The analytic limit of the n-point pressure family

Verdict: the family saturates short. Its limit is exactly H = 0.6725007036794116, and its supremum over all n is at most 0.675142509660254, against a configuration ceiling of 0.6818286874638. The deficit is 0.0093280 in the limit and at least 0.0066862 at the family's best n, which is 71.7% of the whole distance from H to the ceiling. The n-point pressure method cannot reach the ceiling, and the obstruction is not a numerical accident: it is the identity Phi_n <= H + H c - (n-1)/p, in which the pressure term cancels exactly against a configuration of total length (n-1)/H.

This file has been through an independent adversarial audit and did not come out unchanged. A different model (OpenAI Codex, gpt-5.6-sol), working in an isolated directory outside this repository from a self-contained brief, with no access to this repository or to any prior implementation, was asked to refute the claim. It found that two steps of the inequality chain in section 2.1 are invalid as written and exhibited admissible counterexamples to both. It then repaired them with a case split, replaced the tiled-witness coverage with an explicit period-37 construction valid for every n, and came out with a sharper bound than this file had claimed. The full audit, its brief, its report and its scripts are in audit/; its provenance, including the commit that fixed the brief before any work began, is in audit/PROVENANCE.md. Section 2.1a states the repair, section 2.4 states the new coverage, and section 6 says what the audit changed.

Every figure below is labelled VERIFIED (recomputed here from first principles, or matched against a published arbitrary-precision value), MEASURED (float optimisation, an upper bound on an infimum), DERIVED (algebra, checked numerically), or INFERRED (extrapolation or model, with the gap stated). Figures that originate in the audit carry AUDIT and say how the audit established them; where they were recomputed here with this repository's own evaluator they also carry VERIFIED here.

Notation throughout: k = n - 1 gaps, q = floor(1/c), m = k + q the cap, theta = 1/c - q in [0,1), H = 3/2 - (1/sqrt2) cot(1/sqrt2). For a gap vector g, S(g) = sum g_i and

W(g) = sum_{s=1}^{k} (2/(k+1-s)) sum_{i=1}^{k+1-s} w(g_i + ... + g_{i+s-1}), F(g,p) = W(g) + S(g)/p.

W is the pressure-free part of the functional. Because the coefficient 2/(n-s) divides by the number of windows of length s, W is twice the sum over s of the mean of w over windows of length s, that is, an energy per point, not an extensive energy. That single observation is what makes the whole limit computable.

0. Reproduction control

The functional was reimplemented here from the definition (K(x) = int_{-1/2}^{1/2} cos(sqrt2 t) cos(2 pi x t) dt, in the closed form K(x) = (1/2)[sin(u)/u + sin(v)/v], u = pi x - 1/sqrt2, v = pi x + 1/sqrt2, K(0) = sqrt2 sin(1/sqrt2) = 0.9187253698655684) and evaluated at the three published argmins:

VERIFIED. The first six positive zeros of k recomputed as 1.0572782910088552, 2.030067530128161, 3.0202429921714815, 4.015235607036755, 5.0122084484991545, 6.010182789398035: VERIFIED, and matching the two zeros the minimisers are built from.

A second, fully independent reproduction. The audit derived the two-sinc closed form from the integral itself, checked it against direct high-precision quadrature at seven points from x = 0 to x = 19 (largest discrepancy 4.02e-102 at 100 digits), and reported

K(0) = 0.91872536986556843778423152512466175181017247999457... H = 0.67250070367941164573437979080329518859340302862626...

identical to this file's K(0) = 0.9187253698655684 and H = 0.6725007036794116 in every digit either states. It also recomputed the two W levels the barrier is stated against: 0.6751676068/H - 1 = 0.00396565104838863604939... and 0.6818286874638/H - 1 = 0.01387059334414481885584..., confirming this file's rounded 0.0138706. AUDIT; VERIFIED here at 60 digits by f7_point_check.py, which prints the same K(0) and H to 50 places. Two implementations that share no code agree on the constants, so nothing downstream turns on a transcription.

1. The lead survives, with one correction

1.1 The reduction is exact only when 1/c is an integer

Clearing the cap into the fraction,

Phi_n = [H m - k (m-1)/p] / (m - c q), m = k + q, q = floor(1/c).

