C6 consequences: sharp Poisson obstruction, conditional cluster laws and r^6 Poissonization

.github/workflows/c6-rare-cluster-laws.yml

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+name: C6 rare-cluster consequence checks
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c6_rare_cluster_laws_20260929/**'
+      - '.github/workflows/c6-rare-cluster-laws.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-and-source:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+          fetch-depth: 0
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Fetch immutable public premise object
+        run: |
+          set -euo pipefail
+          git fetch --no-tags --depth=1 origin 78ac1100217279e7b692f0959f86eee34cd1910e
+      - name: Verify identities, both Python modes and negative controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/c6_rare_cluster_laws_20260929/verify_packet.py
+          git diff --exit-code

frontiers/c6_rare_cluster_laws_20260929/PROOF.md

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+# Rare-cluster consequences of the sharp C6 moment bounds
+
+**Object:** OA-C6-RARE-CLUSTER-20260929-v1. **Author:** OpenAI / ChatGPT, current owner-authorized session, 29 September 2026.
+**Disposition:** author-side conditional proof candidate; nonauthor review OPEN. **Scientific effect: NONE.** No source proof, status register, Boolean, prize or acceptance disposition is changed. Same GitHub account as the other model lanes: no organizational-independence claim.
+
+## 1. Exact interface and dependency boundary
+
+Use the count N=N_r under the original determinant-tilted pinned law Q_r^W of [C6, Section 1.1]: all-index critical points, excluding the two conditioned endpoints, with heights strictly between b-k r^3 and b. Fix d>=2, the torus side L, compact birth marks and compact positive gap marks k>=k_->0. The source covariance, conditioning, full normalizer and frames are unchanged. All constants below inherit this scope; none is numerical or uniform as d,L or the marks escape it.
+
+Put e=r^3. The ONLY analytic inputs to this note are
+
+    (H1) E N <= A e,
+    (H2) P(N>=2) >= a e, where a>0,
+    (Hq) E (N)_q <= A_q e for every fixed integer q>=2.
+
+Here (N)_q=N(N-1)...(N-q+1), and A_1=A. [D5, Theorem G_d], summed over the fixed d+1 indices, gives H1. [C6, Corollary Theta] gives H2, and [C6, Theorem Q] gives Hq. Thresholds may be decreased for each finite collection of moment orders; no uniform-in-q assertion is needed. These premises imply 2a<=A.
+
+Both sources are pinned to commit 78ac1100217279e7b692f0959f86eee34cd1910e, with full byte identities in SOURCE_PINS.json. Their recorded nonauthor reviews do not constitute a review of this new note. The parent Gaussian proofs are not revalidated here. A defect in an input blocks the corresponding application, not the elementary abstract implication.
+
+[C6] frontiers/c6_palm_route_20260929/PROOF.md, Theorem Q and Corollary Theta.
+[D5] frontiers/d5_dimension_lift_20260929/PROOF.md, Theorem G_d.
+[LOCAL] frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md, Section 7, already proves the exact-mean Bernoulli-configuration obstruction. That result is prior in-project work, not a new claim of this note.
+
+For probability laws, dTV is the supremum over measurable events, equivalently half the l1 distance on this countable state space. Norms of finite or signed measures written ||.||_1 mean full l1, without the factor 1/2.
+
+## 2. Conditional counts and size-biased counts are different
+
+Write p=P(N>0), mu=E N, let J have law F=Law(N | N>0), and let Y have size-biased law P(Y=n)=n P(N=n)/mu. Then
+
+    a e <= p <= A e,             2a e <= mu <= A e.                 (2.1)
+
+For each integer k>=1, define B_k=sum_{j=1}^k S(k,j) A_j, where S(k,j) are the nonnegative Stirling numbers of the second kind. The pointwise identity n^k=sum_j S(k,j)(n)_j gives
+
+    E N^k <= B_k e,
+    E J^k <= B_k/a,              E Y^k <= B_{k+1}/(2a).            (2.2)
+
+Consequently both families are tight and every fixed power is uniformly integrable (use the next power in Markov's tail bound). Moreover
+
+    P(J>=2) >= a/A,
+    E J-1 >= a/A,
+    2a/A <= E(Y-1)=E(N)_2/mu <= A_2/(2a).                          (2.3)
+
+The exact difference between the two excess means is
+
