Furstenberg–Sárközy exponent for square-difference-free sets

Upper bound, last-listed row

| $1$ | Trivial |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $0.75806770413$ (exact $75806770413/10^{11}$) | [Jon2026b] | Naslund's criterion and components, reallocated: the same nine pairwise coprime components — Paley chains at $p = 3, 7, 11, 31, 59, 103$ (lengths $2, 3, 4, 7, 9, 11$, equal widths $1/t$ on one digit, the other digit free), the two composite three-low-digit codes at $215^6$ and $437^6$ ($4913$ and $19683$ retained words, free multiplicities $215^3$ and $437^3$), and the 25-state binary recursion with its transition rules unchanged (depth $10^{15}$, new rational row weights, certified growth factor $a = 1430119207986461/(5 \cdot 10^{14})$ with $\log\_2 a > 2\alpha$) — with several retained words of the $19 \cdot 23$ code replaced, intervals repositioned, rational widths changed, and all nine moment powers reallocated: $f = 0.15494199\ldots$ (binary), $0.18194473\ldots, 0.08579838\ldots, 0.10723242\ldots, 0.08916535\ldots, 0.04217280\ldots, 0.00239690\ldots$ (chains), $0.02653617\ldots, 0.06787893\ldots$ (codes); $\sum\_i f\_i - \alpha = 38652/10^{15}$. The bound is unconditional with an explicit constant, $r(N) \ge c\,N^{\alpha}$ for every $N \ge 1$, from a criterion proved for every $k \ge 1$ (the same repository gives $D\_4(N) \ge c\,N^{0.9142}$ and $D\_6(N) \ge c\,N^{0.95295}$ for fourth- and sixth-power differences). Not an asserted optimum of the method. Lean 4 formalization of the criterion and of the full value, registered as Palomar entry PALOMAR-2026-09-21-000004 (standard axioms only; the certificate data are checked by proved Lean verifiers with kernel replay); no paper, the mathematical account is the repository's `PROOF.md`. |value changed 2026-09-26

Checked against material for its cited source

• Another row, not one shown above: bound row $\approx 0.752796$ line 20 citing [Kra2026] — the claim we checked: that the cited paper, arXiv:2608.01325, states the lower bound ≈ 0.752796 from ten Paley chains at p = 3, 7, 11, 19, 23, 31, 43, 59, 71, 103 with lengths t = 2, 3, 4, 5, 5, 7, 7, 9, 9, 11, glued by the Chinese remainder theorem. What we read supports it. This verdict does not cover the row's formula for alpha, beyond what the paper itself states. source link (read in the body: arXiv v1 (the only version, and the edition the entry cites), Theorem 1 on p. 2 and the proof on p. 5, read from page images). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.