Since c q = 1 - c theta, the denominator is m - 1 + c theta, so

Phi_n = [H m - k (m-1)/p] / (m - 1 + c theta). (DERIVED)

At theta = 0 this is exactly the lead's Phi = H m/(m-1) - k/p. For theta > 0 the lead overstates Phi; because c q <= 1 always, the lead's form is a rigorous upper bound on Phi_n for every c. VERIFIED at the three known peaks:

The overshoot is 6% to 31% of the spacing between adjacent peaks (2.4e-5 from n=7 to 8, 1.9e-5 from n=8 to 9). The lead is safe as a bound and safe as a model, but must not be used to rank adjacent n, the floor defect is the same size as the effect being ranked.

Separately VERIFIED at all three peaks by exhaustive search over m in [k, k+q]: m at the cap is the maximiser, so nothing is lost by taking m = k + floor(1/c).

1.2 Monotonicity in p inside one minimiser family, in closed form

Within one minimiser family the pair (W, S) is essentially constant, recomputed from the published argmins, W varies by less than 4.4e-7 across a family's whole pressure range, so c(p) = W + S/p is a straight line in 1/p. Writing gain = Phi - H and using the theta = 0 model,

gain(c) = H c / (1 + (k-1) c) - k (c - W)/S. (DERIVED)

This is concave in c with a single stationary point

D* = ( sqrt(H S / k) - 1 ) / (k - 1), which exists iff S > k/H. (DERIVED)

gain rises with p while c > D* and falls while c < D*. So within one family Phi is monotone in p, in the direction fixed by the sign of c - D*, the lead was right that it is monotone, but the direction is family-dependent, and both directions actually occur in the existing sweep:

MEASURED (W, S from the published argmins), DERIVED (D*, direction). In every case the family's own stationary pressure lies outside the pressure interval on which that family is the floor. That is the lead's second claim, and it holds.

1.3 The optimal pressure is exactly a family crossover: closed form

Two families (W1,S1) and (W2,S2) exchange the floor at

p_x = (S2 - S1) / (W1 - W2). (DERIVED)

Evaluated on the sweep's own argmins:

Every predicted p_x lands inside the exact grid interval in which the sweep's winning word changes, and the peak value at p_x exceeds the best gridded value. The lead's peak condition is confirmed: the optimum sits at a crossover, not at a stationary point, and the crossover is given in closed form by the two families' (W,S).

MEASURED. Caveat: p_x is computed from (W,S) averaged over each family's sweep rows; the gaps relax slightly with p, which moves p_x by order 1 in absolute pressure.

1.4 What the peak condition means

Both 1.2 and 1.3 collapse into one statement. Order the minimiser families by mean gap gbar = S/k. The crossover chosen by the maximisation is the one that brackets

gbar = 1/H = 1.4869872919 ,

and the peak value is H times the chord of the (mean gap, energy-per-point) ladder evaluated at gbar = 1/H, damped by 1/(1+(k-1)c):

The damped model reproduces the exact excess to 4.7e-8, 1.7e-7 and 3.3e-8, the residual is the theta floor defect of section 1.1. DERIVED, checked MEASURED.

So the whole reach of the family above H is: H times the convexified minimal energy per point of a one-dimensional configuration at density H. Section 2 turns that sentence into a bound.

2. The barrier

2.1 The chain

For k >= 2, with c any valid uniform floor for F_k, m at the cap, and in the branch Phi_n > H, the two conditions the chain needs, both supplied by the case split in 2.1a:

Phi_n = [H m - k(m-1)/p] / (m - c q) <= H m/(m-1) - k/p (A) since c q <= 1 AND the numerator is positive here <= H + H c/(1 + (k-2) c) - k/p (B) since m-1 = k-1+q >= k-2+1/c <= H + H c - k/p (C) <= H + H W(g) + (H S(g) - k)/p (D) since c <= F(g,p) for all g >= 0

Step (D) holds for every nonnegative gap vector g, because c must be a floor for F_k everywhere, hence at g.

VERIFIED: the four inequalities were evaluated on all 36 rows of the existing n=7,8,9 sweep using each row's own argmin as g. Zero violations.

But the chain above is not valid as a chain, and the two conditions it needs are stated in 2.1a. The audit found both. As written here, this section carried one of them as a parenthetical that rescued the conclusion without repairing the chain, "step (A) needs the numerator nonnegative; when it is negative Phi_n < 0 and every bound holds trivially", and carried the other, the move to m at the cap, on nothing but the five spot checks in 2.2(i). Neither is a proof. What follows is.

2.1a The two conditions, and the case split that supplies them (AUDIT)

Write q0 = k = n-1, d = m - q0 >= 1, N = H m - q0(m-1)/p, D = m - c d, so Phi_n = N/D. The cap c d <= 1 gives D >= m - 1 >= q0 >= 2 > 0, so the denominator never vanishes and never changes sign. The numerator can.

Defect 1, step (A) reverses when N < 0. Replacing D by the smaller m-1 raises N/D only when N >= 0. AUDIT, with an admissible counterexample at n = 3, c = 0.01, m = 3, p = 1:

Phi = -0.663042772228015 claimed step-(A) bound = -0.991248944480883

VERIFIED here (chain_repair_check.py), reproducing the audit's digits exactly. The audit also checks that c = 0.01 really is a floor for F_3 on all nonnegative g, so the triple is admissible and not a straw man.

Defect 2, the chain evaluates at the cap without proving Phi increases in m. For fixed (n, c, p), Phi is a Mobius function of m with positive denominator, and the sign of its increment is the sign of

p c H - 1 - c (q0 - 1) . (DERIVED)

That sign is not always positive, so Phi_n(m) <= Phi_n(m_max) is not free. AUDIT, with a second admissible counterexample at n = 3, c = 0.01, p = 2: Phi = 0.005853548842219 at m = 3, but -0.320840873511881 at m_max = 102, while the step-3 bound H + Hc - (n-1)/p = -0.320774289283794 is below the value at m = 3. VERIFIED here.

The repair is a case split, and it costs nothing.

• If Phi_n <= H, then Phi_n <= H(1 + W(g)) at once, for any g, since W >= 0.
• If Phi_n > H, then subtracting directly,

Phi_n - H = [H c d - q0 (m-1)/p] / D > 0 ,

and since (m-1)/d = 1 + (q0-1)/d and 1/d >= c,

p c H > q0 + q0(q0-1)/d >= q0 + q0 c (q0-1) > 1 + c (q0 - 1) ,

which is exactly the condition above. So in the only branch that can threaten a bound above H, Phi does increase in m, and moving to m_max = q0 + floor(1/c) is legitimate. At m_max, Phi_n(m_max) >= Phi_n > H > 0 with D > 0 forces N > 0, so (A) has the right direction; and m_max - 1 = q0 - 1 + floor(1/c) > 1/c + q0 - 2 >= 1/c for q0 >= 2, which is what (B) needs. Steps (A) through (D) then go through and give Phi_n <= H(1 + W(g)).

DERIVED (the audit's argument, re-derived line by line here). VERIFIED here by chain_repair_check.py: the increment-sign formula was checked against 6,475 direct increments over 3 <= n < 40, eight values of c spanning 1e-4 to 0.9 and seven pressures spanning 1 to 1e6, with no mismatch; and every one of 5,729 admissible triples found with Phi_n > H satisfied the increasing condition, had a positive numerator at the cap, satisfied m_max - 1 >= 1/c, and had its maximum over m at the cap. Zero violations.

What this does and does not change. It changes the argument, not the conclusion: the witness implication (ii) below, the trivial leg (i), and every number in 2.3 and 2.4 rest on the repaired chain and are unaffected in value. What it removes is the pretence that the one-line chain of 2.1 was a proof at every admissible m.

2.2 Two consequences, and the barrier they make

(i) The trivial leg. For m >= k, H m/(m-1) <= H k/(k-1) and -k/p < 0. For 1 <= m < k the denominator 1 - c(1-k/m) = 1 + c(k/m - 1) >= 1 while the numerator is at most H, so Phi_n <= H there. Either way

Phi_n <= H (n-1)/(n-2) for every n >= 3 and every m >= 1. (DERIVED)

This alone is already below the ceiling from n = 75 onward: H*74/73 = 0.6818409912 is above 0.6818286874638, and H*75/74 = 0.6817130421 is below. VERIFIED.

VERIFIED separately, at five (n, p, c) triples including n = 20: maximising Phi over the whole range m in [1, k+floor(1/c)] puts the maximiser at the cap every time, and every m < k gives Phi < H. That was five spot checks, and, the audit's point, five spot checks are not the argument. 2.1a is the argument: in the branch Phi_n > H the maximiser is at the cap for every admissible triple, provably, and in the branch Phi_n <= H nothing above H is at stake. Under that split the trivial leg is in fact slightly sharper than stated, since m_max >= n gives Phi_n <= H n/(n-1); the weaker H(n-1)/(n-2) is kept above because it is what the rest of the file uses.

(ii) The witness leg. In (D), choose g with S(g) <= k/H. Then H S(g) - k <= 0, the pressure term is nonpositive for every p, and

Phi_n <= H (1 + W(g)) for every gap vector g with sum(g) <= (n-1)/H. (DERIVED)

This is the heart of it. The pressure p is the family's only free dial for buying gain, and a configuration of total length exactly (n-1)/H makes the dial cancel itself. What is left is H times an energy per point at density H, a quantity of order 10^-3, while reaching the ceiling would need

W(g) >= (0.6818286874638 / H) - 1 = 0.0138706 ,

an order of magnitude more, for every admissible g simultaneously. It is not close.

(iii) The sharp form. Using (B) instead of (C), and any finite set of witnesses g_j (c <= min_j F(g_j, p) pointwise),

Phi_n <= H + max_{p>0} min_j [ H c_j(p)/(1 + (k-2) c_j(p)) - k/p ], c_j(p)=W_j+S_j/p.

Rigorous whatever the witnesses are; better witnesses only tighten it.

2.3 The numbers

Witness ladder used for (iii): for each j = 0..k, the balanced (Sturmian) word with j long letters among k, its total length scaled by 0.90, 0.94, 0.97, 1.00, 1.03, 1.06, 1.10, and W minimised at that fixed length by SLSQP. 7(k+1) witnesses per n.

MEASURED (the bound is rigorous given the witnesses; the witnesses are explicit and the W values are ordinary floating-point evaluations of a finite sum).

For larger n a single tiled witness suffices, with no optimisation in the loop: take the k = 27 constrained minimiser as a period, tile it, rescale to total length k/H, and evaluate W. At every k from 35 to 400, not a sample, every integer, this gives a bound whose worst value is 0.6751676068 (at k = 55, i.e. n = 56); for k >= 401 the trivial leg alone gives H*400/399 = 0.6741861691. MEASURED.

(Restricting the tiling to k >= 74 the worst is 0.6749544944, at k = 82. Running the section-2.3 envelope over 37 <= n <= 74 as well, it was run to n = 51 here and every value came in under 0.67310, would replace the 0.6751676 figure by that one. The number quoted below is the one that holds without that extension, so it stands on the tiled witness alone over 36 <= n <= 401.)

The tiled witness is no longer how the large-n range is covered. It has a scan in it, one evaluation per k from 35 to 400, and a trivial tail bolted on beyond that. The audit replaced the whole apparatus with one word and a closed-form estimate. 2.3a is that replacement; the tiled figures are kept above as what this hunt's own computations gave.

2.3a The period-37 witness: one word, every n (AUDIT)

Define, for every gap index i >= 1,

g_i = 1 + floor(18 i / 37) - floor(18 (i-1) / 37) . (AUDIT)

This is a period-37 word of nineteen 1s and eighteen 2s. Its prefix of length k = n-1 has total length

S_k = k + floor(18 k / 37) <= (55/37) k ,

and 55/37 = 1.4864864864... while 1/H = 1.4869872916545..., so the length constraint S <= k/H holds for every k, uniformly, with a closed-form margin and no per-n check. That single fact removes the scan.

VERIFIED here (period37_check.py), with this repository's own evaluator famlib.Wsum and, independently, at 60 decimal digits, every window sum of this word is a positive integer, and at integer j the kernel collapses exactly to k(j) = (-1)^(j+1)/(2 pi^2 j^2 - 1), so W is a finite sum of exactly representable terms:

The two evaluators agree to 1.3e-17 or better at every row. The n = 8 row reproduces the audit's interval upper bound 0.003527662393159108 / 0.674873059121154543 exactly. VERIFIED here.

Evaluating the same word at every n from 8 to 401 with famlib.Wsum, the worst value is 0.6750627723649344 at n = 9, which reproduces the audit's n = 9 row 0.675062772364934370 to the digit. VERIFIED here.

The uniform tail estimate. For n >= 12 the audit bounds W without evaluating anything, as follows. At least 5/12 of the prefix gaps are 2s, so the scale-1 contribution is at most 2[(7/12) w(1) + (5/12) w(2)]. Every scale-s window has integer length at least s and w decreases at the integers, so scale s contributes at most 2 w(s); and w(s) <= w(1)/s^4, because 2 pi^2 s^2 - 1 >= s^2 (2 pi^2 - 1) for s >= 1. Summing s >= 2 gives 2 w(1) (pi^4/90 - 1). Hence for every n >= 12

W <= 2[(7/12) w(1) + (5/12) w(2)] + 2 w(1) (pi^4/90 - 1) < 0.003928331920529310 , H (1 + W) < 0.675142509660253902 . (AUDIT)

DERIVED (re-derived here step by step) and VERIFIED here: period37_check.py recomputes the closed form at 60 digits as W <= 0.0039283319205293097915 and H(1+W) <= 0.675142509660253901156, matching the audit's figure. The 5/12 constant is tight and not a round number: the minimum of floor(18k/37)/k over 11 <= k <= 20000 is exactly 5/12, attained at k = 12. VERIFIED here.

Three implementations therefore agree on this number: this file's famlib.Wsum, the 60-digit integer-window sum in period37_check.py, and the audit's own directed-interval run, audit/periodic_certificate.py was re-executed here and returns simple_W_upper = 0.003928331920529309791528358237 and simple_bound_upper = 0.675142509660253901156405110373, byte-identical to its committed output. VERIFIED here.

The estimate is deliberately coarse, the word's actual period-averaged energy is about 0.00317879602211, and the evaluated worst over 8 <= n <= 401 is 0.6750627723649344. The coarseness is the price of a bound that needs no evaluation at all beyond n = 11, and it is what fixes the headline figure in 2.4.

Small n. The word alone does not clear 0.6751676068 at n = 3, 4, 5, 7, and the audit closes those with separately polished witnesses, interval-checked in the safe (upper) direction with mpmath.iv at 100 digits:

AUDIT, interval-checked there; not recomputed here. This range is covered redundantly by this hunt's own section-2.3 envelope, which runs at every n from 3 to 36 and whose worst value over 3 <= n <= 11 is 0.6735202493 at n = 5, a looser bound than the audit's, in the same direction, from a different implementation.

2.4 The barrier, stated

Every n is covered, with no gaps, and after the audit, in two pieces instead of three:

sup_n Phi_n <= 0.675142509660254 < 0.6818286874638 = the configuration ceiling.

This is the figure the file now carries. It is fixed entirely by the coarseness of the uniform tail estimate for n >= 12, not by any witness: every witness actually evaluated, at every n from 3 to 401, comes in below 0.67507. The gain over the number this hunt originally claimed is small, 0.6751676068 -> 0.675142509660254, a move of 2.5e-5, and the real gain is structural: the scan and the trivial tail are gone. One explicit word, whose length constraint holds for every k in closed form, plus five small-n witnesses, now cover every n >= 3. Nothing has to be evaluated at 367 values of k and nothing is patched at the end.

What this hunt's own computations support on their own, without the audit, is the weaker sup_n Phi_n <= 0.6751676068 from the three-instrument table this section used to carry: envelope over n = 3..36 (max 0.6735202493), tiled witness over n = 36..401 (max 0.6751676068 at n=56), trivial leg for n >= 402 (max 0.6741861691). That statement is unchanged and still stands; it is simply superseded.

Deficit at the family's best: at least 0.0066862, 71.7% of the whole distance from H to the ceiling.

This is a BARRIER. The n-point pressure certificate family, at any n and any pressure and with any valid floor c, cannot reach the configuration ceiling. Increasing n does not help; the family's own supremum is short by more than 0.0066, which is 71.7% of the whole distance from H to the ceiling.

2.5 The limit

Upper: Phi_n <= H(n-1)/(n-2) -> H. Lower: for any n, sending p -> infinity sends the floor c*(p) -> 0, hence m -> infinity, H m/(m-1) -> H and k/p -> 0; so sup_p Phi_n >= H for every n.

lim_{n -> infinity} Phi_n = H = 0.6725007036794116 (DERIVED, exact)

There is no uncertainty in Phi_inf: it is the constant H itself, not an estimate. The family does not merely stall below the ceiling, it climbs a little, turns over, and comes back down to H. The measured climb from n=7 to n=9 (+4.3e-5) is the front edge of a bump of total height at most 0.675142509660254 - H = 0.0026418, on a curve that ends at H. The measured climb so far, best floor of the two independent searches at each pressure: 0.6730300 (n=7), 0.6730537 (n=8), 0.6730728 (n=9), 0.6730803 (n=10), 0.6730912 (n=14), 0.6730929 (n=20), increments 2.4e-5, 1.9e-5, 7.5e-6, 1.1e-5, 1.7e-6 against a grid that gets coarser relative to the peak as n grows, so each is a lower bound on that n's true peak.

The rate of the decay is Phi_n - H <= H/(n-2), which is the trivial leg; the actual decay is faster, because the witness envelope in 2.3 already falls below H/(n-2) from n = 7 on. INFERRED: from the envelope's behaviour the excess appears to decay like C/n with C of order 10^-2, but the constant is not pinned here.

2.6 Why W stays of order 1e-3: the infinite-chain energy, exactly

The witness bound is only decisive because the minimal W at density H is small but bounded away from anything like 0.0139. That quantity has an exact form. Write f(t) = cos(sqrt2 t) 1_{[-1/2,1/2]}(t), so K = Fourier transform of f and w = FT(G)/K(0)^2 with G = f * f supported on [-1,1]:

G(u) = (1-|u|) cos(sqrt2 u)/2 + sin(sqrt2 (1-|u|))/(2 sqrt2), |u| <= 1, else 0.

Poisson summation over a period-T lattice therefore terminates at a finite Fourier order, only |nu| <= T survives, because G(nu/T) = 0 beyond that, and the energy per particle of an infinite configuration with P points per period is

W_inf = (1/(P T K(0)^2)) sum_{|nu| <= T} G(nu/T) |sum_j exp(-2 pi i nu x_j / T)|^2 - 1.

DERIVED. VERIFIED against a brute-force lattice sum over 800001 images: agreement 7.9e-9, which is the brute force's own truncation error, not the formula's.

Minimising this over P <= 6 and the positions gives the W_inf(gbar) curve in artifacts/periodic-energy-curve.json. It is violently non-monotone in gbar, commensurability with the kernel zeros, which is exactly why the finite-n ladder's convex envelope, not the curve itself, is what section 1.4 evaluates at gbar = 1/H. MEASURED, and an upper bound on the true minimum since the period is capped at 6.

The finite-n W values are substantially below the infinite-chain ones (7.98e-4 at k = 6, gbar = 1.514 against 4.37e-3 for the corresponding infinite chain), because W_k truncates the sum over window lengths at s = k. That is a real finite-size effect, not a numerical discrepancy, and it is the reason the measured peaks rise with n at all: W_k grows toward W_inf while the k/p charge and the 1/(1+(k-1)c) damping grow too, and the second pair eventually wins. INFERRED as the mechanism; the two rigorous legs of section 2.2 are what actually bound the outcome.

3. Independent numerical check

modal_family_limit.py re-derives the floor by multistart global minimisation, independently of any prediction in this file, at n = 7, 10, 14, 20 over the pressure grid 1200, 1800, 2600, 3000, 3400, 4400, 6000, 8500, 13000, 20000 (the existing sweep covers only 2000..6400, so this also tests whether the peak leaves that window at larger n). 320 cells, 8 shards per cell, 2 CPU each; well inside the compute budget.

Control passed, and the residual 2.0e-13 is not a disagreement, it is the quoted number's argmin. n = 7, p = 3000 returned 0.0038262312113044716, which is 2.0e-13 below the 0.0038262312115073 this laboratory quotes for that cell. The audit, minimising the same functional from its own implementation with an 80-digit Newton solve, landed at 0.00382623121130447424828548285770795421, the same value to twelve digits. Two independent searches below a "floor" is the shape of a bad floor, so it was chased down.

hunts/ainta_seven_point/RESULTS.md states the quoted figure correctly and this file was reading it wrongly. There it is the upper end of a bracket: 0.003826 <= inf F6 <= 0.0038262312115073, with "Upper end: the Arb evaluation at the point." artifacts/modal-results.json shows which point: F6_at_point at the argmin rounded to six decimals, (1.046081, 1.989132, 1.986415, 1.041603, 1.977024, 1.045002). A value at a point is an upper bound on the infimum, never the infimum, and the six-decimal rounding is worth exactly this much.

VERIFIED here (f7_point_check.py), evaluating F_{7,3000} at three points in 60-digit arithmetic:

The first reproduces the Arb string 0.00382623121150731291497... to seventeen digits; the third reproduces the audit's reported value to twenty-five. The spread between the first and the third is 2.03e-13, the whole discrepancy, and all of it argmin refinement.

So: nothing is wrong with the Arb computation, the bracket, or RESULTS.md. What was wrong is the word "floor" applied to 0.0038262312115073, here, and in this hunt's MISSION.md under required_oracles, where it is called "the published n=7 arbitrary-precision floor". It is a rigorous upper bound at a rounded point. The correct control for a minimiser is that it must come in at or below it, which is what happened. The audit's own statement of the same caveat is stronger and should be read: it reports inf F_{7,3000} <= 0.003826231211304474248285482857707954213 and says plainly that it has compelling evidence but no formal global certificate: 562 broad local starts, four differential-evolution runs, 1,710 integer-lobe starts producing 953 distinct local minima, next stationary value 4.20e-5 higher, all six Hessian eigenvalues positive, none of which supplies a lower bound over the full six-dimensional domain. Neither does anything in this file. AUDIT for the value; VERIFIED here for the explanation of the gap.

Agreement with the predicted floors. The prediction under test is c_pred(p) = min_j [W_j + S_j/p] over this file's witness ladder, which by construction is an upper bound on the true floor. At n = 7, 10, 14 the Modal search came in below the prediction at every one of the thirty pressures, by 4.5e-6 to 4.2e-4, the prediction is valid as a bound and loose by the expected amount (the ladder is Sturmian and balanced; the true minimisers relax off it).

Disagreement at n = 20, and it goes the informative way. At n = 20 and p = 3400, 4400, 6000, 8500 the ladder came in below Modal, by up to 2.4e-4:

At k = 19 the Modal job could only enumerate 16000 of the 2^19 two-letter words per shard, and its multistart missed basins the ladder reaches directly. Modal's raw Phi_20 = 0.6731391381 is therefore built on a floor that is not a floor, and is not a valid certificate value; with the better of the two searches at each pressure the n=20 peak is 0.6730928938 at p = 8500. That correction matters: the raw Modal number sits above this file's envelope bound for n=20 (0.6731099910), and the corrected one sits below it. The apparent violation was a search failure, and the bound caught it, which is exactly the use a bound of this kind has.

Peaks, best floor of the two searches at each pressure (MEASURED; grid-limited, so each is a lower bound on that n's true peak):

Every measured value lies below the corresponding bound. The climb continues past n=9 but is decelerating hard: +5.1e-5 from n=7 to 10, +1.1e-5 from 10 to 14, +1.7e-6 from 14 to 20. INFERRED from that deceleration and from the bounds above it: the bump tops out in the low 0.67310s. Not pinned; the witness bound is what is rigorous here, and it is 0.675142509660254.

Artifacts: artifacts/family-limit-modal.json, artifacts/compare.json.

4. What would refute this

The claims here fail if any of the following is exhibited.

1. A configuration below a claimed floor. The witness bound (ii) is only as good as the arithmetic in W(g). Exhibit a g with sum(g) <= (n-1)/H whose W(g) this file reports too high, the bound then moves, though in the safe direction (a smaller W makes the barrier stronger, never weaker). The dangerous direction is the opposite: if W(g) is reported too low, the bound is invalid. Recompute W in interval arithmetic to close this; the sum is finite and exactly the kind of thing Arb settles. Partly closed by the audit, which evaluated every one of its witnesses' cumulative sums, K, w, W, caps and final bounds with mpmath.iv at 100 digits and printed directed endpoints, so its W values in 2.3a are upper bounds in the unsafe direction rather than floats. That is an interval backend, not a proof assistant, and it says nothing about global minimality, but the direction that could make a false barrier look true is now controlled for those witnesses. The period-37 word's W is additionally a finite sum of exactly representable integer-argument terms (2.3a), which removes quadrature from the question entirely.
2. A failure of the chain. This one fired. Steps (A) and the move to the cap were both invalid as written, and the audit exhibited admissible counterexamples to each; section 2.1a is the repair, and the conclusion survived it intact. What remains load-bearing is unchanged and untested here: the chain rests on m being capped at k + floor(1/c) and on c being a floor for F_k on all nonnegative gap vectors. If the proved theorem admits an m above that cap, or admits a c valid only on a restricted set of gap vectors, (A) and (D) respectively fail and the barrier goes with them. This should be checked against n_point_bound line by line, and has not been.
3. A direct minimisation returning Phi_n above the barrier. Any (n, p, g) with the resulting Phi_n > 0.675142509660254 refutes section 2.4 outright. The Modal job in section 3 is exactly this test, run at four values of n and ten pressures; the audit ran it again independently at n = 7, 8, 9, 10, 12, 14, 16, 20, 30, 56, 100 with 91 to 191 starts per n, full binary-skeleton enumeration up to n = 10 and the best 300 to 400 of up to 262,144 skeletons above that, and found nothing above either threshold, its largest best-found constrained W was at n = 100, W < 0.001404828486362211, giving H(1+W) < 0.673445451825039116 (AUDIT, interval-checked there). A wider or deeper run is still the cheapest way to attack the claim.
4. A different n-dependence in the theorem. Everything here is driven by the two n-dependent pieces (n-1)(m-1)/(pm) and the cap m <= (n-1)+floor(1/c). A variant of the certificate with a cap growing faster than floor(1/c), or with the pressure term charging less than (n-1)/p, breaks the cancellation in (ii) and would have to be re-analysed from scratch. The barrier is a statement about this family, not about pressure certificates in general.
5. The exact-arithmetic tail. All optimisation here is float. The floors are upper bounds on infima, so the Phi values built from them are optimistic; that direction is safe for the barrier (a true floor is lower, so the true Phi is lower). But the W values used as witnesses are also float, and there the safe direction is the other one. The margin is 0.0066 against float errors of order 1e-12, so this is not a live risk, but it is the right thing to certify if the barrier is ever leaned on.

5. What is not claimed

• No claim that H is the limit of any other certificate family. Section 2 is about the formula in n_point_bound and nothing else.
• No claim about how to reach the configuration ceiling. This file says one route does not, and says why. It proposes no replacement.
• No claim of exact-arithmetic status for any number in this file. Everything is float, with the reproduction control of section 0 and the direction-of-error analysis of section 4.5.
• The n = 10 .. 16 sweep running in parallel is a different measurement and is not duplicated or pre-empted here; its measured peaks should land below the section-2.3 bounds at the same n, which is a further check on both.
• No claim that the audit closed the formal question. It found a real defect in the argument and supplied a repair; it did not, and does not claim to have, produced a machine-checkable proof of anything. Its own weakest point, stated by it and not by us, is the word "global" applied to inf F_{7,3000}.
• No claim that the audit's W values were recomputed here beyond the period-37 word. The n = 3..7 witness table in 2.3a is the audit's, interval-checked in its implementation, and is carried as such.

6. What the adversarial audit changed

An independent model (OpenAI Codex, gpt-5.6-sol), in an isolated directory outside this repository, from the self-contained brief in audit/BRIEF.md, committed before any work began, see audit/PROVENANCE.md, with no access to this repository, this file, famlib.py or any prior implementation of the kernel. It derived the closed form of K itself and checked it against 100-digit quadrature. It worked 44m55s and was told to refute, not confirm.

What it broke.

1. Step (A) of the chain in 2.1, invalid when its numerator is negative, with an admissible counterexample at n = 3, c = 0.01, m = 3, p = 1. This file had a parenthetical about the negative numerator, but a parenthetical that rescues the conclusion is not a repair of the chain.
2. The move to m at the cap in 2.1, invalid without a monotonicity argument this file never made, with an admissible counterexample at n = 3, c = 0.01, p = 2 where Phi decreases in m. This file's only support for the move was five spot checks, which it presented as VERIFIED. That was the real defect and it was ours.

What it built.

1. The case split of 2.1a, which repairs both and costs nothing.
2. The period-37 word of 2.3a, which replaces a 367-value scan and a bolted-on trivial tail with one explicit construction whose length constraint holds for every k in closed form.
3. A sharper supremum, 0.675142509660254 against this file's 0.6751676068.
4. Interval upper bounds, in the unsafe direction, for every witness it reports.

What it could not break. No numerical counterexample, at any n it was asked for or any it chose. The claim survived.

What it flagged and left open. inf F_{7,3000}: compelling evidence, no formal global certificate. Its value sits 2.0e-13 below the figure this laboratory quotes; section 3 explains why, and the explanation is that the quoted figure was mislabelled here as a floor when hunts/ainta_seven_point/RESULTS.md correctly presents it as the upper end of a bracket, evaluated at a six-decimal argmin.

What did not change. Sections 1, 2.5 and 2.6, and every measured number in them. The verdict, the limit H, and the mechanism are as they were.

Everything in audit/ is the audit's own work, unaltered except for one lexical substitution that this repository's reserved-vocabulary gate forces; audit/PROVENANCE.md lists every occurrence.