On 17 September 2026 ζ(5) was proved irrational with integer polynomials built from Hankel determinants of a positive moment functional. The obvious next targets are ζ(7) and Catalan's constant, and two claimed proofs of them are already in circulation. We computed the method's polynomials exactly, divided out every common factor, and measured how fast they shrink at the constant: the true margin, not a provable bound on it. ζ(3) and ζ(5) clear zero; ζ(7) and Catalan's G do not, in any of the 323 families tried. Every headline number was re-run on this machine from the lifted engine and agrees with the bench to the last bit.
Local working document, machine-derived, not peer-reviewed. Per instance the margins are exact and ball-certified: P is an exact integer polynomial, its content an exact rational, log P(ξ) an interval, and every headline row was re-run here. The wall — no family reaches ζ(7) — is a measurement across 323 families at h ≤ 99, extrapolated in h and in k; it is not a theorem and excludes nothing outside the families scanned. The claimed proofs for ζ(7) and Catalan's constant, and the reported problems with them, are race context we have not verified.
For an integer k ≥ 2 the functional μk,X acts on rational functions with simple poles at −j² (Calegari's normalisation of Fauzan): on monomials through Bernoulli numbers, on poles through the harmonic sums Hj(k) = Σv≤j v−k:
At X = ζ(k) it is integration against a positive weight, so the Hankel matrix G(X) = [μX(W ti+l)], W = DNr/DK with Dm(t) = Πj≤m(t + j²), is a Gram matrix there and Δ(X) = det G(X) = det(A + XB) is positive at ζ(k). Δ is computed exactly in ℚ[X] and divided by its content; what is left, P, is the smallest integer polynomial the irrationality lemma can use. If log P(ξ) ≤ −c·h² along a family, the family proves ξ irrational once its arithmetic is established, so log P(ξ)/h² is the family's true margin. Papers prove upper bounds on the denominators; this is the exact content. A second weight, 1/sinh(πy), with half-integer poles gives X = β(k), and β(2) is Catalan's G.
Each family below is the one the bench's notes name for its constant, and each was re-run here at every h the bench recorded, 51 instances in all with Calegari's K = 12n family. From h = 14 on every one keeps its sign (Catalan G starts on the other side at the smallest h). The ± is the margin's ball: P(ξ) evaluated in arb until its radius is below 2−60 of its value, the logarithm taken in arb, the endpoints rounded outward; the bench's own value is good to about 10−9, which is the width of the comparison.
| constant | family | h | margin, re-run here | ball (±) | bench | agreement | verdict |
|---|---|---|---|---|---|---|---|
| ζ(3) | Z · integer poles · K = 4m, N = m, r = 4 | 39 | −1.3797 | ± 5.6e-16 | −1.3797 | agree · bit-identical | clears zero |
| ζ(5) | Z · integer poles · K = 8m, N = m, r = 6 | 35 | −0.1851 | ± 5.6e-17 | −0.1851 | agree · bit-identical | clears zero |
| ζ(7) | Z · integer poles · K = 8m, N = m, r = 4 | 35 | +1.0157 | ± 3.3e-16 | +1.0157 | agree · bit-identical | above zero |
| Catalan G | E · half-integer poles · K = 8m, N = m, r = 1 | 35 | +0.4284 | ± 1.4e-16 | +0.4284 | agree · bit-identical | above zero |
Take one family, K = 60, N = 5, r = 6 (h = 55), and compare k = 5 with k = 7. The decay of Δ itself barely moves: log Δ(ξ) is −2625 at k = 5 and −2313 at k = 7. What moves is the content. Going from 5 to 7 adds two powers of p to Hj(k) at every node j ≥ p, a naive price of 2(K − p) per prime. Prime by prime the measured cost scatters around that price — above it for p = 11, 13, 17, 19, 23, below it for p = 29, 31, 37, equal for p = 41, 43, 47, 53, 59 — and weighted by log p over p > √K it totals 2160 against a naive 2200, 98%. The primes 2, 3, 5, 7 add 969 more, because they lose common factors ζ(5) keeps. So log P(ξ) goes from −567 to +2874.
