diff --git a/.github/workflows/spectral-cluster-closure.yml b/.github/workflows/spectral-cluster-closure.yml
new file mode 100644
index 000000000..9fed2e14d
--- /dev/null
+++ b/.github/workflows/spectral-cluster-closure.yml
@@ -0,0 +1,36 @@
+name: Spectral cluster closure checks
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/spectral_cluster_closure_20260929/**'
+      - '.github/workflows/spectral-cluster-closure.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-and-source:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+          fetch-depth: 0
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Fetch exact public non-main source objects
+        run: |
+          set -euo pipefail
+          git fetch --no-tags --depth=1 origin bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8
+          git fetch --no-tags --depth=1 origin f0d11805854b1938a2f89235ec590a37b04a7cb9
+      - name: Verify every source and replay both Python modes
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/spectral_cluster_closure_20260929/verify.py
+          git diff --exit-code
diff --git a/.github/workflows/two-scale-cluster-geometry.yml b/.github/workflows/two-scale-cluster-geometry.yml
new file mode 100644
index 000000000..b0ac3a4fe
--- /dev/null
+++ b/.github/workflows/two-scale-cluster-geometry.yml
@@ -0,0 +1,36 @@
+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
diff --git a/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md b/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md
new file mode 100644
index 000000000..23dadc6a3
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/AUDIT_AND_REPAIR.md
@@ -0,0 +1,29 @@
+# Source-bound audit and explicit replacement
+
+Scientific effect NONE. The old mathematical bytes are preserved. This document does not rewrite another author's proof or change a global disposition.
+
+## Audited object
+
+Math-#159, author Anthropic Claude, head f0d11805854b1938a2f89235ec590a37b04a7cb9, PROOF.md blob1ce3769e76c540ab03b42f2deb6f2d1124459a81. Full byte identity is AUDITED-159 in SOURCES.json and is checked by the verifier although its results are NOT consumed.
+
+OpenAI source-exposed review5359646638 gave AMEND REQUIRED. Exposure correction5901054091: RC is Claude-authored, not OpenAI-authored as inadvertently grouped in the initial exposure sentence. The relevant OpenAI author-side exposure is LM/LP/RM/RCL. The mathematical findings remain unchanged; no independent revalidation of the parents is claimed.
+
+## Findings and replacement steps
+
+1. Proposition4.4 includes a finite limit for r^-3 P(N_near=0). Since P(N_near=0)=1-O(r^3), that quantity diverges. PROOF.md excludes count0 from the intensity measure and states explicitly that its mass must be centered before scaling.
+
+2. Lemma4.3 uses r/|det D|^2 <= r/eta^2 on |det D|<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.
diff --git a/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md b/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md
new file mode 100644
index 000000000..3f63edb54
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/GLOBAL_DERIVATIVE_WEIGHTED.md
@@ -0,0 +1,161 @@
+# Addendum: the global derivative-weighted rare-count bound
+
+Object: OA-SPECTRAL-CLUSTER-CLOSURE-20260929-GD1.
+Author: OpenAI / GPT-6 Astra Pro, 29 September 2026.
+Disposition: author-side conditional derivation; nonauthor review REQUIRED.
+Scientific effect NONE. The main PROOF.md is unchanged and does NOT depend on
+this addendum. This closes a separately proposed sufficient route if its source
+interfaces and the new insertion argument below are accepted.
+
+## 1. Exact statement and inputs
+
+Use the ORIGINAL count N_r, Q_r, W_r, full Z_r, and Q_r^W of PROOF.md Section1.
+Fix d>=2 and L. Birth and positive-gap marks may range over fixed compact sets,
+with k bounded away from zero, and the axial frame over O(d). For each fixed
+p>=0 there are C_p,r_p>0 such that
+
+    E_(Q_r^W)[ N_r (1+||f||_C4)^p ] <= C_p r^3,   0<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.
diff --git a/frontiers/spectral_cluster_closure_20260929/MANIFEST.json b/frontiers/spectral_cluster_closure_20260929/MANIFEST.json
new file mode 100644
index 000000000..9d6c6eefd
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/MANIFEST.json
@@ -0,0 +1,97 @@
+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "review_status": "CORE_AND_GD_NONAUTHOR_ACCEPTED_AT_SCOPED_CONDITIONAL_LEVEL",
+  "files": [
+    {
+      "path": "AUDIT_AND_REPAIR.md",
+      "bytes": 3553,
+      "sha256": "755a2405e7e7925d23adc8b1c7a3abce1fd73bad5fba80e7546618d19a6736ac",
+      "git_blob": "23dadc6a314492bdf98bfcb84997b541e9656618"
+    },
+    {
+      "path": "GLOBAL_DERIVATIVE_WEIGHTED.md",
+      "bytes": 8813,
+      "sha256": "a445607be26cc2863682c404d702798a984fa2b8994ed0dc5b57fb9c4da153eb",
+      "git_blob": "3f63edb54ea5422cdc328cc54fb093c8a5d84be1"
+    },
+    {
+      "path": "PLAN.md",
+      "bytes": 1851,
+      "sha256": "531d71ad7009195d25fd42d8ed55593267eb4c8e476e79f106cb3d2512d191ee",
+      "git_blob": "12f03d8f5082a2c5e7edfbec040804a69a1de3fb"
+    },
+    {
+      "path": "PROOF.md",
+      "bytes": 25006,
+      "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb",
+      "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a"
+    },
+    {
+      "path": "README.md",
+      "bytes": 5084,
+      "sha256": "b1e2a0722945a6189d8338c2e5b1f1a74b3b5646b8dc10fe29168c165508342c",
+      "git_blob": "895dded8aa8e405ad8e9640d3ffebc6449a108d0"
+    },
+    {
+      "path": "RECONNAISSANCE.md",
+      "bytes": 1458,
+      "sha256": "c3af589197c895d2521721c41ac7866d0945ec3c9de1c853d348bdc1a0ce42a1",
+      "git_blob": "aec1f588dcafa4f1605f4fd53feef04cd350bae5"
+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 338,
+      "sha256": "f7ebe3d9a6767d967a8b0139de02b3d18e9081f36c9f85b847c6f3f01d6ce529",
+      "git_blob": "7c564dacf616f1b6099c93e05daf564e5c36ec8f"
+    },
+    {
+      "path": "REVIEW_RECORD.md",
+      "bytes": 5629,
+      "sha256": "6031d36697789097185a8e65bfbdf80f447e229689eac5a706ee8b0558db64a5",
+      "git_blob": "9bcae183fecdf0ffe4d612a209438a4fb1fa0edd"
+    },
+    {
+      "path": "SOURCES.json",
+      "bytes": 5677,
+      "sha256": "c1e84ef7ccf749b7eb19c713d2b0fe4f4547c03ce56ef18c9858010f99929d9e",
+      "git_blob": "8daf24b7ea01e889102170dcd056634b4f67e911"
+    },
+    {
+      "path": "VALIDATION.md",
+      "bytes": 3082,
+      "sha256": "602ab8a58b2866b63de06923f0fb259b813b031a1e7c0cb6fc995a269be726f5",
+      "git_blob": "bf00357c5c7bfb6a019c1edc85e37e865a73d804"
+    },
+    {
+      "path": "check.py",
+      "bytes": 5877,
+      "sha256": "d24e07af792fc0538f977489e7af24a7815413487577f94f1a74cb229c8dec94",
+      "git_blob": "e747e1c81779e665cdb614562579206237720c76"
+    },
+    {
+      "path": "test_check.py",
+      "bytes": 6510,
+      "sha256": "b094624c46756ab771465ce1eeb73c315e3f36e66e61ff8f116d0e07d88e1f46",
+      "git_blob": "33fdbff05bbae19f9cb9700ba0d7b1ecea5e05c4"
+    },
+    {
+      "path": "test_derivative_weighted.py",
+      "bytes": 2922,
+      "sha256": "39c11a235a6186189d694794078fa9b29b26f1b530b7b79c423c8fff63b5211b",
+      "git_blob": "5389a7400feebae73dbaeeb96cd0f1940c3bab66"
+    },
+    {
+      "path": "test_verify.py",
+      "bytes": 4439,
+      "sha256": "16bccfbe377e87ecfbfba6db895fe508dfa1f2eab9d9be807a3d310750b6d0bf",
+      "git_blob": "d813c34e90a77c9eb15248299380dc8184892cfa"
+    },
+    {
+      "path": "verify.py",
+      "bytes": 4729,
+      "sha256": "d3d499b7b4dc1e7ba7414bcd452d7302e001719fccd6abaceff62a748bb4c3f5",
+      "git_blob": "db4260655b433c393c62b7abe411de2f416b382c"
+    }
+  ]
+}
diff --git a/frontiers/spectral_cluster_closure_20260929/PLAN.md b/frontiers/spectral_cluster_closure_20260929/PLAN.md
new file mode 100644
index 000000000..12f03d8f5
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/PLAN.md
@@ -0,0 +1,30 @@
+# Spectral cluster closure implementation plan
+
+Goal: publish an additive, test-backed conditional replacement for the failed
+unbounded-shear domination, proving the fixed-d global count law without assuming
+a global derivative-weighted estimate. Preserve all source proofs.
+
+Architecture: PROOF.md contains vector exclusion, orthogonal spectral coordinates,
+residual-target absorption and the nonempty-measure limit. check.py implements
+finite exact identities only; test_check.py compares separate formulations.
+verify.py and test_verify.py authenticate every declared source and inventory.
+SOURCES.json binds consumed/credited sources; MANIFEST.json binds packet files.
+The workflow checks the full source objects and leaves existing gates unchanged.
+
+Tech stack: Python standard library, exact Fraction arithmetic, Git, Markdown.
+
+Tasks:
+1. Audit #159 at its exact head and give reproducible blockers; keep a repair log.
+2. Write the full replacement proof and enumerate the remaining non-claims.
+3. Write finite tests before implementation, observe intended RED, implement and
+   replay normal/-O, and reject semantic mutations and unknown labels.
+4. Test custody with real temporary Git repositories, including wrong hash,
+   missing source, duplicate identity, unsafe path and symlink rejection.
+5. Generate manifests and exact results. Authenticate remote tree blobs; publish
+   a new branch/PR, never overwrite peer proof bodies.
+6. Request distinct-provider analytic review of three scoped slices. Do not infer
+   acceptance or a merge from passing tests. Check exact-current-head hosted gates.
+
+Review focus: near-zero hard spectra; the exceptional absorption branch; rotation
+versus isotropy; uncentered zero counts; compact-only cross-term rates; actual
+versus limiting compound-Poisson parameters. Finite controls do not prove analysis.
diff --git a/frontiers/spectral_cluster_closure_20260929/PROOF.md b/frontiers/spectral_cluster_closure_20260929/PROOF.md
new file mode 100644
index 000000000..16c56821b
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/PROOF.md
@@ -0,0 +1,491 @@
+# Orthogonal spectral closure of the rare window-count law
+
+Object: OA-SPECTRAL-CLUSTER-CLOSURE-20260929-v1.
+Author: OpenAI / GPT-6 Astra Pro, 29 September 2026.
+Disposition: AUTHOR-SIDE CONDITIONAL PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED.
+Scientific effect NONE: no existing proof, acceptance record, global register,
+Boolean or prize is changed. Shared GitHub identity is not organizational
+independence. All new analytic steps below require their own review.
+
+## 1. Model, inputs, and the claimed closure
+
+Fix d>=2, m=d-1, torus side L>0, birth b, gap mark k>0, and an orthonormal
+axial/transverse frame. Use exactly [P]'s variance-one periodized Gaussian
+covariance, original endpoint observations
+
+    M=(-r/2,0), S=(r/2,0),
+    f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0,
+
+and endpoint regression Q_r. Let W_r=F_d(H_M)F_(d-1)(H_S), Z_r=E_Q W_r and
+Q_r^W=(W_r/Z_r)Q_r. N_r counts ALL additional critical points with heights in
+(b-k r^3,b), excluding the two pins. Constants below depend on these fixed
+parameters, d, and L; where stated they also depend on R or rho. No uniformity
+in varying dimension, volume, or unbounded/vanishing marks is claimed.
+
+Exact source commits, paths, sizes and hashes are in SOURCES.json. The inputs are:
+
+[P] parent Sections2-5: positive Fourier spectrum, nonsingular contact frame U_r,
+its target v_r, smooth finite-jet regression, every fixed C^q moment, and
+Z_r/r^2 -> z0>0. The endpoint-gradient identity gives W_r/r^2 <= C K^(2d), where
+K>=1 is a fixed multiple of 1+||f||C4 bounding operator derivatives through four.
+
+[D5] the fixed-d global first moment and height-window shell estimate
+
+    E_W N_r <= C r^3,
+    E_W N({Rr<|x|<rho}) <= C r^3(R^-2+rho^2), R>=4.       (1)
+
+[RM,RC] for fixed rho>0, on D_rho={|x|>=rho}, with Lambda the SUM of the
+index-specific normalized contact kernels,
+
+    E_W N(D_rho)=k r^3 integral_Drho Lambda+O_rho(r^4),
+    E_W (N(D_rho))_2=O_rho(r^5).                          (2)
+
+[C6] E_W (N_r)_q<=C_q r^3 for every fixed q>=2. With (1), Stirling's identity
+also gives E_W N_r^q=O_q(r^3). We use this only for moment uniform integrability
+and consequences, not for the new near-event domination or mixed-event estimate.
+
+[CUB] the exact deterministic classifier in #158, Theorem C, at its xAI-reviewed
+algebraic scope. Its Gaussian sector theorem is NOT consumed. [DEEP] is credited
+for the planar exclusion construction; Section3 proves the vector version here.
+[RCL] the previously reviewed generic Poisson/compound-Poisson consequences.
+The defective global proof in #159 is NOT a premise. This note gives an alternative
+spectral-coordinate proof, not a retrospective acceptance of its shear argument.
+
+Theorem (conditional global cluster law). There are explicitly specified finite
+positive numbers a1,a2,beta_far such that, putting
+
+    nu1=a1+beta_far, nu2=a2,
+
+for every fixed nonnegative integer q,
+
+    sum_(n>=1) n^q |r^-3 Q_r^W(N_r=n)-nu1*1{n=1}-nu2*1{n=2}| -> 0.  (3)
+
+The coefficients are (17) and (22) below. Thus the full limiting nonempty count
+measure is unique and supported on {1,2}. This is not a joint law of clusters
+from distinct pin pairs, and not a spatial independence statement. Unscaled
+Law(N_r) tends to delta_0. The zero mass must be CENTERED before multiplying
+by r^-3; r^-3 Q_r^W(N_r=0) diverges.
+
+## 2. Physical jets, residual coupling, and exact orthogonal coordinates
+
+Let A=D_y^2 f(0) be the m-by-m midpoint transverse Hessian and T the list of
+independent third derivatives at 0 EXCEPT f_xxx. Set J=(A,T). At contact,
+(U_0,J) is the complete nonrepeated jet list through degree three: U_0 supplies
+f, all first derivatives, f_xx, f_xyi, and f_xxx. Positive Fourier spectrum
+therefore gives a positive-definite covariance. For sufficiently small r, the
+joint covariance and its inverse are uniformly bounded. The conditional density
+h_r(A,T) under Q_r converges locally uniformly to h_0 and obeys
+
+    h_r(A,T)<=C exp[-c(||A||F^2+|T|^2)].                   (4)
+
+The bounded conditional means have been absorbed by reducing c.
+
+Take an independent copy F of the unconditioned smooth field. For each raw target
+j, construct the conditional field by regression on V_r=(U_r,J):
+
+    f_r^j=F+Cov(F,V_r)Cov(V_r)^-1((v_r,j)-V_r(F)).         (5)
+
+All coefficient functions are bounded in C4 and converge there. Averaged contact
+functionals in U_r are bounded by a constant times ||F||C3; this uses the integral
+forms of the pin divided differences, not separate uncontrolled r^-p terms.
+Consequently, with S=C(1+||F||C4), independent of the target j,
+
+    K(f_r^j)<=C(S+|j|),       E S^p<infinity for every p.  (6)
+
+The same random F can be used at all small r; no independence of the field's
+internal derivatives is asserted. Disintegrating first over j and then F gives
+exactly Q_r. Formula (6) is a pathwise bound under this construction.
+
+Diagonalize -A with ORDERED eigenvalues lambda1<...<lambdam. Repeated eigenvalues
+are a Lebesgue-null set. Write
+
+    lambda1=-r s, h=(h2,...,hm)=(lambda2,...,lambdam),
+    A=O diag(r s,-h2,...,-hm) O^t, O in O(m).              (7)
+
+Rotate the complete tensor T by diag(1,O), calling its coordinates tau. This
+linear tensor representation has absolute determinant one: its determinant in
+absolute value is a continuous homomorphism from a compact group to the positive
+reals, whose only compact subgroup is {1}. The joint coordinate Jacobian is
+block triangular, so the dependence of the tensor rotation on O does not add a
+factor. The field covariance is NOT assumed orthogonally invariant. The density
+h_r(A,T) is always evaluated at the actual rotated raw arguments.
+
+Specify the spectral constant without any convention ambiguity. With raw
+independent upper-triangular Lebesgue coordinates dA and normalized Haar dO,
+
+    integral g(A)dA
+      = c_m integral_(l1<...<lm) integral_O(m)
+           g(O diag(l)O^t) product_(i<j)(lj-li) dO dl.
+
+The positive constant c_m is fixed exactly by
+
+    c_m = (2*pi)^(m/2) * pi^[m(m-1)/4] / H_m,
+    H_m = integral_(l1<...<lm) exp(-sum li^2/2)
+                                      product_(i<j)(lj-li) dl.   (8)
+
+This also is an explicit finite integral definition; c_1=1. To prove the change
+of variables, differentiate O diag(l) O^t: the off-diagonal infinitesimal in
+an ij rotation is (lj-li) times the rotation increment. Their product is the
+Vandermonde; invariance of matrix Lebesgue measure leaves normalized Haar times
+a constant. The Gaussian test integral fixes the constant as (8). This is a
+Lebesgue-coordinate identity, not a GOE assertion about h_r.
+
+After lambda1=-r s, the spectral Jacobian is r c_m J_r, where
+
+    J_r=product_(j=2..m)(h_j+r s) product_(2<=i<j<=m)(h_j-h_i),
+    domain: -r s<h2<...<hm.                              (9)
+
+For m=1, h is absent, every product is 1, and there is no ordering constraint.
+Haar sign choices are retained; invariant observables are unchanged by the
+corresponding reflection of tau and of the soft coordinate. No arbitrary global
+choice of an eigenvector is needed.
+
+Put U=1+S+|h|+|tau|. Orthogonality gives, uniformly in O,
+
+    K<=C(U+r|s|),
+    h_r(A,T)<=C exp[-c(|h|^2+|tau|^2)].                  (10)
+
+Unlike an unbounded shear, these coordinates genuinely preserve norm equivalence
+with constants depending only on the fixed dimension.
+
+## 3. Vector transverse exclusion, and the absorption dichotomy
+
+Use the cylinder |x|<=Rr, |y|<=Rr, R>=1, and let K bound the operator norms of
+all derivatives through four. Let lambda=lambda_min(-A). If
+
+    rK<=k/(4R), lambda>(8RK+56RK^2/k)r,                  (11)
+
+there are only the two pinned critical points in the cylinder.
+
+Proof. Throughout the cylinder f_yy<=-lambda/2, since its variation from A is at
+most 2RKr. Put g(x)=f_y(x,0). The pin gradients give g(-r/2)=g(r/2)=0. Applying
+the scalar two-node interpolation estimate to every unit projection of g gives
+||g(x)||<=KR^2r^2, also outside the pin interval. No vector-valued Rolle zero is
+assumed. Likewise
+
+    g'(x)=r^-1 integral_(-r/2)^(r/2)[g'(x)-g'(t)]dt
+
+implies ||g'(x)||<=K(R+1/2)r. For rho=4R^2Kr^2/lambda<Rr/2, the radial derivative
+f_y(x,y).y is negative on |y|=rho, because it is at most
+KR^2r^2 rho-(lambda/2)rho^2. The strictly concave transverse function attains its
+maximum inside that ball. Its unique critical point zeta(x) depends C3-smoothly
+on x. Strong concavity gives uniqueness throughout |y|<=Rr.
+
+On the graph, ||f_xy||<=2RKr and ||zeta'||<=4RKr/lambda<1/2. For
+psi(x)=f(x,zeta(x)), differentiating f_y=0 gives
+
+    psi'''=f_xxx+3 f_xxy[zeta']+3 f_xyy[zeta',zeta']
+                                      +f_yyy[zeta',zeta',zeta'].
+
+The terms containing zeta'' cancel because f_xy+f_yy zeta'=0. Thus
+|psi'''-f_xxx|<=28RK^2r/lambda<k/2. The exact two values and two axial-gradient
+pins imply f_xxx(eta,0)=12k for some eta between the pins, by Hermite interpolation
+and generalized Rolle. Hence |f_xxx(x,zeta)-12k|<=2RKr<=k/2, so psi'''>=11k>0.
+The strictly convex function psi' already has the two pinned zeros and cannot
+have a third. All critical points in the cylinder lie on its graph. This proves
+(11) in every fixed dimension.
