SC-34: cycle ergodicity on energy shells — map named at every site; false for the exact linear rule, unsupported for the rounded rule by amaybaum · Pull Request #971 · amaybaum/incompleteness

amaybaum · 2026-10-27T23:04:55Z

Source-correction unit SC-34, from gate #946, with its research target tracked on #970 track 4. The owner authorized this unit once the finding's scope and counterexample were independently confirmed; they now are.

Stacked. This branch merges SC-28 (#962, head 7a4e4a8d) and SC-32 (#969, head 6d32ce51), because all three edit the same SM §4.2 lines. The merge commit is 23a2bdab, and SC-34's own change is 0f4f1604. This PR merges after #962 and #969. Against main, its diff therefore includes theirs.

The finding, by map

(a) The exact ℤ/q-linear rule, x(t+1) = αΣ_nn x − x(t−1) (SM Lemma; SM §4.2). It conserves E(u,v) = u·u + v·v − α u·Av, and further quadratic invariants Q_B and P_C. Cycle ergodicity on energy shells is false.
- Exhaustive counterexamples: in 1D (L = 5, q = 3) the E = 1 shell splits into 2,160 cycles; in 3D (L = 2, q = 3) it splits into 1,218,942.
- Exact certificates on the 3D L = 3 torus for p = 2, 5, 7.
- Class theorem, proved on paper and checked in 113 cases: for q = p prime with p ∤ L and a translation-invariant coupling, F^{p(p^{2m}−1)} = I with m = ord_L(p). No shell larger than that order is a single cycle.
- Not claimed for composite q, for p | L, or for couplings that are not translation-invariant.
(b) The rounded bijection of SM §4.2. It conserves no quadratic energy exactly, and its evidence is numerical chaos. Exact small censuses put its largest cycle at ≤ 1.9% of phase space. Shell ergodicity is unsupported, not shown false.
(c) The state-dependent-geometry map of SM §3.1. The corpus makes no ergodicity claim for it.

What changes

Every site now names its map.
New: SM Remark (cycle structure of the exact linear rule).
Now conditional on an open hypothesis about the realized bijection's stationary law:
- the Gaussian-on-shell measure and the pseudofermion measure match;
- gaps (i) and (ii) Part B;
- SM §6.5's gauge-measure choice;
- the "essentially unique" bijection (SM :1412, Juno).
Gap (ii) Part A holds exactly for the uniform shell measure, since E∘T = E. The residual is identifying the realized measure with that shell measure: false for (a), open for (b).
Removed: "Level 2 closed", "we adopt the former implicitly", "present in any faithful finite realization", and Main :754's "excluded by the structural assumptions already in force".
Main :720 and :752 and Substratum :135 are scoped to the rounded rule.
No coverage level changes. One fingerprint is re-affirmed: MAIN:C-past-hypothesis-replacement-rate-condition.

Record: verification/audits/manuscript/intro-corpus-audit-20261026/tracks/SC34-cycle-ergodicity/. It holds scripts s1–s7 with their outputs, and a README with the scope by map, the class theorem's proof, the site table, evidence levels and the open questions.

Verification

Release gate: passes except lean-axioms, which runs in CI. coverage_check, edge_rigidity_probe, the mirror check and the voice check pass.
Scans: the added-lines scan and the secrets and local-path scan are clean.
Script replay: s4–s7 byte-identical; s1–s3 identical apart from appended timing lines.
Builds: SM 150 pages, Main 96, Juno 22, Substratum 53.
Independent review: pending.

Draft: merge only at an exact SHA the owner approves, by normal merge commit.

