Research audit: tensor E8 rate and finite-depth center-copy correction

.github/workflows/tensor-e8-rate-audit.yml

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+name: Tensor E8 paid-rate audit
+on:
+  pull_request:
+    paths:
+      - 'research/tensor-e8-rate-audit/**'
+      - '.github/workflows/tensor-e8-rate-audit.yml'
+  push:
+    paths:
+      - 'research/tensor-e8-rate-audit/**'
+      - '.github/workflows/tensor-e8-rate-audit.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  verify:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 5
+    strategy:
+      fail-fast: false
+      matrix:
+        python: ['3.11', '3.13']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: ${{ matrix.python }}
+      - run: python -B -I -O research/tensor-e8-rate-audit/verify.py --check research/tensor-e8-rate-audit/RESULTS.json
+      - run: python -B -I -O research/tensor-e8-rate-audit/test_verify.py
+      - run: git diff --exit-code

research/tensor-e8-rate-audit/CLAIM.json

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+{
+  "track": "Adversarial mathematical audit",
+  "seed": 66,
+  "seed_source": "OS randomness via Python secrets.randbelow(100)",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "started_at": "2026-10-13T03:44:57Z",
+  "claimed_at": "2026-10-13T03:48:14.740784+00:00",
+  "expected_finish": "2026-10-13T04:30:00Z",
+  "object": "PR647 PROVIDER-PROOF.md sections 8\u20139: full tensor-depth paid histogram and limit 9^n*kappa_n^* = 28/(2739 log 3)",
+  "source_head": "956bb581e7f97328e1b53baa5aca3d2490f2f607",
+  "method": "Independently aggregate helper lifetimes and solve histogram recurrences; exact rational log/moment bounds and a quantitative analytic error bound. Depths 1\u201366, seed selects the maximum audited depth; no random search.",
+  "deliverable": "Scoped audit proof, exact reproducible receipt, negative controls, source pins and attribution; no new multiplication kappa.",
+  "candidates": [
+    "Tensor E8 paid-profile asymptotic coefficient (#647), chosen because existing packaged verifier covers depth one only",
+    "Square-root assembly wall pass/accounting assumptions (#657), not selected",
+    "Dirty echo complete recurrence (#673), not selected because neighbouring #663 already audits the underlying binary interface"
+  ],
+  "overlap": "497 open PR titles inspected; relevant #647/#657/#673 bodies, files and comments read; latest #677 checked immediately before claim. No pinned issues and no sprint coordination issue; local claim only.",
+  "stopping_rule": "One bounded audit; stop expansion by 04:18 UTC and reserve the final quarter for independent checks, overlap refresh and handoff.",
+  "finish_target_revised": "2026-10-13T04:04:00Z after user requested faster turnaround",
+  "verification_started": "2026-10-13T03:56:54Z",
+  "result": {
+    "updated_at": "2026-10-13T03:58:51.513691+00:00",
+    "status": "Completed analytic profile audit; packaging PR",
+    "finding": "Corroborated leading coefficient; derived center-copy correction plus O(n/9^n) error with explicit finite-depth bounds. All depths 1\u201366 and nine negative controls pass.",
+    "link": "https://github.com/swbiggart/integer-mult-bounds/tree/research/seed66-mathematical-audit/research/tensor-e8-rate-audit",
+    "overlap_refresh": "501 open PRs through #681; PR647 unchanged; no pinned issues. No overlapping tensor-rate audit identified."
+  }
+}

research/tensor-e8-rate-audit/NOTICE.md

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+# Attribution and license
+
+Prepared by Scott Woolsey Biggart (@swbiggart), with substantial OpenAI Codex
+assistance in the mathematics, implementation, review and writing. No independent
+human referee or formal proof is claimed. New material is under the repository's
+Apache-2.0 license; see ../../LICENSE.
+
+This audit depends on the paid tensor profile in Artus / @artuskg's PR647,
+head 956bb581e7f97328e1b53baa5aca3d2490f2f607. That contribution credits OpenAI
+Pro for the retained provider proof, Codex for its recovery and scrutiny,
+and a separate Codex referee for its depth-one checker. The limit
+28/(2739 log 3), the construction and all original count formulas belong to
+that predecessor. Our additions are the independent general-depth accounting,
+explicit finite-depth bounds and the center-copy correction expansion.
+No predecessor implementation is imported by our checker.
+
+Retain the precursor credits recorded in PR647: Jacob Sussman's E8 unit and
+phase-frame/cover work; Chafik Boukhalfa's explicit programs; eumemic's batched
+recursion; Ryan Shea's transfer work; and the P390/P401 and paired-cube lineage.
+OpenAI's exact-Fourier source is part of PR647's transfer, which this audit
+does not use or revalidate. See the immutable PR647 NOTICE.md and source hashes
+in SOURCE.json for its complete provenance and retained notices.
+
+The elementary atanh/logarithm and exponential bounds are classical. No
+literature-priority claim is made for them, tensorization or moment methods.