+
+On the typed support H_M<0, A has largest eigenvalue less than rK/2 by the
+Hessian Lipschitz bound, so s<K/2. If N_R, the window count in |x|<=Rr, is positive,
+(11) implies, outside the event rK>k/(4R),
+
+    |s|<=C_R K^2<=a_R U^2+a_R r^2 |s|^2, a_R>=1.         (12)
+
+Here K>=1, k is fixed, and (10) was used. The elementary absorption dichotomy is
+
+    x<=a U^2+a r^2 x^2  implies
+    x<=2a U^2  OR  x>=1/(2a r^2), x>=0.                 (13)
+
+If x>2aU^2, subtracting aU^2 from (12) proves the second alternative.
+On the second alternative |A|op>=r|s|>=c_R/r, hence K>=c_R/r. The first excluded
+event also implies K>=c_R/r. All exceptional mass can therefore be bounded in
+ORIGINAL Q_r coordinates, before any spectral rescaling. For every p,M>=0,
+
+    E_W[K^p;exceptional] <= C E_Q[K^(p+2d);K>=c_R/r]
+                         <= C_(p,M,R) r^M.              (14)
+
+The last bound uses one higher fixed Q_r moment. The normalizer is the full
+Z_r>=z_*r^2 and W_r<=Cr^2K^(2d). There is no need to posit a conditional Gaussian
+tail that is uniform in an unbounded scaled target. The exceptional branch has
+not been discarded; it is negligible to every fixed power of r.
+
+## 4. One global majorant, including multiple soft eigenvalues
+
+At each pin the axial Hessian column has norm <=rK/2 by the vector gradient
+average. The TRANSVERSE part of the midpoint column in the soft eigenvector direction is
+r s times that eigenvector; its endpoint transverse perturbation is bounded by
+rK/2. Its axial entry at a pin is already bounded by the short axial column.
+Thus the full soft column is bounded by r(|s|+K), up to a fixed harmless constant.
+The axial and soft columns correspond to orthogonal coordinate directions. Hadamard therefore gives
+
+    W_r/r^4 <= C K^(2d-2)(|s|+K)^2.                     (15)
+
+This does not divide by any hard eigenvalue. On the regular near-occurrence
+branch |s|<=2a_R U^2, (10) gives K<=C_R U^2 (r<=1), so (15)<=C_R U^(4d).
+Writing v=m(m-1)/2, (9) gives |J_r|<=C_R U^(2v). Thus the density of the rescaled
+NONEMPTY near measure, in spectral variables and the coupling variable F, is
+bounded by
+
+    G_R=C_R exp[-c(|h|^2+|tau|^2)]
+                  U^(4d+2v) 1{|s|<=2a_R U^2}.          (16)
+
+Indeed disintegration gives r^-3 times r from (9), times r^4 from the weight,
+divided by Z_r: the prefactor is c_m r^2/Z_r, which is bounded. Integrating s in
+(16) contributes only another U^2. Gaussian h,tau integration and every fixed
+moment of S make G_R integrable. Multiplication by any fixed polynomial in
+K, |s|, |h|, |tau| remains integrable on the regular branch. Formula (14) controls
+the weighted exceptional branch as well: for a scaled-jet power, use |s|<=K/r
+and choose the arbitrarily large power M in (14) to absorb that extra r^-p.
+
+Consequences: the r^-3 near-occurrence measures are tight in all the spectral
+jet variables; their tails outside increasing compact sets tend to zero uniformly
+as r->0. Their occurrence-weighted K moments are O(r^3). If m>=2, the limsup of
+scaled near-occurrence mass with |h2|<=eta tends to zero as eta->0, directly by
+integrating G_R over that strip. The same holds near repeated hard spectra. No
+inverse determinant or 1/eta bound appears. This proves the needed multiple-soft
+exclusion IN MEASURE, without a pointwise assertion that two soft directions
+cannot occur.
+
+Using C6 only now, Cauchy-Schwarz also yields
+E_W[N_R K^p]<=sqrt(E_W N_R^2)*sqrt(E_W[K^(2p);N_R>0])=O_(R,p)(r^3).
+Thus the LOCAL derivative-weighted bound is established, not assumed. The global
+analogue is not needed for the rest of this proof.
+
+## 5. Pointwise normal form, typed-boundary control, and the near limit
+
+Fix spectral variables s,h,O,tau with 0<h2<...<hm, and a realization of F with
+finite C4 norm. In the orthonormal frame (axis, soft eigenvector, hard eigenvectors),
+write coordinates (x,z,w). Conditional targets are exactly
+A=diag(rs,-h) and the prescribed cubic tensor tau. With x=rX,z=rZ,w=r^2V, the pin
+Taylor identities from [P] give, on each fixed bounded (X,Z,V) set,
+
+    [f(rX,rZ,0)-b]/r^3 -> P_s,tau(X,Z) in C2,
+    r^-2 grad_w f(rX,rZ,r^2V) -> -diag(h)V+Q_tau(X,Z) in C1,
+
+where
+
+    P_s,tau=2kX^3-3kX/2-k/2+sZ^2/2
+             +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3,
+    a=tau_xxz, beta=tau_xzz, c=tau_zzz.
+
+Q_tau contains the hard-index cubic terms. For fixed positive h the hard
+Hessian stays negative definite for small r. On the full physical Rr ball,
+grad_w f=-diag(h)w+O(Kr^2)+O(Kr|w|). Every critical point therefore has
+|w|<=C_(h,R,F,tau) r^2. The hard implicit function is unique and smooth, and
+substitution into f/r^3 changes the planar potential by O(r) in C2 on fixed
+sets. Its inverse-hard-eigenvalue constants are used ONLY for pointwise
+convergence, never in (16).
+
+At the endpoints the soft Hessian blocks divided by r converge to
+
+    M_M=[[-6k,-a/2],[-a/2,s-beta/2]],
+    M_S=[[ 6k, a/2],[ a/2,s+beta/2]].
+
+A Schur complement across the fixed hard block, or the determinant expansion,
+gives W_r/r^4 -> product h_j^2 * w(s,a,beta), where
+
+    w=[-6k(s-beta/2)-a^2/4]_+ [a^2/4-6k(s+beta/2)]_+.
+
+It is positive exactly on s<-|B|/2, B=beta-a^2/(12k), and equals
+9k^2(4s^2-B^2) there. Outside this open typed domain its limiting value is zero;
+on the typed boundary the determinant limit is also zero. Consequently no count
+convergence is needed at a point with zero limiting weight. This is the explicit
+handling of the typed boundary, not an assumed compact exclusion from it.
+
+Within the typed domain use [CUB]'s exact classifier n(s,a,beta,c) in {0,1,2}.
+Let n_R count those additional cubic critical points in X^2+Z^2<R^2 and the
+strict height window. For almost every jet, all relevant roots are nondegenerate,
+none has boundary height, and none lies on the observation circle. The last null
+set can be checked without an informal resultant assertion: on a circle R>=1,
+Z!=0 at any critical root, and the two gradient equations solve for (beta,s)
+as smooth functions of (X,Z,a,c). A circle coordinate plus a,c is three parameters,
+so its image in four-dimensional (s,a,beta,c) space is null. Cover by countably
+many compact charts with |Z| bounded away from zero. Height boundary relations
+are the nonzero polynomial threshold of [CUB]. The pins are nondegenerate and
+remain EXACT, so they cannot split into new counted roots.
+
+Choose disjoint neighborhoods of relevant zeros. C2 convergence and the inverse
+function theorem preserve precisely one root in each; on the rest of the compact
+set in a slightly enlarged height window the gradient is bounded away from zero.
+No condition on lower-height degeneracies is necessary. Hard slaving then proves
+N_R -> n_R pointwise for these fixed variables. The hard coordinates vanish on
+the r spatial scale. The extra points have full index d-1.
+
+For precision, extend the coefficient kernel by zero off ordered positive h and
+the typed domain. With T=Rot_O(tau), define
+
+    dM(s,h,O,tau) = (c_m/z0) h_0(O diag(0,-h)O^t,T)
+             * product_(j=2..m) h_j^3
+             * product_(2<=i<j<=m)(h_j-h_i)
+             * w(s,a,beta) ds dh dO dtau.                (17)
+
+Every angular variable remains inside this integral. On each fixed R, the
+nonempty joint spectral-jet/count measure converges in full variation to
+1{n_R=j}dM on j>=1. To see this, on the regular branch sum the absolute conditional
+errors over the count index j: it is bounded by the weight error plus twice
+the weighted probability of a count mismatch. Pointwise convergence gives zero
+and (16) dominates. Cases of zero limiting weight give zero directly. On a fixed typed jet with n_R>0 and finite S, K is bounded as r->0, so the
+exceptional branch is eventually absent and the regular-branch indicator is
+eventually one. No part of the proposed limit is lost by that split. The
+exceptional mass is o(r^3) by (14). This proves
+
+    r^-3 Q_W(N_R=j) -> a_j^R=integral 1{n_R=j}dM, j=1,2,
+    r^-3 Q_W(N_R>=3) -> 0.                               (18)
+
+There is NO finite uncentered n=0 formula. For fixed R, the right side is a
+finite measure only on nonempty counts.
+
+## 6. Integrability at arbitrary scaled radius
+
+For the planar classifier put D=(c-a beta/(4k)+a^3/(72k^2))/2 and
+T_*=-(s-B)^2(B+2s)/(12k). It says: s>B gives n=2 below D^2=T_* and n=1 at or above;
+s<=B gives n=1 strictly above that threshold and n=0 otherwise. On n>=1, x=-s>0
+satisfies
+
+    x<=2|B|+(48kD^2)^(1/3).
+
+Indeed s>B forces x<|B|. Otherwise D^2>T_*, and if x>=2|B| then
+T_*>=(x/2)^2*x/(12k)=x^3/(48k). Since w<=36k^2 x^2,
+
+    integral_s w 1{n>=1} <=384k^2 |B|^3+2304k^3 D^2.      (19)
+
+This is a polynomial bound in the cubic coordinates at fixed k. The remaining
+factor in (17) is a nonnegative polynomial in h times a uniformly Gaussian-decaying
+function of h,tau, by ORTHOGONAL norm preservation. Therefore
+
+    a_j=integral 1{n=j}dM, j=1,2                         (20)
+
+are finite. They are strictly positive: take the open two-point cubic sector
+(s,a,beta,c)=(-3k/2,0,-2k,0), or the one-point sector (-k,0,0,2k), choose distinct
+positive hard h in a compact box, and arbitrary remaining cubic entries in a
+small box. Gaussian densities are positive, and the spectral/weight factors are
+positive on an open set. For m=1 the empty hard products are 1.
+
+For every fixed jet, n_R=n eventually as R->infinity. Exact-count indicators
+{n_R=1} need NOT be monotone, but are bounded by 1{n>=1}; (19) supplies an
+integrable majorant. Dominated convergence proves a_j^R->a_j. No inverse shear
+parameter is suppressed by integrating out variables that a spatial observable
+still uses.
+
+## 7. Remote singleton coefficient and vanishing mixed event
+
+By (1) and [RM]'s fixed-remote mean limit, for 0<rho'<rho<=s0,
+
+    k integral_(rho'<=|x|<rho) Lambda(x) dx <= C rho^2.    (21)
+
+Apply the shell estimate with lower scale rho' held fixed as r->0; equivalently
+choose R_r=rho'/r, which is permitted because the shell constant is uniform in
+R_r>=4. The kernel is nonnegative and the annulus is fixed, so its mean limit
+is supplied by [RM]. Thus
+
+    beta_far=k integral_(X\{0}) Lambda(x)dx              (22)
+
+exists, is finite, and differs from the fixed-rho integral by O(rho^2).
+It is positive by positivity on any fixed remote region of positive volume.
+Corollary D of [RC] and the mean limit give
+r^-3 Q_W(N_far^rho=1)->k integral_Drho Lambda and
+Q_W(N_far^rho>=2)=O_rho(r^5).
+
+Fix R and rho. To control the mixed event, first restrict the spectral jets
+(s,h,tau) to a fixed compact box H, retaining all O. Under the conditional law
+(5) with these targets, the covariance of (f(x),grad f(x)), |x|>=rho, has a
+uniform positive lower bound. At r=0 the conditioning is the COMPLETE degree-three
+jet at the origin and x is a distinct site; positive Fourier spectrum gives
+full joint rank. Compactness in x, O, and the target-independent covariance,
+plus continuity through r=0, proves the bound. Conditional Hessian moments are
+bounded on H and on the height window. The usual unweighted Gaussian Kac-Rice
+formula, integrated over the window of length k r^3, gives
+
+    E[N_far^rho | U_r,J]<=C_(H,rho) r^3.                 (23)
+
+On H, (15) and (6) give a uniform second moment of W_r/r^4. Conditional
+Cauchy-Schwarz therefore gives
+
+    E[(W_r/r^4)1{N_far^rho>0} | U_r,J]<=C_(H,rho) r^(3/2).
+
+Disintegrating over H with (9), the full normalizer, and bounded density and
+Vandermonde gives Q_W(spectral jets in H,N_far^rho>0)=O_(H,rho)(r^(9/2)).
+In particular this bounds the compact-jet near/far intersection.
+Outside H, its probability is bounded by Q_W(N_R>0,jets outside H). By (16)
+and (14), its scaled limsup tends to zero as H exhausts the spectral variables.
+Thus
+
+    Q_W(N_R>0,N_far^rho>0)=o_(R,rho)(r^3).                (24)
+
+The full mixed term has no asserted r^(9/2) rate: that rate was proved only on
+a fixed compact jet box. This proof avoids conditioning on a near WITNESS whose
+observation frame degenerates. It uses fixed-origin physical jets instead.
+
+## 8. The full count limit and all polynomial weights
+
+Write N=N_R+N_mid+N_far. For fixed R,rho, the bad events N_mid>0,
+{N_R>0,N_far>0}, N_far>=2, and N_R>=3 have probabilities bounded by (1), (24),
+(2), and (18), respectively. On their complement N equals either the near
+count in {0,1,2} or a remote singleton, but not both. Therefore, with an absolute
+constant C independent of R,rho,
+
+    limsup |r^-3 Q(N=2)-a_2^R| <= C(R^-2+rho^2),
+    limsup |r^-3 Q(N=1)-a_1^R-k integral_Drho Lambda| <= C(R^-2+rho^2),
+    limsup r^-3 Q(N>=3) <= C(R^-2+rho^2).                 (25)
+
+First r->0 at fixed R,rho, then R->infinity and rho->0, using (20)-(22).
+This proves coordinate convergence to nu1=a1+beta_far and nu2=a2, and vanishing
+scaled tail mass. For a fixed integer q>=0, C6 supplies E N^(q+1)<=C r^3; hence
+
+    r^-3 E[N^q;N>M]<=C/M.
+
+Finite-coordinate convergence plus this uniform tail proves (3). No interchange
+of a nonuniform radius limit and r limit has been assumed. No finite unscaled
+polynomial moment theorem alone would imply (3); the new domination and regional
+sandwich are the essential ingredients.
+
+In particular
+
+    E N/r^3 -> nu1+2nu2,
+    E (N)_2/r^3 -> 2nu2,
+    E (N)_q/r^3 -> 0 for every fixed q>=3.                (26)
+
+The positive-conditioned count converges in TV to
+(nu1 delta1+nu2 delta2)/(nu1+nu2), and its size-biased version to
+(nu1 delta1+2nu2 delta2)/(nu1+2nu2). The two laws are different. Their formulas
+use the normalized full tilt, not a restricted normalizer.
+
+For independent replicas with floor(t/r^3) counts, the identified compound limit
+is P1+2P2, with independent P1~Pois(t nu1), P2~Pois(t nu2), as follows either by
+the exact generating function or [RCL]'s proved independent-replica bound.
+There is no independence claim for regions of a single field.
+
+The sharp [RCL] Poisson comparison also now has coefficients:
+optimal ordinary-Poisson TV error/r^3 -> nu2 and mean-matched error/r^3 ->2nu2,
+so their ratio tends to 2 (not just liminf>=2). The canonical approximant using
+the ACTUAL conditional mark law still has Theta(r^6) error. Replacing its mark
+law by the identified LIMIT does not inherit that rate: without a convergence
+rate for the coefficients, this replacement is only o(r^3) in one-count TV.
+
+## 9. What is solved here, conditional on review, and what is not
+
+The proof supplies a replacement for scaled jet tails, typed-boundary control,
+nearly singular hard blocks, the near/far mixed event and the radius sandwich.
+It works for every FIXED d>=2 via orthogonal spectral coordinates. It does not
+require the stronger global derivative-weighted insertion estimate proposed in
+#157; Section4 proves the local version as a consequence and the global law does
+not need the global version. Thus that former sufficient route is bypassed, not
+silently assumed true.
+
+The local coefficients in d=2 reduce exactly to #158's alpha1,alpha2, and its
+contact birth factorization remains a local statement. In higher dimensions the
+Haar/stable integration may couple birth-dependent even jets to orientation;
+no higher-dimensional or global birth cancellation is asserted.
+
+No convergence rate, numerical coefficient enclosure, unbounded-mark theorem,
+growing-volume limit, inter-pin-pair cluster independence, elder selection of the
+extra points, or full Lean formalization is claimed. The input Gaussian and
+regional sources remain conditional premises at their exact stated scopes.
+Theorem (3) is a new author-side candidate, not accepted merely because finite
+controls or existing formal gates pass.
+
+Review division: A, vector exclusion, absorption, and orthogonal spectral measure;
+B, integrable majorant, typed/hard-boundary handling and pointwise normal form;
+C, compact-jet remote conditioning, mixed event, radius sandwich and consequences.
+The prior #159 diagnosis and the distinction between that proof and this replacement
+are retained in AUDIT_AND_REPAIR.md. Source preservation is mandatory.
diff --git a/frontiers/spectral_cluster_closure_20260929/README.md b/frontiers/spectral_cluster_closure_20260929/README.md
new file mode 100644
index 000000000..895dded8a
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/README.md
@@ -0,0 +1,41 @@
+# Orthogonal spectral closure of the window-count law
+
+OpenAI / GPT-6 Astra Pro. Core conditional proof has completed scoped nonauthor A/B/C acceptance. The separate global derivative-weighted addendum has its own Claude nonauthor acceptance5360082678. All acceptance is at the exact stated conditional scope. Scientific effect NONE. No self-merge, parent-proof modification, or global status/Boolean change. Shared GitHub identity is not organizational independence.
+
+The new argument proposes the unique limiting nonempty count measure in every FIXED dimension d>=2: nu1 delta1 + nu2 delta2, with explicit positive finite coefficients. Near coefficients are an orthogonal Haar/spectral/Gaussian jet integral; the far contribution is the reviewed remote singleton kernel. The new proof does not consume the audited v1.0 global argument of #159 or the proposed marked Fourier theorem of #157.
+
+The main new steps are the vector transverse-exclusion lemma; the absorption split into a regular soft layer and a separately controlled exceptional raw-field tail; a polynomial majorant with NO inverse hard eigenvalues; weighted typed-boundary and multiple-soft control; pointwise stable-direction elimination; compact-origin-jet near/far decoupling; and a radius sandwich. Existing C6 moments provide polynomial uniform integrability after, not instead of, this new limit argument.
+
+Consequences, conditional on the new theorem: complete polynomially weighted count convergence; higher factorial moments o(r^3) for q>=3; identified conditional and Palm laws; independent-replica limit Pois(t nu1)+2 Pois(t nu2); exact limiting ratio 2 between mean-matched and optimized ordinary-Poisson error. No convergence rate or numerical coefficient is supplied. Replacing the actual conditional law in the canonical compound approximant by its limit does not preserve the old O(r^6) rate without new rate information.
+
+## Read and replay
+
+Read PROOF.md and AUDIT_AND_REPAIR.md. PLAN.md records the task split. SOURCES.json includes all consumed, credited, and audited identities, including the historical #159 proof; no recorded source is silently skipped.
+
+In a full checkout containing all pinned commits:
+
+    python -B -S frontiers/spectral_cluster_closure_20260929/verify.py
+
+In the standalone directory:
+
+    python -B -S verify.py --local-only
+
+The latter explicitly reports source_pins_checked=false. Local Git fixtures test verifier behavior, not the truth of the actual project sources. The hosted workflow performs the actual source-object checks and full packet replay; existing downstream/formal gates remain separate.
+
+Current local suite: 55 tests in normal Python and 55 with -O; deterministic result equality; eight semantic mutants rejected with exit1 in each mode and unknown labels with exit2. Controls cover exact algebra only, not Gaussian convergence. The initial12-test stub failed all12 intended assertions. An expanded positive fixture mistakenly violated the absorption premise; that fixture was corrected, not the valid implementation. Source verifier had its own fail-first fixture suite. See VALIDATION.md for the exact scope.
+
+## Actual scoped core reviews
+
+A: Claude5359968447 accepts vector exclusion, absorption and the exact spectral/Haar measure.
+B: Claude5359845136 accepts the majorant, typed/hard-boundary control, normal form and coefficient integrals.
+C: OpenAI/Codex5359879627 accepts the remote conditioning, mixed-event exhaustion, radius sandwich and consequences; this is same-provider nonauthor review, not independent-provider review.
+
+All bind the unchanged core PROOF.md. The separate OpenAI/Codex A check5360029463 is corroboration, not an additional independent vote. Parent-input acceptance and organizational independence are not renewed by these reviews. A code pass is not their substitute. No Lean formalization of the new proof is claimed.
