[Analysis] The binomial-tail underflow bound was at ~790 sites — audit what else it capped

[LightChainr]

Follow-up to a bug a 1M pilot found by accident. This ticket is about what the bug had been quietly preventing.

Correction (2026-09-05). An earlier version of this ticket said "the fix is in". That was wrong on main. The fix lives only in PR #564, commit 844173c, which is not merged. Nothing below can be checked against main until it is. Read every claim here as conditional on that PR.

What was wrong

analyze_p48_retrospective.tail_and_derivative started its binomial recurrence at (1-p)**n and walked up. At the percolation threshold 1-p ≈ 0.407, so that first term underflows to exactly zero somewhere near 790 sites, and the multiplicative recurrence then stays at zero for every later term. The tail, its derivative, and every channel built on them came back as 0.0 with nothing raised.

project_size then bisects on a quantity that is −1 everywhere and walks off the end of (0,1). That exception is the only reason this surfaced at all — otherwise it would have been a plausible-looking zero.

Fixed (in #564) by anchoring the recurrence at the binomial mode: the largest term is at least 1/(n+1) and so is always representable. Checked against exact Fraction arithmetic at small n, and against the old implementation on real N=290 data, where it agrees to 2e-13 relative at every p the analysis visits. N = 4000 now analyzes cleanly. See tests/test_p48_retrospective.py.

What this ticket is for

The bound had never been noticed because every size this project has run is below it. It nevertheless capped every large-N plan in the repository, and nobody knew, which means some past decision may have been made against a ceiling that was numerical rather than physical.

Four things worth checking, none of them urgent:

1. Was any geometry ever rejected as "too large"? docs/ROADMAP.md item 4 stops adding replicas to N130/N170 and asks for "the smallest larger geometries where R = 2, 4, 8 are injective". [P2] High-memory transfer-matrix resource probe before frontier extension #31 is a high-memory transfer-matrix resource probe. If any size decision was bounded by a silent zero rather than by cost, it should be revisited now.
2. Is the same pattern anywhere else? Any q**n, p**n or factorial-scaled start term in a loop over lattice sites is a candidate. scripts/ has several analyzers that consume the same histogram format. A grep for ** n, **(n, math.factorial and comb( over scripts/ is the whole audit.
3. Above p ≈ 0.9 the old path was already inexact at N=290 — 8.9e-07 relative at p = 0.92. That was harmless, because the bisection reads only a sign up there and the reported channels are evaluated at p₀ ≈ 0.593. Worth confirming that no other consumer evaluates these tails in the upper tail and keeps the value.
4. Which committed artifacts no longer reproduce bit-for-bit? The fix is a reassociation of the same sum, so every artifact derived from tail_and_derivative moves in its last few digits. This is known for results/server-20260828/P49-fullcurve-doubling-100m/analysis/score.json, where it was measured leaf by leaf and recorded in predictions/p49_fullcurve_doubling_semantic_gate_20260830.json under replay_reproducibility: 0 structural differences, 565 of 598 float leaves moved, worst 6.8e-9 relative, worst χ² move 2.4e-10. Three other scorers import the same routine — score_axis_pair_annihilator.py, score_intrinsic_functional_cocycle.py, score_v14_scalar_root_projector.py. Their tests pass, but that only means their tests do not replay a committed artifact through the real numerics. Check whether any of their committed outputs are now stale, and if so measure the move rather than regenerating a dated reveal.

What this does not need

Not a rewrite, not a general numerical-robustness framework, and not a tamper test on the fix. If the grep comes back clean and no size decision was affected, close it saying so — that is a complete answer.

Done looks like

Either a short note in notes/ recording that nothing else was affected, or the specific places that were, with fixes.

[LightChainr]

Queued for Go-quota refill. Analysis of what the binomial-tail underflow at ~790 sites capped; conditional on unmerged PR #564. No new production. Draft PR against main when assigned.

[LightChainr]

ASSIGNED_MACHINE: local

Analysis only. Conditional on unmerged PR #564. No new production, no Huawei. Draft PR against main. Do not edit STATUS. Do not close #570.

[LightChainr]

First CLI hit --max-turns 80 with no PR. Retrying locally with 120 turns. Still conditional on unmerged PR #564. No production, no Huawei, draft against main.

[LightChainr]

Done — draft PR: #687 (notes-only, draft against main; ticket left open).

Status since the last attempts: PR #564 is now merged (via #573), so the audit was re-run unconditionally on current main, and the one gap in the 2026-09-05 note was closed by measurement:

1. Rejected geometries — none capped by the silent zero: N=1300 was abandoned on affordability, ROADMAP 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. Same qn pattern** — one latent duplicate only (score_issue43_full_curve.py's private float tail, safe at N=185/265, underflows only past ~790 sites); everything else is mpmath/exact/q-series.
3. Upper-tail consumers — none keep a high-p value; bisections read signs only.
4. Stale artifacts — P43 primary_score.json replays byte-identical; P50 score.json was missed by the 09-05 note and is tail-derived — replay moves 72/104 leaves, worst 1.25e-3 on a non-decision diagnostic, every verdict leaf intact to reporting precision. Measured, not regenerated.

[LightChainr]

PR: #687

Underflow audit re-run on merged main; P50 staleness measured, verdicts intact. Ticket stays open. Not merging. No new dispatch until the current cluster exits.