research/tensor-e8-rate-audit/PROOF.md

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+# Quantitative audit of the tensor E8 paid moment
+
+This audits sections 4.2, 8 and 9 of PR647's `PROVIDER-PROOF.md` at
+`956bb581e7f97328e1b53baa5aca3d2490f2f607`. It corroborates the stated leading
+constant and adds explicit finite-depth bounds and a first correction term.
+It assumes the provider's listed register paths and three-stage assembly.
+It does not independently prove their scalar, frame, RAM or Fourier realization.
+
+Let architecture depth n >= 1, h=9^n, v=120^n, r=8^n,
+R=(885v-1081r)/14, W=R+2v, m=3h and D=2v-3hr. Let H(s) be the complete
+global paid histogram. Define
+
+    Phi(k) = sum_s H(s) (s/m)^(1-k).
+
+The root k_n of Phi(k_n)=W is a supremum for this profile's strict moment
+condition, not an admitted equality witness or a multiplication exponent.
+
+## Independent accounting
+
+`lifetime_profile` aggregates chronological dimension changes for outputs,
+pair routes, consumed inputs, gathers, decoders and returned child outputs.
+It counts eight equal recursive subtrees by multiplicity. Root input loads,
+root output finishes, both data endpoints, every copied-center descent and
+the final two pivot lines are separately paid. These are dimension invoices;
+checking a dimension path does not establish the actual frame inclusions.
+
+An independent expression unrolls the recursion by level. At an internal level
+j=0,...,n-2, put d=9^j and c=v/15^j. The **local** contributions are:
+
+| Rank | Count |
+| --- | ---: |
+| h-d | c |
+| d | 113c |
+| h-2d | 56c |
+| 8d | 2c |
+| h-9d | 2c |
+| 9d | 2c/15 |
+
+At the terminal level d=h/9, c=v/15^(n-1), they are
+(d,112c), (7d,56c), (8d,2c), (h,c/15).
+Finally add (1,2v), (h-1,2v), (h,r). Multiply the local histogram by three,
+then add (2,2v) for the global endpoints. Equal ranks are combined.
+The terminal level must be treated separately: it has no decoder subtree.
+
+The helper allocations obey R_1=6968 and R_n=59*120^n+8R_(n-1).
+Summing the displayed counts and their ranks, or summing the register paths,
+gives the source identities
+
+    S = (1334v-1726r)/7,
+    sum H = 3S+2v = (4016v-5178r)/7,
+    sum sH(s) = mW-D.
+
+These follow for every n by geometric sums/induction. The executable comparison
+covers every integer n from 1 through 66. Finite agreement is additional evidence,
+not a proof by enumeration of the universal identities.
+
+Since D/v=2-3(3/5)^n >= 1/5, Phi(0)<W. All ranks satisfy 0<s<=h<m,
+so Phi is strictly increasing. The child-count formula gives Phi(1)>W.
+There is exactly one k_n in (0,1).
+
+## A root enclosure without root finding
+
+Write
+
+    M = Phi'(0)/v = sum_s [sH(s)/(mv)] log(m/s),
+    delta = (W-Phi(0))/v = D/(mv),
+    A = h*delta = D/(3v),    U = log(m).
+
+For 0<=z<1, z <= exp(z)-1 <= z/(1-z). Therefore
+
+    delta/(M+delta*U) <= k_n <= delta/M.                 (1)
+
+For the upper endpoint, Phi(k)-Phi(0) >= v*k*M holds for every k>=0.
+For the lower endpoint k=delta/(M+delta*U), kU<1, and
+
+    (Phi(k)-Phi(0))/v <= k*M/(1-kU) = delta.
+
+Thus there is no numerical root-search stopping condition hidden in (1).
+With rational bounds M_lo<=M<=M_hi and U<=U_hi, the reported interval is
+
+    A/(M_hi+(A/h)*U_hi) <= h*k_n <= A/M_lo.              (2)
+
+Logarithms use the atanh series after reducing the argument to [1,2), with a
+geometric remainder. Specifically, for u=(x-1)/(x+1),
+
+    2 sum_(j=0..63) u^(2j+1)/(2j+1) <= log x
+    <= that sum + 2u^129/[129(1-u^2)].
+
+Binary reduction also needs the same formula for log 2. Endpoints are rounded
+outwards to multiples of 2^-180. All subsequent arithmetic is rational. The
+36-place decimal strings in the receipt are floors for display; the fraction
+fields, not those strings, are the certified endpoints.
+
+## An explicit error bound for the asymptotic derivative
+
+Let t=3/5, L=(913/14)log 3 and B_n=sum_(j=0..n-2)t^j, with B_1=0.
+The following bounds hold for every n>=1:
+
+    L-E_high <= M <= L+E_low,
+
+    E_high = [131 B_n + 113 t^(n-1) + 2]/h
+             + (87/14) log(3) / 15^(n-1),
+
+    E_low = [(651/5) B_n + (603/5)t^(n-1) + 10/3] log(3h)/h
+            + log(3)/15^n.                             (3)
+
+Here is a direct derivation. Put f(x)=x log(1/x). Assign the output rank h-d,
+pair rank h-2d, internal input/decoder ranks h-9d, and root data rank h-1 to
+the high-rank group. At the terminal level assign one of the two 8d entries
+to that group (the output); assign the other (input growth) to the low group.
+Every high-rank x=s/(3h) lies in [7/27,1/3]. On this interval
+0<f'(x)=log(1/x)-1<1, so replacing x by 1/3 changes f by at most 1/3-x.
+
+The ideal high-rank counts per v, summed to infinite depth, are
+3*59*sum_(j>=0)15^-j+6=2739/14. Their f(1/3) contribution is L.
+The high-rank distance losses are at most
+[131 B_n+113t^(n-1)+2]/h. At the terminal level the two h-9d entries
+are absent; adding these plus the infinite tail costs
+[2+59/14]log(3)/15^(n-1). This gives E_high.
+
+All remaining terms are nonnegative. Bounding every log(3h/s) by log(3h),
+their internal rank coefficients are 113+2*8+(2/15)*9=651/5 and their
+terminal coefficients are 112+8+9/15=603/5. Root rank-one loading and the
+global rank-two endpoints contribute 2+4/3=10/3. The copied-center term
+is exactly log(3)/15^n. This gives E_low. Thus (3) does not rely on an
+unquantified dominated-convergence argument. It also handles n=1.
+
+Combining (2) and (3), M=L+O(n/9^n). The gap between the two exact-log
+bounds in (2) is O(n/9^n), because A is bounded, M is bounded away from
+zero eventually, and U=O(n). Consequently the refined expansion is
+
+    9^n k_n = [28/(2739 log 3)] [1 - (3/2)(3/5)^n]
+              + O(n/9^n).                             (4)
+
+In particular the leading coefficient asserted in PR647 is correct under
+its paid profile. The copied-center correction decays as (3/5)^n, much
+more slowly than n/9^n. The ordinary exponent itself still tends to zero.
+No search for an improved provider or new multiplication kappa is implied.
+
+## Scope and failure detection
+
+The executable checks all 66 profiles, their counts, masses, analytic
+bounds (3), and rational root brackets. At depth 66 it certifies
+
+    0 < (C-9^66 k_66)/C < 4*10^-15,
+    C = 28/(2739 log 3).
+
+Omitting the centers or rank-two endpoints fails at depths 1,2,6,66.
+A separate mutation replaces one rank-1 and one rank-9 child by one rank-2
+and one rank-8 child at depth two. It preserves both child count and rank
+mass, but changes the logarithmic derivative and fails the histogram
+comparison. Thus mass-only validation would be insufficient.
+
+This is a mathematical profile audit and analytic consequence. The pinned
+source's all-size RAM theorem, physical program and supplied-root cyclic
+transfer remain outside this audit; it is not a proof-assistant result.

