d6g8k5htny-coder · Oct 12, 2026 · Oct 12, 2026 · Oct 12, 2026
diff --git a/docs/RESEARCH_INDEX.md b/docs/RESEARCH_INDEX.md
@@ -20,6 +20,8 @@ Checked on 12 October 2026 against main `8977619e86d2f395b8e4947cf9a5e34082cc2c2
 
 6. **Formal verification: Lean statements with kernel checks, alignment separate.** The [cap-first-exit packet](https://github.com/d6g8k5htny-coder/Math-/blob/79cb314d5b9f6b8b410bfd0df200414314f6b8c7/frontiers/cap_first_exit_lean_20261002/README.md) (author Anthropic Claude) has 152 Lean targets in its [manifest](https://github.com/d6g8k5htny-coder/Math-/blob/79cb314d5b9f6b8b410bfd0df200414314f6b8c7/frontiers/cap_first_exit_lean_20261002/MANIFEST.json). They formalize the deterministic cap step of D1 component III (CAP §§2–5, as steps L1–L13 and L16–L22), which identifies `S` as the global elder partner of `M` on the good event; a green run is kernel evidence for those Lean statements only. Outside it: whether the field satisfies the source setting, the reading of source coordinates as Lean objects, persistence-module bookkeeping, steps L14, L15 and L23, P §8, and every probabilistic statement. The manifest claims neither an alignment review nor mathematical acceptance; [ALIGNMENT.md](https://github.com/d6g8k5htny-coder/Math-/blob/79cb314d5b9f6b8b410bfd0df200414314f6b8c7/frontiers/cap_first_exit_lean_20261002/ALIGNMENT.md) records the earlier scoped read of the v1.4 statements and what a non-Claude review must cover. Main's own formal layer is under [Open work](#open-work).
 
+7. **Zeta-zero heat-flow folds and a CUE short-bar control: exploratory packet; scientific effect NONE.** The [zeta fold exploration packet](../experiments/zeta_fold_exploration/README.md) (author Anthropic Claude), landed through [main #373](https://github.com/d6g8k5htny-coder/main/pull/373) after the main commit checked above, checks an external note's proposed links between zeta zeros and the project's short-bar law: near the collision time, nondegenerate zero collisions of the de Bruijn–Newman heat flow follow the generic cubic fold, and a finite-window short-bar statistic of a Hardy-Z primitive is compared with CUE and one-dimensional Gaussian controls. It makes no claim of progress toward the Riemann Hypothesis. OpenAI/Codex nonauthor reads of its slices A/D (mathematics, wording) and B (citations) returned AMEND, and their delta readbacks returned scoped PASS; independent numerical reproduction (slice C) has not been performed.
+
 The [public-text catalog](PUBLIC_MATHEMATICS.md) counts 2,138 files at its declared source commits: a file count, not a count of distinct theorems.
 
 ## Start with the task
@@ -31,6 +33,7 @@ The [public-text catalog](PUBLIC_MATHEMATICS.md) counts 2,138 files at its decla
 | Review or claim work | [Task queue and claims, main #275](https://github.com/d6g8k5htny-coder/main/issues/275), the [Agent Message Board, main #307](https://github.com/d6g8k5htny-coder/main/issues/307), and the relevant issue or PR; [Open work](#open-work) gives each front's current records and remaining obligations |
 | Locate exact source bytes | [Catalog](https://github.com/d6g8k5htny-coder/meta-framework/blob/main/registry.json) and [query tool](https://github.com/d6g8k5htny-coder/query-) |
 | Work on the larger numerical tree | [Hardening branch](https://github.com/d6g8k5htny-coder/main/tree/chatgpt/drive-github-hardening-20260919), which holds the earlier research software and evidence (last merge 27 September 2026, `a01c72f1`), not this default-home checkout. Numerical work on `main` is in [`experiments/periodic_h0/`](../experiments/periodic_h0/README.md) and [`experiments/universality/`](../experiments/universality/README.md) |
+| Explore zeta zeros, de Bruijn–Newman folds and a CUE short-bar control | [Zeta fold exploration packet](../experiments/zeta_fold_exploration/README.md): exploratory, scientific effect NONE, no claim of progress toward the Riemann Hypothesis |
 
 ## Gaussian persistence
 
@@ -57,7 +60,7 @@ For the numerical study, [confirmation readiness](../experiments/periodic_h0/CON
 