Generated by Claude Code

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

…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

…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

…rate SM §4.3 displayed center independence as equivalent to exact staggered chirality. Both proof lines show one direction, and only on Theorem 1's normalization alpha = 1/d. - Lemma 3a: the rule's two-step operator is K(C,alpha) = B^2 - mu, with mu = C - 2(1 - d alpha), {B, eps} = 0, and the staggered mass fixed within the isotropic family. - Theorem 3 (ID kept): at alpha = 1/d, C = 0 gives the massless chiral operator, and a mass gives C = +-m^2. - Proposition 3b: at fixed alpha = 1/d, chiral => C = 0. - Remark: masslessness is C + 2 d alpha = 2, Corollary 1b's constant-preserving normalization. C = 2(1-d) at alpha = 1 is chiral and center-dependent, and C = 0 at alpha = 1 has m^2 = 2(d-1). Ledger: SM:T-3 re-affirmed at K1 (was K2); SM:L-3a and SM:P-3b added at K1; no ID retired. The ROADMAP §D bullet is corrected. Exact rational checks, with their output, are in the SC32 track. SM .tex/.pdf rebuilt. 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

Applies the independent review of 97d5b37 (accept with changes). - Family qualifier (blocking). F = K T0^-1 Pi_- + T0 Pi_+ is local, anticommutes with eps and squares to K for every (C, alpha), so chirality constrains C only within Lemma 3a's family. The alpha = 1 center-free mass statement now carries "within Lemma 3a's family" and "for d >= 2" in the SM Remark, the ROADMAP, the track README and the script verdict. New check G6 verifies F exactly at d = 2 and 3. - Proposition 3b states the family hypothesis, lambda != 0, characteristic 0 and even extents >= 4. - Lemma 3a: extents >= 4 for the coefficient identification. - Theorem 3: C = +-m^2 with a0 = 1. - Remark: "this lemma" is named as Lemma 3a or Theorem 3. - SM §4.2, "the two roles of the dynamics": the Susskind factorization is massless at alpha = 1/d and massive at the unit coefficient within the family. Added to the dependents table. - "Scope in the chain": the closing sentence becomes a plain status statement with the same content, which SC-17's ch05 :95 cites. The ledger re-affirms SM:L-3a, SM:T-3 and SM:P-3b at their new fingerprints, with levels unchanged. The track output is regenerated and byte-identical under -O. SM .tex/.pdf rebuilt. The two #966 fixes are listed as pending the owner's route decision and are not applied. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

… 1 (not only (B-m)(B+m)) (#946) SM :314 dropped 'only': over C, (B + m eps)^2 with m^2 = -2(d-1) also equals K. Both forms have m^2 = +-lambda mu != 0 for d >= 2, so the conclusion stands. SM :296 and ROADMAP name the (B-m)(B+m) form for m^2 = 2(d-1). SM rebuilt (148 pages). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

Owner decision (owner, 2026-10-27): SC-32's theorem correction is the route; #965 is credited as the first step of Proposition 3b; #966's two documentation fixes are adopted; #966's SM replacement is not adopted. SM:T-3 is downgraded K2 -> K1, and no upgrade is made without a corresponding machine-checked theorem. - OI_Structural_Core.lean:413, center_anticommutator docstring: "zero iff the center terms vanish" -> "an identity, not coefficient uniqueness" (#966's text). Comment only: with comments and docstrings stripped the code is byte-identical, and the file has 551 lines before and after. - structural_chain_probe.py, C3 docstring and print, take #966's text. The AST is identical excluding the module docstring and print calls. The output under python3 -I differs only in the C3 print line. - Track README: records the owner decisions with author and time, and the precise proof coverage. Staggered.factorization covers unit coefficients and abstract generators. center_anticommutator is an identity with {D, eps} = 0 as a hypothesis. SM:L-3a and SM:P-3b have no kernel witness of the statement. The fixes are credited to #966. - LEDGER.json: the SM:T-3 note states the same coverage and the owner decision. Level K1 and the ID are kept, and no other row changes. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

Applies the confirmation review of 86140ac. - README: Staggered.factorization is (sum_{i<n} A_i)^2 = sum_{i<n} S_i^2, with S_i = t_i - s_i, unit coefficients and abstract generators, no t·s = 1 and no site parity; A_sq is named. For SM:T-3, {B, eps} = 0 is not kernel-checked (only Chiral.anti_add) and the generator-parity anticommutation is exact finite algebra (G1). mass_square's hypotheses are stated exactly. - LEDGER: the SM:T-3 note carries the same coverage. The SM:L-3a note records that Staggered.factorization is never instantiated on the concrete lattice (only on Unit). Levels are unchanged, and SM:P-3b is untouched. - OI_Structural_Core.lean mass_square docstring: states the hypotheses and (D + m·eps)^2 = D^2 + m·m. It no longer quotes manuscript wording. Comment only: with comments stripped the code is byte-identical, and the file has 551 lines before and after. - structural_chain_probe.py :137 print: describes the first-order identities tested. The AST is identical excluding docstrings and print calls, and the output differs only in that line. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

