mu|. (G8) + +Take R->infinity and then delta->0 (or the reverse, after the r limit). This +proves unweighted l1 convergence and identifies the UNIQUE intensity law at +fixed parameters. It does not claim a convergence rate or uniformity under +varying parameters. + +For a fixed p, source P's all-fixed-order factorial bounds and the first moment +imply sup_r sum n^(p+1) nu_r(n)M} n^p nu_r(n)<=C_{p+1}/M. Combine finite-dimensional convergence with +this uniform integrability to prove the weighted l1 assertion in(G4). No moment +series or exponential moment is needed. QED. + +## 7. Conditional numerical-free consequences + +All statements in this section retain the conditional status of Theorem G. +Writing lambda=alpha_1+alpha_2+mu and m1=alpha_1+2alpha_2+mu, + + P_r(N>=1) ~ lambda r^3, E_Pr N ~ m1 r^3, + P_r(N>=2) ~ alpha_2 r^3, E_Pr(N)_2 ~2 alpha_2 r^3, + E_Pr(N)_q=o(r^3) for each fixed integer q>=3. (C1) + +The last conclusion is NEW conditional information supplied by exhaustion, +not an inference from previous O(r^3) upper bounds. It is compatible with the +existing two-point lower event and does not assert N<=2 for finite r. + +Conditional on N>=1, the limiting probabilities at1,2 are +(alpha_1+mu)/lambda and alpha_2/lambda. Under size bias they are +(alpha_1+mu)/m1 and2alpha_2/m1. These are not the same law. + +For independent replicas, floor(t/r^3) counts sum to a unique compound-Poisson +limit with jumps1 and2, equivalently X+2Y for independent Poisson variables of +means t(alpha_1+mu), t alpha_2. This follows from the standard generating-function +identity already credited in the landed #153 packet. It is NOT a spatial Poisson +limit of points in one field. + +The birth factorization of source C's LOCAL coefficients does not automatically +extend to the GLOBAL law: mu in(G2) is a different remote integral. No global +birth independence, numerical alpha/mu, dimensional extension, explicit error +rate or selected-persistence pairing law is asserted. + +## 8. Review targets, provenance and limits + +Direct review slices: N1's marked norm insertion with all regional factors; +T3--T9's graph reduction including the three mixed derivative terms; J2/J3's +NONEMPTY tightness and typed-boundary control; M1's physical-band/height/weight +powers. Conditional closure slice: G5's uniform shell use, the choice C then R', +G7's signed-measure error, remote coefficient k, and polynomial uniform +integrability. A verdict on these implications does not accept source C's +classifier or compact-sector theorem on the reviewer's behalf. + +All local constants in the deterministic lemma are explicit; all Gaussian +constants remain existential. Large R is taken after small r. Large derivative +cutoffs are taken after the limsup in the mixed event. Gaussian full-field +marks are Borel through derivative limits of the smooth field and are truncated +before monotone convergence. Every mean/count uses original W and Z exactly once. + +The finite controls prove algebra, normalization, conservative constants and +finite-measure inequalities only. They cannot establish the Gaussian bounds, +compactness or any of the analytic theorem premises. No missing historical +program is recreated. Consensus was actually requested and quota-exhausted; +limited primary-source reconnaissance is recorded separately. No novelty or +priority claim is made for classical Schur complementation, convexity, Markov, +Kac--Rice, Gaussian regression, or elementary compound Poisson limits. diff --git a/frontiers/local_cluster_tightness_20260929/README.md b/frontiers/local_cluster_tightness_20260929/README.md new file mode 100644 index 000000000..4c84cbe0f --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/README.md @@ -0,0 +1,33 @@ +# Local cluster tightness: source-bound companion to #158 + +Author-side mathematics, nonauthor review required. Scientific effect NONE. + +The direct field-norm lemma proves a rare-intensity bound by inserting K4^p into +the LANDED Palm regional argument. It does not consume #157's still-new count +exponential theorem. The deterministic transverse argument is the SAME mechanism +already written by #158's author; it is credited and byte-bound, not claimed as +an independent discovery. The new work combines it with strict typed-boundary +control, a mixed local/remote estimate, and a carefully ordered exhaustion. + +Sections6-7 state a CONDITIONAL closure of the planar global count law using +#158's exact, still-author-side compact-sector theorem. The proposed coefficients +are nu(1)=alpha1+mu_remote and nu(2)=alpha2, with higher values zero in the scaled +limit. This is NOT acceptance of #158, finite-r support {1,2}, a spatial Poisson +limit, or a scientific-register transition. The remote coefficient includes k. + +Run all finite controls in both Python modes: + + python -B -S verify.py --output /path/to/new-output + python -B -O -S verify.py --output /path/to/other-new-output + +Each invocation checks exact packet membership, sizes/SHA256/Git blobs, executes +12 mathematical and6 synthetic custody tests in EACH Python mode, and rejects8 +semantic variants on the intended assertions. With a complete Git checkout, +add --repo /path/to/repository to verify all8 actual historical source objects +and every required current source. C and D are pinned draft-branch sources, +not required to exist on default main; their exact historical bytes are checked. +The hosted workflow performs that full-repository check. A packet-only local +replay explicitly records upstream_verified_count=null, not fictitious custody. + +The code verifies finite identities, conservative constants and source guards. +It does not verify Gaussian analytic estimates, compactness or Kac--Rice. diff --git a/frontiers/local_cluster_tightness_20260929/RECONNAISSANCE.md b/frontiers/local_cluster_tightness_20260929/RECONNAISSANCE.md new file mode 100644 index 000000000..107fc068f --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/RECONNAISSANCE.md @@ -0,0 +1,22 @@ +# Focused reconnaissance — 29 September 2026 + +Consensus.search was actually called with Gaussian weighted Kac--Rice, field +norms, rare events and transverse reduction. It returned the exhausted monthly +30-search quota with October1 reset. No Consensus result is fabricated. + +Direct primary source: Armentano, Azais and Leon, arXiv:2304.07424v3, +https://arxiv.org/abs/2304.07424v3 and its HTML version. Abstract/version record +and the main-results introduction were inspected. The tool's exact-heading +searches for 'Theorem 7.1' and 'Remark 8' did not match the HTML numbering, so no +fresh full verification of those named statements is claimed. The actual formula +consumed is the landed project P Lemma5.1 with its canonical kernel and two-measure +proof. Gass--Stecconi arXiv:2305.17586 and publisher record were read at abstract +scope only; their ordinary moment-finiteness theorem is not a substitute for the +rare-intensity estimate N1. + +The companion is a targeted response to #158 author comment5900564419. The +subsequent comment5900634812 identified its already-written deterministic addendum +D atbfcc67dc, which was fully read and is credited. The alternate constants in +Section3 are a self-contained corroboration, not a separate discovery or review. +The existing #153 replica argument is also credited and pinned. This is not an +exhaustive novelty or priority audit, and no external theorem is silently imported. diff --git a/frontiers/local_cluster_tightness_20260929/RESULTS.json b/frontiers/local_cluster_tightness_20260929/RESULTS.json new file mode 100644 index 000000000..3c07c1a98 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/RESULTS.json @@ -0,0 +1,9 @@ +{ + "tests_per_mode": 18, + "mathematical_tests": 12, + "custody_fixture_tests": 6, + "semantic_mutants_per_mode": 8, + "paired_outputs_identical": 9, + "scientific_effect": "NONE", + "analytic_acceptance": false +} diff --git a/frontiers/local_cluster_tightness_20260929/SOURCE_FILES.json b/frontiers/local_cluster_tightness_20260929/SOURCE_FILES.json new file mode 100644 index 000000000..caac28553 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/SOURCE_FILES.json @@ -0,0 +1,66 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "files": [ + { + "path": "PROOF.md", + "bytes": 20344, + "sha256": "95f619ee5d9c1c1ac5e87397ce06de20fda164d528d6a90a16c4416322d9f269", + "git_blob": "adffa6916d0fb57fede0486b5eb3bc6afc127c07" + }, + { + "path": "README.md", + "bytes": 1966, + "sha256": "2cc45c898f14132d6b611a5972902c96fc34e23787e8347ff2a04eace3f10b82", + "git_blob": "4c84cbe0f69b3951eb93eb2a2828e619a7fd8ecb" + }, + { + "path": "RECONNAISSANCE.md", + "bytes": 1423, + "sha256": "7aeec094d74c9629943f86819e80c5f64f823d522268bfa567d5a43b7662f851", + "git_blob": "107fc068fe73cfb85196afbf4e5d1b7bf4e008d5" + }, + { + "path": "RESULTS.json", + "bytes": 215, + "sha256": "ee2471145f3d3926023b4179b8f16dd56c5210497e15e154cf2fc97b52c36394", + "git_blob": "3c07c1a98102d14ebcb097af3c584b5e3afdbf78" + }, + { + "path": "SOURCE_MAP.json", + "bytes": 3772, + "sha256": "4b976cfa3156c41211e01be177e281f37744a153dbc1ce5c09a7209545f29102", + "git_blob": "c5b3717b955102a1fc73eec6e60ec460436a4c04" + }, + { + "path": "controls.py", + "bytes": 2156, + "sha256": "0fc13e5f00f42784197be6191edd60a7ce8a3dcc8a649dbd88fe3af83ccddf02", + "git_blob": "e6e783368104bad617007b17867e104df16883ee" + }, + { + "path": "custody.py", + "bytes": 2060, + "sha256": "9830fa6684dda1efbced351952210ed46ae40703336acbd4ef8a89f02ae92421", + "git_blob": "24bb6abd86bb0973221f7cbe9a8055166ba40ae2" + }, + { + "path": "test_controls.py", + "bytes": 3411, + "sha256": "992f8af8029efddba79ab31efbf5dfab962e1bd31675927df80d68d966b95e4d", + "git_blob": "6caf8f76e9984a6d6529d9d15be09c82f65598e0" + }, + { + "path": "test_custody.py", + "bytes": 2215, + "sha256": "e167b439cff7480d1c109ef31168d7f7885f8c3270d53964035cc0466dc18ed4", + "git_blob": "c983be24eaa43dab08a95ab19477bf42f78cc7c1" + }, + { + "path": "verify.py", + "bytes": 4206, + "sha256": "977f1e2bfcb3fc727be4d6d3556275aa07052f3c160136ef5cd33fcc2a5b0c8e", + "git_blob": "71a65ccc047c322e62f1498150f48c24ff5565cd" + } + ] +} diff --git a/frontiers/local_cluster_tightness_20260929/SOURCE_MAP.json b/frontiers/local_cluster_tightness_20260929/SOURCE_MAP.json new file mode 100644 index 000000000..c5b3717b9 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/SOURCE_MAP.json @@ -0,0 +1,87 @@ +{ + "scientific_effect": "NONE", + "sources": [ + { + "id": "P", + "path": "frontiers/c6_palm_route_20260929/PROOF.md", + "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b", + "git_blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "current_required": true, + "role": "Landed marked Kac-Rice, conditional norm moments and regional bounds.", + "bytes": 53364 + }, + { + "id": "L", + "path": "frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md", + "sha256": "e6272e8709835ca6290fecb7034fa84832dde08c3c4cc990a48b8019e89e7c8d", + "git_blob": "b5c18aac5529c5f975ee5bb056b304debafc6697", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "current_required": true, + "role": "Planar ten-jet rank and endpoint Taylor identities.", + "bytes": 13100 + }, + { + "id": "DL", + "path": "frontiers/d5_dimension_lift_20260929/PROOF.md", + "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80", + "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "current_required": true, + "role": "Landed shell estimate and global first moment.", + "bytes": 53727 + }, + { + "id": "RM", + "path": "frontiers/remote_window_20260924/PROOF.md", + "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", + "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf", + "current_required": true, + "role": "Fixed-remote singleton kernel; 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No Gaussian checker.""" +from fractions import Fraction as F +from math import ceil, prod + +MUTANT=None + +def positive(*xs): + if any(F(x)<=0 for x in xs): raise ValueError('positive parameters required') + +def reduced_third(A,U,V,W,q): + A,U,V,W,q=map(F,(A,U,V,W,q)) + return A-(2 if MUTANT=='wrong-schur-derivative' else 3)*q*U+3*q*q*V-q**3*W + +def hermite_constant(r,k): + positive(r,k) + return (6 if MUTANT=='half-hermite' else 12)*F(k) + +def soft_threshold(R,r,K,k): + positive(R,r,K,k);R,r,K,k=map(F,(R,r,K,k)) + if R<1 or K=1 and K>=k required') + return 64*(R+1)**2*r*K**(1 if MUTANT=='linear-norm-threshold' else 2)/k + +def model_extra_x(k,lam,a): + positive(k,lam) + if not a: raise ValueError('nonzero mixed coefficient required') + return -12*F(k)*F(lam)/F(a)**2 + +def norm_tail(r,C,p,M): + positive(r,C,M) + if type(p) is not int or p<1: raise ValueError('positive integer moment required') + return F(C)*F(r)**(0 if MUTANT=='lose-rare-scale' else 3)/F(M)**p + +def mixed_power(): + return 4+1+3-(4 if MUTANT=='normalizer-twice' else 2) + +def boundary_volume(J,eps): + positive(J,eps) + return 2*F(eps)*(2*F(J))**3 + +def global_rates(a1,a2,remote): + positive(a1,a2,remote) + return {1:F(a1)+(0 if MUTANT=='omit-remote-singletons' else F(remote)),2:F(a2)} + +def factorial_coefficient(rates,p): + if type(p) is not int or p<1: raise ValueError('positive integer order required') + return sum((mass*prod(n-j for j in range(p)) for n,mass in rates.items() if n>=p),F(0)) + +def decomposition_error(middle,mixed): + if min(F(middle),F(mixed))<0: raise ValueError('nonnegative errors required') + return 3*((0 if MUTANT=='ignore-middle-region' else F(middle))+F(mixed)) + +def nonempty_double(rates): + return F(rates[2])/sum(rates.values()) + +def palm_excess(rates): + denom=sum(rates.values()) if MUTANT=='palm-equals-nonempty' else sum(n*p for n,p in rates.items()) + return sum(n*(n-1)*p for n,p in rates.items())/denom + +def absorption_order(loss,target,a): + positive(a) + return max(0,ceil((F(loss)+F(target))/F(a))) diff --git a/frontiers/local_cluster_tightness_20260929/custody.py b/frontiers/local_cluster_tightness_20260929/custody.py new file mode 100644 index 000000000..24bb6abd8 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/custody.py @@ -0,0 +1,48 @@ +"""Fail-closed current/historical source byte checks; no analytic acceptance.""" +import hashlib,json,pathlib,re,subprocess + +def require(ok,message): + if not ok:raise ValueError(message) + +def identity(b): + return {'bytes':len(b),'sha256':hashlib.sha256(b).hexdigest(),'git_blob':hashlib.sha1(f'blob {len(b)}\0'.encode()+b).hexdigest()} + +def loads(s): + def pairs(items): + d={} + for k,v in items: + require(k not in d,'duplicate JSON key');d[k]=v + return d + return json.loads(s,object_pairs_hook=pairs,parse_constant=lambda _:(_ for _ in ()).throw(ValueError('nonfinite JSON'))) + +def safe_path(s): + p=pathlib.PurePosixPath(s) + require(not p.is_absolute() and str(p)==s and '..' not in p.parts and p.parts,'unsafe path') + return p + +def unique_paths(entries): + paths=[e['path'] for e in entries] + require(len(paths)==len(set(paths)),'duplicate source path') + for p in paths:safe_path(p) + +def read_checked(root,e): + root=pathlib.Path(root);rel=safe_path(e['path']) + require(not root.is_symlink(),'root symlink forbidden') + for i in range(1,len(rel.parts)+1): + require(not root.joinpath(*rel.parts[:i]).is_symlink(),'source symlink forbidden') + path=root.joinpath(*rel.parts);require(path.is_file(),'missing regular file') + b=path.read_bytes();actual=identity(b) + require(all(actual[k]==e[k] for k in actual if k in e),'source identity mismatch') + require('sha256' in e and 'git_blob' in e,'incomplete identity') + return b + +def upstream(repo,entries): + unique_paths(entries);repo=pathlib.Path(repo) + for e in entries: + require(re.fullmatch('[0-9a-f]{40}',e['commit']) is not None,'invalid pinned commit') + obj=e['commit']+':'+e['path'] + b=subprocess.check_output(['git','-C',str(repo),'show',obj],stderr=subprocess.PIPE,timeout=20) + actual=identity(b) + require(all(actual[k]==e[k] for k in actual if k in e),'historical source differs: '+e['id']) + if e.get('current_required',True):read_checked(repo,e) + return len(entries) diff --git a/frontiers/local_cluster_tightness_20260929/test_controls.py b/frontiers/local_cluster_tightness_20260929/test_controls.py new file mode 100644 index 000000000..6caf8f76e --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/test_controls.py @@ -0,0 +1,65 @@ +"""Exact scalar/finite-measure controls, not a Gaussian theorem checker.""" +from fractions import Fraction as F +from itertools import product +from math import prod +import unittest +import controls as m + +class Controls(unittest.TestCase): + def test_reduced_third_derivative(self): + for B,C,A,U,V,W in product([F(-2),F(1,3)], [F(-3),F(2)], [F(12)], [F(3)], [F(-2)], [F(5)]): + q=B/C; hp=-q + by_chain=A+U*hp-2*q*(U+V*hp)+q*q*(V+W*hp) + self.assertEqual(m.reduced_third(A,U,V,W,q),by_chain) + def test_hermite_integral_normalization(self): + for r,k in product([F(1,2),F(1,7)], [F(1,3),F(2)]): + h=r/2 + # Integral (h^2-x^2)*12k = 2*k*r^3. + self.assertEqual(m.hermite_constant(r,k)*F(4,3)*h**3,2*k*r**3) + def test_deep_constants_uniform(self): + for R,k,K,j in product([F(1),F(2),F(5)],[F(1,5),F(1)],[F(13),F(37)],[1,4,16]): + r=k/(2*(R+1)*K*j);lam=m.soft_threshold(R,r,K,k) + q=2*(R+1)*r*K/lam + lower=12*k-(R+1)*r*K-K*(3*q+3*q*q+q**3) + self.assertGreater(lower,10*k) + self.assertLessEqual((R+1)**2*K*r*r/lam,r/8) + self.assertGreaterEqual(lam,4*R*r*K) + def test_threshold_quadratic_norm(self): + self.assertEqual(m.soft_threshold(2,F(1,100),20,1),64*9*F(1,100)*400) + self.assertEqual(m.soft_threshold(2,F(1,100),40,1)/m.soft_threshold(2,F(1,100),20,1),4) + def test_explicit_graph_extra_zero(self): + # f=2kx^3-3kr^2*x/2-kr^3/2-lam*z^2/2 + a*(x^2-r^2/4)*z/2. + for k,lam,a in product([F(1),F(2)],[F(1),F(3)],[F(1),F(2)]): + r=F(1,10);x=m.model_extra_x(k,lam,a);z=a*(x*x-r*r/4)/(2*lam) + self.assertEqual(6*k*(x*x-r*r/4)+a*x*z,0) + self.assertEqual(-lam*z+a*(x*x-r*r/4)/2,0) + def test_rare_norm_tail_keeps_scale(self): + for r in [F(1,4),F(1,16)]: + self.assertEqual(m.norm_tail(r,7,3,5),7*r**3/F(125)) + def test_mixed_local_remote_ledger(self): + self.assertEqual(m.mixed_power(),6) + for r in [F(1,4),F(1,16)]: + self.assertEqual(r**4*r*r**3/r**2,r**m.mixed_power()) + def test_boundary_tube_volume(self): + self.assertEqual(m.boundary_volume(3,F(1,10)),F(1,5)*6**3) + def test_global_rates_and_factorials(self): + rates=m.global_rates(F(2,3),F(5,7),F(11,13)) + self.assertEqual(rates,{1:F(2,3)+F(11,13),2:F(5,7)}) + self.assertEqual(m.factorial_coefficient(rates,2),F(10,7)) + for q in range(3,8):self.assertEqual(m.factorial_coefficient(rates,q),0) + def test_missing_middle_is_not_automatic(self): + self.assertGreater(m.decomposition_error(F(1,10),0),0) + # For deterministic A=B=C=1, l1(delta_3-2delta_1)=3. + self.assertGreaterEqual(m.decomposition_error(1,1),3) + def test_nonempty_and_sizebiased_differ(self): + rates={1:F(3),2:F(2)} + self.assertEqual(m.nonempty_double(rates),F(2,5)) + self.assertEqual(m.palm_excess(rates),F(4,7)) + self.assertNotEqual(m.nonempty_double(rates),m.palm_excess(rates)) + def test_exponential_absorption_for_each_fixed_power(self): + # e^{-c/u^a} <= n! (u^a/c)^n; choose an integer n to absorb a fixed loss. + for loss,target,a in product(range(3,17),[2,5],[F(1,4),F(2,3),F(2)]): + n=m.absorption_order(loss,target,a) + self.assertGreaterEqual(a*n-loss,target) + +if __name__=='__main__': unittest.main() diff --git a/frontiers/local_cluster_tightness_20260929/test_custody.py b/frontiers/local_cluster_tightness_20260929/test_custody.py new file mode 100644 index 000000000..c983be24e --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/test_custody.py @@ -0,0 +1,37 @@ +"""Synthetic source-custody regressions; no upstream proof verification claim.""" +import hashlib,json,pathlib,tempfile,unittest +import custody as c + +class Custody(unittest.TestCase): + def setUp(self): + self.tmp=tempfile.TemporaryDirectory();self.addCleanup(self.tmp.cleanup) + self.root=pathlib.Path(self.tmp.name);(self.root/'sub').mkdir() + (self.root/'sub'/'proof.txt').write_bytes(b'proof\n') + self.entry={'path':'sub/proof.txt',**c.identity(b'proof\n')} + def test_valid_source(self): + self.assertEqual(c.read_checked(self.root,self.entry),b'proof\n') + def test_changed_or_missing(self): + (self.root/'sub'/'proof.txt').write_bytes(b'amended\n') + with self.assertRaises(ValueError):c.read_checked(self.root,self.entry) + (self.root/'sub'/'proof.txt').unlink() + with self.assertRaises((ValueError,FileNotFoundError)):c.read_checked(self.root,self.entry) + def test_three_identity_fields(self): + for field in ('bytes','sha256','git_blob'): + bad=dict(self.entry);bad[field]=0 if field=='bytes' else '0'*len(bad[field]) + with self.assertRaises(ValueError):c.read_checked(self.root,bad) + def test_unsafe_relative_path(self): + for path in ('../proof','/proof','sub/../proof','sub//proof.txt'): + bad=dict(self.entry,path=path) + with self.assertRaises(ValueError):c.read_checked(self.root,bad) + def test_file_and_parent_symlinks(self): + p=self.root/'sub'/'proof.txt';p.rename(self.root/'real');p.symlink_to(self.root/'real') + with self.assertRaises(ValueError):c.read_checked(self.root,self.entry) + p.unlink();(self.root/'sub').rmdir();(self.root/'real').unlink() + (self.root/'target').mkdir();(self.root/'target'/'proof.txt').write_bytes(b'proof\n') + (self.root/'sub').symlink_to(self.root/'target',target_is_directory=True) + with self.assertRaises(ValueError):c.read_checked(self.root,self.entry) + def test_duplicate_json_and_source_paths(self): + with self.assertRaises(ValueError):c.loads('{"a":1,"a":2}') + with self.assertRaises(ValueError):c.unique_paths([self.entry,self.entry]) + +if __name__=='__main__':unittest.main() diff --git a/frontiers/local_cluster_tightness_20260929/verify.py b/frontiers/local_cluster_tightness_20260929/verify.py new file mode 100644 index 000000000..71a65ccc0 --- /dev/null +++ b/frontiers/local_cluster_tightness_20260929/verify.py @@ -0,0 +1,64 @@ +"""Exact packet replay with optional real upstream Git verification.""" +import argparse,io,json,pathlib,subprocess,sys,unittest +import custody +ROOT=pathlib.Path(__file__).resolve().parent +MUTANTS={ + 'wrong-schur-derivative':'test_reduced_third_derivative', + 'half-hermite':'test_hermite_integral_normalization', + 'linear-norm-threshold':'test_threshold_quadratic_norm', + 'lose-rare-scale':'test_rare_norm_tail_keeps_scale', + 