 A [spectral tail bound](../experiments/periodic_h0/GAUSSIAN_TAIL.md) encloses the ideal field's omitted modes. A [finite-word coupling](../experiments/periodic_h0/GAUSSIAN_COUPLING.md) constructs a separate sampler under an explicit independent uniform-input model, and a [deterministic example](../experiments/periodic_h0/WORD_POLYNOMIAL.md) carries one such polynomial to exact barcode bounds. A [uniform finite-cutoff count bound](../experiments/periodic_h0/FINITE_COUNT_LOSS.md) and the [fixed-sample](../experiments/periodic_h0/COUNT_CONFIDENCE.md) and [mean-adaptive](../experiments/periodic_h0/MEAN_ADAPTIVE_CONFIDENCE.md) conditional sampling-error methods support planning a prospective sample; they report no observed confidence interval.
 
-The [exact-sample shrinking-bin lemma](../experiments/periodic_h0/EXACT_SAMPLE_SHRINKING_BIN.md) and its [spectral-sample extension](../experiments/periodic_h0/spectral_shrinking_bins/README.md), both for the planar torus of side 24, give conditions under which grid counts recover continuum counts in shrinking lifetime bins, in relative expectation, under inherited density hypotheses. Both are OpenAI/Codex results. Each has OpenAI/Codex and Anthropic Claude technical reviews recorded as `PASS_TECHNICAL_SCOPED`; the lemma's Anthropic review is [main #229 comment 5976979978](https://github.com/d6g8k5htny-coder/main/issues/229#issuecomment-5976979978). Certifying the stored pilot samples, coupling stored coefficients to the ideal field, a justified lifetime range, infinite-field count moments, a numerical remainder budget and a held-out design remain separate requirements before the study can be read as coefficient confirmation. The [fifty-law universality suite](../experiments/universality/README.md) is exploratory and has consumed no held-out seeds; its [protocol](../experiments/universality/PROTOCOL.md) lists the inputs a blinded confirmation needs.
+The [exact-sample shrinking-bin lemma](../experiments/periodic_h0/EXACT_SAMPLE_SHRINKING_BIN.md) and its [spectral-sample extension](../experiments/periodic_h0/spectral_shrinking_bins/README.md), both for the planar torus of side 24, give conditions under which grid counts recover continuum counts in shrinking lifetime bins, in relative expectation, under inherited density hypotheses. Both are OpenAI/Codex results. Each has OpenAI/Codex and Anthropic Claude technical reviews recorded as `PASS_TECHNICAL_SCOPED`; the lemma's Anthropic review is [main #229 comment 5976979978](https://github.com/d6g8k5htny-coder/main/issues/229#issuecomment-5976979978). Certifying the stored pilot samples, coupling stored coefficients to the ideal field, a justified lifetime range, infinite-field count moments, a numerical remainder budget and a held-out design remain separate requirements before the study can be read as coefficient confirmation. The [fifty-law universality suite](../experiments/universality/README.md) is exploratory and has consumed no held-out seeds; its [protocol](../experiments/universality/PROTOCOL.md) lists the inputs a blinded confirmation needs. The [zeta fold exploration packet](../experiments/zeta_fold_exploration/README.md) adds a further exploratory comparison, a finite-window short-bar statistic of a Hardy-Z primitive beside CUE and one-dimensional Gaussian controls; its estimates are not a limiting exponent and are not evidence about the project's Gaussian theorems.
 
 ## RN counting
 

diff --git a/experiments/zeta_fold_exploration/MANIFEST.json b/experiments/zeta_fold_exploration/MANIFEST.json
@@ -9,7 +9,7 @@
  },
  "generated_by": "make_manifest.py",
  "note": "committed_files lists every file meant to be committed (this manifest excluded). raw_files are regenerable samples written to results/raw/ and not committed; their hashes are from the exploratory run; regeneration is expected to be byte-identical only with OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 (README section 8).",
- "committed_total_bytes": 543383,
+ "committed_total_bytes": 545985,
  "committed_files": [
   {
    "path": "CLAIMS.md",
@@ -18,8 +18,8 @@
   },
   {
    "path": "README.md",
-   "bytes": 96518,
-   "sha256": "067cbc24c527cd273397a923dfcf2dca5bed68781072e563c242b3ce541b56b9"
+   "bytes": 99120,
+   "sha256": "d9416dc35e198de3a1c2799b6c5fd23f5e93a16d054b35c403604a3a2728913c"
   },
   {
    "path": "dbn/check_collision.py",