No Lean theorem proves Proposition 3b or a consequence of it. The row keeps its probe (G4 T1, T2, G5) and records Chiral.center_anticommutator in its delta as the identity used in the proof's first line. Kernel coverage 32/118 -> 31/118. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…eses of center_anticommutator in the ledger; {B, eps} = 0 named in L-3a's delta; K1 basis for T-3 stated (#946) Optional findings (a)-(d) of the confirmation at 4f7ca2d. Record and ledger text only; no grade changes (T-3 K1, L-3a K1, P-3b P; kernel 31/118). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…t, full hypotheses) (#946) Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…8 @ 7a4e4a8) papers/SM.md and verification/coverage/LEDGER.json merged cleanly; each adds exactly SC-32's own diff over the merge base 0d91766. The generated papers/SM.tex and papers/SM.pdf conflicted and were regenerated from the merged source with `sh ./build.sh SM` (148 pages, no dropped glyphs). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…5, §7.5, §8.3; Main §4.5–4.6; Juno; Substratum) Every site now names the map it concerns. - Exact linear rule over Z/q: energy shells are not single cycles. SM §4.2 gains a Remark with the conserved E, Q_B, P_C and the order bound F^{p(p^{2m}-1)} = I for prime q = p, p not dividing L, translation-invariant coupling (proved; checked in 113 cases), with exhaustive instances. - Rounded bijection: chaos is numerical; it conserves no quadratic energy exactly and single-cycle sampling is not shown (largest cycle <= 1.9% of phase space in exact censuses). - Measure reduction, gap (i), gap (ii) Part B, the pseudofermion match, the §6.5 gauge-measure choice and the "essentially unique" bijection become conditional on the open stationary-law hypothesis. Gap (ii) Part A is exact for the uniform shell measure (E o T = E); its residual is identifying the realized law with that measure (false for the exact rule, open for the rounded one). - SM two-roles paragraph: mod-q reduction is linear; the nonlinearity is the floor of the rounded map. "present in any faithful finite realization" and the "not in tension" sentence removed. SC-32's text kept verbatim. - Main (EM) scope (ii): the structural assumptions do not exclude many short orbits; the exact rule fails that necessary condition. Posit (v) and the typical-orbit equivalence narrowed. Substratum ETH argument scoped to the rounding nonlinearity. Juno constraint lists follow SM. - Register neutralized on four touched lines (no claim change). - Coverage: MAIN:C-past-hypothesis-replacement-rate-condition fingerprint re-affirmed (level GAP, delta unchanged). - Evidence: confirmation scripts s1-s7 and outputs retained under tracks/SC34-cycle-ergodicity with README; all replayed under python3 -I (s4-s7 byte-identical; s1-s3 identical except the appended `time` lines). - Rebuilt SM (150 pp), Main (96), Juno (22), Substratum (53). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

Blocking: - B1 SM:298: the rounded map's nonlinearity includes the alphabet-boundary term (order q/3 where a neighbour sum crosses it); "bounded" dropped. SM:130's per-component threshold nonlinearity checked consistent. - B2 Main:754: cycle structure does not settle (EM); the necessary-condition claim is replaced by the four-state transposition example (K_Sigma eigenvalues 1, 0, verified) and (EM) is open for both maps. - B3 SM:298, Main:720: the exact rule does not sample a shell and ETH does not apply to it; the hypothesis's ETH motivation concerns the rounding. - B4 SM:1412/:1414, Juno:397: no derivation of the discretization-scheme parametrization or "essentially unique" exists in the corpus (searched); the assertions are removed and the Layer 2 consequence stated as an "if". Required: A1 45.5-56.1%; A2 91 distinct cases (113 rows; 20 d=1 duplicates and 2 at d=3, p=2 where 1/3 = 1 mod 2); A3 nonconstant invariant, "in these instances"; A4 "in the instances computed" at SM:292/:807; A5 "in the exhaustive instances". Also: SM:292 requires a positive-definite energy, gives few-site Gaussian marginals, and states E's definiteness by alpha on the real lift. SM:294: the unreplicated 0.0092 value, N_levels independence and residual ratio are removed; the reproducer's lambda_max = 0.47 +/- 0.16 (3 of 3 seeds, re-run) is stated with lambda_max > 0. README: real-valued energies not covered by the F_p test, vacuous P_C at L=2, post-hoc T1c threshold, rate follow-up; s4 removed from the track (supports no SC-34 claim). Rebuilt SM (150 pp), Main (96), Juno (22). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

