The second term for Sidon subsets of [N]

• Slug: sidon-second-term
• List: P07
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort max
• Status: dent of Hou–Zhao 0.9435 and of the folder 0.94325. Independent 11-kernel certificate (free histograms, m=48, L=6) has √(ab)=0.943006169985179, so C<0.94301. Erdős–Turán not claimed.
• Area: Additive combinatorics
• Sources: Erdos Problem #30; Green 100 #31
• Started: 2026-08-17

Statement

A Sidon set has all pairwise sums distinct. F(N) is the size of a largest Sidon subset of [N]. Constructions give F(N) >= sqrt(N)+O(1). The best general upper bound is F(N) <= sqrt(N)+0.98183 N^{1/4}+O(1). Any strict improvement of either side for infinitely many N is open.

Tonight

A finite Sidon record that does not improve an infinite-family bound is not a new bound. Hunt one of:

1. An explicit infinite-family construction with second term beating sqrt(N)+O(1) for infinitely many N, with a verifier.
2. A certified improvement of the 0.98183 coefficient that holds for infinitely many N.
3. A reusable exact lemma that strictly tightens one of those two sides.

Do not claim the Erdos-Turan conjecture. Isolated F(N) tables are an incomplete search unless they feed (1)-(3).

After the fetch (2026-08-17)

Erdős #30 is still open (page edited 2026-04-06). Green #31 is still open. The PROBLEM.md line 0.98183 is CHO25. Hou–Zhao, arXiv:2607.01169v2 (5 Jul 2026), already has F(N) ≤ √N + 0.9435 N^{1/4} + O(1) with an eight-kernel rational certificate. That is the published record this folder has to beat.

This folder independently verifies Hou–Zhao, then certifies an L=6 lift of their kernels: F(N) ≤ √N + 0.94349251 N^{1/4} + O(1), which is 8.22×10^{-8} below their exact γ0. No growing lower-bound second term. Erdős–Turán not claimed. Replay in compute/README.md.

After 2026-08-27

Hou–Zhao is still the published record to beat: 0.9435, with exact γ0 = 0.943492590713545. The L=6 lift does not change the four-decimal statement. A later search re-optimizes the eight kernels as free symmetric histograms at L=6 (the published R≥4 shapes were six-mode profiles). The rational certificate compute/q1/certs/joint_r8_L6.json has

F(N) ≤ √N + 0.94325 N^{1/4} + O(1)

with exact √(ab) = 0.9432425309706136. That is a dent of the published 0.9435, not only of γ0. Replay in compute/q1/README.md. Erdős–Turán not claimed. No growing lower-bound second term.

After q2 (2026-08-27)

q1 leftover refine / dropped-symmetry never finished. An L-lift of those kernels saturates. A later search grows free histograms from the q1 mix and reshapes them at $m=48$. The rational certificate compute/q2/certs/r11_m48_L6.json has

F(N) ≤ √N + 0.94301 N^{1/4} + O(1)

with exact √(ab) = 0.943006169985179. That is below the folder 0.94325 and below Hou–Zhao 0.9435. Replay in compute/q2/README.md. Erdős–Turán not claimed. No growing lower-bound second term.