+    E(Y-1) - E(J-1) = Var(J)/E J >= 0.                            (2.4)
+
+Indeed P(Y=n)=n F(n)/E J, so E Y=E J^2/E J. This proves (2.4). It does not assert that the two sampling laws are equal: J taking 1 and 3 with probabilities 1/2 each has mean excess 1, while Y has mean excess 3/2. The new use of all-order C6 bounds is (2.2) and its compactness consequences, not merely the distinction already stated in C6's Corollary P.
+
+## 3. Sharp ordinary-Poisson obstruction, including optimized intensity
+
+**Theorem O.** Under H1 and H2 alone, for all sufficiently small r,
+
+    (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. (3.1)
+
+The same order holds for lambda=mu. Thus an ordinary Poisson law cannot achieve o(r^3) error, even by fitting an arbitrary intensity. Absolute distance still tends to zero; this is an obstruction at the rare-event scale, not a claim that the distance is bounded away from zero.
+
+**Proof.** Take e<=min{a/(4A^2),1/(4A)}. If lambda<=2Ae, then
+
+    P(Pois(lambda)>=2) <= lambda^2/2 <= 2A^2 e^2,
+
+by the second factorial moment of a Poisson variable. Testing {n>=2} gives distance at least ae/2. If lambda>2Ae, test {n>0}; monotonicity and 1-exp(-x)>=x-x^2/2 give
+
+    1-exp(-lambda)-p >= 1-exp(-2Ae)-Ae
+                         >= Ae-2A^2 e^2 >= Ae/2 >= ae/2.
+
+The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p. For mean matching, the triangle inequality through delta_0 gives at most p+1-exp(-mu)<=2Ae, and the same lower bound applies. This proves the theorem. Notice that Hq is not needed.
+
+Conditioning a mean-matched Poisson variable on positivity does not cure the discrepancy: its probability of being >=2 tends to zero as mu->0, whereas F([2,infinity))>=a/A. Therefore the conditional TV discrepancy has liminf at least a/A.
+
+## 4. Canonical r-dependent compound Poissonization: sharp r^6 error
+
+Let C_r have the compound Poisson law CP(p,F): K~Pois(p), independent positive integer marks J_1,J_2,... with law F, and C_r=sum_{i=1}^K J_i. The empty sum is zero.
+
+**Theorem C.** Under H1 and H2, for p<=1,
+
+    p^2/3 <= dTV(Law N, Law C_r) <= p(1-exp(-p)) <= p^2.           (4.1)
+
+In particular the error for THIS approximant is Theta(r^6), with bounds a^2 r^6/3 and A^2 r^6. Its mean is exactly mu.
+
+**Proof.** Law N=(1-p)delta_0+pF. Apply the same mark-summing probability kernel to a Bernoulli(p) count and a Pois(p) count. Total variation contracts under that kernel. The Bernoulli law has its sole positive difference at count 1, of mass p-p exp(-p), so their TV distance is exactly p(1-exp(-p))<=p^2. Positivity of all marks means P(C_r=0)=exp(-p). Testing the zero event gives
+
+    dTV >= exp(-p)-1+p >= p^2/2-p^3/6 >= p^2/3.
+
+The Taylor inequalities follow from their signed integral remainders on p>=0. Finally E C_r=p E J=E N.
+
+**Essential limitation.** F is the actual, generally unknown, r-dependent conditional count law. This is a universal re-expression/Poissonization of a rare occurrence, not an explicitly identified cluster law, an asymptotic independence theorem, or a spatial Poisson-cluster process. No novelty is asserted for this elementary generic probability construction. The cluster rate is p, NOT mu: replacing it by mu changes the mean to mu E J in general. The r^6 lower bound is for CP(p,F), not a minimax lower bound over every conceivable compound Poisson parameterization.
+
+## 5. Factorial-moment matching to second rare-event order
+
+Let a_j=E(N)_j. Under Hq, for each fixed integer q>=1,
+
+    E(C_r)_q = sum_{pi in Partitions({1,...,q})} product_{B in pi} a_{|B|}. (5.1)
+
+For q=1 this equals mu. For q>=2,
+
+    0 <= E(C_r)_q-E(N)_q = O_q(e^2)=O_q(r^6).                    (5.2)
+
+**Proof.** Conditional on K, expand the ordered selections of q distinct items in the sum of its independent integer-valued marks. A set partition of the q labels specifies which labels belong to the same mark; if it has j blocks its expectation is E(K)_j times the product of the mark factorial moments. Since E(K)_j=p^j and p E(J)_b=a_b, summing gives (5.1). All terms are nonnegative and moments finite, so Tonelli applies. No moment-generating-function analyticity is assumed. The single-block partition contributes a_q. Every other partition has at least two blocks; each a_b=O(e), and there are finitely many partitions for fixed q, proving (5.2). In particular the second-moment correction is exactly mu^2.