R1 SM:292: E on the real lift is positive definite for |alpha| < 1/d (per mode u^2 + v^2 - a uv, a = 2 alpha sum cos k_j; verified). The semidefinite shell at alpha = 1/d is non-compact along its null modes; hypothesis (ii) and gap (i) at SM:1221 now name a compact shell of a positive-definite energy. R2 SM:294: the separation sentence states what the run shows. New diagnostic s8 (retained with output) re-derives it: order-one separation for 25/49/36 steps, a one-step jump to ~q/3 (factor 10^2.6-10^2.9) at the six neighbours of the first boundary-straddling site, spreading at about one site per step to saturation of the perturbed parity sublattice; reproducer rates 0.25/0.50/0.64 reproduced and set by the window's position on the jump; power-law and exponential residuals within 1.3x. The "Lyapunov-relevant diagnostic" parenthetical is removed; chaos rests on the four diagnostics, with values re-run from mixing_battery.py (0.931 vs 0.126, 0.019 vs 0.543, 8,110 vs 20; its own Lyapunov fit finds no window on four seeds). Follow-ons at SM:294, SM:298, Main:496. No other corpus site cites Lyapunov or lambda_max. Optional: duplicate "(EM) concerns the one-step compression." removed (Main:754); SM:1414 "which element of the class realizes our universe". Rebuilt SM (150 pp), Main (96). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…d lambda_max > 0 removed from Main :496 and the reproducer docs (#946) Main :496 no longer cites the separation measurement (which SM :294 says does not carry the chaos conclusion). The orbit non-fragmentation diagnostic has no retained reproducer and its no-merges half is vacuous for a bijection, so SM :294 lists three diagnostics. lyapunov_v3.py's docstring and print, papers/main_repro/README.md and verification/README.md no longer state lambda_max > 0 as the paper's claim; the script's computation is unchanged. SM and Main rebuilt (150, 96 pages). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

…ositive fitted rate and saturation) (#946) Optional findings of the confirmation at 5c0d297; text only, computation unchanged. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

amaybaum · 2026-11-03T06:48:24Z

Closed unmerged. SC-34 and the stacked SC-28 are on main, with stricter text:

Reconcile recurrence and map-specific stationary-law claims on current main #999 (c9036cff). Its record supersedes the stale SC34/SC28 README rows and corrects this PR's covariance statement: action kernel M⁻¹ gives covariance M.
Workstream D: consolidated minimal source corrections #1019 D-2 (d655b319).

The SC34 track and the SC28 README are on main, blob-identical. The cycle-ergodicity question stays open at #970 track 4. The stacked SC-32 hunks are #969's (#970 orphan 25). This head was not independently accepted; main's replacements were.

Branch corpus/sc34-cycle-ergodicity is kept.

Generated by Claude Code

claude added 19 commits October 27, 2026 17:20

SC-28: way out (i) requires non-integrability, shown numerically as c…

7a4e4a8

…haos (SM :294) (#946) Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

SC-32 record: T-3's K1 basis worded as in the ledger (proof ingredien…

6d32ce5

…t, full hypotheses) (#946) Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb

amaybaum closed this Nov 3, 2026

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SC-34: cycle ergodicity on energy shells — map named at every site; false for the exact linear rule, unsupported for the rounded rule - #971

SC-34: cycle ergodicity on energy shells — map named at every site; false for the exact linear rule, unsupported for the rounded rule#971
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amaybaum commented Oct 27, 2026

The finding, by map

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