+
+The initial hosted nested-directory source-path defect is repaired with git ls-tree --full-tree and a real-Git regression. See VALIDATION.md; a predecessor failure is not concealed by local test success.
+
+## Separate global derivative-weighted addendum
+
+GLOBAL_DERIVATIVE_WEIGHTED.md derives E[N_r(1+||f||C4)^p]<=C_p r^3 directly from the earlier C6 regional Kac-Rice proof, in fact with C6 norm. It retains the one full normalizer, adds the conditional-moment cost (1+beta)^p, and writes all eight regional absorptions. It is not a consequence assumed by the core spectral theorem. Nine additional finite ledger tests are included. The addendum has separate Slice D acceptance5360082678; it was not inferred from the earlier core reviews.
+
+REVIEW_RECORD.md records the completed core A/B/C and separate D coverage, exact byte bindings and provider exposure. GD1 is not an input to the core theorem. The unchanged manuscripts retain their author-time candidate headers to preserve reviewed bytes; this record supplies the later dispositions. Integration still requires successful current-head checks and a nonauthor integrator. The author will not self-merge or change a global scientific register.
diff --git a/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md b/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md
new file mode 100644
index 000000000..aec1f588d
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/RECONNAISSANCE.md
@@ -0,0 +1,10 @@
+# Narrow primary-source reconnaissance, September29 2026
+
+Consensus.search was actually called with “Gaussian random fields critical point rare clusters Palm spectral eigenvalue boundary layer Kac Rice domain:math”. It returned the exhausted monthly30-search quota and October1 reset. No Consensus result is claimed.
+
+Primary arXiv metadata/abstract pages were opened through web:
+
+- https://arxiv.org/abs/2304.07424v3 — Diego Armentano, Jean-Marc Azais and Jose Rafael Leon, On a general Kac-Rice formula for the measure of a level set. The repository cites v3 (December5 2023); the page reports a later v4 dated April22 2025. This pass did not call v3 latest or audit changes between versions. The parent Kac-Rice interface is retained at its recorded scope.
+- https://arxiv.org/abs/2305.17586 — Louis Gass and Michele Stecconi, The number of critical points of a Gaussian field: finiteness of moments, v2 July19 2023. The abstract concerns finite critical-count moments under jet nondegeneracy. It does not substitute for the rare tilted O(r^3) bounds or the identified cluster law proposed here.
+
+Access depth: metadata and abstracts only, not full-text theorem audits. The orthogonal spectral Jacobian is derived in the manuscript and its normalization is fixed by an explicit Gaussian test integral. No external unverified theorem is imported as a replacement for a missing project premise, and no exhaustive novelty or priority claim is made.
diff --git a/frontiers/spectral_cluster_closure_20260929/RESULTS.json b/frontiers/spectral_cluster_closure_20260929/RESULTS.json
new file mode 100644
index 000000000..7c564dacf
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/RESULTS.json
@@ -0,0 +1,16 @@
+{
+  "absorption_cases": 141,
+  "groups": [
+    "spectral-jacobian",
+    "hard-weight",
+    "absorption",
+    "cutoff-direction",
+    "nonempty-measure",
+    "vector-third-derivative",
+    "full-normalizer"
+  ],
+  "mathematical_acceptance": false,
+  "passed": true,
+  "scientific_effect": "NONE",
+  "scope": "finite exact controls only"
+}
diff --git a/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md b/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md
new file mode 100644
index 000000000..9bcae183f
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/REVIEW_RECORD.md
@@ -0,0 +1,97 @@
+# Completed scoped reviews — spectral core and derivative-weighted addendum
+
+Scientific effect NONE. This is a factual record of actual nonauthor reviews,
+not a self-issued acceptance, governing-register transition, or organizationally
+independent endorsement. All lanes share one GitHub account. Source reviews do
+not revalidate every parent result or extend beyond the stated parameter scope.
+
+## Exact immutable mathematical objects
+
+Core PROOF.md:25006 bytes; SHA256
+e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb;
+Git blob16c56821b52fd76b0be791622b9c3809eafde75a, unchanged since
+41814598d37de8446dd538ce3cf2d57982a4077f.
+
+GLOBAL_DERIVATIVE_WEIGHTED.md:8813 bytes; SHA256
+a445607be26cc2863682c404d702798a984fa2b8994ed0dc5b57fb9c4da153eb;
+Git blob3f63edb54ea5422cdc328cc54fb093c8a5d84be1, unchanged since
+3892aeed64601350472566c657d954e84ffdd78d. It is NOT an input to the core proof.
+The manuscripts retain their historical author-time headers; the later actual
+review dispositions are recorded below without changing their mathematical bytes.
+
+## Actual core coverage
+
+A — Anthropic Claude5359968447, submitted2026-09-30T00:07:07Z at7a44130:
+ACCEPT Sections2-3, complete-jet coupling, ordered spectral/Haar coordinates,
+vector exclusion, both absorption branches and the original-Q exception bound.
+The reviewer declares no authorship of the consumed sources or competing159.
+Different provider from the author; no organizational-independence credit.
+
+B — Anthropic Claude5359845136, submitted2026-09-29T23:46:11Z at41814598:
+ACCEPT Sections4-6, majorant, typed/hard-boundary control, normal form and
+coefficient integrals, conditional on A and the pinned classifier. That reviewer
+authored consumed C6/D5/RC and competing159; this is source-exposed scrutiny,
+not a renewal of those inputs or organizational independence.
+
+C — OpenAI/Codex5359879627, submitted2026-09-29T23:51:40Z at7a44130:
+ACCEPT Sections7-8, compact-origin-jet far conditioning, full mixed o(r^3), radius
+sandwich and probability consequences, conditional on A/B and exact inputs.
+Different author session but SAME PROVIDER. Independent-provider and
+organizational credit are zero. Its read-only challenges are not extra votes.
+Comment5901273120 clarifies arbitrary fixed real b, not b>0.
+
+Additional OpenAI/Codex5360029463 corroborates A at the unchanged bytes. It has
+the same-provider limitation and is not an additional independent vote.
+
+Combined core scope: fixed d>=2,L>0,k>0,b in R and frame, with exact imported
+Gaussian/regional interfaces. No numerical coefficient or convergence rate,
+growing-volume or vanishing-mark uniformity, or distinct-pin-pair independence.
+
+## Separate D acceptance
+
+Anthropic Claude5360082678, submitted2026-09-30T00:26:33Z at3892aeed:
+ACCEPT GD1-GD5, exactly one full Z^-1, conditional K^(3d+p) moment insertion,
+all eight regional absorptions with height window retained, dyadic integration,
+and the global rare-intensity derivative tail. This reviewer declares no
+authorship of C6 or the other pinned sources; different provider from the author,
+but the same shared GitHub account and zero organizational independence.
+
+Scope: every fixed d>=2,L, compact birth/positive-gap marks bounded away from
+k=0, O(d) frames, and each fixed p, with existential constants. The verdict is
+conditional exactly on earlier C6 Lemma4.4/5.1 and regional interfaces, not a
+revalidation of them. It does not accept an exponential derivative moment or rate.
+The earlier core reviews were never used as substitutes for this D review.
+
+Nonblocking D reading clarifications adopted in author comment5901687461:
+choose K=max(source K,1), which preserves its moment bound; R3a is r^3 s^d
+<=r^3 s^2; the unmarked comparison is the underlying DL/RM bound before C6's
+extra Psi mark; R4 uses Lemma4.4 with RM's bounded beta. GD4's displayed constant
+is a convenient valid upper bound, not a claim of the smallest possible constant.
+Core clarifications: the soft direction is largest for A/smallest for -A;
+||f_xy||<=K(R+1/2)r+K|zeta|<=2RKr; norms are invariant or uniformly equivalent
+coordinate norms, never an assumption of field isotropy; N_R is the Euclidean
+Rr-ball count inside the exclusion cylinder.
+
+## Verification and integration boundary
+
+Initial target36646437664 failed because git ls-tree inherited the nested cwd.
+A real-Git fail-first fixture reproduced it; --full-tree fixed it at7a44130.
+Codex P1 comment4139440699 is answered4139467347 and RESOLVED. All identity,
+path, symlink and negative controls remain; no source check was bypassed.
+
+Target/full runs36646903471/36646903884 at7a44130,36649121320/36649121574 at3892aeed,
+and36650263358/36650263498 atc433a592 all succeeded. These observations are not
+automatically rebound to later metadata-only commits. Final-head evidence belongs
+in the PR handoff after a fresh readback.
+
+Local packet:55 tests per normal/optimized mode,141 exact absorption probes,
+eight semantic mutants rejected per mode and unknown-label rejection. Eleven
+actual consumed/credited/audited Git-source objects are checked by hosted replay.
+The local container did not obtain a full upstream checkout; its explicit
+local-only mode never reports actual project source pins checked. Existing Lean
+package success does not formalize the new analytic proofs.
+
+Both manuscripts and all executables/source pins are unchanged by this record.
+All four requested analytic review slices are complete at their stated scope.
+A nonauthor integrator must still read the final diff, review threads and
+current-head/current-base checks. No self-merge or global-status fold is implied.
diff --git a/frontiers/spectral_cluster_closure_20260929/SOURCES.json b/frontiers/spectral_cluster_closure_20260929/SOURCES.json
new file mode 100644
index 000000000..8daf24b7e
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/SOURCES.json
@@ -0,0 +1,129 @@
+{
+  "schema": 1,
+  "repository": "d6g8k5htny-coder/Math-",
+  "scientific_effect": "NONE",
+  "source_policy": "Every entry, including credit and audited historical bytes, is mandatory for full source verification. Identity is not mathematical acceptance.",
+  "sources": [
+    {
+      "id": "P",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "author": "OpenAI",
+      "role": "consumed",
+      "use": "Sections2-5, contact jet rank, full-field regression, derivative moments, full normalizer."
+    },
+    {
+      "id": "D5",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/d5_dimension_lift_20260929/PROOF.md",
+      "bytes": 53727,
+      "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80",
+      "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc",
+      "author": "Anthropic Claude",
+      "role": "consumed",
+      "use": "Fixed-d first moment and height-window shell estimate, exact scope."
+    },
+    {
+      "id": "C6",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/c6_palm_route_20260929/PROOF.md",
+      "bytes": 53364,
+      "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b",
+      "git_blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5",
+      "author": "Anthropic Claude",
+      "role": "consumed",
+      "use": "All fixed factorial upper moments, only for uniform integrability and local weighted-count consequence."
+    },
+    {
+      "id": "RM",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/remote_window_20260924/PROOF.md",
+      "bytes": 18355,
+      "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7",
+      "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84",
+      "author": "OpenAI",
+      "role": "consumed",
+      "use": "Fixed-remote first moment and continuous positive normalized contact kernel."
+    },
+    {
+      "id": "RC",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/remote_collision_20260928/PROOF.md",
+      "bytes": 26002,
+      "sha256": "b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2",
+      "git_blob": "7b48a88e2af54e759e89a8c8219573bb450a65ea",
+      "author": "Anthropic Claude",
+      "role": "consumed",
+      "use": "Corollary D, all-index fixed-remote factorial bound. The actual source author is Claude."
+    },
+    {
+      "id": "LM",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md",
+      "bytes": 13100,
+      "sha256": "e6272e8709835ca6290fecb7034fa84832dde08c3c4cc990a48b8019e89e7c8d",
+      "git_blob": "b5c18aac5529c5f975ee5bb056b304debafc6697",
+      "author": "OpenAI",
+      "role": "prior_credit",
+      "use": "Original exact pinned cubic and Taylor mechanism, credited and re-derived in spectral coordinates."
+    },
+    {
+      "id": "CUB",
+      "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8",
+      "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md",
+      "bytes": 19889,
+      "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0",
+      "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758",
+      "author": "OpenAI",
+      "role": "consumed",
+      "use": "Only exact deterministic Theorem C/classifier, xAI review5359488967; not Gaussian sector acceptance."
+    },
+    {
+      "id": "DEEP",
+      "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8",
+      "path": "frontiers/planar_cubic_cluster_20260929/DEEP_TRANSVERSE_EXCLUSION.md",
+      "bytes": 6174,
+      "sha256": "95c03e59262cb9607a21a296a6837932839dd47ec8e15e67bdfa46782adbcd5a",
+      "git_blob": "309a38cab16ef67dd909351f40a48e1d83ba2c28",
+      "author": "OpenAI",
+      "role": "prior_credit",
+      "use": "Planar exclusion construction; vector extension is fully proved as new work here."
+    },
+    {
+      "id": "RCL",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/c6_rare_cluster_laws_20260929/PROOF.md",
+      "bytes": 11811,
+      "sha256": "c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb",
+      "git_blob": "2ab625cedfc2e47575c4a7fa4853413418c532da",
+      "author": "OpenAI",
+      "role": "consumed",
+      "use": "Generic conditional/Palm, compound approximation and independent-replica consequences, not claimed anew."
+    },
+    {
+      "id": "RCL-A",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/c6_rare_cluster_laws_20260929/SHARP_POISSON_COEFFICIENT.md",
+      "bytes": 4712,
+      "sha256": "f96e7a02ceb711fbaf10142aff7ee5dbc5fb908ac24f695a289180a1e094d14f",
+      "git_blob": "9f96f7d619a3486d0b125a4f4a3528e213f149d6",
+      "author": "OpenAI",
+      "role": "consumed",
+      "use": "Exact optimized/mean-matched Poisson error inequalities."
+    },
+    {
+      "id": "AUDITED-159",
+      "commit": "f0d11805854b1938a2f89235ec590a37b04a7cb9",
+      "path": "frontiers/c6_cluster_law_20260929/PROOF.md",
+      "bytes": 51313,
+      "sha256": "7941ae8062aed14e3085583903867ee3612ca34c1b636c69671a9541c2dd4801",
+      "git_blob": "1ce3769e76c540ab03b42f2deb6f2d1124459a81",
+      "author": "Anthropic Claude",
+      "role": "audited_not_consumed",
+      "use": "Historical proof diagnosed in review5359646638; no theorem or domination from it is imported."
+    }
+  ]
+}
diff --git a/frontiers/spectral_cluster_closure_20260929/VALIDATION.md b/frontiers/spectral_cluster_closure_20260929/VALIDATION.md
new file mode 100644
index 000000000..bf00357c5
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/VALIDATION.md
@@ -0,0 +1,29 @@
+# Verification record at author publication
+
+Scientific effect NONE; finite evidence is not analytic acceptance.
+
+Actual local execution:
+- Initial mathematical stub:12 tests,12 intended assertion failures. Expanded stub checks followed.
+- Initial implementation replay exposed one erroneous positive fixture: x=2,U=1,r=1/10,a=1 violates the premise x<=aU^2+ar^2x^2 since2>1.04. Corrected the test to x=1, without weakening the premise or modifying the valid implementation.
+- Mathematical suite including CLI checks:32 tests pass in normal and optimized modes.
+- Source-custody placeholder:13 tests produced17 expected assertion failures including subtests; valid fixture passed. Implemented complete Git-object/mode and byte identity checks.
+- Full local suite:45 tests pass in normal and optimized modes. The deterministic checker output matches RESULTS.json. Eight named false variants exit1 in each mode; unknown labels exit2. All packet identities are generated from exact local bytes.
+
+Custody fixtures use real isolated Git repositories. They cover wrong size/SHA256/blob, missing path/commit, unsafe paths, duplicate source IDs, empty inventories, duplicate JSON, Git leaf and parent symlinks, and packet symlinks. Source verification authenticates the pinned tree/blob rather than following a worktree link.
+
+The container could not obtain a complete remote checkout, so actual project Git-object verification is not claimed local. Default hosted replay is configured to fetch and verify EVERY declared source, including historical audit and prior-credit entries. Hosted success and full repository/formal checks must be read at the actual current PR head before any integration. No Lean formalization of the new analytic proof is claimed.
+
+## Hosted nested-directory repair
+
+Initial hosted run36646437664 failed at source verification, job109670396383, because `git ls-tree` interpreted the pinned repository-root path relative to the nested packet cwd. Both source fetches and inventory authentication had succeeded; the error was not a missing upstream proof. A new real-Git nested-packet regression failed before repair (None!=1). Adding `--full-tree` makes root-relative source paths independent of cwd. The mathematical PROOF.md and all premise identities remain unchanged. Latest local suite:46 tests per mode. The original failing hosted run is retained; the successor still needs its own hosted result.
+
+## Addendum and review-metadata completion
+
+The separate GD1 addendum adds nine exact finite ledger tests. Fresh complete
+local replay passes55 tests in normal and55 in optimized Python, with identical
+RESULTS and all eight semantic mutations rejected per mode. These test the
+normalizer/radius/penalty ledgers, not conditional Gaussian integration.
+At addendum head3892aeed64601350472566c657d954e84ffdd78d, hosted target36649121320
+and full36649121574 both succeeded. Later metadata-only commits must be checked
+at their own exact PR test merge. Core PROOF.md, GD1, all source pins and every
+executable remain unchanged by this final review-metadata update.
diff --git a/frontiers/spectral_cluster_closure_20260929/check.py b/frontiers/spectral_cluster_closure_20260929/check.py
new file mode 100644
index 000000000..e747e1c81
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/check.py
@@ -0,0 +1,155 @@
+"""Exact finite controls for orthogonal spectral cluster closure.
+
+Standard library only. These controls do not prove Gaussian convergence.
+"""
+import argparse
+import json
+from fractions import Fraction as F
+from math import prod
+
+MUTANTS=('drop-spectral-r','drop-soft-vandermonde','wrong-hard-power',
+         'drop-far-branch','reverse-cutoff','include-zero',
+         'drop-mixed-term','swap-full-normalizer')
+
+
+def rat(x):
+    if type(x) not in (int,F): raise ValueError('exact int/Fraction required')
+    return F(x)
+
+
+def absorption(x,u,r,a):
+    x,u,r,a=map(rat,(x,u,r,a))
+    if x<0 or u<1 or r<=0 or a<=0: raise ValueError('invalid absorption arguments')
+    if x>a*u*u+a*r*r*x*x: raise ValueError('absorption premise does not hold')
+    if x<=2*a*u*u: return 'near'
+    if x<1/(2*a*r*r): raise ValueError('absorption conclusion failed')
+    return 'far'
+
+
+def vandermonde(values):
+    values=list(map(rat,values))
+    if any(b<=a for a,b in zip(values,values[1:])):
+        raise ValueError('strictly ordered spectrum required')
+    return prod((values[j]-values[i] for i in range(len(values))
+                  for j in range(i+1,len(values))),start=F(1))
+
+
+def spectral_factor(r,s,hard):
+    r,s=map(rat,(r,s)); hard=list(map(rat,hard))
+    if r<=0: raise ValueError('positive radius required')
+    return r*vandermonde([-r*s]+hard)
+
+
+def limit_hard_factor(hard):
+    hard=list(map(rat,hard))
+    if any(x<=0 for x in hard): raise ValueError('positive hard spectrum required')
+    return prod((x**3 for x in hard),start=F(1))*vandermonde(hard)
+
+
+def typed_weight(k,s,a,beta):
+    k,s,a,beta=map(rat,(k,s,a,beta))
+    if k<=0: raise ValueError('positive gap required')
+    return max(F(0),-6*k*(s-beta/2)-a*a/4)*max(F(0),a*a/4-6*k*(s+beta/2))
+
+
+def nonempty_intensity(law,epsilon):
+    epsilon=rat(epsilon)
+    if epsilon<=0 or not isinstance(law,dict) or not law:
+        raise ValueError('positive scale and PMF required')
+    out={}
+    for n,p in law.items():
+        if type(n) is not int or n<0: raise ValueError('invalid count')
+        p=rat(p)
+        if p<0: raise ValueError('negative probability')
+        if n>0 and p>0: out[n]=p/epsilon
+    if sum(map(rat,law.values()))!=1: raise ValueError('PMF must sum to one')
+    return out
+
+
+def reduced_third(xxx,xxy_v,xyy_vv,yyy_vvv):
+    xxx,xxy_v,xyy_vv,yyy_vvv=map(rat,(xxx,xxy_v,xyy_vv,yyy_vvv))
+    return xxx+3*xxy_v+3*xyy_vv+yyy_vvv
+
+
+def majorant_degree(d):
+    if type(d) is not int or d<2: raise ValueError('integer d>=2 required')
+    return 4*d+(d-1)*(d-2)
+
+
+def invalid_cutoff_bound(r,determinant,eta):
+    r,determinant,eta=map(rat,(r,determinant,eta))
+    if not 0<determinant<eta or r<=0: raise ValueError('invalid cutoff probe')
+    return r/determinant**2<=r/eta**2
+
+
+def det(matrix):
+    """Exact determinant, with empty determinant equal to one."""