+
+## 6. Subsequential finite cluster-intensity measures, not a unique limit
+
+Define a finite measure on the positive integers by
+
+    nu_r(n)=r^(-3) P(N_r=n),       lambda_r=nu_r(N_+)=p/r^3.
+
+Then a<=lambda_r<=A, nu_r({2,3,...})>=a, and sum n^k nu_r(n)<=B_k for k>=1 (use B_0=A for mass).
+
+**Theorem S.** Every sequence r_i->0 has a subsequence and a finite measure nu on N_+ such that for every fixed nonnegative integer k,
+
+    sum_{n>=1} n^k |nu_{r_i}(n)-nu(n)| -> 0.                    (6.1)
+
+Its mass lambda is in [a,A], and nu({2,3,...})>=a. Along that subsequence the positive conditional laws converge to nu/lambda in TV and all fixed moments. The size-biased laws converge to n nu(n)/sum_m m nu(m), also in TV and all fixed moments.
+
+**Proof.** Each coordinate is bounded by A. Diagonal selection gives coordinatewise convergence. Fatou preserves the moment bounds. For every k>=0 and M>=1,
+
+    sum_{n>M} n^k nu_r(n) <= B_{k+1}/M,
+
+and the same holds for nu. On the finite set n<=M coordinatewise convergence gives weighted l1 convergence. First take the subsequence limit and then M->infinity. The SAME selected subsequence works for every k because it was selected only by coordinates and has all these tail bounds. Mass and the multiple-count lower bound pass to the limit by k=0 convergence. Denominators for the two normalized laws stay bounded below by a and 2a, respectively, so their claims follow from (6.1), using one extra power for size bias.
+
+There is also the exact signed-measure identity
+
+    Law N_r = delta_0 + r^3 (nu_r-lambda_r delta_0).               (6.2)
+
+Thus (6.1) gives a first-order signed-measure expansion along each selected subsequence. It gives neither uniqueness nor a rate for nu_r->nu.
+
+**Sharp non-implications.** Set P(N_r>0)=r^3 and let the positive count alternate between 2 and 3 on successive dyadic intervals of r. H1,H2 and every Hq hold, but nu_r alternates between delta_2 and delta_3. Taking N_r=2 Bernoulli(r^3) also shows that no positive lower bound for the third or higher factorial moments follows. Finally, all polynomial moments need not imply any exponential moment: let P(J=2^j) be proportional to 2^(-j^2), j>=1. Every polynomial moment is finite, whereas E exp(tJ)=infinity for each t>0. The rare mixture with probability r^3 satisfies H1,H2,Hq. These examples concern what the premises alone imply, not counterexamples to the specific Gaussian model having additional structure.
+
+## 7. Independent-replica limit, with an explicit extra assumption
+
+Take m_r=floor(t/r^3) INDEPENDENT copies of the entire count N_r, for fixed t>=0, and let T_r be their sum. Along a subsequence from Theorem S,
+
+    dTV(Law T_r, CP(t nu))
+       <= t A^2 r^3 + A r^3 + t ||nu_r-nu||_1 -> 0.              (7.1)
+
+Here CP(t nu) denotes the compound Poisson law with finite Levy measure t nu, with no jump of size zero.
+
+**Proof.** Couple each copy to its Section 4 approximant: the total error is at most m_r p^2<=t A^2 r^3. The sum of the independent approximants has Levy measure m_r r^3 nu_r. Two finite Levy measures eta,zeta can be coupled by sharing their pointwise minimum and independent residual Poisson jumps; their compound count laws have TV distance at most ||eta-zeta||_1. Apply
+
+    ||m_r r^3 nu_r-t nu||_1 <= r^3 A + t ||nu_r-nu||_1.
+
+No independence of windows or spatial cells in ONE field has been proved. Such a use needs a new dependence estimate. Equation (7.1) is a theorem about deliberately independent experiments, not a new spatial claim about the original torus.
+
+## 8. Review and verification boundary
+
+Requested nonauthor slices: (A) optimized-intensity Poisson bound and exact binary-kernel constants; (B) factorial partitions, weighted-l1 compactness and the independent-replica coupling; (C) source interface and non-claims. The checker covers exact finite identities, rational probes and explicit false variants only. Neither it nor a green workflow proves H1/H2/Hq, validates all infinite-dimensional arguments, or promotes any register. There is no Lean formalization of this packet. External primary-source reconnaissance is recorded separately, without claiming a comprehensive novelty audit.