diff --git a/experiments/zeta_fold_exploration/README.md b/experiments/zeta_fold_exploration/README.md
@@ -8,11 +8,19 @@ de Bruijn–Newman constant Λ beyond known results. None should be read into it
 - Author: Anthropic Claude (delegated AI work for the project owner, Dylan Roy).
 - Nonauthor review from a different provider: slices A and D read, verdict **AMEND** (review
   5486843735); slice B (citations) read, verdict **AMEND** (review 5486879914); both applied
-  here (§9, §10); slice C open. Dylan's personal reading: **PENDING**.
-- Object: `ZETA-FOLD-EXPLORATION-20261012`, draft v2 (v1, which carried the same-provider passes,
-  plus the slice A, B and D amendments, §10).
+  here (§10). Delta readbacks at `0969b327`: slices A/D **PASS (scoped)** (review 5487707101);
+  slice B **PASS (scoped)** (review 5487692150). Integrated into main as `3e589ae0` on
+  12 October 2026 by a third OpenAI lane, distinct from both reviewers (§9). Automatic Codex
+  review: **NOT_RUN** (quota). Slice C (independent numerical reproduction): **NOT_PERFORMED;
+  still open** (§9). Dylan's personal reading: **PENDING**.
+- Object: `ZETA-FOLD-EXPLORATION-20261012`, draft v2 as read back at `0969b327` and landed at
+  `3e589ae0` (README blob `b367aecd`; v1 carried the same-provider passes, v2 adds the slice A,
+  B and D amendments, §10), plus post-integration edits that neither readback covers: the §6.2
+  coefficient wording (A/D delta note, §10.1), review-status currency (this block, §6.3, §9)
+  and Markdown escapes that fix GitHub rendering in §1.3 and §4.
 - Project sources read at main `8977619e` (the cited main files are unchanged from `03acb7be`
-  through the current base `7588716d`) and Math- `79cb314d`.
+  through `7588716d`, the base recorded in `MANIFEST.json`, and through `3e589ae0`) and Math-
+  `79cb314d`.
 
 ## In brief
 
@@ -193,7 +201,7 @@ Let Θ(u) = Σ_{n∈ℤ} e^{−π u n²}.
   integrand behaves like u^{(s−3)/2}, so the integral diverges for Re s ≤ 1; continuation needs
   Riemann's split at u = 1.
 - **Higher dimensions.** For d ≥ 2 and Re w > d/2,
-  ∫_0^∞ (Θ^d − 1) u^{w−1} du = π^{−w} Γ(w) Z_d(w), where Z_d(w) = Σ'_{n∈ℤ^d} |n|^{−2w} is the
+  ∫_0^∞ (Θ^d − 1) u^{w−1} du = π^{−w} Γ(w) Z_d(w), where Z_d(w) = Σ'\_{n∈ℤ^d} |n|^{−2w} is the
   Epstein zeta function of ℤ^d, with Λ_d(w) = Λ_d(d/2 − w). Two cases:
   Z_2 = 4 ζ(w) L(w, χ_{−4}) and Z_4 = 8 (1 − 4^{1−w}) ζ(w) ζ(w−1).
 - **Theta structure plus a symmetry does not place zeros on the symmetry line.** Z_4 has zeros
@@ -530,8 +538,8 @@ were checked numerically. No literature search was done, and no novelty is claim
    midpoint offset is 2(B/A)τ + O(τ²), and the asymmetry of q makes the second piece
    (B/A) r²/20 + O(r⁴), not 0. The relative Taylor remainder is at most
    sup|w''| (r/2 + |m − z₀|)²/(2w(z₀)).
-   For smooth w the ratio is even in √τ, and ℓ_w/(w(z₀)ℓ) = 1 + [(12/5)(B/A)(w'/w) +
-   (1/5)(w''/w)](z₀) τ + O(τ²) (exact series in `test_exact.py`).
+   For smooth w the ratio is even in √τ, and ℓ_w/(w(z₀)ℓ) = 1 + \[(12/5)(B/A)(w'/w) +
+   (1/5)(w''/w)\](z₀) τ + O(τ²) (exact series in `test_exact.py`).
 
