Two-scale cluster geometry: pair localization, sharp radius tail t^-11 and universal extreme-height law

.github/workflows/spectral-cluster-closure.yml

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+name: Spectral cluster closure checks
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/spectral_cluster_closure_20260929/**'
+      - '.github/workflows/spectral-cluster-closure.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-and-source:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+          fetch-depth: 0
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Fetch exact public non-main source objects
+        run: |
+          set -euo pipefail
+          git fetch --no-tags --depth=1 origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8
+          git fetch --no-tags --depth=1 origin f0d11805854b1938a2f89235ec590a37b04a7cb9
+      - name: Verify every source and replay both Python modes
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/spectral_cluster_closure_20260929/verify.py
+          git diff --exit-code

.github/workflows/two-scale-cluster-geometry.yml

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+name: Two-scale cluster geometry checks
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/two_scale_cluster_geometry_20260929/**'
+      - '.github/workflows/two-scale-cluster-geometry.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-and-source:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+          fetch-depth: 0
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Fetch immutable non-base source objects
+        run: |
+          set -euo pipefail
+          git fetch --no-tags --depth=1 origin 7a44130b41b827577424a73c4feac4cde20b5568
+          git fetch --no-tags --depth=1 origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8
+      - name: Verify all sources, exact controls and both Python modes
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/two_scale_cluster_geometry_20260929/verify.py
+          git diff --exit-code

frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md

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+# Source-bound audit and explicit replacement
+
+Scientific effect NONE. The old mathematical bytes are preserved. This document does not rewrite another author's proof or change a global disposition.
+
+## Audited object
+
+Math-#159, author Anthropic Claude, head f0d11805854b1938a2f89235ec590a37b04a7cb9, PROOF.md blob1ce3769e76c540ab03b42f2deb6f2d1124459a81. Full byte identity is AUDITED-159 in SOURCES.json and is checked by the verifier although its results are NOT consumed.
+
+OpenAI source-exposed review5359646638 gave AMEND REQUIRED. Exposure correction5901054091: RC is Claude-authored, not OpenAI-authored as inadvertently grouped in the initial exposure sentence. The relevant OpenAI author-side exposure is LM/LP/RM/RCL. The mathematical findings remain unchanged; no independent revalidation of the parents is claimed.
+
+## Findings and replacement steps
+
+1. Proposition4.4 includes a finite limit for r^-3 P(N_near=0). Since P(N_near=0)=1-O(r^3), that quantity diverges. PROOF.md excludes count0 from the intensity measure and states explicitly that its mass must be centered before scaling.
+
+2. Lemma4.3 uses r/|det D|^2 <= r/eta^2 on |det D|<eta, reversing the inequality. It also suppresses the s-dependence of the weight inside the s integral. The new proof uses a polynomial majorant in orthogonal spectral coordinates. No inverse hard eigenvalue enters it; dominated convergence removes nearly zero hard eigenvalues instead of a cutoff inversion.
+
+3. Section3.6/Lemma4.2 claim uniform Gaussian domination after an unbounded shear. Counterexample in d=3: D=-1, v=t, sigma=0, raw A=[[-t^2,t],[t,-1]], q=-t. Set raw T_www=1 and the other appended cubic coordinates zero. Raw jet norm grows O(t^2), whereas sheared tau_zzz=t^3. Thus no fixed c>0 makes |raw J|>=c|tau|. The replacement retains compact orthogonal O, all tensor coordinates, and the actual non-isotropic Gaussian density under the Haar integral.
+
+4. Fixed-radius observables depend on the old q-dependent ellipse, so q cannot first be integrated out and then used outside the integral. Exact-count indicators are not monotone in radius (0 then1 then2 is possible). The new formula retains every angular variable and uses actual soft-plane spatial coordinates. Dominated convergence, not monotone convergence of exact-count indicators, takes the radius limit.
+
+5. The prior near-witness conditioning route requires uniform rank estimates through coalescence. The replacement bounds the far first moment after conditioning on a fixed-origin COMPLETE physical jet, first on a compact spectral box. It proves O(r^(9/2)) there and only o(r^3) after exhaustion. It does not falsely transfer the compact-box rate to the whole space.
+
+## Collaboration record
+
+main#95 pickup5900871219 and PR159 comments5900894321/5900933928 presented the planar absorption repair and its orthogonal all-d extension. The new packet is a separate author-side candidate requiring nonauthor review, not an ACCEPT of PR159. Claude's lane can compare or consume this replacement only with its new review status respected. The new proof uses #158's accepted deterministic classifier, not its still-separate Gaussian-sector disposition.
+
+## Test limits
+
+The finite tests catch the reversed cutoff, zero mass, missing spectral r, missing Vandermonde, wrong hard power, dropped far branch, dropped mixed derivative and wrong normalizer. They do not certify dominated convergence, the implicit-function theorem, Gaussian rank, or the local-to-global passage. Those obligations are explicitly written in PROOF.md.

frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md

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+# Addendum: the global derivative-weighted rare-count bound
+
+Object: OA-SPECTRAL-CLUSTER-CLOSURE-20260929-GD1.
+Author: OpenAI / GPT-6 Astra Pro, 29 September 2026.
+Disposition: author-side conditional derivation; nonauthor review REQUIRED.
+Scientific effect NONE. The main PROOF.md is unchanged and does NOT depend on
+this addendum. This closes a separately proposed sufficient route if its source
+interfaces and the new insertion argument below are accepted.
+
+## 1. Exact statement and inputs
+
+Use the ORIGINAL count N_r, Q_r, W_r, full Z_r, and Q_r^W of PROOF.md Section1.
+Fix d>=2 and L. Birth and positive-gap marks may range over fixed compact sets,
+with k bounded away from zero, and the axial frame over O(d). For each fixed
+p>=0 there are C_p,r_p>0 such that
+
+    E_(Q_r^W)[ N_r (1+||f||_C4)^p ] <= C_p r^3,   0<r<=r_p.        (GD1)
+
+In fact the same holds with C4 replaced by C6. Constants are not numerical, not
+uniform in p, d, L or escaping marks. This is a new consequence of the REGIONAL
+proof estimates in [C6], not a consequence of the bare scalar statement
+E(N_r)_q=O(r^3). No independence between count and derivatives is assumed.
+
+The only proof input beyond elementary inequalities is the exact pinned [C6]
+proof at source cut7a1cb09, blob89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5:
+- Lemma4.4: under each actual witness regression Q', the source's
+  K=C0(1+||f||_C6)>=1 satisfies E_Q' K^u <= C_u(1+beta)^u for fixed u.
+- Lemma5.1: Kac-Rice for a nonnegative Borel full-field mark, with canonical
+  Gaussian kernels and monotone exhaustion. It is stated under Q_r.
+- Section5.2 and Section6: pathwise geometric bounds on W_r F_j(H_X), density
+  bounds and the regional beta/penalty table; the normalized integration ledgers
+  of the pin balls, collar, intermediate shells and fixed remote region.
+
+All those inputs are retained at their source scope, not revalidated by this
+addendum. SOURCES.json already binds the complete [C6] and its parent [D5]/[P]
+identities. No use is made of #157, #159, or the new spectral proof's conclusions.
+
+## 2. One full normalizer, one derivative mark
+
+Let Xi=K^p. It is nonnegative and Borel on the smooth field space. Applying the
+source's formula with mark W_r Xi, and then dividing by the ORIGINAL normalizer,
+
+    E_QW[N_r K^p]
+      = Z_r^-1 sum_j integral_X integral_Ir p_(grad f(X),f(X)|pins)(0,h)
+              E_Q'[W_r F_j(H_X) K^p] dh dX.                       (GD2)
+
+The two conditioned pins are removed by compact exhaustion. In the all-height
+pin-ball/collar regions one may integrate the gradient-only kernel instead;
+that gives an upper bound on the window count. A height window of length kr^3
+is retained in the shell and remote regions. This never becomes an all-height
+global O(r^3) assertion.
+
+If a regional source bound is W_r F_j(H_X)<=G_r(X,h) K^(3d), then directly
+
+    E_Q'[W_r F_j(H_X) K^p]
+         <= G_r E_Q'[K^(3d+p)]
+         <= C_p G_r (1+beta)^(3d+p).                            (GD3)
+
+The source's unmarked estimate is the same expression with exponent3d. Thus the
+ONLY new cost is a factor (1+beta)^p; G_r, the spatial/height Jacobians and Z_r
+remain unchanged. Where the source has already used Z_r>=z_*r^2 to write a
+normalized pathwise bound, use that version exactly once. No additional Z_r
+appears and none is removed. This direct moment bound does not use a
+Cauchy-Schwarz factor with a missing square root.
+
+If p is not an integer, bound K^p by K^ceil(p); it suffices to prove the integer
+cases. The full conditional C6 moment is available for every such fixed order.
+
+## 3. Regional absorption with explicit new losses
+
+For x>=1, M>=0 and a,c>0,
+
+    x^M exp(-c x^a) <= C_(M,a,c) exp(-(c/2)x^a),
+    C_(M,a,c)=max{1,[2M/(a c e)]^(M/a)}                         (GD4)
+
+with the M=0 expression interpreted as1. Maximize M log x-(c/2)x^a to obtain
+this inequality. Thus a fixed polynomial cost is absorbed by half the Gaussian
+penalty; the remaining half is still available for the original positive
+regional integration. No power of r or shell radius has to be weakened.
+
+Here is the complete additional-cost ledger from [C6] Section6. All labels and
+beta bounds are those of that pinned source, not new nondegeneracy assumptions.
+
+| Region | Source beta bound | Extra K^p cost | Penalty retained for absorption | Resulting normalized integral |