+    a=[list(map(rat,row)) for row in matrix]; n=len(a)
+    if any(len(row)!=n for row in a): raise ValueError('square matrix required')
+    value=F(1)
+    for i in range(n):
+        pivot=next((j for j in range(i,n) if a[j][i]),None)
+        if pivot is None: return F(0)
+        if pivot!=i: a[i],a[pivot]=a[pivot],a[i]; value=-value
+        p=a[i][i]; value*=p
+        for j in range(i+1,n):
+            t=a[j][i]/p
+            for k in range(i+1,n): a[j][k]-=t*a[i][k]
+    return value
+
+
+def require(ok,message):
+    if not ok: raise ValueError(message)
+
+
+def run_checks(mutant=None):
+    factor=spectral_factor(F(1,10),-1,[2,4])
+    if mutant=='drop-spectral-r': factor/=F(1,10)
+    if mutant=='drop-soft-vandermonde': factor=F(1,10)*vandermonde([2,4])
+    require(factor==F(741,500),'spectral r and soft Vandermonde')
+    factor=limit_hard_factor([2,4])
+    if mutant=='wrong-hard-power': factor=2**2*4**2*vandermonde([2,4])
+    require(factor==1024,'hard powers include weight squared and spectral factor')
+    branch=absorption(200,1,F(1,10),1)
+    if mutant=='drop-far-branch': branch='near'
+    require(branch=='far','exceptional absorption branch retained')
+    bound=invalid_cutoff_bound(F(1,10),F(1,100),F(1,10))
+    if mutant=='reverse-cutoff': bound=True
+    require(not bound,'inverse cutoff direction')
+    law={0:F(7,8),1:F(1,16),2:F(1,16)}
+    intensity=nonempty_intensity(law,F(1,8))
+    if mutant=='include-zero': intensity[0]=F(7)
+    require(intensity=={1:F(1,2),2:F(1,2)},'nonempty intensity excludes zero')
+    reduced=reduced_third(1,2,3,4)
+    if mutant=='drop-mixed-term': reduced-=3*F(2)
+    require(reduced==20,'all reduced third-derivative terms')
+    ledger=1+4-2
+    if mutant=='swap-full-normalizer': ledger=1+4-5
+    require(ledger==3,'physical jet / two soft columns / full normalizer')
+    cases=0
+    for a in (F(1),F(3)):
+        for u in (F(1),F(2),F(5)):
+            for r in (F(1,100),F(1,10),F(1,2)):
+                for x in map(F,(0,1,2,5,10,20,100,200,1000,20000)):
+                    if x<=a*u*u+a*r*r*x*x:
+                        branch=absorption(x,u,r,a)
+                        require(x<=2*a*u*u if branch=='near' else x>=1/(2*a*r*r),
+                                'absorption dichotomy grid')
+                        cases+=1
+    return {'passed':True,'scientific_effect':'NONE','mathematical_acceptance':False,
+            'scope':'finite exact controls only','absorption_cases':cases,
+            'groups':['spectral-jacobian','hard-weight','absorption','cutoff-direction',
+                      'nonempty-measure','vector-third-derivative','full-normalizer']}
+
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--mutant',choices=MUTANTS)
+    args=parser.parse_args()
+    try: result=run_checks(args.mutant)
+    except ValueError as exc:
+        print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True)); return 1
+    print(json.dumps(result,indent=2,sort_keys=True)); return 0
+
+
+if __name__=='__main__': raise SystemExit(main())
diff --git a/frontiers/spectral_cluster_closure_20260929/test_check.py b/frontiers/spectral_cluster_closure_20260929/test_check.py
new file mode 100644
index 000000000..33fdbff05
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/test_check.py
@@ -0,0 +1,136 @@
+import unittest
+from fractions import Fraction as F
+import check as c
+
+class InitialTests(unittest.TestCase):
+    def test_absorption_near(self):
+        self.assertEqual(c.absorption(1,1,F(1,10),1),'near')
+    def test_absorption_far(self):
+        self.assertEqual(c.absorption(200,1,F(1,10),1),'far')
+    def test_absorption_invalid(self):
+        with self.assertRaises(ValueError): c.absorption(5,1,F(1,10),1)
+    def test_vandermonde(self):
+        self.assertEqual(c.vandermonde([1,3,5]),16)
+    def test_spectral_factor(self):
+        self.assertEqual(c.spectral_factor(F(1,10),-1,[2,4]),F(741,500))
+    def test_empty_hard_factor(self):
+        self.assertEqual(c.spectral_factor(F(1,10),-1,[]),F(1,10))
+    def test_limit_hard_factor(self):
+        self.assertEqual(c.limit_hard_factor([2,4]),1024)
+    def test_weight(self):
+        self.assertEqual(c.typed_weight(1,F(-3,2),0,-2),45)
+    def test_zero_filtered(self):
+        self.assertEqual(c.nonempty_intensity({0:F(7,8),1:F(1,16),2:F(1,16)},F(1,8)),{1:F(1,2),2:F(1,2)})
+    def test_reduced_derivative(self):
+        self.assertEqual(c.reduced_third(1,2,3,4),20)
+    def test_majorant_degree(self):
+        self.assertEqual(c.majorant_degree(3),14)
+    def test_bad_cutoff(self):
+        self.assertFalse(c.invalid_cutoff_bound(F(1,10),F(1,100),F(1,10)))
+
+
+class AlgebraTests(unittest.TestCase):
+    def test_absorption_grid(self):
+        count=0
+        for a in (F(1),F(3)):
+            for u in (F(1),F(2),F(5)):
+                for r in (F(1,100),F(1,10),F(1,2)):
+                    for x in map(F,(0,1,2,5,10,20,100,200,1000,20000)):
+                        if x<=a*u*u+a*r*r*x*x:
+                            branch=c.absorption(x,u,r,a)
+                            self.assertTrue(x<=2*a*u*u if branch=='near' else x>=1/(2*a*r*r))
+                            count+=1
+        self.assertGreater(count,80)
+    def test_float_rejected(self):
+        with self.assertRaises(ValueError): c.absorption(1.0,1,F(1,10),1)
+    def test_bad_dimension(self):
+        with self.assertRaises(ValueError): c.majorant_degree(1)
+    def test_bool_dimension(self):
+        with self.assertRaises(ValueError): c.majorant_degree(True)
+    def test_unordered_spectrum(self):
+        with self.assertRaises(ValueError): c.spectral_factor(F(1,10),-30,[1,2])
+    def test_hard_zero_rejected(self):
+        with self.assertRaises(ValueError): c.limit_hard_factor([0,1])
+    def test_weight_boundary_zero(self):
+        self.assertEqual(c.typed_weight(1,-1,0,-2),0)
+    def test_weight_shear(self):
+        self.assertEqual(c.typed_weight(1,F(-3,2),6,1),45)
+    def test_bad_probability(self):
+        with self.assertRaises(ValueError): c.nonempty_intensity({0:F(1,2)},F(1,10))
+    def test_zero_mass_diverges(self):
+        for j in (10,100,1000):
+            e=F(1,j)
+            self.assertEqual((1-e)/e,j-1)
+            self.assertNotIn(0,c.nonempty_intensity({0:1-e,2:e},e))
+    def test_full_mixed_third_identity(self):
+        for g in (F(1),F(12),F(24)):
+            for vd in (F(2),F(5)):
+                for tensor in (F(3),F(7)):
+                    terms=(g-3*vd-tensor,vd+tensor,-tensor,tensor)
+                    self.assertEqual(c.reduced_third(*terms),g)
+                    for index in (1,2,3):
+                        changed=list(terms); changed[index]=F(0)
+                        self.assertNotEqual(c.reduced_third(*changed),g)
+    def test_determinant(self):
+        self.assertEqual(c.det([[2,1],[1,3]]),5)
+        self.assertEqual(c.det([]),1)
+    def test_weyl_local_jacobian_2(self):
+        for co,si in ((F(1),F(0)),(F(3,5),F(4,5)),(F(5,13),F(12,13))):
+            for l1,l2 in ((F(-2),F(3)),(F(1,3),F(7))):
+                gap=l2-l1
+                jac=[[co*co,si*si,2*co*si*gap],
+                     [co*si,-co*si,-(co*co-si*si)*gap],
+                     [si*si,co*co,-2*co*si*gap]]
+                self.assertEqual(abs(c.det(jac)),gap)
+    def test_two_soft_column_determinant(self):
+        # Exact d=3 Schur expansion, keeping the r^3 mixed correction.
+        for r in (F(1,100),F(1,10),F(1),F(2)):
+            matrix=[[-6*r,-r,2*r],[-r,-2*r,3*r],[2*r,3*r,-5]]
+            expected=-55*r*r+50*r**3
+            self.assertEqual(c.det(matrix),expected)
+    def test_orthogonal_norms(self):
+        for co,si in ((F(3,5),F(4,5)),(F(5,13),F(12,13))):
+            l1,l2=F(-2),F(7)
+            a=co*co*l1+si*si*l2; b=co*si*(l1-l2); d=si*si*l1+co*co*l2
+            self.assertEqual(a*a+2*b*b+d*d,l1*l1+l2*l2)
+    def test_unbounded_shear_counterexample(self):
+        # D=-1,v=t,A00=-t^2, raw T_www=1, transformed tau_zzz=t^3.
+        for t in (10,100,1000):
+            raw_norm_upper=F(t*t+2*t+2)
+            self.assertGreater(F(t**3)/raw_norm_upper,F(t,2))
+    def test_deep_constants(self):
+        for R in (F(1),F(3)):
+            for K in (F(1),F(12)):
+                for k in (F(1,10),F(1),F(3)):
+                    r=k/(4*R*K); lam=2*(8*R*K+56*R*K*K/k)*r
+                    rho=4*R*R*K*r*r/lam; slope=4*R*K*r/lam
+                    self.assertLess(rho,R*r/2)
+                    self.assertLess(slope,F(1,2))
+                    self.assertLess(28*R*K*K*r/lam,k/2)
+                    self.assertEqual(2*R*K*r,k/2)
+    def test_near_far_scaling(self):
+        self.assertEqual(F(3)+F(3,2),F(9,2))
+    def test_higher_factorials_vanish_in_limit(self):
+        from math import prod
+        for q in range(3,9):
+            for n in (1,2): self.assertEqual(prod(n-j for j in range(q)),0)
+
+class ExecutionTests(unittest.TestCase):
+    def test_both_modes_and_semantic_controls(self):
+        import subprocess,sys
+        from pathlib import Path
+        command=str(Path(c.__file__).resolve())
+        reference=None
+        for flags in (['-B','-S'],['-B','-O','-S']):
+            run=subprocess.run([sys.executable,*flags,command],capture_output=True,timeout=30)
+            self.assertEqual(run.returncode,0,run.stderr)
+            if reference is None: reference=run.stdout
+            self.assertEqual(run.stdout,reference)
+            for label in c.MUTANTS:
+                run=subprocess.run([sys.executable,*flags,command,'--mutant',label],capture_output=True,timeout=30)
+                self.assertEqual(run.returncode,1,(label,run.stdout,run.stderr))
+                self.assertEqual(run.stderr,b'')
+            run=subprocess.run([sys.executable,*flags,command,'--mutant','unknown'],capture_output=True,timeout=30)
+            self.assertEqual(run.returncode,2)
+
+if __name__=='__main__': unittest.main()
diff --git a/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py b/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py
new file mode 100644
index 000000000..5389a7400
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/test_derivative_weighted.py
@@ -0,0 +1,73 @@
+"""Exact finite ledgers for GD1. They do not prove conditional Gaussian bounds."""
+from fractions import Fraction as F
+from math import ceil, factorial
+import unittest
+
+
+class DerivativeWeightTests(unittest.TestCase):
+    def test_one_full_normalizer(self):
+        # Remote normalized moment: endpoint short columns, normalizer, height.
+        self.assertEqual(2-2+3,3)
+        self.assertNotEqual(2+3,3)
+        self.assertNotEqual(2-2-2+3,3)
+
+    def test_pin_and_collar_spatial_powers(self):
+        for d in range(2,10):
+            self.assertEqual((3-d)+d,3)
+
+    def test_exponential_absorption_budgets(self):
+        # These overestimate by retaining the older source's cap costs as well.
+        for d in range(2,9):
+            for p in range(10):
+                for alpha,cost in ((F(2),F(6+3*d+p)),
+                                   (F(2,3),F(15+39*d+10*p)),
+                                   (F(1,4),F(40*d+5+10*p))):
+                    n=ceil(cost/alpha)
+                    self.assertGreaterEqual(alpha*n,cost)
+                    self.assertLess(alpha*(n-1),cost)
+
+    def test_inverse_angle_budgets(self):
+        for d in range(2,9):
+            for p in range(10):
+                for cost in (16*d+10+4*p,22*d+15+6*p):
+                    n=ceil(F(cost,2))
+                    self.assertGreaterEqual(2*n,cost)
+                    self.assertLess(2*(n-1),cost)
+
+    def test_positive_exponential_series_term(self):
+        # exp(y)>=y^n/n!: retain the n-th term of a strictly positive polynomial.
+        for y in (F(1,2),F(1),F(3),F(10)):
+            for n in (1,2,5,9):
+                partial=sum((y**j/F(factorial(j)) for j in range(n+1)),F(0))
+                self.assertGreater(partial,y**n/F(factorial(n)))
+
+    def test_shell_cross_term(self):
+        for r in (F(1,100),F(1,20),F(1,10)):
+            for s in (4*r,F(1,2),F(1)):
+                if r>s/4: continue
+                value=(r+s*s)**2/(s*s)
+                self.assertEqual(value,r*r/(s*s)+2*r+s*s)
+                self.assertLessEqual(value,2*(r*r/(s*s)+s*s))
+
+    def test_full_strip_volume(self):
+        for d in range(2,10):
+            for s in (F(1,100),F(1,3),F(1,2),F(1)):
+                self.assertLessEqual(s**d,s*s)
+
+    def test_dyadic_sums(self):
+        for levels in range(1,13):
+            r=F(1,2**(levels+3))
+            scales=[4*r*2**j for j in range(levels)]
+            self.assertLess(sum((r/s)**2 for s in scales),F(1,12))
+            self.assertLess(sum(s*s for s in scales),F(4,3)*scales[-1]**2)
+
+    def test_count_weighted_tail_is_pointwise(self):
+        for count in (0,1,2,9):
+            for norm in (F(1),F(3),F(10)):
+                for threshold in (F(1),F(2),F(5)):
+                    for p in range(5):
+                        left=count if norm>threshold else 0
+                        self.assertLessEqual(left,count*norm**p/threshold**p)
+
+
+if __name__=='__main__': unittest.main()
diff --git a/frontiers/spectral_cluster_closure_20260929/test_verify.py b/frontiers/spectral_cluster_closure_20260929/test_verify.py
new file mode 100644
index 000000000..d813c34e9
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/test_verify.py
@@ -0,0 +1,80 @@
+"""Source-custody tests against a real isolated Git repository, not mocked Git."""
+import copy
+import hashlib
+import json
+from pathlib import Path
+import subprocess
+import tempfile
+import unittest
+import verify as v
+
+class CustodyTests(unittest.TestCase):
+    def setUp(self):
+        temp=tempfile.TemporaryDirectory(); self.addCleanup(temp.cleanup)
+        self.root=Path(temp.name); self.data=b'proof fixture\n'
+        (self.root/'nested').mkdir(); (self.root/'nested'/'source.md').write_bytes(self.data)
+        (self.root/'leaf.md').symlink_to('nested/source.md')
+        (self.root/'alias').symlink_to('nested',target_is_directory=True)
+        self.git('init','-q'); self.git('add','nested/source.md','leaf.md','alias')
+        self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid','commit','-qm','fixture')
+        self.entry={'id':'S','commit':self.git('rev-parse','HEAD').decode().strip(),
+                    'path':'nested/source.md','bytes':len(self.data),
+                    'sha256':hashlib.sha256(self.data).hexdigest(),
+                    'git_blob':self.git('rev-parse','HEAD:nested/source.md').decode().strip()}
+    def git(self,*args):
+        return subprocess.run(['git',*args],cwd=self.root,check=True,capture_output=True,timeout=15).stdout
+    def run_pins(self,entries):
+        p=self.root/'SOURCES.json'; p.write_text(json.dumps({'sources':entries}))
+        return v.verify_sources(self.root)
+    def test_valid_source(self):
+        self.assertEqual(self.run_pins([self.entry]),1)
+    def test_valid_source_from_nested_packet(self):
+        packet=self.root/'frontiers'/'packet'; packet.mkdir(parents=True)
+        (packet/'SOURCES.json').write_text(json.dumps({'sources':[self.entry]}))
+        try:
+            result=v.verify_sources(packet)
+        except (ValueError,subprocess.CalledProcessError):
+            result=None
+        self.assertEqual(result,1,'repo-root source identity must work from nested packet')
+    def test_wrong_size(self):
+        e=copy.deepcopy(self.entry); e['bytes']=0
+        with self.assertRaises(ValueError): self.run_pins([e])
+    def test_wrong_sha(self):
+        e=copy.deepcopy(self.entry); e['sha256']='0'*64
+        with self.assertRaises(ValueError): self.run_pins([e])
+    def test_wrong_blob(self):
+        e=copy.deepcopy(self.entry); e['git_blob']='0'*40
+        with self.assertRaises(ValueError): self.run_pins([e])
+    def test_missing_path(self):
+        e=copy.deepcopy(self.entry); e['path']='missing.md'
+        with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e])
+    def test_bad_commit(self):
+        e=copy.deepcopy(self.entry); e['commit']='0'*40
+        with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e])
+    def test_unsafe_paths(self):
+        for path in ('../source.md','/source.md','',':source.md','nested/../source.md','nested//source.md'):
+            e=copy.deepcopy(self.entry); e['path']=path
+            with self.subTest(path=path),self.assertRaises(ValueError): self.run_pins([e])
+    def test_duplicate_source_ids(self):
+        with self.assertRaises(ValueError): self.run_pins([self.entry,self.entry])
+    def test_missing_inventory(self):
+        with self.assertRaises(ValueError): self.run_pins([])
+    def test_git_leaf_symlink(self):
+        e=copy.deepcopy(self.entry); e['path']='leaf.md'
+        e['git_blob']=self.git('rev-parse','HEAD:leaf.md').decode().strip()
+        data=b'nested/source.md'; e['bytes']=len(data); e['sha256']=hashlib.sha256(data).hexdigest()
+        with self.assertRaises(ValueError): self.run_pins([e])
+    def test_git_parent_symlink(self):
+        e=copy.deepcopy(self.entry); e['path']='alias/source.md'
+        with self.assertRaises((ValueError,subprocess.CalledProcessError)): self.run_pins([e])
+    def test_duplicate_json(self):
+        p=self.root/'dup.json'; p.write_text('{"a":1,"a":2}')
+        with self.assertRaises(ValueError): v.read_json(p)
+    def test_packet_leaf_symlink(self):
+        packet=self.root/'packet'; packet.mkdir()
+        (packet/'file.md').symlink_to('../nested/source.md')
+        e={k:value for k,value in self.entry.items() if k in ('bytes','sha256','git_blob')}; e['path']='file.md'
+        (packet/'MANIFEST.json').write_text(json.dumps({'files':[e]}))
+        with self.assertRaises(ValueError): v.verify_inventory(packet)
+
+if __name__=='__main__': unittest.main()
diff --git a/frontiers/spectral_cluster_closure_20260929/verify.py b/frontiers/spectral_cluster_closure_20260929/verify.py
new file mode 100644
index 000000000..db4260655
--- /dev/null
+++ b/frontiers/spectral_cluster_closure_20260929/verify.py
@@ -0,0 +1,102 @@
+"""Fail-closed spectral packet replay. --local-only excludes project Git objects."""
+import argparse
+import hashlib
+import json
+from pathlib import Path, PurePosixPath
+import re
+import subprocess
+import sys
+
+
+def read_json(path):
+    def unique(pairs):
+        result={}
+        for key,value in pairs:
+            if key in result: raise ValueError('duplicate JSON key: '+key)
+            result[key]=value
+        return result
+    return json.loads(Path(path).read_text(encoding='utf-8'),object_pairs_hook=unique)
+
+
+def safe_path(path):
+    if not isinstance(path,str) or not path or '\\' in path or ':' in path or '\x00' in path:
+        raise ValueError('unsafe source path')
+    p=PurePosixPath(path)
+    if p.is_absolute() or '..' in p.parts or '.' in path.split('/') or str(p)!=path:
+        raise ValueError('unsafe source path')
+    return path
+
+
+def identity(data,entry):
+    blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest()
+    if (type(entry['bytes']) is not int or len(data)!=entry['bytes']
+        or hashlib.sha256(data).hexdigest()!=entry['sha256'] or blob!=entry['git_blob']):
+        raise ValueError('identity mismatch: '+entry['path'])
+
+
+def verify_sources(root):
+    entries=read_json(root/'SOURCES.json')['sources']
+    ids=[e['id'] for e in entries]
+    if not entries or len(ids)!=len(set(ids)):
+        raise ValueError('missing or duplicate source inventory')
+    for entry in entries:
+        commit,path=entry['commit'],safe_path(entry['path'])
+        if not re.fullmatch('[0-9a-f]{40}',commit): raise ValueError('invalid source commit')
+        # The pinned tree, not a followed worktree symlink, is the authority.
