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SC-36 (PR-E): black-hole status and the scope of the Page-curve theorem - #978

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SC-36 (PR-E): black-hole status and the scope of the Page-curve theorem#978
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@amaybaum amaybaum commented Oct 28, 2026

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Source-correction unit for #946 (the gate) and #918 (the Introduction). It comes from the full-corpus audit of #918 at 60c976fe: PR-E, R-01 / OD-9.

  • Base: main at 0d917665. Head: 021850d9.
  • The independent review returned ACCEPT on this exact head after three rounds.
  • The unit applies the owner's G11 decision.

Findings

  • Theorem A.8 (Page curve) has no instance. GR App. A Theorem A.8 counts configuration initial states, and its bipartition is aligned to the configuration coordinates. In that setting every state on the orbit is a product state, so H-scramble fails for every bijection. Theorems A.8–A.9 as stated therefore have no instance. The book and appendix A reported it as a conditional result about black holes.
  • Theorem A.7 was false as stated. A superposed initial state of D loses its coherences while its configuration stays frozen. Exact counterexample: φ(v,b,d) = (v, b⊕d, d) from |0⟩|0⟩|+⟩ gives purity(ρ_VB) = ½ after one step.
  • Further defects:
    • GR §8(iii) reported A.8's vacuous count.
    • The 30 M_⊙ figures applied A.5's chain to a black hole that does not have that chain.
    • The convention factors were stated against the wrong denominators.
    • A stale Tier 1 count.
    • ch18 presented information conservation, a premise, as an achievement and called the paradox "dissolved".

Corrections

  • A.7 is restated with D starting in a configuration d₀. Its proof has a trace-distance bound, ‖ρ_VB − |ψ⟩⟨ψ|‖₁ ≤ 2w_t, and a Remark explains why a superposed D is not enough. The exact check K3c covers 576 bijections, and the reviewer's own check covers 564,840 cases with 0 violations. The countercontrol K3b fires.
  • A.8 states H-scramble's operative bound and says the theorem as stated has no instance. GR:609 is rewritten, and e^{−10^77} now appears only inside A.8's proof.
  • GR:785–789 become "Orders of magnitude", stated as not derived for this black hole.
  • Convention factors are corrected at GR:366, ch07:193 and appendix-a:117.
  • ch18:130 and :146: information is conserved by the bijectivity premise, the Page curve is conditional, and the paradox is not resolved. The heading is now "The second component". The other ch18 headings belong to SC-37, which applies :146 identically.
  • Owner decision on G11: the row keeps its ID and is marked "Open research target (not counted)". Counts are updated to "5 C + 2 L. Total: 7" and "seven gravitational(-sector) entries" at 22 sites. Tier 1 has four entries and the three tiers total nineteen.
  • Smaller fixes: ch09:108, ch07:288, book README:64, and the BHa REPORT post-review list with its SHA256SUMS.
  • Coverage: GR:T-a.7 is re-affirmed at GAP. No Lean change.

Checks (re-run by the reviewer under python3 -I)

  • sc36_checks.py: output identical to the committed file.
    • K7 checks every count in all three tiers and the appendix-A totals.
    • The reviewer ran 8 mutation tests, and K7 failed with the correct message on each.
  • sc36_scan.py: 0 hits.
  • coverage_check OK. edge_rigidity_probe passes. release_gate passes everything except lean-axioms, which runs in CI.
  • Builds: GR 90 pages (median steady), book 558.

Reconciliation with SC-37 (by merits, §A.39)

ch09 conflicts with SC-37. The verified merged text has:

  • Tier 1 = 2, Tier 2 = 9, Tier 3 = 8, total nineteen;
  • "Two universality-class entries tested";
  • SC-36's information-conservation sentence followed by SC-37's dark-sector sentence.

A wholesale "theirs" resolution would silently restore the black-hole row. So K5, K7 and SC-37's §2c must all be re-run on the merged tree, and the owner requires that the black-hole row deletion survive.

For r32: Introduction L298 must add "from an initial state in which D is in a definite configuration".

The per-claim record is in verification/audits/manuscript/intro-corpus-audit-20261026/tracks/SC36-page-curve-scope/README.md.

No merge is authorized; that needs owner approval at an exact SHA, as a normal merge commit.

Generated by Claude Code

claude added 3 commits October 28, 2026 04:24
@claude
SC-36 (PR-E): black-hole status and the scope of the Page-curve theorem
2eab918 
GR App. A.8 counts initial configurations at a bipartition aligned with
them, where H-scramble fails for every bijection (CON-01, OD-9, route b):
the statement now says it has no instance there, and a scope paragraph
proves the entropy budget S(rho_R) <= H(p0) under any bijection and
states what a Page value requires. A.9, GR §8 (iii), App. A intro and the
QES comparison follow; the "reproducible from a finite deterministic
substratum" and "can be combinatorial" readings are removed.

Book (CON-02): the paradox is not resolved; information conservation is
the bijectivity premise. ch07 §7.6, ch09 Tier 1 table and count, ch18,
ch19 §19.2 heading/status/table, appendix A G11 and :102, glossary,
all mirrored into FULL.

Outside-slice items, checked and corrected: A.5's coupling chain is not
the A.1 geometry (scope paragraph); A.7 needs an initial VB|D product
(hypothesis added, proof step made explicit); the 10^122 convention
factor range is replaced by stated conventions (1/pi, 1/(96 pi^2),
1/(384 pi^4 Omega_L) ~ 3.9e-5) in GR, ch07 and appendix A.