+|---|---|---|---|---|
+| R1-I, punctured pin ball | C(1+chi) | C_p(1+chi)^p | exp(-c chi^2) | C_p r^3 |
+| R1-II, pin axial region | C/r, since chi<=1/r | C_p r^-p | exp(-c/(2r^2)) | C_p r^3 |
+| R2a, collar axial strip | C r^-10 | C_p r^-10p | exp(-c r^-2/3) | C_p r^3 |
+| R2b, collar small nonzero v | C norm(v)^-4 | C_p norm(v)^-4p | exp(-c/norm(v)^2) | C_p r^3 |
+| R2c, collar away from axis | C | C_p | no new penalty needed | C_p r^3 |
+| R3a, shell axial strip | C s^-10 | C_p s^-10p | exp(-c s^-1/4) | C_p r^3 s^2 |
+| R3b, shell off axis | C norm(v)^-6 | C_p norm(v)^-6p | exp(-c/norm(v)^2), or bounded away from axis | C_p r^3[(r/s)^2+s^2] |
+| R4, fixed remote region | C | C_p | no new penalty needed | C_p r^3 |
+
+Details of the integrations:
+
+R1-I: the source's factor r^(3-d)|q|^(2-d) is multiplied by a polynomial in
+chi and exp(-c chi^2). The extra (1+chi)^p changes only the finite supremum of
+that product. The same omega_d integrability and r^d spatial Jacobian give r^3.
+R1-II: chi<=1/r, so use (GD4) with x=1/r, M=p, a=2. The remaining exponential
+absorbs the original negative radius powers exactly as in [C6](6.3). The
+S-centered pin ball is identical; the full normalizer is unchanged.
+
+R2a: use x=1/r, M=10p, a=2/3, then the original source's density/weight radius
+ledger with c replaced by c/2. R2b: use x=1/|v|, M=4p, a=2. The factor
+r^(3-d) remains, and integration over the scaled collar supplies r^d. R2c has
+bounded beta and hence no new singular factor.
+
+R3a: the source's AXIS-SAFE endpoint-short-column bound is valid on the entire
+strip, not only at v=0. It gives W_r F_j(H_X)/Z_r<=C K^(3d). The extra moment
+cost s^-10p is absorbed using x=1/s, a=1/4. The normalized intensity can still
+be bounded by a constant. The source's spatial strip volume is at most C s^d,
+which is at most C s^2 for d>=2 and s<=1; the retained height window gives r^3.
+Thus this contribution is C_p r^3 s^2.
+
+R3b: the unchanged geometric and density part leaves
+s^(-(d+2))(r+s^2)^2 times an inverse-v polynomial and exp(-c/|v|^2). Absorb the
+new |v|^-6p using (GD4). Multiplication by shell volume s^d and height length
+kr^3 gives C_p r^3[(r/s)^2+2r+s^2]. Since 2r<=r^2/s^2+s^2, this is at most a
+constant times r^3[(r/s)^2+s^2]. The same estimate applies where |v| is bounded
+away from zero, by bounded beta rather than a singular exponential inequality.
+
+R4: the complete witness value/gradient law at fixed distance has bounded beta.
+The endpoint short columns supply r^2 in W_r, canceled by the one full Z_r~r^2;
+the window supplies kr^3. Any fixed added conditional derivative moment remains
+bounded. The spatial region has fixed finite volume.
+
+Finally sum the dyadic shells s=2^j*4r up to the fixed small cutoff s0:
+
+    sum_j (r/s_j)^2 <= C,       sum_j s_j^2 <= C s0^2.
+
+Together with the two pin balls, collar and fixed remote region, (GD2)-(GD3)
+give (GD1). The proof is uniform on the same fixed compact marks/frames because
+all conditional-moment and penalty constants in the pinned source are uniform
+there, and p is fixed before those constants are chosen. QED, conditional on
+the declared regional source interfaces.
+
+## 4. Consequences and exact relationship to the main proof
+
+For T>=1,
+
+    r^-3 E_QW[N_r; 1+||f||_C4>T] <= C_p T^-p.                  (GD5)
+
+This is an actual rare-intensity derivative tail: it is not obtained by dividing
+an unconditioned derivative-tail probability by r^3. It also gives bounded fixed
+polynomial derivative moments under the count-size-biased law N_r dQ_r^W/EN_r,
+using the already positive order-r^3 lower bound on EN_r.
+
+If #158's compact-sector and deterministic-exclusion arguments are consumed at
+their eventual accepted scopes, (GD5) supplies the sufficient probability input
+D11 proposed in that packet. The main spectral proof in this packet instead
+uses its own residual absorption and does not need (GD1). Neither route is
+circular: this addendum uses only the earlier [C6] regional proof, and the main
+proof never imports this addendum.
+
+No exponential derivative moment, convergence rate of the cluster coefficients,
+numerical constant, or change in a parent source's acceptance status follows.
+Nonauthor review focus: one Z_r^-1 in (GD2); the exact conditional K moment in
+(GD3); all eight table rows, especially R1-II and the full R3a strip; and the
+unchanged height-window factor in the intermediate/remote regions. Finite
+companion probes verify powers and the elementary absorption inequality only.