+        out=subprocess.run(['git','ls-tree','--full-tree','-z',commit,'--',path],cwd=root,
+                           check=True,capture_output=True,timeout=60).stdout
+        records=[x for x in out.split(b'\0') if x]
+        if len(records)!=1: raise ValueError('missing or ambiguous pinned path: '+path)
+        header,actual=records[0].split(b'\t',1)
+        mode,kind,blob=header.decode().split()
+        if actual.decode()!=path or mode!='100644' or kind!='blob':
+            raise ValueError('pinned source must be a regular nonexecutable blob: '+path)
+        if blob!=entry['git_blob']: raise ValueError('tree blob mismatch: '+path)
+        data=subprocess.run(['git','cat-file','blob',blob],cwd=root,check=True,
+                            capture_output=True,timeout=60).stdout
+        identity(data,entry)
+    return len(entries)
+
+
+def verify_inventory(root):
+    if (root/'MANIFEST.json').is_symlink(): raise ValueError('manifest symlink')
+    entries=read_json(root/'MANIFEST.json')['files']
+    names=[e['path'] for e in entries]
+    if len(names)!=len(set(names)): raise ValueError('duplicate packet path')
+    if sorted(p.name for p in root.iterdir())!=sorted(names+['MANIFEST.json']):
+        raise ValueError('packet inventory mismatch')
+    for entry in entries:
+        name=safe_path(entry['path']); path=root/name
+        if PurePosixPath(name).name!=name or path.is_symlink() or not path.is_file():
+            raise ValueError('flat regular packet file required')
+        identity(path.read_bytes(),entry)
+    return len(entries)
+
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--local-only',action='store_true')
+    args=parser.parse_args(); root=Path(__file__).resolve().parent
+    verify_inventory(root)
+    source_count=0 if args.local_only else verify_sources(root)
+    expected=(root/'RESULTS.json').read_bytes()
+    mutants=['drop-spectral-r','drop-soft-vandermonde','wrong-hard-power',
+             'drop-far-branch','reverse-cutoff','include-zero',
+             'drop-mixed-term','swap-full-normalizer']
+    for flags in (['-B','-S'],['-B','-O','-S']):
+        subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'],cwd=root,
+                       check=True,timeout=120)
+        cmd=[sys.executable,*flags,str(root/'check.py')]
+        out=subprocess.run(cmd,check=True,capture_output=True,timeout=30)
+        if out.stdout!=expected or out.stderr: raise ValueError('deterministic replay mismatch')
+        for mutant in mutants+['unknown']:
+            out=subprocess.run(cmd+['--mutant',mutant],capture_output=True,timeout=30)
+            if out.returncode!=(2 if mutant=='unknown' else 1):
+                raise ValueError('negative control not rejected: '+mutant)
+    verify_inventory(root)
+    print(json.dumps({'passed':True,'source_pins_checked':not args.local_only,
+                      'source_count':source_count,'scientific_effect':'NONE',
+                      'mathematical_acceptance':False},sort_keys=True))
+
+
+if __name__=='__main__': main()
diff --git a/frontiers/two_scale_cluster_geometry_20260929/MANIFEST.json b/frontiers/two_scale_cluster_geometry_20260929/MANIFEST.json
new file mode 100644
index 000000000..89b7780e9
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/MANIFEST.json
@@ -0,0 +1,79 @@
+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "review_status": "NONAUTHOR_ACCEPTED_SCOPED_CONDITIONAL_GEOMETRY",
+  "files": [
+    {
+      "path": "RADIAL_TAIL.md",
+      "bytes": 12533,
+      "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973",
+      "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4"
+    },
+    {
+      "path": "README.md",
+      "bytes": 3850,
+      "sha256": "4e2de660b46fa3c0107a06062de42ee581bade186daac35d666bbfa4a1f95aa3",
+      "git_blob": "8ee02224f4a5edeec0050fb54c3cc36f85c1070b"
+    },
+    {
+      "path": "RECONNAISSANCE.md",
+      "bytes": 2643,
+      "sha256": "b9af79b2ee56f151f947e233a684c6a91b0b306cd564dcefa722e43d3cb975e4",
+      "git_blob": "d6e523e5f92ae9ea710d6da54acb5bf4213a39c9"
+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 337,
+      "sha256": "3277199a22d6f187354e6f912361648e737bc8aa65107d1d7863a099c176ca8d",
+      "git_blob": "99678cd670b9f8fee2f02d73fdb091705b825491"
+    },
+    {
+      "path": "REVIEW_RECORD.md",
+      "bytes": 5121,
+      "sha256": "5295c373a16b6557584e1a633e28fd5962f3e96023945bcc068819e83a0d2cad",
+      "git_blob": "9f7c7f6fc59ead440258887eb113135ab1dc3463"
+    },
+    {
+      "path": "SOURCES.json",
+      "bytes": 2923,
+      "sha256": "de09c2cf3c9082a09d168f451f78a1a60b150f9f55c0c6b9bfe94ac26652dfe0",
+      "git_blob": "59d11113e208965fefeaadd1bd5866585e77dc82"
+    },
+    {
+      "path": "TWO_SCALE_LAW.md",
+      "bytes": 17139,
+      "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769",
+      "git_blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c"
+    },
+    {
+      "path": "VALIDATION.md",
+      "bytes": 3023,
+      "sha256": "7600037a03b43c6d20e50002cc0afed912cf6c5bc71e51a490959b3725cfaec0",
+      "git_blob": "33f00592e26a8f0a652c6554b6e3ddc389065425"
+    },
+    {
+      "path": "geometry.py",
+      "bytes": 7252,
+      "sha256": "e8582266faf3b4277412c00ea68751aeab0aa4e764ef831c56e34b8dbf68a7e9",
+      "git_blob": "f7fd5b69953d631b8cd9e1e92b3d25f1ce824f69"
+    },
+    {
+      "path": "test_custody.py",
+      "bytes": 7574,
+      "sha256": "f46073d3ff276cd0fdc571bf76e1cda8e0e4b20d93f3f7ab52a43d936258b325",
+      "git_blob": "d055261a2ff56d038fcf1ce803e0167c0e04d95b"
+    },
+    {
+      "path": "test_geometry.py",
+      "bytes": 5999,
+      "sha256": "ad87486bec154bf2ed9eb427c2527912579878ae7cd3462082c76bfd57acfa4d",
+      "git_blob": "a787f6e6bfe90385b51ce41d3709bf08af636d04"
+    },
+    {
+      "path": "verify.py",
+      "bytes": 6002,
+      "sha256": "335dd00fabb70db04498a84eef926266faa9f34486458724d3b0655d10b4f4ca",
+      "git_blob": "e401a78a1a28200f3f7e473b9260f36919ab69f8"
+    }
+  ]
+}
diff --git a/frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md b/frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md
new file mode 100644
index 000000000..0f14417ef
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md
@@ -0,0 +1,277 @@
+# An eleven-power microscopic radius tail and its universal extreme-height law
+
+Object: OA-TWO-SCALE-CLUSTER-GEOMETRY-20260929-R1.
+Author: OpenAI / GPT-6 Astra Pro, foreground recovery, 29 September 2026 (Central).
+Disposition: AUTHOR-SIDE CONDITIONAL PROOF; NONAUTHOR REVIEW REQUIRED.
+Scientific effect NONE. No scientific-status transition or numerical Gaussian
+coefficient is asserted. This is an analytic calculation for the explicit
+limiting MICROSCOPIC measure M of TWO_SCALE_LAW.md (M1), not for the entire
+finite-r global field or its remote singleton component.
+
+## 1. Which radius and which sampling law?
+
+Use the exact source conventions and fixed d,L,b,k,frame of TWO_SCALE_LAW.md.
+For each additional root (X_i,Z_i) of the unsheared planar cubic, its physical
+rescaled position is
+
+    v_i=X_i e_axis+Z_i e_soft(O),        R_i=|v_i|.
+
+Define the finite point-intensity measure by integrating the sum over these roots
+against M. Its tail mass is
+
+    F(t)=integral sum_i 1{R_i>t} dM.                         (R1)
+
+This is point-intensity sampling: a cluster with two qualifying points contributes
+two, not one. Every assertion below is a statement about the limiting microscopic
+measure. In an application to the Gaussian field, r->0 is taken before t->infinity.
+No finite-r uniform spatial moment or rate is inferred.
+
+**Theorem R.** There is an explicit finite positive C_star such that
+
+    F(t) ~ C_star t^-11.                                    (R2)
+
+For p>=0 the microscopic intensity moment integral sum_i R_i^p dM is finite
+exactly when p<11. The same threshold holds for the maximum radius of a nonempty
+cluster under its normalized limiting law. At the boundary, the capped intensity
+moment satisfies
+
+    integral sum_i min(R_i,t)^11 dM ~ 11 C_star log t.        (R3)
+
+A full formula for C_star is given in (R13). No numerical evaluation of its
+Gaussian integral is claimed.
+
+## 2. Resolve one stationary point by an exact change of variables
+
+Retain the canonical shear invariants of CUB:
+
+    u=X+aZ/(12k), B=beta-a^2/(12k),
+    D=(c-a beta/(4k)+a^3/(72k^2))/2, x=-s>0.
+
+At fixed a the inverse raw-jet map is
+
+    beta=a^2/(12k)+B,
+    c=a^3/(144k^2)+aB/(4k)+2D,
+    |d(beta,c)/d(B,D)|=2.                                (R4)
+
+The factor 2 is essential. This is a change in the JET coordinates, separate
+from the determinant-one spatial shear. The remaining third-jet entries, denoted
+xi, are unchanged.
+
+Let
+
+    b0(u)=3-12u^2,       v=xZ^2/k.
+
+For an extra root, the canonical conic and line give the inverse parametrization
+
+    x=kv/Z^2,
+    B=k b0(u)/Z^2,
+    D=k(v-b0(u)u)/Z^3,       Z!=0.                        (R5)
+
+Conversely, every parameter in the region defined below with Z!=0 gives a strict
+window root at (X,Z)=(u-aZ/(12k),Z). The normalized DOWNWARD height is
+
+    eta=-P/k=u+1/2+v/6.                                   (R6)
+
+Typed endpoints require v>|b0(u)|/2. The open height window requires 0<eta<1.
+Combining these gives precisely the bounded region
+
+    D_shape={ -3/2<u<1/2, |b0(u)|/2<v<3-6u }.              (R7)
+
+For completeness, the lower height inequality v>-6u-3 is automatic on R7:
+for -1/2<=u<1/2 its right side is nonpositive; for -3/2<u<-1/2, subtracting it
+from |b0|/2 gives 6(u+1/2)^2>=0. The upper height inequality and v>|b0|/2 are
+compatible exactly on the displayed u interval. Boundaries are excluded.
+
+An explicit determinant computation gives the ORIENTED Jacobian
+
+    det d(x,B,D)/d(v,u,Z)
+       =6k^3(12u^2+4uv-3)/Z^8
+       =-6k^3[b0(u)-4uv]/Z^8.                             (R8)
+
+On R7 both b0(u)-4uv and 4v^2-b0(u)^2 are strictly positive. The first is also
+the saddle determinant sign from CUB; it can be checked directly: for u<-1/2,
+use v>|b0|/2, and for u>=0 use v<3-6u, giving a lower bound
+12(u-1/2)^2, with the remaining interval -1/2<=u<0 immediate. The second follows
+from v>|b0|/2. Thus the Jacobian is nonzero in the open region.
+
+For each fixed (x,B,D) in the typed domain there are at most two extra window
+roots. The usual change of variables with multiplicity is legitimate here:
+cover R7 times (R minus {0}) by countably many neighborhoods where R5 is a
+diffeomorphism, make them disjoint measurably, apply the ordinary change of
+variables to each, and sum the resulting nonnegative integrals. The multiplicity
+is exactly the number of counted roots, rather than an omitted or extra factor
+of two. Degenerate/boundary jets are algebraically null and not in the strict
+window. This justifies applying R5 to a point-intensity sum.
+
+Define
+
+    Q(u,v)=[4v^2-b0(u)^2][b0(u)-4uv].                      (R9)
+
+The endpoint weight is w=9k^4[4v^2-b0(u)^2]/Z^4. Combining this with the absolute
+Jacobian R8 and the raw-jet Jacobian 2 of R4 gives exactly
+
+    w ds d beta dc =108 k^7 Q(u,v)|Z|^-12 du dv dZ,         (R10)
+
+in the point-counted integral, with a and all other spectral/tensor variables
+retained. This identity incorporates the root multiplicity through the map,
+not by an additional constant. The two signs of Z will be integrated next.
+
+## 3. Tail scaling, domination and the complete coefficient
+
+Let h,O,xi be the remaining variables of M, and let tau(a,beta,c,xi) denote the
+full rotated cubic tensor in those coordinates. Its raw contact Gaussian density
+remains evaluated at
+
+    h0(A0,Rot_O(tau)),      A0=O diag(0,-h)O^t.
+
+It is NOT replaced by an isotropic density. Combining M1 and R10 gives the exact
+representation of F(t) as the integral over R7 and Z!=0 of
+
+    (108 k^7 c_m/z0) Q(u,v) H(h) |Z|^-12
+    h0(A0,Rot_O tau(a, a^2/(12k)+k b0/Z^2,
+           a^3/(144k^2)+a b0/(4 Z^2)+2k(v-b0 u)/Z^3, xi))
+    *1{ (u-aZ/(12k))^2+Z^2 > t^2 },                     (R11)
+
+with measure du dv dZ da dxi dh dO, ordered positive h, and the empty-product
+conventions for d=2. The coefficient a b0/(4Z^2) in c includes the k cancellation
+in aB/(4k). Formula R11 is often where confusing the spatial and jet Jacobians,
+or integrating out a before testing the actual radius, gives a wrong exponent
+or coefficient.
+
+Put Z=t z and gamma(a)=sqrt(1+(a/(12k))^2). Then |Z|^-12 dZ=t^-11 |z|^-12 dz.
+At each fixed nonzero z the raw jet tends to the cusp slice
+
+    beta_cusp=a^2/(12k),         c_cusp=a^3/(144k^2),
+
+and the radius indicator tends almost everywhere to 1{gamma(a)|z|>1}.
+For t>=3 and |u|<=3/2, the indicator in R11 implies
+
+    |z|>1/[2gamma(a)]
+
+by the triangle inequality t<|u|+gamma|Z|. Orthogonal norm equivalence and SC's
+Gaussian density bound imply, uniformly in t,z,u,v,O,
+
+    h0(A0,Rot_O tau(a,beta,c,xi))
+                   <= C exp[-c(|h|^2+a^2+|xi|^2)].
+
+Dropping beta and c from this upper bound is legitimate; it never introduces
+an inverse hard eigenvalue. Integration in z over |z|>1/(2gamma) costs at most
+a constant times gamma^11. Q is bounded on the bounded region R7, H(h) is a
+nonnegative polynomial, and Gaussian integration in h,a,xi dominates gamma^11.
+Dominated convergence therefore applies to t^11 F(t).
+
+Let
+
+    J_cusp=integral H(h) gamma(a)^11
+       h0(A0,Rot_O tau(a,a^2/(12k),a^3/(144k^2),xi))
+                                      da dxi dh dO.       (R12)
+
+This is finite by the same majorant and positive by positivity of the Gaussian
+density and H on open sets. Integrating both signs of z gives
+integral_(gamma|z|>1)|z|^-12 dz=2 gamma^11/11. Finally,
+
+    I_shape=integral_Dshape Q(u,v) du dv=246528/35,
+
+    C_star=(216/11) k^7 (c_m/z0) I_shape J_cusp
+          =(53250048/385) k^7 (c_m/z0) J_cusp.             (R13)
+
+For an exact check of I_shape, integrate v first using
+
+    (4b0/3)v^3 -4u v^4 -b0^3 v +2u b0^2 v^2.
+
+The strips -3/2<u<-1/2 and -1/2<u<1/2 contribute, respectively,
+231744/35 and 2112/5. Their sum is 246528/35. These are exact rational polynomial
+integrals, not quadrature estimates. The factors in R13 are: 9 from the weight,
+6 from the stationary-point Jacobian, 2 from dc/dD, and 2/11 from both radius
+signs and the tail integral.
+
+This proves R2 in every fixed d>=2 under the source interface. The exponent 11
+is dimension-independent here because the hard directions affect only the
+finite integral J_cusp; the soft jet weights are s,B of degree -2, D of degree -3,
+and endpoint weight of degree -4, for total 2+2+3+4=11. This explanatory homogeneity
+is not substituted for the dominated-convergence proof.
+
+For p>0, Tonelli gives the moment identity integral R^p=p integral_0^infinity
+t^(p-1)F(t)dt. Finite total point mass handles t<=1 and R2 handles infinity.
+This proves the threshold in Theorem R and the boundary formula R3. If R_max
+is the maximum radius of a nonempty cluster, its unnormalized tail lies between
+F(t)/2 and F(t), since there are at most two points. Its normalized law therefore
+has the same exact moment threshold, although no exact maximum-tail coefficient
+is claimed. QED.
+
+## 4. A universal law for an intensity-selected far local point
+
+Normalize the point-intensity measure restricted to R_i>t. For that selected
+point retain its canonical shape variables (u,v), downward height eta, radial
+ratio q=R_i/t, sign epsilon=sign Z, and the remaining variables (a,h,O,xi).
+
+**Theorem H (extreme local marks).** As t->infinity, the joint law converges
+weakly to the product of:
+
+1. shape density Q(u,v)/I_shape on R7;
+2. a Pareto radial ratio q>=1 with density 11q^-12;
+3. a symmetric sign epsilon in {-1,1};
+4. the parameter measure with density H(h) gamma(a)^11 times the cusp Gaussian
+   in R12, divided by J_cusp.
+
+In particular eta=u+1/2+v/6 is independent of the limiting radial ratio and the
+remaining parameter measure. Its probability density on 0<e<1 is H_eta(e)/I_shape,
+where
+
+    H_eta(e)=(110592/7)[e^(7/2)-(1-e)^(7/2)+1]
+                      +13824[e^4-2e^3+5e^2-4e].          (H1)
+
+This density, unlike the amplitude C_star or orientation law, is independent of
+d,L,b,k and the fixed axial frame. It is a universality statement only for the
+intensity-selected LARGE-RADIUS tail of the limiting microscopic process.
+The full local height law and the global height law are not asserted universal.
+
+**Proof.** Apply the dominated-convergence argument of Section 3 with any bounded
+continuous function of the listed variables. In the limit use the radial change
+q=gamma(a)|z|. The measure |z|^-12 dz becomes gamma(a)^11 q^-12 dq on each sign,
+and the threshold is q>1. The limiting Gaussian depends only on (a,h,O,xi), while
+Q depends only on (u,v). Division by R13 gives exactly the four factors above.
+No claim of spatial independence in the original field is involved.
+
+To obtain H1, substitute v=6e-6u-3, with dv=6de. Conditions R7 become
+
+    -1/2-sqrt(e) < u < 1/2-sqrt(1-e),    0<e<1.
+
+Thus an explicitly nonnegative integral form of H_eta is
+
+    H_eta(e)=6 integral_(-1/2-sqrt(e))^(1/2-sqrt(1-e))
+                                  Q(u,6e-6u-3) du.       (H2)
+
+Expanding this polynomial integral yields H1. This also proves its positivity
+on (0,1), despite cancellations in the closed form. Its mass and first two
+moments, computed exactly either from H1 or directly on R7, are
+
+    integral_0^1 H_eta(e)de=I_shape,
+    E eta=5771/7062,
+    E eta^2=2440/3531.                                   (H3)
+
+For integer j>=0 a reproducible rational formula before dividing by I_shape is
+
+    integral e^j H_eta(e) de
+      =(110592/7)[1/(j+9/2)-B(j+1,9/2)+1/(j+1)]
+          +13824[1/(j+5)-2/(j+4)+5/(j+3)-4/(j+2)],        (H4)
+
+with B(j+1,9/2)=j!/product_(l=0..j)(9/2+l). No special-function numeric library
+is needed to check these moments. Seven such moments are checked independently
+against exact double polynomial integration on R7. QED.
+
+## 5. Evidence boundary and useful falsifiers
+
+The finite companion independently checks the oriented root Jacobian, the inverse
+raw-jet map and its factor 2, stationary heights and gradients at 648 rational
+parameter choices, the two exact shape integrals, H4 versus direct integration,
+and the k^7/radius-power ledgers. Eight explicit false variants must fail.
+These controls do not prove the Gaussian bound, change-of-variables multiplicity,
+weak convergence or domination; those are the written analytic obligations above.
+
+The result would fail if the contact slice lost full positive Gaussian density,
+if the physical radius were incorrectly replaced by |Z| before retaining a,
+or if the invariant density were assumed isotropic rather than kept under Haar.
+A finite-r spatial-moment convergence claim, a global tail including remote
+singletons, an ordinary configuration-selected version of H1, or a numerical
+Gaussian value of C_star would require new work and is not supplied here.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/README.md b/frontiers/two_scale_cluster_geometry_20260929/README.md
new file mode 100644
index 000000000..8ee02224f
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/README.md
@@ -0,0 +1,67 @@
+# Two-scale cluster geometry and an exact eleven-power tail
+
+OpenAI / GPT-6 Astra Pro. Author-side conditional mathematical packet, nonauthor
+review COMPLETE at the stated conditional scope: Claude5360227991 (A),
+5360178611 (B), and5360216551 (C). See REVIEW_RECORD.md. Scientific effect NONE;
+no source proof or global status is changed.
+No self-merge or organizational-independence claim.
