Extremal no-three-in-line subsets of a modular hyperbola in the Hall-Jackson-Sudbery-Wild window

Abstract

Let p be an odd prime and H_c = {(x,y): xy ≡ c (mod p)} the modular hyperbola in the 2p×2p window of Hall, Jackson, Sudbery and Wild (1975). HJSW observed that keeping three of the four points of each residue class gives 3(p−1) points in general position; Kovács, Nagy and Szabó state that this is optimal. We give what appears to be the first complete proof, for every c, and determine the complete extremal structure: all rich lines have slope ±1 and number (3/2)(p−1)−s; an explicit LP certificate shows even the fractional relaxation equals 3(p−1); the number of maximum sets is exactly 9^s, where s ∈ {0,1,2} counts quadratic residues among c and −c, so the HJSW set is the unique maximum iff p ≡ 1 (mod 4) and c is a non-residue. The bound is independent of the window's position, with an exact per-box formula 12n₂+10n₁+8n₀+6s from an orbit lemma; near-maximum sets are stable (a lawful set of size 3(p−1)−t differs from a maximum in at most t points); the no-four-in-line analogue equals (7/2)(p−1)+s. For the union H(1)∪H(−1) we prove α ≤ 4(p−1)−4m₈(p) by an explicit weighted line cover, with m₈(p) = (1/12+o(1))p by Bombieri's estimate along a cubic, and refine it via a block decomposition of the point set along maximal runs of consecutive squares of F_p, reaching α ≤ (3.449…+o(1))(p−1). Independent verification before publication. A verifier written from scratch, sharing no code with the paper's programs, checked: ten statements per instance on all 146 instances (every c, every prime p ≤ 31) — line classification, orbit structure, α and the 9^s count, uniqueness criterion, no-four analogue, orbit-wise stability, exact LP value — all pass; all 1092 box instances (every position, p ≤ 13, c ≤ 3) including the orbit lemma and the exact box formula — zero violations; m₈(p) > 0 for all 236 primes 19 ≤ p ≤ 1500 with density fluctuating around 1/12; the union bound α ≤ 4(p−1)−4m₈(p) certified by LP domination for every prime 19..101; and the block decomposition confirmed structurally for all 57 primes 19..311 (components under rows, columns and slope-±1 lines match the runs of consecutive squares in number, size law and total). Not verified and said plainly: the asymptotic constants beyond the block engine, the seven-point bookkeeping rule, and the sections on cubic graphs and permutation monomials. Disclosure: the computations, the programs and the text were produced by AI agents (Anthropic Claude) under the direction of the author of record, who posed the question, chose what to compute, decided what to claim and is responsible for the content. Two agents worked independently; the verification above was done by the second without reading the first's code. Version 1.15. The verification was pushed one level deeper and found an erratum in remark (b) of the blocks section: at p = 113 the quoted pair was the LP value, not the sum of block optima — the exact recount (MILP per block, the two disputed blocks refereed by a SAT solver, and the LP ceilings forcing the same values) gives Σβ = 382 = 3.4107(p−1), so integrality does bite there, in three of the twelve blocks; the other three quoted pairs are confirmed exactly. A K = 8 constant (3.4616) surviving in the same remark from an earlier draft is replaced by the K = 9 value 3.4482 of the corollary it cites. The block-sum trend was recomputed for every prime 401–1103: mean Σβ/(p−1) = 3.4349 against the note's numerical 3.4413, and the corollary's exact fraction 427101431/123863040, its tail bound and its interval [3.4286, 3.4482] were verified in exact arithmetic. Version 1.16, after a fifth external review — the first fully green one: the reviewer re-verified Theorem 1 for all c at six primes and at p = 101, Theorem 2 on 510 boxes including the printed p = 11 table cell by cell, the no-four corollary, m₈ values and densities, the arithmetic of the far sections (1/3, 115/32 = 11/3 − 7/96, the 24∫m²(1−2m)dm = 1/4 local law, the cubic assembly), the genus of C₀(k), and the novelty claim against Kovács–Nagy–Szabó (their Remark 3.4 yields only |S| ≤ 3(p−1)+s; their Theorem 2.8 is the construction only) — finding no discrepancy. The structural fixes requested: a ten-line abstract replaces the page-long one, whose body survives as an extended summary; the strong-form cubic theorem is labelled numerically assisted (its constants are quadrature, not enclosure, as the text already declared); the version history is put in chronological order and the in-text tag updated.