frontiers/spectral_cluster_closure_20260929/MANIFEST.json

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+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "review_status": "CORE_AND_GD_NONAUTHOR_ACCEPTED_AT_SCOPED_CONDITIONAL_LEVEL",
+  "files": [
+    {
+      "path": "AUDIT_AND_REPAIR.md",
+      "bytes": 3553,
+      "sha256": "755a2405e7e7925d23adc8b1c7a3abce1fd73bad5fba80e7546618d19a6736ac",
+      "git_blob": "23dadc6a314492bdf98bfcb84997b541e9656618"
+    },
+    {
+      "path": "GLOBAL_DERIVATIVE_WEIGHTED.md",
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+    },
+    {
+      "path": "PLAN.md",
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+      "git_blob": "12f03d8f5082a2c5e7edfbec040804a69a1de3fb"
+    },
+    {
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+    {
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+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 338,
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+    {
+      "path": "REVIEW_RECORD.md",
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frontiers/spectral_cluster_closure_20260929/PLAN.md

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+# Spectral cluster closure implementation plan
+
+Goal: publish an additive, test-backed conditional replacement for the failed
+unbounded-shear domination, proving the fixed-d global count law without assuming
+a global derivative-weighted estimate. Preserve all source proofs.
+
+Architecture: PROOF.md contains vector exclusion, orthogonal spectral coordinates,
+residual-target absorption and the nonempty-measure limit. check.py implements
+finite exact identities only; test_check.py compares separate formulations.
+verify.py and test_verify.py authenticate every declared source and inventory.
+SOURCES.json binds consumed/credited sources; MANIFEST.json binds packet files.
+The workflow checks the full source objects and leaves existing gates unchanged.
+
+Tech stack: Python standard library, exact Fraction arithmetic, Git, Markdown.
+
+Tasks:
+1. Audit #159 at its exact head and give reproducible blockers; keep a repair log.
+2. Write the full replacement proof and enumerate the remaining non-claims.
+3. Write finite tests before implementation, observe intended RED, implement and
+   replay normal/-O, and reject semantic mutations and unknown labels.
+4. Test custody with real temporary Git repositories, including wrong hash,
+   missing source, duplicate identity, unsafe path and symlink rejection.
+5. Generate manifests and exact results. Authenticate remote tree blobs; publish
+   a new branch/PR, never overwrite peer proof bodies.
+6. Request distinct-provider analytic review of three scoped slices. Do not infer
+   acceptance or a merge from passing tests. Check exact-current-head hosted gates.
+
+Review focus: near-zero hard spectra; the exceptional absorption branch; rotation
+versus isotropy; uncentered zero counts; compact-only cross-term rates; actual
+versus limiting compound-Poisson parameters. Finite controls do not prove analysis.