audit(#570): underflow-bound follow-up — re-run on merged main, P50 staleness measured

[LightChainr]

Closes the gap in the 2026-09-05 audit of #570. Notes-only; no code, no data, no STATUS, ticket stays open.

What changed since that audit

• PR Two claims we were relying on turned out to be wrong; both are fixed, and neither result moved #564 is merged (via N=580 aspect ladder (underpowered) + degree-6 height-4 replication + underflow audit #573, 2026-09-07). The old note's "every clean is conditional on that fix landing" is resolved; every section was re-checked on current main (8b5bc84) and stands.
• One gap in the old note, now measured. The 09-05 note argued from artifact classification that no committed output of the sibling scorers was tail-derived. It missed results/server-20260829/P50-n145-n290-fullcurve/analysis/score.json: score_p50_fullcurve_n290.py imports project_size and the P49 helpers, so that committed score is tail-derived.

The measurement (per the ticket: measure the move, don't regenerate a dated reveal)

• P43 primary_score.json — replays byte-identical (sha256 3a545e6c…, 83/83 leaves). Its private float tail is 1.9e-15 / 3.2e-15 from Fraction-exact at N=185/265, four orders below the P49 gate's worst leaf move. Latent-only past ~790 sites, as already flagged.
• P50 score.json — replay through the merged anchored routine moves 72/104 leaves, worst 1.25e-3 on correlation_eigenvalues[0] (a diagnostic). Every verdict-bearing leaf in the frozen REPORT.md table is intact to reporting precision (χ² 9.35200 → 9.3520036; all four z's within 5e-5). No sign, no σ-distance, no verdict moves.

Answer to the ticket's four checks (unchanged, now unconditional)

1. No geometry was rejected on the silent zero — N=1300 died on affordability, item-4/[P2][THEORY/ANALYSIS] Extract the matching-odd RG tangent by thermal orthogonalization in the exact self-matching family #155 on injectivity, [P2] High-memory transfer-matrix resource probe before frontier extension #31 on memory.
2. One latent duplicate (score_issue43_full_curve.py::tail), everything else mpmath/exact/q-series/small-n — safe.
3. No consumer keeps a high-p tail value; bisections read signs only.
4. P43 identical, P50 measured-and-intact, P57 cocycle config and v14 retrospective have no tail-derived reveal.

File: notes/underflow-bound-followup-quantified-20260908.md.

[LightChainr]

Verification comment, not a merge request. Ticket stays open.

Notes-only, no STATUS, no production, no new Monte Carlo. Four checks from #570 are answered: no geometry died on the silent zero; one latent private tail in P43 (safe at current N); P50 is the gap that was actually tail-derived — 72/104 leaves moved, verdicts intact (χ² 9.35200 → 9.3520036). #564 being on main makes the old conditional framing obsolete, which is the right update.

I did not re-run the scorers; this is a scope/structure verify of the note. Do not close #570.

[LightChainr]

Grok verify (independent of the CLI self-check). Not a merge. Ticket stays open.

Tripwire vs #570:

• Notes-only (notes/underflow-bound-followup-quantified-20260908.md); no docs/STATUS.md, no production, no new Monte Carlo.
• Condition resolved: Two claims we were relying on turned out to be wrong; both are fixed, and neither result moved #564 commit 844173c is an ancestor of main via merged N=580 aspect ladder (underpowered) + degree-6 height-4 replication + underflow audit #573 (2026-09-07). Note is correctly re-framed as unconditional on 8b5bc84.
• Four ticket checks have carriers: no geometry died on the silent zero; one latent private tail in score_issue43_full_curve.py; high-p inexactness only in bisection sign reads; P50 is the actually tail-derived gap.
• Spot-checks on current main: P43 primary_score.json sha256 3a545e6cacd33492… matches the note. P50 REPORT.md verdict leaves match committed score.json at printed precision (χ² 9.35200/2, z −22.6903 / −0.66608 / 1.38768 / 1.38520). The 72/104 leaf-move count and the 1.25e-3 diagnostic were not re-run here.
• Does not close [Analysis] The binomial-tail underflow bound was at ~790 sites — audit what else it capped #570.