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Main §2.3, continuous-time extension: P-indivisibility now follows from a stated hypothesis, near-total return at a common time. That return is the counterpart of Γ_L = I for a bijection of finite order. Poincaré recurrence does not supply it, since a mixing flow has T_ii(t) → μ(cell) < 1, so the general-flow case is open. Main :174 now says "finite-recurrence argument". Main §2.3, fast-bath remark: the JC and Davies–Merkli examples lie outside the finite-recurrence hypothesis, and (C2) is not necessary for recurrence-cycle P-indivisibility. The §3.4 theorem bounds conditional memory under the mixing hypothesis, not P-divisibility. Main §4.4 and GR §6.3: no type II algebra, no entropy identification (G_sub is undefined in the corpus), and no dS phase computation are derived. Reproducing the cancellation is open. The HUZ dimension is a finite partition count, and the AAIL scaling is not derived. The evidential self-assessment is removed. The GR §6 heading is kept per the documented decision at 9ff2771. SM §7.5 gap (ii) Part A: uniqueness of the φ-invariant distribution needs a single cycle, and P-indivisibility does not give it. ρ∘T = ρ is now conditional on the cycle ergodicity of gap (i). The nonexistent "Main Theorem 4" citation is dropped. Bridge (c) now says "a stationary distribution". Mirrors corrected: SM §8.5 and the Explainer FAQ (self-assessment), and book ch01 §1.5, ch07 §7.7 and ch18 §18.7, each with its FULL.md copy. Regenerated Main, GR, SM, Explainer and the book. Checks: the release gate passes except lean-axioms, which has no lake toolchain locally. edge_rigidity_probe passes. The added-lines register scan has 0 hits. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
…ecurrence terms Owner decisions and the ACCEPT WITH CHANGES review of 42bc405. - Main §4.4, GR §6.3, ch07 §7.7 (+FULL), R1: the finite-dimensional statement is scoped to each finite region. The quasilocal C*-completion (GR §3.3) is cited, with no state or representation selected. No von Neumann closure, relation to CLPW or entropy identification is derived. Whether the construction reproduces CLPW's algebra and its entropy is open. - GR §6.3, R2: the horizon entropy is σ per boundary mode over the N_modes of §5. It recovers S_dS under the §3 calibration. - SM §7.5 gap (ii), (6b) and R4: the whole gap is labelled conditional on cycle ergodicity. The φ-invariance of ρ∘T is unconditional. The form E(a,b) = a·a + b·b − a·Nb of the §4.2 rule is conserved and even under T, so its level sets are φ- and T-invariant. Both parts carry gap (i)'s residual. Dropped "does not bear on this". - SM §4.2 :294, (6a): the diagnostics show chaos numerically, within the sizes, ensembles and protocols sampled. No single-cycle theorem is established. "chat-track" is removed there and at Structure :1115. - Main :176: the theorem uses no form of (C2), and (C2-structural) holds by reversibility. Adds the range condition ε ≤ 1 − 1/n_V. - Main :706: Slagle-Preskill comparison narrowed to the proved equivalence. The continuous-time law is additional structure, and operational QM needs the quantum-complete extension. - ch18 §18.7 (+FULL): the route needs Γ_n = I. The SM §4.2 rule fixes the zero configuration, so no single cycle covers S. Exact finite-bijection recurrence is now named "finite recurrence" or "exact recurrence" at ch18 :174 and :193, ch09 :106 and ch02 :32. - Record: corrected the G_sub reason, the survivor counts and the provenance of the heading decision. Rebuilt Main, GR, SM, Structure and the book. Release gate: every step passes except lean-axioms (no lake locally). edge_rigidity_probe passes. The added-lines register scan and the secrets scan are clean. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
Book ch18 §18.7 called lcm(1..|S|) "Landau's function" and gave N ~ e^{|S|}
for a "generic permutation". It then presented e^{e^{10^{122}}} as the
period, although that number is only an upper bound.
The period of an arbitrary bijection now has three separate quantities:
- every order divides lcm(1..n), with ln lcm = psi(n) ~ n and
psi(n) < 1.03883 n (Rosser-Schoenfeld 1962);
- the maximum order is Landau's g(n), with ln g ~ sqrt(n ln n) (Landau 1903);
- the typical order of a uniform permutation follows Erdos-Turan: ln ord
~ (ln n)^2/2 in probability (1965), with a normal law of variance
(ln n)^3/3 (1967). This applies only under a uniform model, and the
corpus adopts no such model (ch03 §3.2).
The framework's linear rule (A5) has its own bound. An invertible linear
map of (Z/q)^D has order at most q^D - 1 = |S| - 1 for every q >= 2, by
Cayley-Hamilton. On a ring the wave rule's period is qL (Theorem B.3.1).
The framework bounds the 3D period but does not compute it.
ch01 §1.5 said "N can be as large as m!" and "t_R ~ e^{10^{122}} years".
It now gives the g(n) and lcm bounds and the linear-rule bound N < n. The
Explainer gets the same correction. "Infinite observation time" becomes
the finite bound.
Summaries now quote the linear-rule bound: ch18 :20, :28 and the closing
summary, App A F8, App C, and book/README. The bibliography adds Landau,
Erdos-Turan I and III, and Rosser-Schoenfeld.