+
+TWO_SCALE_LAW.md is a conditional consumer of the exact spectral proof in
+Math-#162. It upgrades the count interface to a weak law of whole two-scale
+configurations: microscopic roots are scaled by 1/r, remote singletons keep
+physical torus positions. It proves localization of all leading ordered pair
+mass, describes remote height marks, and shows why spatial total variation of
+double clusters is impossible in d>=3 even though weak convergence holds.
+
+RADIAL_TAIL.md computes the tail of the resulting limiting MICROSCOPIC point
+intensity. The tail is C_star t^-11 with a finite positive Gaussian coefficient;
+its geometric factor is exactly 53250048/385 times k^7. The radial moment threshold
+is p<11. An intensity-selected point far out in this limiting local law has an
+independent Pareto(11) radial ratio and a universal downward-height density, with
+mean 5771/7062. This is NOT a finite-r uniform tail or a global law including
+remote singletons; the order is r->0 first, radius->infinity second.
+
+The exact scalar classifier of #158 is consumed only at its algebraic scope.
+The #159 competitor is not a premise, and the #157 exponential-moment theorem
+is not consumed. Source-condition failures block the corresponding applications.
+A green check does not decide any mathematical review.
+
+## Replay
+
+    python -B -S verify.py
+
+Run inside a full Git checkout with the immutable commits in SOURCES.json
+available. This verifies all six named source objects (including ancestors'
+Git modes), the flat packet inventory, 52 unit tests in each Python mode,
+deterministic output, eight semantic negative controls and unknown-label refusal.
+The workflow fetches the two non-base source commits explicitly.
+
+    python -B -S verify.py --local-only
+
+This explicitly excludes project source-object authentication and reports
+source_pins_checked=false. Local isolated-Git tests still exercise the checker;
+they do not establish the actual project source identities. The new arguments
+are not Lean-formalized; the repository's existing formal gate is separate.
+
+Exact controls include 648 rational stationary-point substitutions, the signed
+three-variable Jacobian, raw-jet inverse, two shape-strip integrals, seven height
+moments computed in two independent ways, localization inequalities and source
+custody regressions. No Gaussian coefficient is numerically estimated.
+
+## Completed scoped reviews
+
+A. Two-scale configuration convergence, arbitrary cutoff, remote marks, counted
+pair localization and the d>=3 spatial-TV obstruction.
+B. Root change of variables with multiplicity, all Jacobian factors, Gaussian
+majorant with the shear parameter retained, tail coefficient and moment boundary.
+C. Extreme intensity-selected mark law, universal height density and exact
+rational companion. All three reviews bind the unchanged mathematical files
+at7c82252533c3fe14ee262f7f87c0549c10968592. One Claude session reviewed these
+three disjoint scopes; this is not three independent-provider votes. Current
+head/base verification and nonauthor integration remain separate requirements.
+
+The custody successor disables Git replacement objects and requires the declared
+source identity to be a commit object, not merely a forty-hex tree-ish name. Four
+real-Git red/green regressions cover forged/legitimate replacement bindings and
+tree/tag substitution. Both mathematical manuscripts and all mathematical inputs
+remain byte-identical to the initial reviewed head.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md b/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md
new file mode 100644
index 000000000..d6e523e5f
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/RECONNAISSANCE.md
@@ -0,0 +1,45 @@
+# External reconnaissance — foreground recovery, 29 September 2026 Central
+
+Scope: primary-source context for Gaussian critical-point geometry, not an
+exhaustive novelty, priority or full-text literature audit. No external theorem
+is imported as an unverified substitute for the source-bound arguments.
+
+Consensus.search was actually called for Gaussian random field persistence,
+rare critical clusters, spatial scaling, cubic saddles and marked processes.
+It returned the monthly 30-search quota error (reset October 1), not papers.
+
+Primary arXiv abstract/metadata pages actually opened:
+
+1. Diego Armentano, Jean-Marc Azais and Jose Rafael Leon, On a general Kac-Rice
+formula for the measure of a level set, arXiv:2304.07424v3.
+https://arxiv.org/abs/2304.07424v3
+The page distinguishes v3 (5 December 2023) from later v4 (22 April 2025). The
+in-project sources cite v3; this memo does not silently replace that version.
+Only metadata and abstract were read here, not an independent framework audit.
+
+2. Dmitry Beliaev, Valentina Cammarota and Igor Wigman, No repulsion between
+critical points for planar Gaussian random fields, arXiv:1911.03455v3
+(27 November 2020). https://arxiv.org/abs/1911.03455
+The abstract concerns near-diagonal two-point correlations for isotropic
+stationary fields and index-sensitive behavior. It is relevant context, not the
+same determinant-tilted, coalescing-pinned law or the new limiting radial tail.
+No priority claim follows from that distinction.
+
+3. Job Feldbrugge and coauthors, Stochastic Homology of Gaussian vs. non-Gaussian
+Random Fields: Graphs towards Betti Numbers and Persistence Diagrams,
+arXiv:1908.01619 (2019). https://arxiv.org/abs/1908.01619
+The abstract supplies random-field topology and persistence context. This pass
+does not claim to have searched every occurrence of a cubic normal form in its
+full text or the wider literature.
+
+The new calculation gives its stationary-root change of variables, integrable
+Gaussian majorant, shape integrals and marked convergence arguments in full.
+The sources in SOURCES.json keep their own review conditions. The generic
+measure-theoretic tools are not claimed novel, and the exact radius/height
+result is an author-side candidate pending separate nonauthor review.
+
+For the later custody repair, the official Git documentation was searched and
+read at https://git-scm.com/docs/git and the primary git-replace manual at
+https://www.kernel.org/pub/software/scm/git/docs/git-replace.html. They document
+replacement refs and the --no-replace-objects override. This external check
+supports the engineering repair only, not any mathematical claim.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json b/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json
new file mode 100644
index 000000000..99678cd67
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/RESULTS.json
@@ -0,0 +1,11 @@
+{
+  "mathematical_acceptance": false,
+  "passed": true,
+  "radial_factor_at_k1": "53250048/385",
+  "scientific_effect": "NONE",
+  "scope": "exact finite identities, not Gaussian or limiting-law verification",
+  "shape_integral": "246528/35",
+  "stationary_map_cases": 648,
+  "tail_exponent": 11,
+  "universal_height_mean": "5771/7062"
+}
diff --git a/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md
new file mode 100644
index 000000000..9f7c7f6fc
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/REVIEW_RECORD.md
@@ -0,0 +1,92 @@
+# Scoped review and custody repair record
+
+Scientific effect NONE. This is a record of actual reviews, not a new scientific
+status register or a self-issued acceptance. Same GitHub login gives no
+organizational-independence credit.
+
+## Completed analytic Slice B
+
+Anthropic Claude review5360178611, submitted2026-09-30 00:42:15Z:
+https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360178611
+
+Binding: head7c82252533c3fe14ee262f7f87c0549c10968592, RADIAL_TAIL.md
+blob0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4, SHA256
+250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973.
+
+ACCEPT scope: R4-R13, the tail F(t)~C_star*t^-11 of the already-formed limiting
+microscopic point intensity, the complete coefficient, p<11 radial-moment
+threshold and capped-eleventh-moment asymptotic, at fixed d,L,b,k,frame and
+conditional on the exact SC/CUB interfaces. The reviewer did not author the
+pinned sources; prior reviewer exposure to SC is disclosed. No source theorem
+is independently revalidated by this consumer review.
+
+Four nonblocking clarifications are adopted without changing the proof bytes:
+R8's sign is chart orientation and R10 uses its absolute value; root multiplicity
+uses CUB Theorem C/C6; the density majorant is SC(10); dimension dependence stays
+in c_m/z0,H,h0 and J_cusp, not the shape integral/exponent.
+
+At the initial B-only record, A and C remained open; the courtesy height check
+was not counted as a verdict. The separate actual A/C reviews below now complete
+the requested mathematical scope, without changing either manuscript.
+
+## Completed analytic Slice C
+
+Anthropic Claude review5360216551, submitted2026-09-30 00:49:42Z:
+https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360216551
+
+Binding: the same mathematical head and RADIAL_TAIL.md blob as B. ACCEPT of
+Theorem H and H1-H4: the four-factor limit under point-intensity sampling, the
+universal downward-height density, its exact normalization and moments. This
+uses the already formed microscopic measure and then radius->infinity, not a
+finite-r global distribution or uniformly sampled configuration. The parameter
+and orientation factors need not be universal. Courtesy algebra checks in B
+were not substituted for this full C review.
+
+Its two nonblocking notes are reading clarifications: the bounded test uses
+q=R_i/t at finite t before passing to gamma(a)|z|, and the H4 moment index is a
+nonnegative integer with the second bracket coming from the polynomial density.
+
+## Completed analytic Slice A
+
+Anthropic Claude review5360227991, submitted2026-09-30 00:52:00Z:
+https://github.com/d6g8k5htny-coder/Math-/pull/166#pullrequestreview-5360227991
+
+Binding: head7c82252533c3fe14ee262f7f87c0549c10968592, TWO_SCALE_LAW.md
+bloba32fd5f7d941bbe1fe943df045b1e0fbec8d691c, SHA256
+e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769.
+
+ACCEPT of Theorem T, Theorem L, ordered-pair law L3, the sampling/order-of-limits
+statements and spatial-TV obstruction V1, conditional on the exact SC/CUB/RM/C6/P
+interfaces. The review checks entire-configuration functionals, both spatial
+scales, arbitrary admissible cutoff, count-weighted tail control, uniform remote
+height marks and the absolutely-continuous versus collinear-support argument.
+It does not revalidate the consumed Gaussian parent proofs or supply a rate.
+
+Three nonblocking notes are retained as reading clarifications: near configuration
+convergence holds for almost every (theta,F), with hard coordinates O(r^2)/r->0;
+the configuration space is Polish with locally constant cardinality; separated
+spatial exhaustion suffices for V1 since pins and coincidence diagonal are null,
+and the latter contains no ordered distinct pair.
+
+## Combined mathematical disposition
+
+All A/B/C requests now have actual nonauthor ACCEPT at their stated conditional
+scope. The same provider-distinct Claude session supplied all three; this is
+not three independent votes or organizational independence. Both manuscripts
+are unchanged throughout custody and metadata successors. No review merges the
+packet, promotes a global register, removes parent conditions, evaluates a
+numerical Gaussian coefficient or establishes finite-r spatial-moment rates.
+
+## Custody repair, separate from mathematical acceptance
+
+Author-side audit reproduced replacement-ref and tree/tag-substitution defects
+with four failing real-Git regression assertions. Source lookup now explicitly
+disables Git replacement objects and requires actual commit type. All52 tests
+pass in each local Python mode; old tests and semantic mutants remain. The
+changed verifier received Codex engineering comment5901897673 at repair commit
+80059a97cd8eb32d1a17a0ad9cb22cb2a0d7b337: no major issues found. That comment
+is not a mathematical verdict or an exhaustive security guarantee. Targeted
+source run36652196578 and full downstream/formal run36652196938 both succeeded
+at the repair commit. A later metadata successor still needs fresh current-head
+checks; no old-head run is automatically rebound. The mathematical files,
+geometry.py, source pins and RESULTS.json have not changed.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json b/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json
new file mode 100644
index 000000000..59d11113e
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/SOURCES.json
@@ -0,0 +1,62 @@
+{
+  "schema": 1,
+  "repository": "d6g8k5htny-coder/Math-",
+  "scientific_effect": "NONE",
+  "condition": "Source identity is not a new review or an automatic status transition. All applications retain the pinned source hypotheses.",
+  "sources": [
+    {
+      "id": "SC",
+      "commit": "7a44130b41b827577424a73c4feac4cde20b5568",
+      "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md",
+      "bytes": 25006,
+      "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb",
+      "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a",
+      "use": "Sections4-8: actual spectral/root limit, domination, radius and mixed-event estimates, full count limit. Conditional input; not revalidated."
+    },
+    {
+      "id": "CUB",
+      "commit": "bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8",
+      "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md",
+      "bytes": 19889,
+      "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0",
+      "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758",
+      "use": "Theorem C and exact shear/height/type identities only. Separate Gaussian sector theorem not consumed."
+    },
+    {
+      "id": "RM",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/remote_window_20260924/PROOF.md",
+      "bytes": 18355,
+      "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7",
+      "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84",
+      "use": "Sections2-5, complete height-disintegrated integrand12 and normalized kernel13; finite-r spatial Kac-Rice framework."
+    },
+    {
+      "id": "C6",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/c6_palm_route_20260929/PROOF.md",
+      "bytes": 53364,
+      "sha256": "aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b",
+      "git_blob": "89eb8adf08fe7afc2cab9662d3a9875c05ae5cc5",
+      "use": "Theorem Q for polynomial count uniform integrability, not a spatial moment bound."
+    },
+    {
+      "id": "P",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "use": "Section2 distinct-site jet rank for finite-r spatial absolute continuity; original model conventions."
+    },
+    {
+      "id": "D5",
+      "commit": "7a1cb09a9d58d179393e6d29146252f714b9c9bf",
+      "path": "frontiers/d5_dimension_lift_20260929/PROOF.md",
+      "bytes": 53727,
+      "sha256": "6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80",
+      "git_blob": "9d82c707fdb17d3072a8930f26dabedf59e456fc",
+      "use": "Global first moment and uniform R-dependent shell interface, as composed in SC."
+    }
+  ]
+}
diff --git a/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md b/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md
new file mode 100644
index 000000000..a32fd5f7d
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md
@@ -0,0 +1,331 @@
+# Two-scale marked clusters and localization of all leading pair mass
+
+Object: OA-TWO-SCALE-CLUSTER-GEOMETRY-20260929-v1.
+Author: OpenAI / GPT-6 Astra Pro, foreground recovery, 29 September 2026 (Central).
+Disposition: AUTHOR-SIDE CONDITIONAL PROOF; NONAUTHOR REVIEW REQUIRED.
+Scientific effect NONE. No source proof, prior verdict, global register, Boolean,
+prize or integration disposition is changed. A shared GitHub account is not
+organizational independence.
+
+## 1. Exact interface, not a silent revalidation
+
+Fix d>=2, L>0, k>0, b in R, and one orthonormal axial/transverse frame. Use the
+original periodized Gaussian field, pins M=(-r/2,0), S=(r/2,0), values b,b-kr^3,
+zero gradients, determinant weight W_r and FULL normalizer Z_r of [SC]. Write
+P_r=Q_r^W. N_r counts all additional critical points in the open height window,
+excluding M,S. No observation set or normalizer is replaced.
+
+All source identities are in SOURCES.json. This paper is a new conditional
+consumer of [SC], not a review of it. In particular:
+
+[SC] frontiers/spectral_cluster_closure_20260929/PROOF.md, exact commit
+7a44130b41b827577424a73c4feac4cde20b5568, blob
+16c56821b52fd76b0be791622b9c3809eafde75a.
+
+[CUB] frontiers/planar_cubic_cluster_20260929/PROOF.md, exact commit
+bfcc67dc8957b3ed1fbb789e5a0305e98d7359c8, blob
+bb446d08db8a944537a743ad550b88c1c2ad5758. Only its deterministic cubic classifier
+is consumed; not its separate Gaussian compact-sector theorem.
+
+[RM] frontiers/remote_window_20260924/PROOF.md, Sections 2-5: the complete
+height-disintegrated integrand in (12) is Lambda_j(x)+O_rho(r), uniformly at
+fixed remote separation rho and throughout the window.
+
+[C6] frontiers/c6_palm_route_20260929/PROOF.md, Theorem Q, together with [SC]'s
+first moment: every fixed ordinary count moment is O(r^3).
+
+[P] imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md,
+Section 2: full distinct-site jet rank. Used only for the finite-r spatial
+absolute-continuity statement in Section 6.
+
+RM, C6 and P are pinned at source cut 7a1cb09a9d58d179393e6d29146252f714b9c9bf.
+A defect or unfulfilled condition in an imported result blocks the corresponding
+application here. Publication, tests and a status string do not remove that
+condition. No theorem from the competing #159 proof or the #157 Fourier packet
+is imported.
+
+The following exact portions of SC are needed, not just its final count law:
+
+* (14)-(18): the integrable full-field/spectral near-occurrence majorant,
+  negligible exceptional branch, and pointwise convergence of the individual
+  roots, heights and indices in each fixed rescaled ball.
+* (19)-(20): a finite spectral measure on nonempty cubic configurations, with
+  positive masses a1,a2 for one and two points.
+* (21)-(25): beta_far=k integral Lambda is finite and positive, remote multiple
+  probability O_rho(r^5), mixed near/remote probability o_(R,rho)(r^3), and
+  middle-window mean <=C r^3(R^-2+rho^2), with the last constant independent
+  of R>=4 and small rho.
+* (3), or its C6 tail argument: polynomially weighted count convergence and
+  E_r N_r^p=O_p(r^3) for every fixed positive integer p.
+
+Let m=d-1. In SC's coordinates theta=(s,h,O,tau), ordered hard eigenvalues are
+0<h2<...<hm, dO is NORMALIZED Haar on O(m), and tau is the full third-derivative
+tensor except f_xxx, rotated with the same orthogonal frame. Denote its soft
+entries by a=tau_xxz, beta=tau_xzz, c=tau_zzz. Set
+
+    A0=O diag(0,-h2,...,-hm) O^t,
+    H(h)=product_(j=2..m) h_j^3 product_(2<=i<j<=m)(h_j-h_i).
+
+With c_m fixed by SC (8), z0 the full normalizer limit, and h0 the RAW contact
+jet density, retain the measure
+
+    dM(theta)=(c_m/z0) h0(A0,Rot_O(tau)) H(h)
+                     w(s,a,beta) ds dh dO dtau.                 (M1)
+
+It is extended by zero off the typed domain s<-|B|/2, where
+B=beta-a^2/(12k) and w=9k^2(4s^2-B^2). Empty hard products are one for d=2.
+The measure M on the whole typed domain need not be finite. Its restriction
+to NONEMPTY cubic configurations is finite; that is the measure used below.
+
+The original, unsheared planar cubic is
+
+    P_theta(X,Z)=2kX^3-3kX/2-k/2+sZ^2/2
+       +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3.                (M2)
+
+Its additional window roots form C(theta), with n(theta) in {0,1,2}. Each has
+index d-1 in the full field limit. In the fixed original frame its limiting
+position is X e_axis + Z e_soft(O), not the sheared coordinate u. This distinction
+is essential for the radius calculation in RADIAL_TAIL.md.
+
+## 2. A configuration space that retains both spatial scales
+
+Choose ANY deterministic positive cutoff delta_r satisfying
+
+    delta_r -> 0,           delta_r/r -> infinity.              (T1)
+
+For sufficiently small r the delta_r ball is embedded. For each additional
+window critical point x put eta=(b-f(x))/(k r^3) in (0,1), and retain its index j.
+Use the following two tags:
+
+    micro:  dist(x,0)<delta_r, with mark (x/r,eta,j);
+    remote: dist(x,0)>=delta_r, with mark (x,eta,j).             (T2)
+
+The micro position uses the unique local lift, expressed in the fixed original
+frame. Remote positions remain actual torus points; they are not divided by r.
+The state space Y is the disjoint topological union
+
+    Y_micro=R^d x (0,1) x {0,...,d},
+    Y_remote=(X minus {0}) x (0,1) x {0,...,d}.
+
+Let Conf_+(Y) be the disjoint union over n>=1 of the finite symmetric products
+Y^n/S_n, with the quotient product topology. Multiplicities are allowed in this
+ambient space; the particular laws below are simple almost surely. Cardinality
+is a continuous, discrete-valued function. This space retains configurations,
+not merely their one-point intensity.
+
+Let Xi_r be the tagged configuration (T2). Define the finite nonempty measure
+
+    L_r(G)=r^-3 E_r[G(Xi_r);N_r>0].                           (T3)
+
+For a spectral jet theta, define the micro-only configuration
+
+    C_micro(theta)=sum_(X,Z in C(theta))
+        delta_(micro, X e_axis+Z e_soft(O), -P_theta(X,Z)/k,d-1).
+
+Let L_micro be the pushforward of 1{n>=1}dM under this map. Let L_remote be the
+singleton-configuration measure induced by
+
+    k sum_(j=0..d) Lambda_j(x) dx d eta,
+         x in X minus {0},   0<eta<1.                       (T4)
+
+Lambda_j are RM's normalized contact kernels, and beta_far is the mass of T4.
+There is no product factorization of the Gaussian endpoint and remote Hessians
+inside Lambda_j.
+
+**Theorem T (two-scale marked law).** Conditional on the exact inputs above,
+for every cutoff satisfying (T1),
+
+    L_r -> L=L_micro+L_remote weakly as finite measures       (T5)
+
+on Conf_+(Y). The convergence also holds when testing G times any fixed
+nonnegative integer power of the total count, for bounded continuous G.
+The mass of L is a1+a2+beta_far=nu1+nu2. In particular, the probability law
+of Xi_r conditioned on N_r>0 converges weakly to L/(nu1+nu2).
+
+This is weak convergence of spatially marked configurations. It is not stated
+as spatial total-variation convergence; Section 6 shows that such a statement
+would be false in general. Count and tag-count marginals, which have a discrete
+state space, do converge in total variation.
+
+### Proof: fixed-radius limits, then the regional sandwich
+
+Fix R>=4 and a sufficiently small rho>0. For small r, Rr<delta_r<rho.