research/tensor-e8-rate-audit/README.md

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+# Tensor E8 rate audit: the center-copy correction controls convergence
+
+The paid profile in PR647's retained tensor-provider proof passes an
+independent count and moment audit at every depth 1–66. Its asserted limit
+is corroborated, and [PROOF.md](PROOF.md) derives the sharper expansion
+
+    9^n k_n = 28/(2739 log 3) * (1 - (3/2)(3/5)^n) + O(n/9^n).
+
+Here k_n is the root of that profile's moment equation. It is not a new
+multiplication saving or a claim to realize a different physical supplier.
+The copied-center correction dominates the other asymptotic profile effects.
+At depth 66, the scaled root is below its limit by less than 4e-15 relatively.
+
+Reproduce from the repository root, Python 3.9+ with no dependencies:
+
+```sh
+python3 -B research/tensor-e8-rate-audit/verify.py --check research/tensor-e8-rate-audit/RESULTS.json
+```
+
+Expected: `PASS_TENSOR_E8_RATE_AUDIT: 66 depths; 9 negative controls; exact rational bounds`.
+Additional independent checks: `python3 -B research/tensor-e8-rate-audit/test_verify.py`.
+Assertions are not used for acceptance, and isolated optimized execution is supported.
+
+Two implementations reconstruct the profile: aggregated chronological register
+lifetimes and a closed level sum. Exact rational log bounds give a rigorous
+root enclosure without numerical root finding. The receipt records all 66
+profiles' hashes, counts, masses and rational bounds. Decimal fields are
+truncated display values; use the fraction fields as bounds. This finite
+replay supports the accompanying algebraic proof and does not formalize it.
+
+The new evidence concerns the general-depth profile retained in #647;
+#647's packaged verifier primarily realizes depth one. Its broader scalar,
+frame, ownership, exact-RAM and Fourier claims are not certified by this audit.
+No earlier closed search is reopened. No selected certificate changes.
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+Audited PR647: `956bb581e7f97328e1b53baa5aca3d2490f2f607`.
+[Source pins](SOURCE.json), [assignment and local claim](CLAIM.json),
+[validation](VALIDATION.md), [attribution](NOTICE.md).