Mirrors: ch01, ch18, App A, App C and the bibliography, each with its
FULL copy. Regenerated the Explainer and the book. The record is at
verification/audits/manuscript/intro-corpus-audit-20261026/tracks/SC31-period-growth/.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
The SC-28 review amendment (df061d6) rewrites ch18 §18.7 :174, :180, :191 and :193 and harmonizes ch02 :32 and ch09 :106. The Markdown sources merged without conflict. Checked by merits: the merge's source diff against df061d6 equals SC-31's own diff, and against 72c347b it equals SC-28's diff. SC-28's sentences are kept as they stand. The book .tex and .pdf were regenerated, not hand-merged. ch18 :20 now says "exact recurrence", matching SC-28's terms. The SC-31 record is updated with the post-merge re-grep and the owner decisions. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
…e, integrability - SM §7.5 gap (ii) Part A (Q1): dropped the E-level-set derivation. The integrable linear rule conserves a whole family of E_p, so its level sets are not single cycles. TφT = φ⁻¹ is now credited to the update form (a,b) ↦ (R(a) − b, a) with T(a,b) = (b,a), which holds for every R. T maps cycles to cycles, and ρ uniform on a cycle C has ρ∘T = ρ exactly when T(C) = C. Part A's label, the bold heading and the closing sentence now read "cycle ergodicity of the realized bijection on a T-invariant set carrying ρ". - Main §4.4, GR §6.3, ch07 §7.7 (+FULL) (Q2): the disclaimer is scoped to "no identification of CLPW's entropy with a substratum quantity". - SM §4.2 :294 (Q3): "Breaking the integrability of the linear rule is necessary for full-shell sampling; chaos does not establish full-shell sampling". - Main :706 (optional): "exact finite operational quantum mechanics". - Record rows updated to match. Rebuilt Main, GR, SM and the book. Release gate: every step passes except lean-axioms (no lake locally). edge_rigidity_probe passes. The register, secrets and model-id scans are clean. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
The SC-28 confirmation-review fixes (c43347f) touch SM :294 and :1211, Main :706, GR :368 and ch07 :199 with its FULL copy. None of these are SC-31 lines. The sources merged without conflict. Checked by merits: SC-31's source diff against c43347f is unchanged from its diff against df061d6, and the SC-28 delta is carried exactly. The book .tex and .pdf were regenerated, not hand-merged. Content is unchanged. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
…requirement Two phrases in the :294 paragraph still made chaos the requirement for full-shell sampling, against the sentence below them. Way out (i) now reads "breaking the linear rule's integrability, which full-shell sampling needs". The closing sentence now reads that the chaos "shows that the rounding breaks the linear rule's integrability as the ergodicity interpretation requires". Rebuilt SM. Release gate: every step passes except lean-axioms (no lake locally). The register and secrets scans are clean. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
51ba665 is a single-parent child of c43347f. It changes only SM :294 and the regenerated SM.tex and SM.pdf. The merge had no conflicts, so nothing was regenerated. Checked by merits: the merge delta against 7114aa2 is exactly SC-28's SM change, and SC-31's source diff against 51ba665 is unchanged. Content is unchanged. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
…#946) R1. e^{10^122} is no longer used as a bound on |S|. The corpus bounds only |C_V x C_B| (Substratum §4), treats |C_D| as gauge, and puts S's information above the Bekenstein-Hawking bound. Bounds are now stated in terms of |S|, and the ch18 numbers are marked as an illustration, conditional on |S| ~ e^{10^122}. R2. Falsifiability is stated for a substratum of given |S| and phi. Non-recurrence within g(|S|) steps contradicts a finite bijection, and within |S| - 1 steps only A5. Since |S| is not fixed, no finite observation falsifies the framework as a whole, and records return with S. R3. The summaries (ch18 :20, :28 and the close, App A F8, App C, book/README and the Explainer) quote the bijection-level bound g(|S|). "< |S|" is scoped to the exactly linear fixed-graph rule. It does not cover the state-dependent-geometry map (SM §3.1) or the rounded bijection (SM §4.2). R4. App B :420 now says omega "can be" astronomically large. O1. ch18 states that the framework supplies no lower bound on N. The Rosser-Schoenfeld constant and its citation are dropped from the manuscript, the bibliography and the record. The check script has 11 controls, all passing. Regenerated the book and the Explainer. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
…haos (SM :294) (#946) Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
This was referenced Oct 27, 2026
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Closing without merge: fully absorbed into All 6 SC-31 corrections, plus the SC-28 hunks inherited from #962, are on
The Erdős–Turán entries are obsolete: #1005 removed the typical-order clause and both references. The branch and history are kept. Generated by Claude Code |
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Source-correction unit SC-31, from audit #942 / gate #946.
Stacked on SC-28 (#962). This branch contains SC-28's head
51ba6652, brought in by normal merges, so againstmainthe diff includes #962. It should merge after #962. Againstcorpus/sc28-assumption-scope, its own change is 16 files.The flaw
Book ch18 §18.7 made three errors:
The correction
Three quantities are now kept distinct:
None of the three is an OI recurrence time without a justified model of φ, and the corpus adopts none.
The exactly linear fixed-graph rule (A5) has N ≤ q^D − 1 = |S| − 1. This bound is stated for that rule only. The state-dependent-geometry map (SM §3.1) and the rounded bijection (SM §4.2) are named as not covered.
How the remaining claims now read:
Sites: ch18, ch01, App A F8, App B :420, App C, book/README, the bibliography, Explainer :175, and the FULL mirror (exact). Explainer and FULL
.tex/.pdfare regenerated.Record:
verification/audits/manuscript/intro-corpus-audit-20261026/tracks/SC31-period-growth/, with README,period_growth_check.pyand its output. The check uses exact integers and a fixed seed, and runs underpython3 -I; 10 controls pass.Verification
beb24b35returned ACCEPT on every item (R1–R4, O1, O3, references, scope). Its one record fix, the control count 11 → 10, is71c96578.lean-axioms, which needs lake and runs in CI.Draft: merge only at an exact SHA the owner approves, by normal merge commit.
Generated by Claude Code