+Write N=N_R+N_mid+N_far using a disjoint near ball Rr, middle annulus and remote
+complement of rho. Boundary spheres have zero expected count for each fixed r
+by the spatial Kac-Rice formula, so choices of open/closed boundaries do not
+matter. Form the bad event
+
+    E_bad={N_mid>0} union {N_R>0,N_far>0}
+                    union {N_far>=2} union {N_R>=3}.
+
+SC's exact bounds imply
+
+    limsup_(r->0) r^-3 P_r(E_bad) <= C(R^-2+rho^2).         (T6)
+
+Only the middle term survives this limsup. The mixed bound here is the full
+o(r^3) result, not a uniform extension of the compact-jet O(r^(9/2)) estimate.
+
+For fixed R, SC's pointwise hard-direction slaving gives convergence of the
+individual micro roots, their normalized heights and their indices for almost
+every spectral jet and field coupling realization. Height and observation-circle
+exceptional sets are null. Its integrable near-occurrence majorant (16), with the
+exceptional branch handled by (14), therefore proves convergence of the entire
+bounded continuous CONFIGURATION functional G, not only sums of one-point
+functions, to the pushforward of 1{n_R>=1}dM. Here n_R counts cubic roots in the
+R ball. All angular variables remain under the integral. No independent root
+labelling is needed: local labellings from root stability induce convergence in
+the symmetric-product topology.
+
+For the remote component, change height variable t to eta=(b-t)/(kr^3) in
+RM (12). Its complete integrand differs uniformly from Lambda_j(x) by O_rho(r).
+The rescaled first-moment measure on D_rho x (0,1) x indices thus converges in
+full variation to k Lambda_j(x) dx d eta. On N_far>=2, the mean number of remote
+points is at most E_r(N_far)_2=O_rho(r^5). Subtracting this multiple-point
+intensity shows that the SINGLETON-configuration measure has the same limit.
+This proves T4 first on D_rho. In particular the limiting eta is uniform on
+(0,1), independent of the remote location and index under the normalized remote
+component; this is a marked consequence of the uniform height disintegration,
+not a field-independence assertion.
+
+On the complement of E_bad the global configuration is precisely either the
+near configuration or a remote singleton, but never both. Testing a bounded G
+and comparing these three finite measures gives an error bounded by a fixed
+multiple of ||G||_infinity P_r(E_bad). Hence T6 bounds the scaled error.
+
+For almost every fixed theta, every cubic root is finite and n_R=n eventually
+as R->infinity. The measure 1{n>=1}M is finite by SC (19), so dominated convergence
+of bounded configuration functionals gives the full micro pushforward. Meanwhile
+SC (21) bounds the missing remote mass by C rho^2. Let r->0 first at fixed R,rho,
+then R->infinity and rho->0. These steps prove T5 for every bounded continuous G,
+including G=1. They apply to any delta_r obeying T1 because only the inequalities
+Rr<delta_r<rho were used.
+
+For a count power q, first truncate to N<=M. The truncation is a continuous
+function on the disjoint-cardinality space. C6 and the first moment give
+
+    r^-3 E_r[N^q;N>M] <= C_q/M,                            (T7)
+
+using the (q+1)-st ordinary moment, including q=0. The limiting count is at most
+two. Let M increase after the finite truncation limit. This proves the weighted
+claim. Total mass tends to the positive value nu1+nu2, so normalization is valid.
+Discrete tag-count convergence follows from the coordinate probabilities and
+the vanishing scaled mass outside the three possible limit sectors. QED.
+
+## 3. What singleton and double-cluster sampling mean
+
+Conditional on N_r=1, Theorem T gives a mixture of a micro singleton, with weight
+
+    a1/(a1+beta_far),
+
+and a remote singleton, with weight beta_far/(a1+beta_far). The first is the
+pushforward of 1{n=1}dM/a1; the second has density T4/beta_far. Conditional on
+N_r=2, it gives the micro pair pushforward of 1{n=2}dM/a2, and both points have
+index d-1. Positivity of a1,a2,beta_far was established by SC and remains an input.
+
+A law obtained by sampling a nonempty configuration is not the same as a law
+obtained by selecting one point from the intensity measure. The latter weights
+clusters by their sizes. RADIAL_TAIL.md uses explicitly the second convention.
+Likewise a first-moment formula of the form E sum phi(point) does not, by itself,
+identify the full configuration law; T5 uses bounded continuous functionals of
+ALL points in each configuration.
+
+Plainly scaling ALL global positions by 1/r and seeking a finite Euclidean
+configuration limit would lose a positive component. For fixed rho>0, the
+remote singleton event has scaled probability tending to k integral_Drho Lambda,
+while its rescaled distance is at least rho/r. Dividing by P_r(N>0)~(nu1+nu2)r^3
+shows that a fixed positive fraction of the conditional mass escapes every
+compact spatial set. The appropriate remedy is the two-scale space T2, not
+silently deleting the remote contribution.
+
+## 4. All leading ordered pair mass is microscopic
+
+Let N_in count the points in the delta_r ball, with any cutoff T1. For q>=2,
+(N)_q-(N_in)_q counts ordered distinct q-tuples not wholly inside that ball.
+
+**Theorem L (factorial localization).** For every fixed integer q>=2,
+
+    r^-3 E_r[(N_r)_q-(N_in)_q] -> 0.                     (L1)
+
+In particular all order-r^3 second-factorial mass is microscopic and its limiting
+coefficient is 2a2. The assertion does not hold for q=1, whose omitted coefficient
+is beta_far>0.
+
+**Proof.** On the good event used in T6, either all points belong to the near ball
+and hence to the delta_r ball, or the only point is a remote singleton. For q>=2
+the difference in L1 is zero in both cases. Everywhere it is between zero and
+N_r^q 1_Ebad. By Cauchy-Schwarz and C6,
+
+    r^-3 E_r[N_r^q 1_Ebad]
+       <= (r^-3 E_r N_r^(2q))^(1/2)
+                                 (r^-3 P_r(E_bad))^(1/2).
+
+Take the small-r limsup, then R->infinity and rho->0 using T6. This proves L1.
+The coefficient 2a2 follows from SC's count law and polynomial moment convergence.
+The identical argument for fixed R,rho and their mixed event gives
+
+    E_r[N_R N_far^rho]=o_(R,rho)(r^3),                    (L2)
+
+using N_R N_far<=N^2 and E N^4=O(r^3). Thus the probability localization also
+controls the COUNTED mixed pair mass; an event bound alone without uniform
+integrability would not have supplied that conclusion. QED.
+
+For a bounded continuous function of an ordered pair of micro position/height/
+index marks, the scaled ordered-pair measure converges weakly to
+
+    integral_(n=2) [delta_(z1(theta),z2(theta))
+                     +delta_(z2(theta),z1(theta))] dM(theta).  (L3)
+
+This is T5 with a count-quadratic functional and T7, followed by L1 to remove
+nonmicroscopic pairs. No arbitrary root ordering enters L3. It gives a precise
+conditional answer to localization of the leading pair mass, without changing
+any existing scientific register or accepting a different proof of that problem.
+
+## 5. The order of limits remains a real restriction
+
+T5 concerns a two-scale weak limit as r->0. The sharp radius tail in the companion
+paper concerns the tail of the already formed MICROSCOPIC limiting measure.
+Neither result asserts a convergence rate or an interchange with a radius that
+grows arbitrarily quickly at finite r. Count-polynomial uniform integrability
+T7 is not spatial-moment uniform integrability. In particular an eleventh radial
+moment conclusion for the limiting local law must not be advertised as an
+asymptotic for the finite-r global eleventh moment, which also has remote points.
+
+## 6. A rigorous obstruction to spatial total-variation convergence in d>=3
+
+In the micro double-cluster limit, both transverse displacement vectors are
+multiples of the same soft eigenvector. Let E_col be the event that a two-point
+configuration has both tags micro and linearly dependent transverse vectors.
+The limiting probability conditional on N=2 is one.
+
+For any fixed r>0, the factorial spatial measure of two distinct additional
+critical points under P_r is absolutely continuous away from the pins and the
+coincidence diagonal. To see this, exhaust that domain by compact separated
+sets; [P]'s distinct-site jet rank gives a nonsingular two-gradient covariance.
+The Gaussian two-point Kac-Rice formula with the single endpoint weight W_r and
+single Z_r gives a Lebesgue density there. Polynomial Gaussian moments make it
+finite on each compact piece. A spatially null set therefore has zero factorial
+measure. The conditional law on N=2 is dominated, as a null-set statement, by
+this nonnegative factorial measure. No unproved uniform near-diagonal density
+bound is needed for this absolute-continuity assertion.
+
+When d>=3, the transverse dimension m>=2. Linear dependence of two transverse
+vectors is a Lebesgue-null algebraic condition: for example the zero set of the
+sum of squares of their 2-by-2 minors, a nonzero polynomial. Hence P_r(E_col|N=2)=0
+whenever that conditioning is defined. SC implies it is defined for all small r
+because P_r(N=2)~a2 r^3 and a2>0. The probability total-variation distance is thus
+
+    dTV(Law(Xi_r|N_r=2), L_micro restricted to n=2 / a2)=1. (V1)
+
+Weak convergence is fully compatible with V1. Latent-jet/count total variation
+from SC must not be pushed through an r-dependent spatial map as if the map were
+fixed. No V1 claim is made in d=2.
+
+## 7. Review scope
+
+New obligations: the marked configuration-space sandwich (not merely intensity),
+remote height disintegration, arbitrary-cutoff localization with counted tails,
+and the finite-r versus limiting-support argument V1. The cubic, SC domination
+and imported Gaussian estimates keep their own review obligations. Exact finite
+controls verify exponent ledgers and count inequalities; they do not certify
+these measure-theoretic or Gaussian steps. The radius/height calculation is
+separate in RADIAL_TAIL.md and has its own nonauthor review request.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md b/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md
new file mode 100644
index 000000000..33f00592e
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/VALIDATION.md
@@ -0,0 +1,49 @@
+# Executed validation and scope
+
+Scientific effect NONE. Finite tests are not analytic acceptance.
+
+The mathematical companion was test-first: ten assertion failures were observed
+against explicit missing-implementation stubs; implementation then passed those
+ten tests and the expanded suite. A separate real-Git custody suite was written
+against stubs: 19 tests produced 24 intended assertion failures (some tests had
+multiple subcases), then all 19 passed after implementation. Logs are retained
+in the conversation evidence bundle, not invented as upstream execution receipts.
+
+Initial standalone suite: 29 mathematical tests plus 19 custody tests = 48 tests
+in each normal/optimized mode. The mathematical executable verifies 648 rational
+stationary substitutions and exact root/weight/height constants. Eight semantic
+mutants must each return1; an unknown mutant must return2. Normal and optimized
+RESULTS.json outputs are byte-identical.
+
+The source checker uses git ls-tree --full-tree for repository-root paths while
+running from the nested packet directory. Fixtures include changed size/SHA/blob,
+missing path, malformed commit, path traversal, leaf and parent Git symlinks,
+missing/duplicate/empty inventories, duplicate JSON, and packet/manifest symlinks.
+All six declared sources are mandatory; no prior or auxiliary source is skipped.
+
+The local environment did not obtain a full network checkout of the project.
+Therefore local-only replay explicitly reports source_pins_checked=false.
+Actual project Git-object identities and the final packet inventory must pass
+hosted replay at the published commit. Full downstream and existing Lean gates
+remain separate and must be read at their current tested head/base. No local
+full-repository run or Lean formalization of these new proofs is claimed.
+
+Mathematical limits not validated by the checker: Gaussian rank/regression,
+spatial root convergence, change-of-variables multiplicity, domination, two-scale
+configuration topology, singular-support TV obstruction and the continuum tail
+asymptotics. These are the written proof obligations for nonauthor reviewers.
+
+## Replacement-object and object-type repair
+
+A later author-side audit reproduced four intended assertion failures in actual
+local Git repositories: replacement refs could forge a commit/path binding,
+replacement refs could incorrectly invalidate a legitimate original pin, and a
+tree or annotated-tag object could pass as a declared forty-hex commit. These
+were verifier weaknesses, not observations of altered project evidence.
+
+The repair uses git --no-replace-objects for every source-object lookup and
+requires cat-file -t to return commit before tree traversal. Four new regressions
+pass after the change. The complete suite is now 29 mathematical plus 23 custody
+tests = 52 per Python mode, with all previous exact and mutant controls retained.
+All mathematical and source-pin bytes remain unchanged. The final successor's
+hosted replay must be checked separately from initial-head runs.
diff --git a/frontiers/two_scale_cluster_geometry_20260929/geometry.py b/frontiers/two_scale_cluster_geometry_20260929/geometry.py
new file mode 100644
index 000000000..f7fd5b699
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/geometry.py
@@ -0,0 +1,217 @@
+"""Exact finite controls for two-scale geometry and the limiting radial tail.
+
+Standard library only. No Gaussian integral is numerically approximated.
+Sparse polynomials have keys (power_of_u, power_of_v) and Fraction values.
+"""
+import argparse
+from fractions import Fraction as F
+import json
+import math
+
+MUTANTS = ('omit-jet-factor-two', 'omit-z-reflection', 'power-ten',
+           'wrong-cusp', 'height-reversed', 'wrong-window', 'wrong-k-power',
+           'half-root-jacobian')
+
+
+def rational(x):
+    if type(x) not in (int, F):
+        raise ValueError('exact int or Fraction required')
+    return F(x)
+
+
+def nonnegative_integer(q):
+    if type(q) is not int or q < 0:
+        raise ValueError('nonnegative integer required')
+    return q
+
+
+def positive_k(k):
+    k = rational(k)
+    if k <= 0:
+        raise ValueError('positive k required')
+    return k
+
+
+def add(p, q):
+    out = dict(p)
+    for powers, c in q.items():
+        out[powers] = out.get(powers, F(0)) + c
+    return {powers:c for powers,c in out.items() if c}
+
+
+def scale(p, c):
+    c = rational(c)
+    return {powers:c*a for powers,a in p.items() if c*a}
+
+
+def mul(p, q):
+    out = {}
+    for (i,j), a in p.items():
+        for (k,l), b in q.items():
+            powers = (i+k,j+l)
+            out[powers] = out.get(powers,F(0))+a*b
+    return {powers:c for powers,c in out.items() if c}
+
+
+def power(p, n):
+    nonnegative_integer(n)
+    out = {(0,0): F(1)}
+    for _ in range(n):
+        out = mul(out,p)
+    return out
+
+
+def integrate_strip(poly, lower, upper, left, right):
+    """Exact integral: u in [left,right], polynomial lower(u)<=v<=upper(u)."""
+    left, right = map(rational,(left,right))
+    if left > right or any(j for i,j in set(lower)|set(upper)):
+        raise ValueError('ordered endpoints and univariate bounds required')
+    out = {}
+    for (i,j), c in poly.items():
+        v_integral = add(power(upper,j+1),scale(power(lower,j+1),-1))
+        out = add(out,mul({(i,0): c/F(j+1)},v_integral))
+    return sum((c*(right**(i+1)-left**(i+1))/F(i+1)
+                for (i,j),c in out.items()),F(0))
+
+
+def shape_polynomial():
+    b = {(0,0):F(3),(2,0):F(-12)}
+    return mul(add({(0,2):F(4)},scale(mul(b,b),-1)),
+               add(b,{(1,1):F(-4)}))
+
+
+def shape_integrals(height_order=0):
+    nonnegative_integer(height_order)
+    b = {(0,0):F(3),(2,0):F(-12)}
+    upper = {(0,0):F(3),(1,0):F(-6)}
+    height = {(0,0):F(1,2),(1,0):F(1),(0,1):F(1,6)}
+    poly = mul(shape_polynomial(),power(height,height_order))
+    return (integrate_strip(poly,scale(b,F(-1,2)),upper,F(-3,2),F(-1,2)),
+            integrate_strip(poly,scale(b,F(1,2)),upper,F(-1,2),F(1,2)))
+
+
+def radial_factor(k):
+    """Coefficient multiplying the remaining spectral/Gaussian cusp integral."""
+    return F(216,11)*positive_k(k)**7*sum(shape_integrals())
+
+
+def tail_power():
+    return 2+2+3+4  # ds,dB,dD and the typed determinant weight
+
+
+def height_moment(q):
+    """Exact integral of eta^q H_eta(eta), without probability normalization."""
+    nonnegative_integer(q)
+    beta = F(math.factorial(q),1)
+    for j in range(q+1):
+        beta /= F(9,2)+j
+    return (F(110592,7)*(1/(F(q)+F(9,2))-beta+F(1,q+1))
+            +13824*(F(1,q+5)-F(2,q+4)+F(5,q+3)-F(4,q+2)))
+
+
+def raw_jet(k,a,B,D):
+    k = positive_k(k)
+    a,B,D = map(rational,(a,B,D))
+    return a*a/(12*k)+B, a**3/(144*k*k)+a*B/(4*k)+2*D
+
+
+def point_map(k,u,v,z,a):
+    """Recover original s,a,beta,c from one canonical stationary point."""
+    k = positive_k(k)
+    u,v,z,a = map(rational,(u,v,z,a))
+    if z == 0:
+        raise ValueError('nonzero transverse coordinate required')
+    b = 3-12*u*u
+    s = -k*v/z**2
+    B = k*b/z**2
+    D = k*(v-b*u)/z**3
+    beta,c = raw_jet(k,a,B,D)
+    return s,a,beta,c
+
+
+def cubic_gradient(k,s,a,beta,c,X,Z):
+    k = positive_k(k)
+    s,a,beta,c,X,Z = map(rational,(s,a,beta,c,X,Z))
+    return (6*k*X*X-F(3,2)*k+a*X*Z+beta*Z*Z/2,
+            s*Z+a*(X*X-F(1,4))/2+beta*X*Z+c*Z*Z/2)
+
+
+def cubic_height(k,s,a,beta,c,X,Z):
+    k = positive_k(k)
+    s,a,beta,c,X,Z = map(rational,(s,a,beta,c,X,Z))
+    return (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)
+
+
+def root_jacobian(k,u,v,z):
+    k = positive_k(k)
+    u,v,z = map(rational,(u,v,z))
+    if z == 0:
+        raise ValueError('nonzero transverse coordinate required')
+    return 6*k**3*(12*u*u+4*u*v-3)/z**8
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def run_checks(mutant=None):
+    raw = raw_jet(1,6,0,0)
+    if mutant == 'wrong-cusp': raw = (raw[0],F(3))
+    require(raw == (F(3),F(3,2)), 'cusp raw-coordinate inverse')
+    exponent = 10 if mutant == 'power-ten' else tail_power()
+    require(exponent == 11,'weighted radial homogeneity')
+    volume = sum(shape_integrals())
+    if mutant == 'wrong-window': volume = shape_integrals()[1]
+    require(volume == F(246528,35),'both typed height-window strips')
+    factor = radial_factor(1)
+    if mutant in ('omit-jet-factor-two','omit-z-reflection','half-root-jacobian'):
+        factor /= 2
+    require(factor == F(53250048,385),'raw-jet, root-Jacobian and sign factors')
+    value = radial_factor(2)
+    if mutant == 'wrong-k-power': value /= 2
+    require(value == 128*factor,'k-to-the-seventh scaling')
+    mean = height_moment(1)/height_moment(0)
+    if mutant == 'height-reversed': mean = 1-mean
+    require(mean == F(5771,7062),'height measured downward from birth')
+    for q in range(7):
+        require(sum(shape_integrals(q)) == height_moment(q),
+                'independent double-integral/height-density moment comparison')
+    cases = 0
+    for k in (F(1,2),F(1),F(2)):
+        for u in (F(-5,4),F(-1),F(-3,4),F(-1,4),F(0),F(1,4)):
+            b = 3-12*u*u
+            for frac in (F(1,4),F(1,2),F(3,4)):
+                v = abs(b)/2+frac*(3-6*u-abs(b)/2)
+                for z in (F(-3),F(-1),F(1,2),F(2)):
+                    for a in (F(-2),F(0),F(6)):
+                        s,a,beta,c = point_map(k,u,v,z,a)
+                        X = u-a*z/(12*k)
+                        require(cubic_gradient(k,s,a,beta,c,X,z)==(0,0),'stationary map')
+                        eta = u+F(1,2)+v/6
+                        require(0<eta<1,'open height-window map')
+                        require(cubic_height(k,s,a,beta,c,X,z)==-k*eta,'stationary height')
+                        require(root_jacobian(k,u,v,z)<0,'nonzero oriented root Jacobian')
+                        cases += 1
+    return {'passed': True, 'stationary_map_cases': cases,
+            'tail_exponent': 11, 'shape_integral': str(volume),
+            'radial_factor_at_k1': str(factor), 'universal_height_mean':str(mean),
+            'scientific_effect':'NONE', 'mathematical_acceptance':False,
+            'scope':'exact finite identities, not Gaussian or limiting-law verification'}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--mutant',choices=MUTANTS)
+    args = parser.parse_args()
+    try:
+        result = run_checks(args.mutant)
+    except ValueError as exc:
+        print(json.dumps({'passed':False,'error':str(exc)},sort_keys=True))
+        return 1
+    print(json.dumps(result,indent=2,sort_keys=True))
+    return 0
+
+
+if __name__=='__main__': raise SystemExit(main())
diff --git a/frontiers/two_scale_cluster_geometry_20260929/test_custody.py b/frontiers/two_scale_cluster_geometry_20260929/test_custody.py
new file mode 100644
index 000000000..d055261a2
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/test_custody.py
@@ -0,0 +1,173 @@
+"""Real isolated Git fixtures for the source/inventory verifier. No network."""
