Research log — second term for Sidon subsets of [N]

Status (accessed 2026-08-27, q2 session)

Opened this session:

• arXiv:2607.01169 abs — still v2 (5 Jul 2026). Theorem $F(N)\le N^{1/2}+0.9435 N^{1/4}+O(1)$. No v3.
• arXiv HTML v2 — Lemma 2.1 requires symmetric kernels; $L$ is free; §5 asks for more kernels / a systematic outer search, and mentions cross-kernel correlations with a positivity constraint (not implemented here).
• Green, 100 Open Problems — Problem 31, Dec 2025 update: still $N^{1/2}+O(1)\le F(N)\le N^{1/2}+N^{1/4}+O(1)$; comments record BFR $0.998$ and CHO $0.98183$. Does not cite Hou–Zhao.
• Tao et al. $C_{5a}$ — published upper bounds still stop at CHO25 $0.98183$; $0.97633$ still unpublished. Does not list $0.9435$.
• Hou–Zhao GitHub — landing page only; local snapshot in compute/refs/ already matches Claim 4.1. Did not clone.
• Erdős #30 — Cloudflare interstitial; page body not readable this session.

Published record to beat is still Hou–Zhao $0.9435$. Folder record is the q1 certificate $C<0.94325$.

Status (accessed 2026-08-27)

Opened this session:

• arXiv:2607.01169 abs — still v2 (5 Jul 2026). Theorem $F(N)\le N^{1/2}+0.9435 N^{1/4}+O(1)$. No v3.
• arXiv HTML v2 — Lemma 2.1 requires symmetric kernels; $L$ is free; Table 1 saturates near $R=8$. §5 asks for more kernels / a systematic outer search, and mentions cross-kernel correlations with a positivity constraint (not implemented here).
• Green, 100 Open Problems — Problem 31, Dec 2025 update: still $N^{1/2}+O(1)\le F(N)\le N^{1/2}+N^{1/4}+O(1)$; comments record BFR $0.998$ and CHO $0.98183$. Does not cite Hou–Zhao.
• Tao et al. $C_{5a}$ — published upper bounds still stop at CHO25 $0.98183$; $0.97633$ still unpublished. Does not list $0.9435$.
• Hou–Zhao GitHub — landing page only; local snapshot in compute/refs/ already matches Claim 4.1. Did not clone.
• Erdős #30 — Cloudflare interstitial; page body not readable this session. Last successful access in this folder was 2026-08-17 (OPEN, edited 2026-04-06).

Published record to beat is still Hou–Zhao $0.9435$. Anything in $(0.9435,0.98183]$ is a weaker constant.

Status (accessed 2026-08-17)

• Erdős Problem #30 — OPEN. Page last edited 2026-04-06. $1000 form:$h(N)=N^{1/2}+O_\varepsilon(N^\varepsilon)$. Zero claimed proofs. Do not claim Erdős–Turán.
• Green, 100 Open Problems, Problem 31 (Dec 2025 update): improve, for infinitely many $N$, $N^{1/2}+O(1)\le F(N)\le N^{1/2}+N^{1/4}+O(1)$. Comments record BFR 0.998 and CHO 0.98183.
• Tao et al., constant $C_{5a}$ — published upper bounds stop at CHO25 0.98183; 0.97633 listed as unpublished AlphaEvolve (Carter–Georgiev–Gómez-Serrano–Hunter–O’Bryant–Tao–Wagner, 2025). Forum note (Tao, 17 Feb 2026) mentions a tentative ~0.947 argument “subject to confirmation”.

Published upper bounds on the $N^{1/4}$ coefficient

Hou–Zhao is later than the Erdős #30 page edit and later than PROBLEM.md’s “best general upper bound 0.98183”. It is the current arXiv record. Code: https://github.com/HbZhao1/sidon-vector-smoothing. The 8-kernel verifier hashes to 957a5afadd849ac4f97c2b71252abb5c796c2db3c91a608ab35097e3c49292a8, matching Claim 4.1.

Independent checks (this folder)

• compute/verify_houzhao.py — PASS. Rebuilt λ, p, w from the integer tables. √(ab) = 0.943492590713545 < 0.9435. a, b match Claim 4.1.
• compute/search_kernels.py --phase replay — Table 1 floats match.
• compute/cho_two_windows.py — CHO Thm 2.1 parameters give b_∞ ≤ 1.990578 (C ≤ 0.99529) on a 400² sample of (w1,w2). Weaker than Hou–Zhao.
• compute/construct_singer.py — Singer unfolds for primes p≤31: size p+1, second term → ½, 0 extra points in [p²+p+1].
• compute/construct_bose.py — Bose–Chowla for primes 3≤p≤79 is Sidon of size p in [p²-1]; greedy extras inside the interval are 0 for all p≥7.

What we certified

Same eight kernels and mixing weights as Hou–Zhao, boundary length L=6 instead of L=4 (a valid instance of their Lemma 2.1). Exact certificate compute/certs/hz_kernels_L6.json:

• a is the Claim 4.1 fraction (kernels unchanged)
• √(ab) = 0.9434925085033526
• so $F(N)\le N^{1/2}+0.94349251\,N^{1/4}+O(1)$
• this is 8.22×10^{-8} below Hou–Zhao’s γ0, and below (0.94349251)²

Verifiers (do not import each other):

python3 compute/verify_certificate.py compute/certs/hz_kernels_L6.json --beat 0.94349259
python3 compute/verify_beat_hz.py

This is a microscopic tightening of the published certificate, not a new argument and not an improvement of the four-decimal statement 0.9435.

2026-08-27 — q2 certificate

compute/q2/certs/r11_m48_L6.json, SHA-256 341cba5bd8364cd315561d1b89ad3e3ba0c9d5160047781d86c40213b53b02c6. Eleven free symmetric histograms, m=48, L=6, grown from the q1 mix. Exact √(ab) = 0.943006169985179 < 0.94301. Parent nested-loop verifier, q2 matrix-vector verifier, and a GMP C nested-sum verifier all PASS. Beats the folder 0.94325 and Hou–Zhao 0.9435.

q1 leftover refine / nosym confirmed incomplete (compute/q2/leftover_check.py). L-lift of the q1 kernels saturates.

2026-08-27 — q1 certificate

compute/q1/certs/joint_r8_L6.json, SHA-256 edcc2c973809c4bb8a3f25233ffc80e6b5ce432a70c4d01697a3ba8ead8beda5. Eight free symmetric histograms, m=32, L=6. Exact √(ab) = 0.9432425309706136 < 0.94325. Parent nested-loop verifier, q1 convolution verifier, and a GMP C nested-sum verifier all PASS. Published record beaten: Hou–Zhao 0.9435.

OEIS / tables (residue only)

• A003022, A143824, A227590 — finite Sidon / Golomb-ruler values. Isolated F(N) tables are not a dent.

Did not beat

• Erdős–Turán / the $1000 form.
• $F(N)\ge\sqrt{N}+\omega(1)$ for infinitely many N. Singer gives +½+o(1); Shakan’s gap 2√p unwraps to another O(1). Greedy extras stay bounded.
• A constant at the 0.942 level or below. L-lifts of both the published mix and the q1 mix saturate; 1-kernel search stays ~0.946; equal-bin splits of a fixed shape are inert. The m=48 reshape of the q1-grown mix is what moved the folder record to C<0.94301.