[amaybaum]

Source-correction unit SC-26 of the #942 audit (gate #946). Base: main 55e3217b, the merged #955 tree. Head: 64c20e7e.

Review history:

Commit	What it is	Review verdict
7c117827	first draft	ACCEPT WITH CHANGES
8d8ca8c6	applies that review	ACCEPT WITH CHANGES; confirmed the earlier fixes, the GR :591 paper proof and SM :112
72eece11	applies the re-review	confirmation: ACCEPT WITH CHANGES (four short ceiling sites, plus the owner ruling on ch15 :134)
64c20e7e	applies the confirmation	—

main has moved. It is now at 4449b0d9, after #958 (SC-27) merged. A test merge resolves every .md automatically. Only generated .tex/.pdf conflict: Main, GR, SM and the FULL book. Reconciliation (merge main, regenerate with sh ./build.sh, then check that the landing equals this PR's own diff) is pending a decision on timing, because it moves the head.

Each residual site that stated P-indivisibility, backflow, a Bell mechanism, a numerical gain or an unconditional ceiling beyond what the corpus proves now states the strongest proved result. Three notions are kept apart:

1. Conditional memory. M_t, or the readback-lemma lower bound, on accessible windows; (C4) supplies it.
2. Recurrence-cycle P-indivisibility. On a finite reversible representative, wherever a rooted visible map is not a permutation.
3. Backflow. A revival of distinguishability. It certifies P-indivisibility on the window, and C1–C4 do not supply it.

Sites

Site	Was	Anchor	Now
Structure :13, :31, :90, :513, :752, :836, :1025, :1041	P-indivisibility as one of four features; "three transport directly"; "necessary under hidden-sector entanglement"	Structure §14.3, §14.5	Feature renamed memory/divisibility. Memory transports; P-indivisibility is not established cross-framework. :1025: necessary for three of the four features. :1041: a separate divisibility analysis is needed
Structure :903, :1306	"(P-indivisibility) Confirmed"; "analog of P-indivisibility"	§14.3	"memory; P-indivisibility not tested by these experiments — Observed"; analog of hidden memory
Complexity :370, :376, :394, :396, :472, :492	CMI "O(1)"; "P-indivisible systems/dynamics"; "follows from the same characterization theorem"	Main §2.3; Complexity §7.2; Main :153	readback-lemma lower bound; memory-bearing; §7.2 consequence; backflow ⇒ P-indivisible
Complexity :484, :488, :490	contradictory memory clause; τ_B/τ_S extension, 100 μs → 1 ms, 10³ ×, nT → pT	the section's own headings and table	memory cases vs revival cases; "the extension factor is not quantified"; "the size of the sensitivity gain is not quantified: it requires an NV-specific model of the bath, readout and control"
Bioinformatics :18, :48, :56, :138, :142, (P4) :148, (P6) :152, :186, (P7) :198, :459, :463, :465, :467, :469	unconditional ceiling; "structural content of P-indivisibility"; "empirically demonstrated"; "made empirically manifest"	Main §3.4: hidden-memory theorem; canonical predictive quotient with certified fiber freedom (Main :616–:618)	the conditional theorem: where a quantity differs between hidden-sector completions sharing the scRNA-seq law, no method using only that law recovers it. The explanatory and predictive uses are prefixed "Where the relevant structure lies in the hidden fibers". Whether each quantity lies in the fibers is open
ch17 :12, :14, :20, :22, :30, :34, :36, :40, :42, :70, :72, :76, :78, :206, :210, (P1) :218 [FULL]	"no-go theorem"; "structural commitment … no amount of transcriptome data can recover"; "predicts and explains"; "structurally calculable amount"	as above	conditional theorem. The strict gap I(phenotype|T) < I(phenotype|T,S) is stated as the framework's hypothesis, open per task, and dependent claims carry "where the strict gap holds". :78's modality gain is "an amount not computed here". :76 reads "outperform transcriptome-only methods"
Main :131, :620, :696	§2.3 theorem "requires only … (C1)"; Bell-class dynamics force (C1) / certify (ii)	Main :74, :622, :628	non-permutation T (C1 at the first step supplies one); Bell checks C1 on the adopted branch; observed memory effects certify (ii)
Main :730 (status row)	"(ii) ⟹ (i): P-indivisible ⟹ …"; "The converse is retired"	Main :532	"(i) under (T) … (ii) or not; P-indivisibility bears only on whether the represented dynamics is nontrivially indivisible …"; "(i) does not imply (ii) …"
SM :112	P-indivisibility "for any L and q", with no partition	SM :106, :112; Main §2.3	site two-slice partition; the hidden x_{L−1}(t) is uniform, so T is not a permutation; holds for L ≥ 4, q ≥ 2; not an accessible-window statement
GR :591	"restoring P-divisibility would require decoupling"; "correlated data ⇒ (C1)"	Main §2.3; Main :74	one-line argument on a finite representative of the constraint surface, invariant under φ, with K_{mL} = I for every m ≥ 0 (below); C1 from Main §1.3's ADM-boundary and constraint coupling
Computation :83; ch12 :232 [FULL]	"genuinely P-indivisible … memory witness"	Main §3.4	nonzero conditional memory ⇒ no faithful Markov model
Explainer :135, :684, :1005, :1033	coupled ⇒ "must exhibit P-indivisibility"; "three of four"; "non-Markovian"; "marking the memory sector"	Main §2.3; Structure §14.5; Main :20	a reversible finite system with non-permutation T ⇒ recurrence-cycle P-indivisibility; the accessible-timescale lemma for windows; conditional memory marks the sector
appendix-c :14; README :100, :76	"P-indivisibility producing factorizability violation"; "outcome-dependence-without-parameter-dependence"; evolution "P-indivisible"	appendix-c :42; Main §3.3; ch12 :40	adopted Bell branch; memory-bearing
preface :3; ch01 :93, :163, :207; ch02 :212 [FULL]	readback forces "… indivisibility"; "Indivisibility … identifies the memory-bearing sector"; Bell route closed by P-indivisibility	Main :20, :141; ch01 :219	readback forces memory, and recurrence-cycle indivisibility is a separate fact on a fixed finite reversible representative; conditional memory identifies the sector; Bell is governed by ontic parameter dependence
ch03 :36, :102, :114, :238, :240 [FULL]	universal P-indivisibility, "no observer can avoid"	Structure :1025, §14.5	"memory/divisibility — accessible memory where the hidden sector reads back its record (C4), and, on any finite reversible substratum, recurrence-cycle P-indivisibility … wherever a rooted visible map is not a permutation"
ch13 :80; ch18 :193 [FULL]	"specifically, information backflow"; "Recurrence together with C1" with no scope	Scheme A; Main §2.3	memory where the record is read back, with backflow a further property; C1 at the first step, holding across the recurrence cycle (the R7-AUDA phrase is kept)
ch15 :16, :18, :49, :75, :85, :87, :91, :99, :116, :122, :128, :130, :132, :134, :138, :244, :260, §15.9 item 2 [FULL]	revival and gain stated as predictions; overhead "three to ten" and 3,000 → 300–1,000; "factor-of-three"; 10³ ×; SQUID comparisons; "concrete numerical predictions"	ch15 :101, :104, :112, :134, §15.9 item 1; appendix-a F2/F3/F4; Complexity :486/:496; owner ruling on counterfactual performance comparisons	memory first, revival a further signature; "proposes" for backflow-conditioned claims; "the size of the gain/reduction is not derived"; "no factor is claimed"; "a sizeable improvement"; the SQUID counterfactual at :134 is deleted; "testable predictions"