| p | v_p(content), ζ(5) | v_p(content), ζ(7) | extra cost | naive 2(K − p) |
|---|---|---|---|---|
| 11 | −5 | −105 | 100 | 98 |
| 13 | −55 | −154 | 99 | 94 |
| 17 | −117 | −221 | 104 | 86 |
| 19 | −131 | −237 | 106 | 82 |
| 23 | −173 | −251 | 78 | 74 |
| 29 | −167 | −201 | 34 | 62 |
| 31 | −170 | −208 | 38 | 58 |
| 37 | −140 | −178 | 38 | 46 |
| 41 | −104 | −142 | 38 | 38 |
| 43 | −90 | −124 | 34 | 34 |
| 47 | −62 | −88 | 26 | 26 |
| 53 | −20 | −34 | 14 | 14 |
| 59 | −2 | −4 | 2 | 2 |
Both profiles were recomputed here from the lifted padic.py and are equal to the bench's, prime by prime. The valuations are exact integers; the percentage is their log-weighted ratio.
The bench scanned the shape (K = am, N = bm) and the zero multiplicity r for both weights and both pole lattices: 300 families in the main scan, 11 in a first scan at larger r, and 12 structural variants — 4,902 exact determinants, 0 errors. The best margin at the largest h computed (h ≥ 30) is below, with the alternating weight beside it; that weight is worse at every k.
| constant | best family (weight Z, or E for G) | margin | families | best with weight E | verdict |
|---|---|---|---|---|---|
| ζ(3) | Z · integer poles · K = 6m, N = m, r = 4 · h = 40 | −1.38 | 30 | +0.15 | clears zero |
| ζ(5) | Z · integer poles · K = 8m, N = m, r = 6 · h = 35 | −0.19 | 60 | +1.41 | clears zero |
| ζ(7) | Z · integer poles · K = 8m, N = m, r = 4 · h = 35 | +1.02 | 85 | +2.59 | above zero |
| Catalan G | E · half-integer poles · K = 8m, N = m, r = 1 · h = 35 | +0.43 | 30 | — | above zero |
The structural variants — zeros above the poles, odd poles only, staircase multiplicities, zeros interleaved with the poles, integer and half-integer poles mixed, a window of zeros inside the poles — are all worse than Fauzan's shape (a contiguous block of poles, integer zeros below it): +0.22 to +2.97 for ζ(5), +1.46 to +5.01 for ζ(7), every one of them positive.
Read across k, the best margins sit on a line: 0.60·h² per unit of k, crossing zero at k ≈ 5.31 (least squares through k = 3, 5, 7; the bench's notes round this to "k ≈ 6"). That is what "the class stops near ζ(5)" means here: a measurement over these families at h ≤ 99, extrapolated. It proves nothing about a family outside them, and no limit in h is proved. ζ(7) by this route needs a structurally new idea — more decay, or arithmetic savings that grow with k — not tuning.
Calegari's reconstruction uses K = 12n, N = n, r = 6, h = 11n and proves P(ζ(5)) ≤ e−3n²/2, which is −0.0124·h². Fauzan's abstract states degree 37n and exp(−139n²/5), −0.0203·h². The true margin of Calegari's own family, re-run here to h = 99, is −0.152·h² — 12 times what his normalisation proves and 7 times Fauzan's rate. The same family at k = 7 sits at +0.992·h². So a much better irrationality exponent for ζ(5) than Fauzan's 260 exists in principle; claiming it needs the arithmetic proved for the better normalisation, which nothing here does.
Both published rates are read from the sources as fetched and pinned on 2026-10-05; that is a comparison of their stated exponents with our exact computation, not a check of their proofs.
Two proofs past ζ(5) are circulating. We have checked neither, and nothing below is a finding of this machine; the sources are pinned in the corpus as fetched on 2026-10-05.
What this page adds is narrower and checkable: within the class of Hankel families measured, the margin at k = 7 is positive, and the reason is arithmetic. A proof of ζ(7) by this method has to leave the class, and a reader of one can ask where it does.
The ledger (every re-run row with its ball, the bench value beside it, the fingerprint of P, the cross-check, the class reading and its label) · the engine and its provenance in instruments/zetahankel/ · the bench's runs in certs/zeta-hankel/runs/ · the sibling lane, the ζ(3) sheet, decided, where the Ramanujan Machine's ζ(3) conjectures are re-decided with certificates.