Results: where the n-point pressure family stops

Bounded outcome of the family_wall hunt. The analysis is FAMILY-LIMIT.md; this page is the one-page verdict and the labels. Labels: VERIFIED means recomputed here from first principles; MEASURED means float optimisation, an upper bound on an infimum; DERIVED means algebra checked numerically; INFERRED means extrapolation with the gap stated; AUDIT means established by the independent adversarial audit in audit/ and says how.

Nothing here is a proof and nothing here is machine-checked.

1. The verdict

lim_{n -> infinity} Phi_n = H = 0.6725007036794116 DERIVED, exact sup_n Phi_n <= 0.675142509660254 AUDIT, VERIFIED here configuration ceiling 0.6818286874638 deficit at the family's best >= 0.0066862: 71.7% of the H-to-ceiling distance

The n-point pressure certificate family, at any n, any pressure, and with any valid floor c, cannot reach the configuration ceiling. Increasing n does not help: the family climbs a little above H, turns over, and returns to H.

The mechanism is one identity. Phi_n <= H + H c - (n-1)/p, and a witness configuration of total length at most (n-1)/H makes the pressure term cancel itself, leaving H(1 + W) with W an energy per point of order 10^-3. Reaching the ceiling would need W >= 0.0138706 for every admissible witness simultaneously. It is not close.

2. Coverage of every n

Redundantly, and from this hunt's own computations alone: witness envelope over n = 3..36 (max 0.6735202493), tiled witness over n = 36..401 (max 0.6751676068 at n=56), trivial leg Phi_n <= H(n-1)/(n-2) for n >= 402 (max 0.6741861691), giving the weaker sup_n Phi_n <= 0.6751676068, which is what this hunt supported before the audit and which still stands. MEASURED.

The period-37 word is g_i = 1 + floor(18i/37) - floor(18(i-1)/37), nineteen 1s and eighteen 2s per period, with S_k = k + floor(18k/37) <= (55/37) k and 55/37 < 1/H, so the length constraint holds for every k in closed form. Evaluated at every n from 8 to 401 with this hunt's own famlib, its worst bound is 0.6750627723649344 at n = 9; the headline 0.675142509660254 is fixed by the coarseness of the uniform tail estimate, not by any evaluated witness. VERIFIED here (period37_check.py).

3. What the audit changed

A different model, in an isolated directory outside this repository, from a brief committed before any work began, with no access to this repository or any prior implementation. Provenance: audit/PROVENANCE.md. Report: audit/results/REPORT.md.

It broke the argument. Two steps of the inequality chain were invalid as written: the step that replaces the denominator reverses when the numerator is negative, and the step that evaluates at the largest admissible m was never given a monotonicity argument. Both were exhibited with admissible counterexamples at n = 3. Section 2.1a of FAMILY-LIMIT.md is the case split that repairs them, re-derived and re-checked here (chain_repair_check.py: 6,475 increment-sign checks and 5,729 admissible triples with Phi_n > H, zero violations).

It did not break the claim. No numerical counterexample at any n. Its largest best-found constrained W was at n = 100: W < 0.001404828486362211, giving H(1+W) < 0.673445451825039116. AUDIT, interval-checked there.

It sharpened the result to 0.675142509660254 and replaced a 367-value scan plus a bolted-on trivial tail with one explicit word.

4. The open items

1. The chain's remaining load-bearing assumption is untested here. It rests on m being capped at (n-1) + floor(1/c) and on c being a floor for F_{n-1} on all nonnegative gap vectors. That should be checked against n_point_bound line by line, and has not been.
2. inf F_{7,3000} has no global certificate. The audit's value, 0.003826231211304474248285482857707954213, rests on 562 broad local starts, four differential-evolution runs and 1,710 integer-lobe starts producing 953 distinct local minima, with the next stationary value 4.20e-5 higher and all six Hessian eigenvalues positive. That is evidence, not a lower bound over the domain. It sits 2.0e-13 below the 0.0038262312115073 this laboratory quotes, and section 3 of FAMILY-LIMIT.md shows why: the quoted figure is an Arb evaluation at an argmin rounded to six decimals, so an upper bound on the infimum. hunts/ainta_seven_point/RESULTS.md states it that way correctly; this hunt had been calling it a floor.
3. Nothing here was run in exact arithmetic on this side. The audit's witnesses carry directed interval bounds in the unsafe direction from its own mpmath.iv runs at 100 digits; this hunt's own numbers are float, with the direction-of-error analysis in section 4 of FAMILY-LIMIT.md.