No change, with reasons:

• "backflow-aware" protocol names: the name states the protocol's target.
• ch01 :282: the swap genuinely revives distinguishability.
• ch01 :201: it follows Main :532.
• Main :622 and Explainer :143/:352: already correct.
• Bioinformatics :84: an RNA-velocity regime reading, not the ceiling.
• Bioinformatics :475, :517, :537: already conditional, or about multi-modal methods.

Excluded, belonging to other units:

• SC-22: ch15 :20, :103, :142–:212, :226 and the "Five quantitative predictions" heading; ch13 :134–:144 and the last sentence of ch13 :80; ch18 :74–:92; Complexity :488's O(τ_S/τ_B) amplitude heuristic and :492.
• SC-28: SM :1211.
• Explainer :767: a separate item.

Findings 4 and 6

GR :591, option (a): the argument is stated in the text.

1. On a finite representative of the constraint surface, invariant under φ, φ is a permutation of finite order L.
2. With the hidden prior on each root's fiber, the rooted maps K_t are square stochastic matrices on the visible values the surface admits, and K_{mL} = I for m ≥ 0.
3. K_s Λ = K_{mL} with mL > s would make Λ a stochastic right inverse of K_s.
4. Then K_s is a permutation, by the stochastic-inverse lemma (Main :133; isPermMatrix_of_stochastic_inverse).
5. No visible prior enters. C1 is taken from Main :74.

The re-review confirmed the argument.

SM :112.

• SM :106 makes (x₀(t−1), x₀(t)) a coordinate projection.
• For L ≥ 4, the hidden x_{L−1}(t) is independent of the other terms, so each row of T is uniform over q values.
• An exact integer/Fraction enumeration (python3 -I -B, not committed) covered (4,2), (4,3) and (5,2). In each case φ is a bijection, T is not a permutation, and the rows are uniform.
• Two countercontrols give permutations: the full state visible, and a variant in which site 0 is decoupled.
• Result: ALL CONTROLS GREEN. Script sha256 26d5fb4a…; output sha256 511ff827….

Checks (at 64c20e7e)

• Builds. sh ./build.sh succeeds for every touched paper and for --book, with no dropped glyphs.
• Release gate. Every step passes except lean-axioms, which needs lake and is absent locally. Coverage is OK at 128 statements, with no fingerprint changed. mirror reports 0 lines absent, and claims passes.
• Edge probe. edge_rigidity_probe.py reports ALL CHECKS PASS.
• Scans. The §A.32/§A.33 added-lines scan is clean; its one hit is the pre-existing "methodological discipline". The secret scan is clean.
• Re-grep, 0 hits: "factor-of-three", "structurally calculable", "made empirically manifest", "predicts and explains", "would compete with", "can be made quantitatively explicit", "three to ten", "SQUID-competitive", "concrete numerical predictions", "no transcriptome-only model can exceed", "Indivisibility instead identifies", "The converse is retired", "no-go theorem", and the earlier phrase lists.

Correctness bands unchanged: every edit narrows a claim to its proved scope, and SM :112 states a property proved for the stated partition.

Links: #942, #946.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb