Independent audit of the barrier claim

Verdict

The supporting argument is false as written. Steps 2 and 3 both omit necessary conditions. Step 2 reverses direction when its numerator is negative. Step 3 moves to the largest allowed m even though Phi is not always increasing in m.

This is a refutation of the stated argument, not a numerical counterexample to the final claim. A case split repairs both defects. After that repair, I found no numerical counterexample. In fact, an explicit gap word gives a uniform bound for every n >= 8, and interval-checked witnesses handle 3 <= n <= 7. The resulting all-n bound is

Phi_n < 0.675142509660254 < 0.6751676068 < 0.6818286874638.

For the requested numerical searches, the largest best-found W was at n = 100:

W < 0.001404828486362211
H(1 + W) < 0.673445451825039116.

Thus none of the requested dimensions breaks either threshold.

Kernel and constants

Put q = 1/sqrt(2) and use sinc(z) = sin(z)/z, continuously extended at zero. Product-to-sum gives

K(x) = 1/2 [sinc(q - pi*x) + sinc(q + pi*x)].

An equivalent form, useful away from the removable singularities, is

K(x) = [q sin(q) cos(pi*x) - pi*x cos(q) sin(pi*x)]
       / [q^2 - pi^2*x^2].

I checked the first formula against direct high-precision quadrature at seven points from x = 0 through x = 19. The largest discrepancy at 100 decimal digits was 4.02e-102.

My independently computed constants are

K(0) = 0.9187253698655684   (16 digits)
H    = 0.6725007036794116   (16 digits)

More digits are

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

The exact W levels corresponding to the two quoted barriers are

0.6751676068 / H - 1
  = 0.00396565104838863604939044560637081256...

0.6818286874638 / H - 1
  = 0.01387059334414481885584355033936444273...

The brief's rounded 0.0138706 is consistent with the recomputed value.

Algebraic audit

Write q0 = n - 1 >= 2, d = m - q0 >= 1, and

D = m - c*d.

The cap gives c*d <= 1, hence

D >= m - 1 >= 2.

The original denominator is at least 1 - 1/m >= 2/3. It cannot vanish or change sign. The numerator can be negative.

Step 1

Valid. Multiplying numerator and denominator by the positive integer m gives

Phi = [H*m - q0*(m-1)/p] / [m - c*(m-q0)].

Step 2

Invalid as stated. Replacing D by its lower bound m-1 is valid only when

N = H*m - q0*(m-1)/p >= 0.

When N < 0, the inequality reverses.

A genuinely admissible counterexample is

n = 3, c = 0.01, m = 3, p = 1.

For this triple,

Phi                         = -0.6630427722280151...
claimed step-2 upper bound  = -0.9912489444808825...

so the claimed inequality is false.

The certificate c = 0.01 is genuine for both p = 1 and p = 2. For n = 3, if S >= 1/2, then F >= S/p >= 1/4. If S < 1/2, restricting the defining integral to |t| <= 1/4 gives

K(S) >= 15/(32*sqrt(2)),
2 w(S) >= 225/1024 > 0.01.

Here the integrand is nonnegative on the full integration interval and K(0) < 1. Thus F >= 0.01 for every nonnegative gap vector, and the cap is also satisfied.

The flaw does not by itself threaten a positive barrier. A negative numerator makes Phi < 0.

Step 3

Invalid without an omitted monotonicity argument. The cap gives only

m <= q0 + floor(1/c).

For fixed n,c,p, the sign of the discrete change in m is the sign of

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

Therefore Phi can increase, stay constant, or decrease with m.

For another genuinely admissible instance, take n = 3, c = 0.01, p = 2. At m = 3,

Phi = 0.005853548842219049...

while step 3 would assert

Phi <= H + H*c - 2/p = -0.3207742892837943...,

which is false. At the largest allowed m = 102, the actual value is -0.3208408735118813..., confirming that Phi decreased.

The numerical inequality m_max - 1 >= 1/c is valid at m_max = q0 + floor(1/c). What was missing was permission to replace the original m by m_max.

Steps 4 and 5

Step 4 is valid once step 3 has been repaired. The certificate condition gives, for every chosen witness,

c <= S/p + W.

Because H > 0, substitution has the claimed direction. Calling c "the floor" is imprecise: admissibility only says that c is an accepted lower bound for the floor.

Step 5 is also valid once step 4 is available. If S <= q0/H, then H*S - q0 <= 0, and division by p > 0 preserves that sign.

Repair

The same final witness implication can be recovered.

If Phi <= H, then W >= 0 immediately gives Phi <= H(1+W). Now suppose Phi > H. Direct subtraction gives

Phi - H = [H*c*d - q0*(m-1)/p] / D > 0.

It follows that

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

This is exactly the condition that makes Phi increase with m. In the only branch that can threaten a bound above H, moving to m_max is therefore legitimate. At m_max, the numerator is positive, step 2 has the correct direction, and m_max - 1 >= 1/c. Steps 3 through 5 then yield

Phi <= H(1+W).

Requested constrained searches for W

The following values are directed interval upper bounds for the exact decimal witnesses, rounded upward to 18 decimal places. S is also rounded upward and the cap downward, so each displayed feasibility comparison is safe.

The largest best-found W is the n = 100 value. Its resulting bound is below the stated ceiling by more than 0.001722154974960884 and below the target by more than 0.008383235638760884.

The complete exact decimal vectors are in WITNESSES.md and the machine-readable values are in search-results.json (search-results.json). None of these bounds exceeds either 0.6751676068 or 0.6818286874638.

Optimisation and local-minimum defenses

I used SciPy SLSQP with exact analytic gradients for the nonnegative simplex constraint, then solved the unconstrained or equality-constrained KKT equations with the exact dense Hessian. The search starts included:

• regular spacings at several total lengths;
• alternating and periods 3, 4, 5, and 7;
• clustered vectors with exact zero gaps;
• edge-heavy and center-heavy vectors;
• random Dirichlet vectors with concentration parameters from 0.15 to 20;
• integer and nearby-kernel-zero skeletons;
• balanced 1/2 words, cyclic phases, and explicit low-low defects.

The base searches used 91 to 191 starts per requested n. For n = 7,8,9,10, every cap-feasible binary 1/2 skeleton was polished. For n = 12, the best 300 of 1,024 skeletons were polished from both integer and nearby-zero starts. For n = 14,16,20, the best 400 skeletons were selected from 4,096, 16,384, and 262,144 feasible binary words respectively. The higher dimensions also received balanced-word and defect seeds.

Every reported point has KKT residual below 1.7e-15. The smallest relevant Hessian or projected-Hessian eigenvalue decreases from 0.2935 at n = 7 to 0.01852 at n = 100, but remains positive at every reported point. This proves strict local minimality in the relevant free directions, not global minimality. Global minimality is unnecessary for the witness direction: any feasible point gives a safe upper bound on the minimum W.

The only active cap among the final requested vectors is n = 30. An initial solve using the binary64 cap produced a decimal sum about 5e-15 above the true cap. The final vector was re-solved with an explicit 1e-10 cap backoff, then interval-checked. This is why its reported W is about 1.5e-14 above the mathematical equality-constrained stationary value.

An explicit all-n construction

There is no uncontrolled large-n tail. Define, for every gap index i >= 1,

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

This is a 37-period word containing 19 ones and 18 twos. For q0 = n-1, its prefix length is

S_q0 = q0 + floor(18*q0/37)
     <= (55/37) q0.

Directed intervals give

55/37 = 1.486486486486486486...
1/H   = 1.486987291654509269...

so every prefix satisfies the length cap.

Every window sum is a positive integer. At integer j >= 1, the kernel simplifies exactly to

k(j) = (-1)^(j+1) / (2*pi^2*j^2 - 1),
w(j) = 1 / (2*pi^2*j^2 - 1)^2.

These values decrease with j. For q0 >= 11, at least 5/12 of the prefix gaps are twos. The scale-one contribution is therefore at most

2 [(7/12) w(1) + (5/12) w(2)].

Every scale-s window has integer length at least s, so its whole contribution is at most 2w(s). Also w(s) <= w(1)/s^4. Consequently, 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.

The final margin below 0.6751676068 is more than 0.000025097139746098. The estimate is deliberately coarse. The computed period-averaged energy is about 0.00317879602211.

The same word directly handles the remaining n = 8,9,10,11 cases:

Separate polished witnesses close n = 3 through 7:

The exact finite-prefix scan through n = 599 in periodic-certificate.json (periodic-certificate.json) agrees: this simple word fails the claimed threshold only at n = 3,4,5,7, precisely the cases replaced above. Together with the repaired algebra, these witnesses establish the stated ceiling for every n >= 3.

F_7,3000

The best-supported numerical value is

F_7,3000 = 0.00382623121130447424828548285770795421241114973478...

at

(1.04608035577143543724508620334,
 1.98913202062119593163024512345,
 1.98641493610882775469506115045,
 1.04160329372111326218470199081,
 1.97702352233741593883693409703,
 1.04500209461784484467937259671).

Its reversal is a second minimizer by symmetry. At this point,

S = 9.08525622317783316927140116178...
W = 0.000797812470245196525195015804...

An 80-digit Newton solve reduced the gradient residual below 5e-102. The Hessian eigenvalues are

0.2008115114, 0.2906542995, 0.4030075399,
0.6826625458, 0.7330771658, 1.3114628294.

The point is therefore a well-conditioned strict local minimum.

The global search used 562 broad local starts, four differential-evolution runs with 120 population members and up to 800 generations, and 1,710 integer-lobe starts. The latter produced 953 distinct local minima up to numerical clustering. The next stationary value found was 0.0038681976920066, about 4.20e-5 higher.

There is also a strong compact-domain reduction. With the exact rational incumbent U = 0.003826231211305, any better point must satisfy

S <= 3000 U = 11.478693633915

and every singleton term forces w(g_i) <= 3U. This sharply restricts each coordinate to the low regions surrounding the kernel zeros.

I do not claim a rigorous global certificate for this value. Differential evolution, basin enumeration, positive Hessian, and high-precision agreement are serious evidence, but none supplies the required lower bound over the full six-dimensional domain. A true certificate would need interval branch-and-bound plus interval Newton isolation of every remaining stationary box. Therefore the honest statement is

inf F_7,3000 <= 0.003826231211304474248285482857707954213,

with strong numerical evidence for equality. I do not use this number as an admissible c, and I found no concrete admissible triple exceeding the claimed ceiling.

Interval status and reproducibility

All reported witness gaps are stored as decimal strings. I evaluated their cumulative sums, K, w, W, caps, and final bounds with mpmath.iv at 100 decimal digits. The printed upper and lower endpoints use explicit decimal ceiling and floor rounding. This certifies the unsafe witness direction relative to the interval backend. It is not a proof-assistant certificate or a global-minimum certificate.

The implementation has independent checks for:

• the two-sinc kernel against direct quadrature;
• analytic first and second derivatives against finite differences;
• the O(n^2) gradient against central differences;
• reversal invariance;
• direct evaluation of the definition.

All six automated tests pass. Main artifacts:

• audit.py: kernel, derivatives, objectives, multistart searches;
• w_binary_challenge.py and w_balanced_challenge.py: structured basin attacks;
• refine_witnesses.py: KKT refinement;
• interval_verify.py: directed point-witness checks;
• periodic_certificate.py: the all-n construction and tail bounds;
• f7_landscape.py and high_precision_f7.py: the F_7,3000 search and refinement;
• constants.json (constants.json), intervals-requested.json (intervals-requested.json), f7-high-precision.json (f7-high-precision.json), and periodic-certificate.json (periodic-certificate.json): machine-readable outputs.

Bottom line

The audit found a real formal defect: steps 2 and 3 are not universally valid. The numerical claim itself survived every attack. After an explicit case split repairs the algebra, the period-37 construction plus five small-n witnesses controls every n and stays strictly below 0.6751676068. The weakest unresolved numerical point is the word "global" for the reported F_7,3000 minimum. I found compelling evidence, not a formal global certificate.