    For a merely C¹ weight only 1 + o(1) holds (in fact 1 + o(r), from the modulus of
    continuity of w'), not O(r²). *Counterexample* (review 5486843735; `test_exact.py`): for
@@ -973,9 +981,14 @@ is part of the conjecture).
   It is not a density statement. The probability measure with mass 2^{−n} at each 2^{−n}
   (n ≥ 1) has ℓ < G(ℓ) ≤ 2ℓ for ℓ ≤ 1/2, hence exponent 1, but no Lebesgue density.
   "Density exponent 0" needs extra regularity or transfer assumptions on the limit.
-- *Coefficient form (not part of the conjecture; only with a caveat).* If μ_{T,H}((0, ℓ]) ~
-  c(T) ℓ at finite T, a comparison of c(T) with a CUE reference must allow for an arithmetic
-  correction of unknown size. In the analogous moment conjectures the arithmetic factor (the
+- *Coefficient form (not part of the conjecture; only with a caveat).* At finite T, μ_{T,H} is
+  a finite empirical measure: μ_{T,H}((0, ℓ]) = 0 below the window's shortest normalized bar,
+  so it has no positive ℓ → 0 asymptotic c(T) ℓ. A finite-T coefficient c(T) can only be a
+  finite-tail approximation μ_{T,H}((0, ℓ]) ≈ c(T) ℓ fitted on a declared range
+  [ℓ_lo(T), ℓ_hi(T)] (fixed cutoffs, frozen in advance as in the protocol below, not the ℓ₀
+  of (i)); it lies outside Conjecture Z. A comparison of such a c(T) with a CUE reference
+  must allow for an arithmetic correction of unknown size. In the analogous moment
+  conjectures the arithmetic factor (the
   Hughes–Keating–O'Connell a(k)) is part of the moment model, not a derived correction for
   this barcode observable. The comparison must also fix in advance which N defines the CUE
   reference: the BBLM finite-height spacing model suggests N_eff ≈ 2.2–2.8 for gaps on
@@ -1008,7 +1021,9 @@ exponent and not RH evidence.
 ### 6.3 Lower priority
 
 - **Proposition 1 and the vineyard birth law** (§3): essentially done; slice A was read
-  (review 5486843735, AMEND), and the applied amendments await its delta readback. Novelty is essentially nil, and it must not be presented as new or as related to RH.
+  (review 5486843735, AMEND), and the delta readback of the applied amendments returned
+  PASS (scoped) (review 5487707101 at `0969b327`). Novelty is essentially nil, and it must not
+  be presented as new or as related to RH.
   Follow-up question: in the project's own scale or heat-flow families, does a PDE likewise
   force folds to be transversal? Deterministic polynomial heat-flow fold controls with
   finite-interval H0 are the scope of the open proposal in main#307 comment 6116121839 (§2),
@@ -1177,35 +1192,48 @@ order), with identical counts, so byte-identity is expected only with thread pin
 A nonauthor read from a non-Anthropic provider was requested, in four slices:
 
 - **A.** Proposition 1 and the κ coefficients. **Read: AMEND** (review 5486843735; the local
-  normal form and all three κ coefficients check; findings applied, §10.1).
+  normal form and all three κ coefficients check; findings applied, §10.1). **Delta readback:
+  PASS (scoped)** (review 5487707101 at `0969b327`).
 - **B.** `CLAIMS.md` and the citations, ideally by a lane with working arXiv/journal access,
   confirming every item marked [snippet] or [memory], and the statement of Lagarias–Montague
   (arXiv:1106.4348). **Read: AMEND** (review 5486879914; the core classical claims are
   supported, citation scopes amended and provenance retagged, §10.2). The originals it did not
   read or audit stay unverified: Saouter–Gourdon–Demichel 2011, the 1988–1992 lower-bound
   papers, Davenport–Heilbronn 1936, de la Vallée Poussin 1896, Jacobi, Mumford, Cerf, Gonek,
   Hejhal and the 2009 Ki–Kim–Lee full proof; Edwards 1974 was identified bibliographically
-  only, its coefficients being left to slice C.
+  only, its coefficients being left to slice C. **Delta readback: PASS (scoped)** (review
+  5487692150 at `0969b327`).
 - **C.** The numerics, re-run independently, with zero counts checked against `nzeros` and a
-  different seed or an independent Riemann–Siegel check. **Open.**
-- **D.** A wording audit of the non-claims. **Read: AMEND** (the same review, §10.1).
+  different seed or an independent Riemann–Siegel check. **NOT_PERFORMED; still open** (offered to
+  GROK-H01 in main#373 comment 6118393130 and again after landing in main#307 comment
+  6123501853; not picked up as of 12 October 2026), so no read from a different provider has
+  reproduced the numerics.
+- **D.** A wording audit of the non-claims. **Read: AMEND** (the same review, §10.1). **Delta
+  readback: PASS (scoped)** (review 5487707101, with slice A).
 
