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

2026-08-17 — literature (do not claim Erdős–Turán)

Fetched:

• Green, 100 Open Problems (update Dec 2025), Problem 31: improve, for infinitely many $N$, the bounds $N^{1/2}+O(1)\le F(N)\le N^{1/2}+N^{1/4}+O(1)$. Comments record Balogh–Füredi–Roy $0.998$ (2021) and Carter–Hunter–O’Bryant $0.98183$ (2023/2025).
• Erdős Problems #30, accessed 2026-08-17 (page last edited 2026-04-06): still OPEN. The page states the \1000 $form$h(N)=N^{1/2}+O_\varepsilon(N^\varepsilon)$and records the published upper bound$h(N)\le N^{1/2}+0.98183,N^{1/4}+O(1)$of Carter–Hunter–O’Bryant, Acta Math. Hungar. 175 (2025), 108–126 (arXiv:2310.20032). Singer gives the matching-order lower bound. Zero claimed proofs. Forum notes an unpublished AlphaEvolve refinement$0.97633$ (Tao, 17 Feb 2026) and a tentative further argument “subject to confirmation”.
• Tao–Davis et al. optimization-constants page $C_{5a}$: published upper bounds stop at CHO25 $0.98183$; $0.97633$ is listed as unpublished.
• Hou–Zhao, arXiv:2607.01169v2 (5 Jul 2026), Vector-valued smoothing for finite Sidon sets: claim $F(N)\le N^{1/2}+0.9435\,N^{1/4}+O(1)$ via an eight-kernel rational certificate. Ancillary code: https://github.com/HbZhao1/sidon-vector-smoothing. This is later than the Erdős #30 page edit and later than the PROBLEM.md “best general upper bound” line. It is the current arXiv record.

Published record I must beat, after the fetch: Hou–Zhao $0.9435$, not CHO25 $0.98183$. Reproducing anything in $(0.9435,0.98183]$ is verification of a weaker constant, not a dent. Isolated small-$N$ Sidon tables are residue.

Do not claim the Erdős–Turán conjecture.

2026-08-17 — tonight’s handle

Three live handles, in order of plausibility:

1. Re-optimize the Hou–Zhao vector-smoothing program (more kernels, finer grid, longer boundary, or a lemma that drops kernel symmetry) and emit an independent rational certificate with $\sqrt{ab}<0.9435$.
2. Independently verify Hou–Zhao’s eight-kernel certificate (so the residue is not “we trusted a PDF”).
3. Lower bound: Singer / Bose–Chowla give $F(N)\ge\sqrt{N}+O(1)$ infinitely often (second term $\to 1/2$ along Singer moduli). A growing second term needs either $\omega(1)$ extra integer points on an infinite family, or an unwrap with a gap $\omega(\sqrt{q})$. Green #32 only guarantees a gap $2\sqrt{p}$ (Shakan), which still produces only $O(1)$ after unwrap. Adding a bounded number of points to Singer is still $O(1)$. This is the historically hard side.

Starting with (2) then (1).

2026-08-17 — Hou–Zhao certificate, independent check

Downloaded HbZhao1/sidon-vector-smoothing into compute/refs/. SHA-256 of sidon_certificate_8kernel.py is 957a5afadd849ac4f97c2b71252abb5c796c2db3c91a608ab35097e3c49292a8, matching the hash printed in arXiv:2607.01169v2 Claim 4.1.

compute/verify_houzhao.py re-reads only the integer tables, rebuilds λ, p, w from the paper’s denominators / η-shift, and checks Lemma 2.1 with fractions.Fraction. Result:

• all 129 covering inequalities hold (min slack = 0 at q = 128)
• least positive slack = 4735171805469436153 / 6250000000000000000000000000000
• a, b match the fractions in Claim 4.1
• √(ab) = 0.943492590713545 < 0.9435

So Hou–Zhao is a real published bound, not a blog claim. The constant to beat is 0.9435. Their two-kernel file sidon_certificate.py only claims C < 0.94601; not needed once the eight-kernel check passed.

Next: independent QP + search (symmetric 1-kernel grid, asymmetric 1-kernel, R=2, R=9 continuation). A floating γ is not a dent.

2026-08-17 — Table 1 replay

search_kernels.py --phase replay re-solves the fixed-kernel QP with our own NNLS dual. The eight floating Table 1 values match Hou–Zhao to all printed digits (R=1: 0.9461473014, …, R=8: 0.94349259006). So the independent solver agrees with theirs.

2026-08-17 — false starts

• 1-kernel (m,L) grid. Best floats sit around 0.9462–0.9468, matching Hou–Zhao’s R=1 row. No path to 0.9435 with one kernel.
• m-refinement of the published histograms (repeat each bin). a is invariant and b does not move. Shape, not grid, is the bottleneck.
• Bose–Chowla + greedy extras (construct_bose.py). Construction checks Sidon for every prime 3≤p≤79. Extra points inside [p²-1]: 1,1 then 0 for all p≥7. Wide-interval extras are +1 and make the second term vs √N worse. Bounded extras are still O(1).
• Singer + greedy extras (construct_singer.py). All 11 primes p≤31 give a Sidon set of size p+1 in [p²+p+1], second term → ½, and zero extras in the interval. Expected: a Singer unfolding already uses every difference in range.
• CHO two-window replay (cho_two_windows.py). Published parameters give b_∞ ≤ 1.990578 on a 400² grid, i.e. C ≤ 0.99529, weaker than Hou–Zhao. Not a dent.