CON-03: BHa REPORT status header aligned with its retained review;
SHA256SUMS updated. Coverage GR:T-a.7 fingerprint re-affirmed (GAP).
GR and book rebuilt. Track README, checks and outputs retained.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
@claude
SC-36 review round: A.7 restricted to a definite D-configuration; G11…
9d11a5d 
… an open research target

Independent review of 2eab918 (REVISE) and the owner decision on G11.

- GR A.7: the VB|D-product hypothesis is not enough (phi(v,b,d) =
  (v, b xor d, d) freezes D's configuration but decoheres a superposed D;
  purity(rho_VB) = 1/2 after one step). Restated for D starting in a
  configuration d0, with the proof made explicit, and a remark giving
  the counterexample. Coverage GR:T-a.7 re-affirmed (GAP).
- GR A.8: statement says the theorem as stated has no instance and gives
  H-scramble's operative bound (a full cycle of configurations is an
  exact state 1-design). GR §8 (iii) no longer reports the count.
- GR A.5: the 30 M_sun figures become orders of magnitude the theorem
  would give, not derived for that black hole (chain B<->V<->D).
- Convention factor: S_dS/(96 pi^2 sigma^2) form in GR and ch07;
  appendix A label rho_QFT/rho_classical.
- Book: ch09 Tier 1 count four, premise wording at ch09:108; ch18:130
  reading and the second-component heading; ch07:288 and book README
  caveats; all mirrored into FULL.
- Owner decision: G11 is an open research target and not counted
  (identifier kept); appendix A counts 5 C + 2 L, total 7; "seven
  gravitational(-sector) entries" at every count site and in FULL.
- BHa REPORT: status-line edit listed under post-review changes;
  SHA256SUMS recomputed.
- Checks: must-fire countercontrol K3b, exact K3c, computed-count check
  K7 (mutation-tested), extended re-grep; GR and book rebuilt.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
@claude
SC-36 round 3: three-tier total, full-tier count check, A.7 pointwise…
021850d 
… clause

Review of 9d11a5d (ACCEPT WITH CHANGES).

- ch09:178 and FULL: "The three tiers contain nineteen entries" (4 + 7 + 8).
- K7 now computes Tier 1, 2 and 3 from their tables in ch09 and in FULL and
  checks every stated tier count and the three-tier total; it fires on
  the former "twenty".
- GR A.7 proof: step (ii) bounds the displacement of every configuration
  pointwise, so the bound on w_t holds for arbitrary psi_VB.
- appendix A:225 (and FULL): duplicated "beside them" removed.
- README: ch18 heading ownership (SC-37 owns :142, :152, :160) and the
  reconciliation note for ch09 against SC-37 (resolve by merits; re-run
  K5, K7 and SC-37's section 2c on the merged tree; the BH-row deletion
  survives).
- GR and book rebuilt; outputs regenerated.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Y87wExth9RoUb3dtbumhTb
@amaybaum Claude

amaybaum commented Oct 28, 2026

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Lean kernel check failed on 021850d9 (run 40636918091) during install, before any Lean check ran. elan's download of lean-4.35.0-linux.tar.zst from releases.lean-lang.org failed with an SSL connect error.

This PR changes no Lean file and no workflow file. The same error has hit #974, #976 and #977 since 04:45 UTC. The host is reachable from outside the runners: it redirects to the GitHub release asset, which downloads normally.

Once this run finishes I will re-run the failed job once. If it fails again, I'll report it as an infrastructure blocker; it won't be treated as passing.

Generated by Claude Code

This was referenced Oct 28, 2026
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amaybaum pushed a commit that referenced this pull request Oct 31, 2026
@claude
fix(D-8): GR Page-curve scope, A.7 configuration hypothesis, A.5 chai…
7192983 
…n, 10^122 factor

#978 (SC-36), GR part. Verified before editing (exact/numerical, scratch scripts):
an aligned-cut bound S(rho_R(t)) <= H(p0) under every permutation (0 violations in
1,209,600 exhaustive checks over all 8! permutations; S = 0 from a configuration
against Page(2,4) = 0.51; a Haar unitary countercontrol violates it); the A.7
counterexample phi(v,b,d) = (v, b xor d, d) from |0>|0>|+> gives Tr rho_VB^2 = 1/2
exactly; A.1 places V between the horizons (chain B<->V<->D) while A.5 assumes
V<->B<->D; S_dS/pi, S_dS/(96 pi^2 sigma^2), S_dS/(384 pi^4 Omega_L) ~ 3.9e-5 S_dS,
outside the stated [1/(96 pi^2), 8/3].

- GR:366 convention-dependent numerical factor, with its three cases.
- GR:603, :609 (iii), :715: Page curve conditional; H-scramble fails for every
  bijection from a configuration at an aligned bipartition, so A.8-A.9 have no
  instance in the setting they count; e^-10^77 no longer reported as a result.
- GR:785-789: A.5's 30 M_sun figures as the orders the theorem gives under its
  chain; new "Scope of the coupling chain" paragraph (A.1 gives B<->V<->D).
- GR A.7: hypothesis "D starts in a configuration d0"; proof replaced (the
  "rho_D constant => product" step is invalid); Remark with the counterexample.
- GR A.8 statement, H-scramble paragraph, proof conclusion "Under H-scramble";
  new "Scope of Theorem A.8" with the bound S(rho_R) <= H(p0); A.9 inherited scope.
- GR:875/:877: "reproducible from a finite deterministic substratum" and
  "it can be combinatorial" made conditional / removed.
- verification/coverage/LEDGER.json: GR:T-a.7 re-affirmed (same ID, GAP),
  fingerprint ef7fd416 -> d1d4eef7, line 811 -> 813.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_013T6pfKuv5mi3MXtKjH2tZb
@amaybaum amaybaum closed this in 141d9c9 Oct 31, 2026
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@amaybaum @claude