'normalizer-twice':'test_mixed_local_remote_ledger', + 'omit-remote-singletons':'test_global_rates_and_factorials', + 'ignore-middle-region':'test_missing_middle_is_not_automatic', + 'palm-equals-nonempty':'test_nonempty_and_sizebiased_differ', +} + +def source_ids(): + mp=ROOT/'SOURCE_FILES.json' + custody.require(not mp.is_symlink() and mp.is_file(),'regular manifest required') + entries=custody.loads(mp.read_text())['files'];custody.unique_paths(entries) + custody.require(sorted(x.name for x in ROOT.iterdir())==sorted([e['path'] for e in entries]+['SOURCE_FILES.json']),'packet membership differs') + for e in entries: + custody.require(pathlib.Path(e['path']).name==e['path'],'packet must be flat') + custody.read_checked(ROOT,e) + return {p.name:custody.identity(p.read_bytes()) for p in sorted(ROOT.iterdir())} + +def child(mutant): + import controls,test_controls,test_custody + controls.MUTANT=mutant + suite=unittest.TestSuite([unittest.defaultTestLoader.loadTestsFromModule(x) for x in (test_controls,test_custody)]) + result=unittest.TextTestRunner(stream=io.StringIO()).run(suite) + print(json.dumps({'tests':result.testsRun,'passed':result.wasSuccessful(), + 'failures':sorted(t.id().split('.')[-1] for t,_ in result.failures), + 'errors':sorted(t.id().split('.')[-1] for t,_ in result.errors)},sort_keys=True)) + return 0 if result.wasSuccessful() else 1 + +def replay(out,repo): + out=out.resolve();custody.require(not out.exists() and not out.is_relative_to(ROOT),'new external output required') + before=source_ids();count=custody.upstream(repo,custody.loads((ROOT/'SOURCE_MAP.json').read_text())['sources']) if repo else None + out.mkdir(parents=True);records={} + for case in ['baseline',*MUTANTS]: + pair=[] + for mode,flags in [('normal',[]),('optimized',['-O'])]: + cmd=[sys.executable,'-B',*flags,'-S',str(ROOT/'verify.py'),'--child'] + if case!='baseline':cmd+=['--mutant',case] + p=subprocess.run(cmd,capture_output=True,timeout=30) + (out/(case+'_'+mode+'.stdout')).write_bytes(p.stdout);(out/(case+'_'+mode+'.stderr')).write_bytes(p.stderr) + r=custody.loads(p.stdout.decode());custody.require(not p.stderr and not r['errors'] and r['tests']==18,'execution error') + if case=='baseline':custody.require(p.returncode==0 and r['passed'],'baseline failed') + else:custody.require(p.returncode==1 and not r['passed'] and MUTANTS[case] in r['failures'],'intended semantic failure missing') + pair.append(p.stdout) + custody.require(pair[0]==pair[1],'Python mode disagreement');records[case]=custody.loads(pair[0].decode()) + summary={'tests_per_mode':18,'mathematical_tests':12,'custody_fixture_tests':6,'semantic_mutants_per_mode':8, + 'paired_outputs_identical':9,'scientific_effect':'NONE','analytic_acceptance':False} + custody.require(summary==custody.loads((ROOT/'RESULTS.json').read_text()),'stored summary differs') + custody.require(before==source_ids(),'sources changed') + (out/'REPORT.json').write_text(json.dumps({**summary,'upstream_verified_count':count,'source_identities':before,'cases':records},sort_keys=True,indent=2)+'\n') + print(json.dumps(summary,sort_keys=True,indent=2));return 0 + +if __name__=='__main__': + p=argparse.ArgumentParser(description=__doc__);g=p.add_mutually_exclusive_group(required=True) + g.add_argument('--child',action='store_true');g.add_argument('--output',type=pathlib.Path) + p.add_argument('--mutant',choices=MUTANTS);p.add_argument('--repo',type=pathlib.Path) + a=p.parse_args();custody.require(a.child or a.mutant is None,'mutant needs child mode') + raise SystemExit(child(a.mutant) if a.child else replay(a.output,a.repo)) 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/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)