C_4b: three 2026 lower bounds (Krachun 0.752796; Jones 0.753742; Naslund 0.758067)

[JD-Jones-ASES]

Three rows appended to the lower-bound table of constants/4b.md, the README cell and "Recent progress" updated, references added. Markdown only. (Updated 20 Sep 2026: the third row, Naslund's 0.75806746, was added after its registration on 19 Sep; the first two rows are unchanged and now stand as the intermediate bounds.)

Row 1 — Krachun (arXiv:2608.01325, 2 Aug 2026), 0.752796… — ten Paley chains glued by CRT; the first bound past 3/4. The page listed nothing after Lewko 2015, so this row is added for the record and because the next two rows build on it.

Row 2 — Jones (23 Aug 2026), 0.753741541837329405… — Krachun's gluing over an eleven-block pool: his nine chains (the chain at 23 dropped because 23 | 299) plus two certificates on the square-free composite moduli 235 and 299, lifted by one new lemma. The Comments column lists both certificates in full (17 and 19 residues, heights 11 and 12) and the closed form, so the value can be recomputed without downloading anything; I recomputed both rows' constants from that data before submitting (including the two longest-path heights).

Row 3 — Naslund (19 Sep 2026), 0.75806746 (exact 37903373/50000000) — the current best. An interval-moment criterion (digits carry ordered intervals; a component in square base b contributes a moment power f) over nine pairwise coprime components: six Paley chains, two composite three-low-digit codes at 215⁶ and 437⁶, and a 25-state recursion at the prime 2 in base 4^(10^10). The Comments column gives the criterion, the nine bases, the chain lengths and the exact surplus Σ fᵢ − α = 25671/10¹². From the paper's parameter table I rechecked the six chains (every forward difference a nonzero square), the six chain-power thresholds and the surplus identity; the composite and binary moments were not recomputed here. The row says so.

Supporting material lives outside this repository, per the contributing guidelines:

• Rows 1–2: note, data, and a stdlib-only verification script: https://github.com/JD-Jones-ASES/fs-lower-bound (commit c2c0687); Lean 4 / Mathlib formalization of the full chain (definitions, the new lemma, Krachun's two lemmas reproved, the liminf statement, and 0.7537 < alphaInf by kernel-checked integer comparison; standard axioms only): https://github.com/JD-Jones-ASES/fs-formal (commit 4e79656); Palomar registry entry PALOMAR-2026-08-26-000004: https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000004&version=1
• Row 3: papers, exact certificate programs and the Lean 4 / Mathlib formalization (Lean 4.32.0; criterion and full value; standard axioms only): https://github.com/enaslund/sarkozy-lower-bound-0.758 (commit e5d6937, Apache-2.0); Palomar registry entry PALOMAR-2026-09-19-000006: https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-19-000006&version=1. Not on arXiv at the time of writing.

The recorded values are the limits of closed forms or exact rationals; each construction is finite and explicit at every stage. If the maintainers prefer the unverified marker for a bound whose write-up is a repository note rather than a paper, please add the asterisk; in each case the registered formalization is offered as the verification.

AI-use disclosure. Rows 1–2: the result is AI-generated with a human managing the workflow (see the repository's DISCLOSURE.md). Row 3: the cited papers disclose that they were written by GPT-6-Astra under the author's supervision and the Lean development with Codex agents. This pull request's text and the three rows were drafted with Claude (Anthropic); the references were checked by the contributor for rows 1–2 and read directly from the registry record and the repository for row 3; the contribution note on the page says so.