2026-08-17 — the L-lift

Hou–Zhao Lemma 2.1 treats L as a free parameter. Their certificate uses L=4. The same (λ,p) with w padded by ones is feasible for every L'>4, so the L' quadratic program can only decrease b. Independent QP:

Saturates at L=6. a is exactly the Claim 4.1 fraction (kernels unchanged). certify_l6.py re-solves L=6, rounds w at 10^{-12}, adds a common η, and writes certs/hz_kernels_L6.json.

Exact check (verify_certificate.py, then verify_beat_hz.py which does not import the other verifier):

• all 193 covering inequalities hold (min slack 0)
• a equals Hou–Zhao Claim 4.1
• √(ab) = 0.9434925085033526 < γ0 = 0.943492590713545
• ab < (0.94349251)²

So

F(N) ≤ √N + 0.94349251 N^{1/4} + O(1)

holds for all large N. This is a strict improvement of Hou–Zhao’s exact certificate constant by 8.22×10^{-8}. It is not a new method, and it does not improve their four-decimal statement 0.9435 (they already had γ0 < 0.9435). It does improve the PROBLEM.md target 0.98183, but that target was already superseded on arXiv in July 2026.

2026-08-17 — leftover searches killed

Multi-scale sanity recovered the equal-m Hou–Zhao value exactly, so the different-width lemma specialises correctly. The outer Powell search (two widths, then HZ+new scale) and the eight-kernel coordinate descent produced no evaluation below the L=6 plateau in ~9 minutes (one QP evaluation of R=8, L=6 is already expensive; Powell never returned). Killed. grid1 died earlier on an infeasible covering row from a degenerate seed.

No search found a constant at the 0.9434 level. The only certified improvement of Hou–Zhao’s γ0 is the L=6 lift.

Replay:

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

2026-08-27 — q1, continue the vector-smoothing search

Published record is still Hou–Zhao arXiv:2607.01169v2: $F(N)\le N^{1/2}+0.9435\,N^{1/4}+O(1)$. Fetched the abs page and the HTML. Still v2 (5 Jul 2026); no v3. Green #31 (Dec 2025 PDF) still cites CHO $0.98183$. Tao $C_{5a}$ still stops at CHO / unpublished $0.97633$ and does not list Hou–Zhao. Erdős #30 was behind Cloudflare this session.

The 2026-08-17 L=6 lift already has $\sqrt{ab}=0.9434925085<\gamma_0$, but it does not change the four-decimal statement $0.9435$. Hou–Zhao §5 asks for a systematic outer search and more kernels. q1 takes the leftover handles with L-BFGS instead of Powell:

1. joint free-histogram re-opt of all eight published kernels at $L=6$
2. continuation $R=9,10,12$ with 12 cosine modes at $L=6$
3. drop kernel symmetry (independent left/right weights)
4. resample to $m=48$ and re-shape (not a bin-split)
5. two-width L-BFGS

A floating $\gamma$ is not a dent. Code: compute/q1/.

Independent replay this session (exact, then floating QP):

• verify_houzhao.py PASS. Hash matches Claim 4.1. All 129 covering inequalities hold. $\sqrt{ab}=0.943492590713545<0.9435$.
• q1 floating Table 1 matches Hou–Zhao to all printed digits (R=1: $0.9461473014$, …, R=8 L=4: $0.94349259006$).
• Same kernels at L=6: $0.94349250848$. Independent left/right weights on the symmetric mix give the same number, as they should.

2026-08-27 — q1 dent: free histograms beat 0.9435

L-BFGS on the eight published kernels, now as free symmetric histograms (16 half-logits each) at $L=6$, dropped the floating $\gamma$ from the L=6 plateau $0.9434925085$ to $0.9432425303$ (maxiter 40, not declared converged). Adding $R=9,\ldots,12$ twelve-mode kernels from the published mix only reached $0.94326$, so the leftover slack was in the six-mode shape class, not in $R$.

rationalize_certificate.py on candidates/joint-lbfgs-hz8-L6.json wrote compute/q1/certs/joint_r8_L6.json. Exact check:

• all 193 covering inequalities hold (min slack 0)
• $\sqrt{ab}=0.9432425309706136$
• $ab<(0.943243)^2<(0.94325)^2<(0.9433)^2<(0.9435)^2$

Three verifiers that do not share covering code all PASS:

python3 compute/verify_certificate.py compute/q1/certs/joint_r8_L6.json --beat 0.94325
python3 compute/q1/verify_q1.py compute/q1/certs/joint_r8_L6.json --beat 0.94325
# gcc -O2 -o compute/q1/verify_q1 compute/q1/verify_q1.c -lgmp
# python3 compute/q1/dump_cert.py ... && ./compute/q1/verify_q1 ... 94325 100000

So

F(N) ≤ √N + 0.94325 N^{1/4} + O(1)

holds for all large $N$. This is a dent of Hou–Zhao’s published four-decimal statement $0.9435$, not only of their $\gamma_0$. Same lemma, new kernels. SHA-256 of the JSON: edcc2c973809c4bb8a3f25233ffc80e6b5ce432a70c4d01697a3ba8ead8beda5.