 The slice A/D read is OpenAI dot / Codex (`/root/review_zeta_fold_mathematics`), delegated AI
 review for Dylan Roy: nonauthor and source-exposed, not blind, with organizational-independence
 credit 0. It is static inspection plus independent symbolic arithmetic. It is not a numerical
-reproduction, a citation-wide audit or an acceptance, and the amendments here await its delta
-readback. The slice B read is OpenAI dot / Codex (`/root/verify_zeta_packet_citations`),
+reproduction, a citation-wide audit or an acceptance. Its delta readback of the applied
+amendments, review 5487707101 at `0969b327`, is PASS (scoped), with one non-blocking note
+(§10.1). The slice B read is OpenAI dot / Codex (`/root/verify_zeta_packet_citations`),
 delegated AI review for Dylan Roy, likewise nonauthor and source-exposed, not blind, with
 organizational-independence credit 0. It inspected public primary papers, author manuscripts
 and official documentation (two scanned papers by local text/OCR inspection), and ran no
 repository program, test, numerical pipeline or Lean proof. It is a
 bounded citation comment, not approval, readiness, numerical reproduction, a full literature
 search, a novelty determination or theorem-wide acceptance. "Supported" there means that a
 source supports the stated claim within its hypotheses, not that its proof was reconstructed.
-Its amendments here likewise await its readback. The same-provider passes (§10.3) were
+Its delta readback, review 5487692150 at `0969b327`, is PASS (scoped). Neither delta readback
+is an acceptance or a numerical reproduction. The same-provider passes (§10.3) were
 Anthropic Claude reading Anthropic Claude work and earn no organizational-independence
 credit.
 
+The packet was integrated into main as `3e589ae0` on 12 October 2026 by a third OpenAI dot /
+Codex lane (`/root/integrate_zeta_exploration_packet`), distinct from both reviewer lanes. The
+automatic Codex review did not run (NOT_RUN, quota), and slice C has not been performed.
+Integration changes no scientific status: the scientific effect stays NONE.
+
 ## 10. Review findings applied
 
 ### 10.1 Nonauthor review 5486843735, slices A and D (applied in v2)
@@ -1218,6 +1246,7 @@ credit.
 | ZF-AD-04 | Real-zero ODE asserted at t = 0 without hypotheses | §1.2 and `CLAIMS.md` 3.4: simple-zero velocity in general, real-zero sum for t > Λ, RH at t = 0 (simplicity of z_j stated separately, not as part of RH), the 300-zero comparison a consistency check; docstrings of `dbn/check_dbn_basic.py` and `dbn/zero_velocity.py` |
 | ZF-AD-05 | Conjecture Z's limit and observable; log-CDF versus density; T-CUE strata | §6.2: iterated limit over all T with the window fixed as H(T) = √T, amplitude of every bar, boundary, positivity, which median, the 2^{−n} example, "well posed" withdrawn; §6.1: nonlocal strata and global selection; `CLAIMS.md` "What is kept" |
 | Info | Λ > 0 forces a finite multiple real zero (Polymath15 Theorem 1.5(i)) | `CLAIMS.md` 6.1, 6.2 and error 5; §3 Remark 3; §7 |
+| Delta note | Non-blocking note in the delta readback (review 5487707101): §6.2 stated μ((0, ℓ]) ~ c(T) ℓ at finite T, but a finite empirical measure has no positive ℓ → 0 asymptotic below its shortest bar | §6.2 coefficient form: a finite-tail approximation fitted on a declared range [ℓ_lo(T), ℓ_hi(T)] of fixed cutoffs (not the ℓ₀ of Conjecture Z (i)), outside Conjecture Z (applied after integration; not read back) |
 
 ### 10.2 Nonauthor review 5486879914, slice B (applied in v2)