+import copy
+import hashlib
+import json
+from pathlib import Path
+import subprocess
+import tempfile
+import unittest
+import verify as v
+
+
+class CustodyTests(unittest.TestCase):
+    def setUp(self):
+        temp = tempfile.TemporaryDirectory()
+        self.addCleanup(temp.cleanup)
+        self.repo = Path(temp.name)
+        self.packet = self.repo/'frontiers'/'fixture'
+        self.packet.mkdir(parents=True)
+        for name in ('a','b','c','d','e','f'):
+            (self.repo/(name+'.md')).write_text('fixture '+name+'\n', encoding='utf-8')
+        (self.repo/'link.md').symlink_to('a.md')
+        (self.repo/'dirlink').symlink_to('frontiers',target_is_directory=True)
+        self.git('init','-q')
+        self.git('add','.')
+        self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid',
+                 'commit','-qm','fixture')
+        self.commit=self.git('rev-parse','HEAD').decode().strip()
+        self.entries=[]
+        for ident,name in zip(('SC','CUB','RM','C6','P','D5'),('a','b','c','d','e','f')):
+            data=(self.repo/(name+'.md')).read_bytes()
+            self.entries.append({'id':ident,'commit':self.commit,
+                                 **self.fingerprint(name+'.md',data)})
+        self.write_sources(self.entries)
+
+    @staticmethod
+    def fingerprint(path,data):
+        return {'path':path,'bytes':len(data),'sha256':hashlib.sha256(data).hexdigest(),
+                'git_blob':hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest()}
+
+    def git(self,*args):
+        return subprocess.run(['git',*args],cwd=self.repo,check=True,
+                              capture_output=True,timeout=20).stdout
+
+    def write_sources(self,entries):
+        (self.packet/'SOURCES.json').write_text(json.dumps({'sources':entries}),encoding='utf-8')
+
+    def test_valid_from_nested_packet(self):
+        self.assertEqual(v.verify_sources(self.packet),6)
+
+    def test_wrong_size(self):
+        e=copy.deepcopy(self.entries[0]);e['bytes']+=1
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_wrong_sha256(self):
+        e=copy.deepcopy(self.entries[0]);e['sha256']='0'*64
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_wrong_git_blob(self):
+        e=copy.deepcopy(self.entries[0]);e['git_blob']='0'*40
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_missing_source(self):
+        e=copy.deepcopy(self.entries[0]);e['path']='missing.md'
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_malformed_commit(self):
+        e=copy.deepcopy(self.entries[0]);e['commit']='HEAD'
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_unsafe_paths(self):
+        for path in ('../a.md','/a.md','a//b','a/./b','','a\\b'):
+            e=copy.deepcopy(self.entries[0]);e['path']=path
+            with self.subTest(path=path),self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_source_symlink(self):
+        e=copy.deepcopy(self.entries[0]);e['path']='link.md'
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_source_parent_symlink(self):
+        e=copy.deepcopy(self.entries[0]);e['path']='dirlink/fixture/a.md'
+        with self.assertRaises(ValueError):v.verify_source_entry(self.packet,e)
+
+    def test_missing_inventory_entry(self):
+        self.write_sources(self.entries[:-1])
+        with self.assertRaises(ValueError):v.verify_sources(self.packet)
+
+    def test_duplicate_inventory_entry(self):
+        self.write_sources(self.entries+[self.entries[0]])
+        with self.assertRaises(ValueError):v.verify_sources(self.packet)
+
+    def test_empty_inventory(self):
+        self.write_sources([])
+        with self.assertRaises(ValueError):v.verify_sources(self.packet)
+
+    def test_duplicate_json(self):
+        path=self.packet/'bad.json';path.write_text('{"x":1,"x":2}')
+        with self.assertRaises(ValueError):v.read_json(path)
+
+    def test_source_inventory_symlink(self):
+        path=self.packet/'SOURCES.json';path.unlink()
+        path.symlink_to(self.repo/'a.md')
+        with self.assertRaises(ValueError):v.verify_sources(self.packet)
+
+    def changed_commit(self):
+        (self.repo/'a.md').write_text('different historical source\n',encoding='utf-8')
+        self.git('add','a.md')
+        self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid',
+                 'commit','-qm','changed historical source')
+        return self.git('rev-parse','HEAD').decode().strip()
+
+    def accepts(self,entry):
+        try:
+            v.verify_source_entry(self.packet,entry)
+        except (ValueError,subprocess.CalledProcessError):
+            return False
+        return True
+
+    def test_replacement_cannot_forge_commit_binding(self):
+        changed=self.changed_commit()
+        entry=copy.deepcopy(self.entries[0]);entry['commit']=changed
+        self.assertFalse(self.accepts(entry))
+        self.git('replace',changed,self.commit)
+        self.assertFalse(self.accepts(entry),'replacement ref forged historical source binding')
+
+    def test_valid_original_ignores_replacement(self):
+        changed=self.changed_commit()
+        self.git('replace',self.commit,changed)
+        self.assertTrue(self.accepts(self.entries[0]),'original immutable object was replaced')
+
+    def test_tree_sha_is_not_a_commit(self):
+        entry=copy.deepcopy(self.entries[0])
+        entry['commit']=self.git('rev-parse',self.commit+'^{tree}').decode().strip()
+        self.assertFalse(self.accepts(entry),'tree identity accepted as a commit')
+
+    def test_annotated_tag_sha_is_not_a_commit(self):
+        self.git('-c','user.name=Fixture','-c','user.email=fixture@example.invalid',
+                 'tag','-a','fixture-tag','-m','tag fixture',self.commit)
+        entry=copy.deepcopy(self.entries[0])
+        entry['commit']=self.git('rev-parse','fixture-tag').decode().strip()
+        self.assertFalse(self.accepts(entry),'tag identity accepted as a commit')
+
+    def inventory_fixture(self):
+        temp=tempfile.TemporaryDirectory();self.addCleanup(temp.cleanup)
+        root=Path(temp.name);data=b'packet fixture\n'
+        (root/'note.md').write_bytes(data)
+        entry=self.fingerprint('note.md',data)
+        (root/'MANIFEST.json').write_text(json.dumps({'files':[entry]}))
+        return root,entry
+
+    def test_valid_packet(self):
+        root,entry=self.inventory_fixture()
+        self.assertEqual(v.verify_inventory(root),1)
+
+    def test_changed_packet(self):
+        root,entry=self.inventory_fixture();(root/'note.md').write_text('changed')
+        with self.assertRaises(ValueError):v.verify_inventory(root)
+
+    def test_extra_packet_leaf(self):
+        root,entry=self.inventory_fixture();(root/'extra').write_text('extra')
+        with self.assertRaises(ValueError):v.verify_inventory(root)
+
+    def test_packet_symlink(self):
+        root,entry=self.inventory_fixture();(root/'note.md').unlink()
+        (root/'note.md').symlink_to(self.repo/'a.md')
+        with self.assertRaises(ValueError):v.verify_inventory(root)
+
+    def test_manifest_symlink(self):
+        root,entry=self.inventory_fixture();(root/'MANIFEST.json').unlink()
+        (root/'MANIFEST.json').symlink_to(self.repo/'a.md')
+        with self.assertRaises(ValueError):v.verify_inventory(root)
+
+
+if __name__=='__main__':unittest.main()
diff --git a/frontiers/two_scale_cluster_geometry_20260929/test_geometry.py b/frontiers/two_scale_cluster_geometry_20260929/test_geometry.py
new file mode 100644
index 000000000..a787f6e6b
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/test_geometry.py
@@ -0,0 +1,125 @@
+import unittest
+from fractions import Fraction as F
+import geometry as g
+
+class InitialTests(unittest.TestCase):
+    def test_volume_parts(self):
+        self.assertEqual(g.shape_integrals(), (F(231744,35), F(2112,5)))
+    def test_total_volume(self):
+        self.assertEqual(sum(g.shape_integrals()), F(246528,35))
+    def test_tail_factor(self):
+        self.assertEqual(g.radial_factor(1), F(53250048,385))
+    def test_k_scaling(self):
+        self.assertEqual(g.radial_factor(2), 128*g.radial_factor(1))
+    def test_height_mass(self):
+        self.assertEqual(g.height_moment(0), F(246528,35))
+    def test_height_mean(self):
+        self.assertEqual(g.height_moment(1)/g.height_moment(0), F(5771,7062))
+    def test_height_second(self):
+        self.assertEqual(g.height_moment(2)/g.height_moment(0), F(2440,3531))
+    def test_point_map(self):
+        # k=1,u=-1,z=2,v=6 => s=-3/2,B=-9/4,D=-3/8.
+        self.assertEqual(g.point_map(1,-1,6,2,0), (F(-3,2),F(0),F(-9,4),F(-3,4)))
+    def test_cusp_curve(self):
+        self.assertEqual(g.raw_jet(1,6,0,0), (F(3),F(3,2)))
+    def test_exponent(self):
+        self.assertEqual(g.tail_power(),11)
+
+
+class AdditionalTests(unittest.TestCase):
+    def test_sparse_algebra(self):
+        p={(1,0):F(1),(0,0):F(1)}
+        self.assertEqual(g.power(p,2),{(2,0):F(1),(1,0):F(2),(0,0):F(1)})
+        self.assertEqual(g.add(p,g.scale(p,-1)),{})
+    def test_exact_rectangular_integral(self):
+        self.assertEqual(g.integrate_strip({(2,1):F(1)},{},{(0,0):F(2)},0,3),18)
+    def test_constant_integral(self):
+        self.assertEqual(g.integrate_strip({(0,0):F(1)},{},{(0,0):F(1)},0,1),1)
+    def test_empty_polynomial(self):
+        self.assertEqual(g.integrate_strip({}, {}, {(0,0):F(1)},0,1),0)
+    def test_bad_order(self):
+        with self.assertRaises(ValueError): g.height_moment(-1)
+    def test_float_rejected(self):
+        with self.assertRaises(ValueError): g.radial_factor(1.0)
+    def test_bool_rejected(self):
+        with self.assertRaises(ValueError): g.radial_factor(True)
+    def test_zero_k_rejected(self):
+        with self.assertRaises(ValueError): g.radial_factor(0)
+    def test_zero_z_rejected(self):
+        with self.assertRaises(ValueError): g.point_map(1,0,1,0,0)
+    def test_invalid_strip(self):
+        with self.assertRaises(ValueError): g.integrate_strip({}, {}, {(0,1):F(1)},0,1)
+    def test_height_moments_independent(self):
+        for q in range(8):
+            self.assertEqual(sum(g.shape_integrals(q)),g.height_moment(q))
+    def test_height_variance_positive(self):
+        mass=g.height_moment(0)
+        self.assertGreater(g.height_moment(2)/mass-(g.height_moment(1)/mass)**2,0)
+    def test_root_jacobian_independent(self):
+        def det(a):
+            return (a[0][0]*(a[1][1]*a[2][2]-a[1][2]*a[2][1])
+                    -a[0][1]*(a[1][0]*a[2][2]-a[1][2]*a[2][0])
+                    +a[0][2]*(a[1][0]*a[2][1]-a[1][1]*a[2][0]))
+        for k in (F(1,2),F(1),F(3)):
+            for u in (F(-5,4),F(-3,4),F(-1,4),F(1,4)):
+                b=3-12*u*u; bp=-24*u
+                v=(abs(b)/2+3-6*u)/2
+                for z in (F(-3),F(1),F(4)):
+                    rows=[[k/z**2,0,-2*k*v/z**3],
+                          [0,k*bp/z**2,-2*k*b/z**3],
+                          [k/z**3,-k*(bp*u+b)/z**3,-3*k*(v-b*u)/z**4]]
+                    self.assertEqual(det(rows),g.root_jacobian(k,u,v,z))
+    def test_raw_inverse(self):
+        for k in (F(1,2),F(1),F(3)):
+            for a in (F(-3),F(0),F(4)):
+                for B,D in ((F(-2),F(1,3)),(F(1),F(-1))):
+                    beta,c=g.raw_jet(k,a,B,D)
+                    self.assertEqual(beta-a*a/(12*k),B)
+                    self.assertEqual((c-a*beta/(4*k)+a**3/(72*k*k))/2,D)
+    def test_root_homogeneity(self):
+        k,u,v,z,a=F(1),F(-3,4),F(3),F(2),F(3)
+        s,a,beta,c=g.point_map(k,u,v,z,a)
+        B=beta-a*a/(12*k); D=(c-a*beta/(4*k)+a**3/(72*k*k))/2
+        for t in (F(2),F(5),F(10)):
+            st,at,betat,ct=g.point_map(k,u,v,t*z,a)
+            self.assertEqual(st,s/t**2)
+            self.assertEqual(betat-a*a/(12*k),B/t**2)
+            self.assertEqual((ct-a*betat/(4*k)+a**3/(72*k*k))/2,D/t**3)
+    def test_factorial_localization_inequality(self):
+        def falling(n,q):
+            out=1
+            for j in range(q):out*=max(n-j,0)
+            return out
+        for near in range(5):
+            for mid in range(5):
+                for far in range(5):
+                    n=near+mid+far
+                    bad=bool(mid or (near and far) or far>=2 or near>=3)
+                    for q in range(2,6):
+                        loss=falling(n,q)-falling(near,q)
+                        self.assertGreaterEqual(loss,0)
+                        self.assertLessEqual(loss,n**q*bad)
+    def test_moment_uniform_integrability_bound(self):
+        for q in range(5):
+            for M in range(1,10):
+                for n in range(M+1,30):
+                    self.assertLessEqual(F(n**q),F(n**(q+1),M))
+    def test_counting_grid(self):
+        self.assertEqual(g.run_checks()['stationary_map_cases'],648)
+    def test_cli_modes_and_mutants(self):
+        import subprocess,sys
+        from pathlib import Path
+        script=str(Path(g.__file__).resolve())
+        reference=None
+        for flags in (['-B','-S'],['-B','-O','-S']):
+            out=subprocess.run([sys.executable,*flags,script],capture_output=True,timeout=30)
+            self.assertEqual(out.returncode,0,out.stderr)
+            if reference is None: reference=out.stdout
+            self.assertEqual(out.stdout,reference)
+            for name in g.MUTANTS:
+                bad=subprocess.run([sys.executable,*flags,script,'--mutant',name],capture_output=True,timeout=30)
+                self.assertEqual(bad.returncode,1,(name,bad.stdout,bad.stderr))
+            bad=subprocess.run([sys.executable,*flags,script,'--mutant','unknown'],capture_output=True,timeout=30)
+            self.assertEqual(bad.returncode,2)
+
+if __name__=='__main__': unittest.main()
diff --git a/frontiers/two_scale_cluster_geometry_20260929/verify.py b/frontiers/two_scale_cluster_geometry_20260929/verify.py
new file mode 100644
index 000000000..e401a78a1
--- /dev/null
+++ b/frontiers/two_scale_cluster_geometry_20260929/verify.py
@@ -0,0 +1,137 @@
+"""Fail-closed local packet and historical Git-source verification.
+
+Default replay verifies all six declared project source objects. --local-only
+excludes those objects explicitly; real local Git fixtures still test the code.
+"""
+import argparse
+import hashlib
+import json
+from pathlib import Path, PurePosixPath
+import re
+import subprocess
+import sys
+
+SOURCE_IDS = frozenset(('SC','CUB','RM','C6','P','D5'))
+MUTANTS = ('omit-jet-factor-two','omit-z-reflection','power-ten','wrong-cusp',
+           'height-reversed','wrong-window','wrong-k-power','half-root-jacobian')
+
+
+def read_json(path):
+    path=Path(path)
+    if path.is_symlink() or not path.is_file():
+        raise ValueError('regular JSON source required: '+str(path))
+    def unique(pairs):
+        result={}
+        for key,value in pairs:
+            if key in result:raise ValueError('duplicate JSON key: '+key)
+            result[key]=value
+        return result
+    return json.loads(path.read_text(encoding='utf-8'),object_pairs_hook=unique)
+
+
+def safe_path(path):
+    if not isinstance(path,str) or not path or '\\' in path or '\0' in path:
+        raise ValueError('unsafe path')
+    pure=PurePosixPath(path)
+    if pure.is_absolute() or str(pure)!=path or any(p in ('.','..') for p in path.split('/')):
+        raise ValueError('unsafe path: '+path)
+    return pure
+
+
+def identity(data,entry):
+    expected_bytes=entry['bytes']
+    if type(expected_bytes) is not int or expected_bytes<0:
+        raise ValueError('invalid source size')
+    sha=hashlib.sha256(data).hexdigest()
+    blob=hashlib.sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest()
+    if len(data)!=expected_bytes or sha!=entry['sha256'] or blob!=entry['git_blob']:
+        raise ValueError('identity mismatch: '+entry['path'])
+
+
+def git(root,*args):
+    return subprocess.run(['git','--no-replace-objects',*args],cwd=root,check=True,capture_output=True,
+                          timeout=60).stdout
+
+
+def tree_entry(root,commit,path):
+    # --full-tree is essential: callers run from a NESTED packet directory.
+    data=git(root,'ls-tree','-z','--full-tree',commit,'--',path)
+    records=[r for r in data.split(b'\0') if r]
+    if len(records)!=1:raise ValueError('missing or ambiguous source: '+path)
+    header,found=records[0].split(b'\t',1)
+    mode,kind,blob=header.decode('ascii').split()
+    if found.decode('utf-8')!=path:raise ValueError('source path substitution')
+    return mode,kind,blob
+
+
+def verify_source_entry(root,entry):
+    commit=entry['commit'];path=entry['path'];pure=safe_path(path)
+    if not isinstance(commit,str) or not re.fullmatch('[0-9a-f]{40}',commit):
+        raise ValueError('immutable forty-hex commit required')
+    if git(root,'cat-file','-t',commit).strip()!=b'commit':
+        raise ValueError('source identity must name a commit object')
+    for length in range(1,len(pure.parts)):
+        prefix='/'.join(pure.parts[:length])
+        mode,kind,blob=tree_entry(root,commit,prefix)
+        if mode!='040000' or kind!='tree':raise ValueError('non-tree source ancestor')
+    mode,kind,blob=tree_entry(root,commit,path)
+    if mode not in ('100644','100755') or kind!='blob':
+        raise ValueError('regular Git blob required: '+path)
+    if blob!=entry['git_blob']:raise ValueError('Git tree/blob identity mismatch: '+path)
+    identity(git(root,'cat-file','blob',blob),entry)
+
+
+def verify_sources(root):
+    entries=read_json(root/'SOURCES.json')['sources']
+    if not isinstance(entries,list) or len(entries)!=len(SOURCE_IDS):
+        raise ValueError('six source entries required')
+    ids=[e['id'] for e in entries]
+    if len(set(ids))!=len(ids) or set(ids)!=SOURCE_IDS:
+        raise ValueError('missing, duplicate or unexpected source identity')
+    for entry in entries:verify_source_entry(root,entry)
+    return len(entries)
+
+
+def verify_inventory(root):
+    entries=read_json(root/'MANIFEST.json')['files']
+    if not isinstance(entries,list) or not entries:raise ValueError('empty packet inventory')
+    names=[e['path'] for e in entries]
+    if len(set(names))!=len(names) or 'MANIFEST.json' in names:
+        raise ValueError('duplicate or self-referential packet inventory')
+    if sorted(p.name for p in root.iterdir())!=sorted(names+['MANIFEST.json']):
+        raise ValueError('packet inventory mismatch')
+    for entry in entries:
+        pure=safe_path(entry['path'])
+        path=root/entry['path']
+        if len(pure.parts)!=1 or path.is_symlink() or not path.is_file():
+            raise ValueError('flat regular packet leaf required')
+        identity(path.read_bytes(),entry)
+    return len(entries)
+
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--local-only',action='store_true')
+    args=parser.parse_args()
+    root=Path(__file__).resolve().parent
+    count=verify_inventory(root)
+    source_count=0 if args.local_only else verify_sources(root)
+    expected=(root/'RESULTS.json').read_bytes()
+    for flags in (['-B','-S'],['-B','-O','-S']):
+        subprocess.run([sys.executable,*flags,'-m','unittest','discover','-v'],cwd=root,
+                       check=True,timeout=120)
+        command=[sys.executable,*flags,str(root/'geometry.py')]
+        result=subprocess.run(command,check=True,capture_output=True,timeout=60)
+        if result.stdout!=expected or result.stderr:
+            raise ValueError('deterministic checker output mismatch')
+        for mutant in MUTANTS+('unknown',):
+            result=subprocess.run(command+['--mutant',mutant],capture_output=True,timeout=60)
+            if result.returncode!=(2 if mutant=='unknown' else 1):
+                raise ValueError('negative control not rejected: '+mutant)
+    verify_inventory(root)
+    print(json.dumps({'passed':True,'packet_leaves':count,'source_count':source_count,
+                      'source_pins_checked':not args.local_only,'scientific_effect':'NONE',
+                      'mathematical_acceptance':False},sort_keys=True))
+
+
+if __name__=='__main__':main()