Adding $R=9$–$12$ twelve-mode kernels on the published mix reached $0.94326$, worse than the free-histogram $R=8$. A longer L-BFGS refine of the certified mix, the dropped-symmetry search, the $m=48$ reshape, and the two-width L-BFGS had not returned a stricter rational when this was written. Those leftover phases are residue, not a bound.

2026-08-27 — q2, leftover check and a different route

Published record is still Hou–Zhao arXiv:2607.01169v2 (opened abs and HTML this session; still v2, 5 Jul 2026, no v3). Green #31 (Dec 2025 PDF) still cites CHO $0.98183$. Tao $C_{5a}$ still stops at CHO / unpublished $0.97633$ and does not list Hou–Zhao. Erdős #30 was behind Cloudflare again.

Independent replay of the folder record this session:

cd compute/q1 && ./run_all.sh

Python nested-loop and convolution verifiers both PASS on q1/certs/joint_r8_L6.json with $\sqrt{ab}=0.9432425309706136$. SHA-256 still edcc2c973809c4bb8a3f25233ffc80e6b5ce432a70c4d01697a3ba8ead8beda5. The GMP C check needed libgmp-dev in this environment; after that install it also PASS (ab < (0.94325)^2). Do not regress this certificate.

q1 search.jsonl has nosym-start and refine-start and no finished nosym-lbfgs-L6 / nosym-meta / refine-lbfgs row. No finer or widths rows at all. Replay: python3 compute/q2/leftover_check.py. Those two phases are residue. q2 does not continue them.

Hou–Zhao Lemma 2.1 still requires symmetric kernels; §5 asks for more kernels / a systematic outer search, and mentions cross-kernel correlations with a positivity constraint (not implemented here). Route in compute/q2/:

1. L-lift of the q1 kernels (L is free; q1 only certified L=6)
2. block coordinate descent, one mix or one histogram at a time
3. grow R by adding free histograms to the q1 mix, not 12-mode cosine profiles on the published mix
4. resample that mix to $m=48$ and re-block

A floating γ is not a dent.

2026-08-27 — q2 L-lift of the q1 kernels

Independent QP on the stored q1 float mix:

Saturates by L=8. The drop from L=6 is $1.3\times 10^{-11}$, smaller than the rationalization gap on the existing certificate. Same shape class; more boundary is not a new four-decimal statement. Kept as a log, not a new cert.

2026-08-27 — q2 dent: m=48 free histograms, C<0.94301

Block coordinate descent on the q1 mix (one mix or one histogram at a time, four cycles) reached floating $\gamma=0.94324234566$. Adding free-histogram kernels $R=9,10,11$ to that mix only saved another $2\times 10^{-8}$. The leftover slack was not a ninth six-mode profile and not more $R$ on the 32-bin grid.

Piecewise-constant resample of the $R=11$ mix to $m=48$ first jumped up to $0.943709$ (the 32-bin shapes are not already continuous-limit optimal on the finer grid). Two block cycles on that 48-bin mix dropped the float to $0.94300616604$.

rationalize_certificate.py on candidates/resample-m48-best.json wrote compute/q2/certs/r11_m48_L6.json. Exact check:

• all 289 covering inequalities hold (min slack 0)
• $\sqrt{ab}=0.943006169985179$
• $ab<(0.94301)^2<(0.94325)^2<(0.9435)^2$

Three verifiers that do not share covering code all PASS:

cd compute/q2 && ./run_all.sh

that is: leftover check, q1 cert still beats $0.94325$, then

python3 compute/verify_certificate.py compute/q2/certs/r11_m48_L6.json --beat 0.94301
python3 compute/q2/verify_q2.py compute/q2/certs/r11_m48_L6.json --beat 0.94301
# gcc -O2 -o compute/q2/verify_q2 compute/q2/verify_q2.c -lgmp
# python3 compute/q2/dump_cert.py ... && ./compute/q2/verify_q2 ... 94301 100000

So

F(N) ≤ √N + 0.94301 N^{1/4} + O(1)

holds for all large $N$. This is a dent of the folder record $0.94325$ and of Hou–Zhao’s published $0.9435$. Same lemma, finer free histograms. SHA-256 of the JSON: 341cba5bd8364cd315561d1b89ad3e3ba0c9d5160047781d86c40213b53b02c6.

Erdős–Turán not claimed. No growing lower-bound second term.