frontiers/c6_rare_cluster_laws_20260929/README.md

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+# C6 rare-cluster consequence candidate
+
+Conditional consequence packet: Claude review 5358565704 accepts the Section 1 interface and Sections 2,5–7; xAI/Grok review 5358623521 accepts Sections 3–4 and the coefficient addendum, at the exact unchanged mathematical bytes. See REVIEW_RECORD.md. Scientific effect NONE. No register, Boolean, proof-index disposition or parent source is changed. Shared GitHub login is not organizational independence.
+
+PROOF.md derives six consequences of the exact C6/D5 count interface: conditional and size-biased moment bounds; a sharp r^3 ordinary-Poisson obstruction for arbitrary fitted intensity; a canonical r-dependent compound-Poisson representation with sharp r^6 error; factorial-moment matching to order r^6; subsequential weighted-l1 cluster-intensity limits; and a compound-Poisson limit for deliberately independent replicas only.
+
+The r^6 approximant uses the actual unknown conditional positive-count law. It is NOT an explicit limiting cluster law. Limits need not be unique from the premises alone. No spatial independence or higher-factorial lower bound is claimed. The ordinary-Poisson and canonical-compound TV bounds need only the first-moment upper and multiple-event lower bounds; C6's all-order upper bounds are used for conditional moments and compactness.
+
+## Replay
+
+    python -B -S verify_packet.py
+
+Run from this directory in a Git checkout containing the exact source commit in SOURCE_PINS.json. The verifier checks source bytes, all packet identities, 40 unit tests in each Python mode, deterministic output, seven rejected semantic mutants per mode, and unknown-mutant rejection. Source objects may be fetched read-only using the exact pinned commit; the workflow does this. Full repository/formal checks are separate and remain mandatory.
+
+    python -B -S verify_packet.py --local-only
+
+This explicitly excludes Git source-object verification and reports source_pins_checked=false. It is useful for the downloadable standalone packet, not a substitute for hosted source verification. Python standard library only. No Lean/formal verification of these claims is asserted.
+
+## Nonauthor review completed at scoped mathematical bytes
+
+Claude Slice B: interface and PROOF Sections 2,5–7. xAI/Grok Harper Slice A: PROOF Sections 3–4 and coefficient addendum A1–A6. These reviews accept the conditional elementary-probability implications, not a fresh validation of the Gaussian premises, a status transition, or a claim of organizational independence. Hosted verification of the final metadata/repair successor remains separate. The author will not self-merge.
+
+SHARP_POISSON_COEFFICIENT.md sharpens the optimized error to P(N>=2)+O(r^6), compares mean matching via E[N;N>=2]+O(r^6), and identifies the subsequential interval of leading-order optimal rates. The first proof and original checker remain byte-identical. Six additional exact unit tests cover the addendum.
+
+The Codex prior-credit pin omission is repaired: the credited LOCAL_MULTIPLICITY source is checked by commit/path/blob as well as the two full premise identities. Six local-Git-fixture regressions cover valid pins and wrong prior blob/path/commit, unsafe paths, and retained premise SHA256 checking.

frontiers/c6_rare_cluster_laws_20260929/RECONNAISSANCE.md

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+# External reconnaissance — 29 September 2026
+
+Scope: locate relevant existing probability frameworks and prevent claiming elementary Poissonization as novel. This is a narrow reconnaissance memo, not an exhaustive novelty audit or an external acceptance of the project's Gaussian estimates.
+
+An actual Consensus search for "Gaussian random fields critical points factorial moments rare clusters Palm conditional distribution compound Poisson approximation domain:math" returned: all 30 monthly searches used, resetting October 1. No result was returned or attributed to Consensus.
+
+Primary arXiv abstract/metadata pages were opened directly:
+
+1. 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 . Proves finite critical-count moments under finite-jet nondegeneracy. Finiteness by itself is not the uniform rare-pinned O(r^3) input used here.
+2. Louis H. Y. Chen and Aihua Xia, *Stein's method, Palm theory and Poisson process approximation*, Annals of Probability 32 (2004), 2545–2569, DOI 10.1214/009117904000000027; arXiv:math/0410169. https://arxiv.org/abs/math/0410169 . Existing Palm/Stein/local-dependence framework. This note does not derive a Stein bound or silently identify positivity conditioning with size bias.
+3. Michael V. Boutsikas and Eutichia Vaggelatou, *A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables*, Bernoulli 16 (2010), 301–330, DOI 10.3150/09-BEJ201; arXiv:1010.1625. https://arxiv.org/abs/1010.1625 . Existing Poisson and compound-Poisson TV approximation for independent or locally dependent nonnegative integer variables. No novel generic Poissonization method is claimed here.
+
+Access depth: these three metadata/abstract pages, not a theorem-by-theorem full-text audit. None is imported as an unverified proof step: PROOF.md gives the elementary implications in full and names the in-project analytic premises separately. No statement that the conditional cluster law is absent from all literature, no spatial independence, and no priority claim follows from this reconnaissance.

frontiers/c6_rare_cluster_laws_20260929/RESULTS.json

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+{
+  "groups": [
+    "conditional-versus-size-biased",
+    "compound-rate-and-factorial-moments",
+    "ordinary-from-factorial-moments",
+    "poisson-obstruction-exact-probes",
+    "compound-TV-bounds-and-r6-ledger",
+    "nonclaims-counterexamples"
+  ],
+  "mathematical_acceptance": false,
+  "passed": true,
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "scope": "finite exact identities and probes only"
+}