Walkthrough — the second term for Sidon subsets of [N]
Discovery notes, not a paper. Empty would mean not done.
0. What was actually missing
The missing degree of freedom was not a new Sidon set and not a new energy identity. Hou–Zhao’s Lemma 2.1 already gives
F(N) ≤ √N + √(ab) N^{1/4} + O(1)
for any covering-feasible (λ, p, w) on a boundary of length L kernel widths. Their certificate fixes L=4. L is free. The same kernels with w padded by ones are feasible for every L'>4, so the L' quadratic program can only decrease b, and a is a function of (λ,p) alone.
The degree of freedom was the boundary length on an already-optimal kernel mix.
1. Named false starts
Finite F(N) tables. Optimal Golomb rulers beat Singer’s second term at small k (27 marks, length 553 is already √N + ~0.7 N^{1/4}). That is a finite picture. It does not give an infinite family with √N+ω(1). Rejected by the house rule.
Singer plus extras. Implemented the F_{p^3} discrete-log Singer set for every prime p≤31. Size p+1 in [p²+p+1], second term climbing toward ½, and zero integers in the interval can be added while staying Sidon. Obstruction: the unfolding already realises every difference in range.
Bose–Chowla plus extras. Field implementation for primes 3≤p≤79. Sidon of size p in [p²-1]. Greedy extras inside the native interval: 1,1, then 0 for all p≥7. One extra point far outside makes N grow faster than |A|, so the second term vs √N gets worse. Bounded extras are still O(1).
1-kernel re-optimisation. Symmetric cosine-mode search over (m,L) reproduces Hou–Zhao’s R=1 value ~0.94615 and does not go below ~0.946. The drop from 0.946 to 0.943 is a multi-kernel effect, not a grid effect.
m-refinement without changing shape. Splitting each of the eight histograms into 2, 3, or 4 equal sub-bins leaves a invariant and does not move b. The published kernels are already at the continuous limit of their shape class.
CHO two-window replay. Their hand parameters give b_∞ ≤ 1.99058 (C ≤ 0.99529). Real, weaker than 0.9435, not a new bound.
Claiming 0.98183 as the record. PROBLEM.md and Erdős #30 (edited April 2026) still say 0.98183. Hou–Zhao is on arXiv 5 July 2026. After the fetch, the record to beat is 0.9435 / γ0 = 0.943492590713545.
2. The useful failure
Singer extras = 0 is the useful failure on the lower side: you cannot nibble a growing second term by adding points to a perfect difference set inside its own interval. Green’s unwrap-with-a-gap produces another O(1) unless the gap is ω(√q), which is Problem 32.
On the upper side, L-lifts saturate. L=4→5 saves ~8×10^{-8}; L=5→6 saves ~1×10^{-9}; L≥6 is flat to 12 digits. More boundary is not an endless well. The published mix is already almost fully relaxed in L.
3. The click
Re-solve the published eight-kernel mix at L=6. a is the same rational as Hou–Zhao Claim 4.1. b drops. The floating γ is 0.94349250848, below their exact γ0 = 0.943492590713545. Round the new weights, add a common η so every covering inequality holds over Q, and the exact γ stays below γ0.
That is the whole new bound: same lemma, same kernels, longer boundary, exact arithmetic.
4. The argument, in the order it was found
- Fetch Green #31 and Erdős #30. Still open. Published C = 0.98183 on those pages.
- Fetch CHO25. Then Hou–Zhao arXiv:2607.01169v2: C < 0.9435 with an eight-kernel rational certificate.
- Independently rebuild their certificate from the integer tables.
Hash matches Claim 4.1. Covering, a, b, and √(ab)<0.9435 all pass.
(
compute/verify_houzhao.py) - Re-implement the fixed-kernel QP. Table 1 replays.
- Notice Lemma 2.1 does not prefer L=4. Run the same (λ,p) at L=4,…,12.
- Certify L=6:
compute/certs/hz_kernels_L6.json. - Two verifiers that do not import each other both accept √(ab) < 0.94349251 < γ0.
Hou–Zhao Lemma 2.1 (scalar form, L free): kernels p^{(r)} are symmetric probabilities on m bins; weights w^{(r)} live on n=Lm bins and equal 1 after that; covering
∑_r λ_r ∑_i p_i^{(r)} W_{q+i}^{(r)} ≥ 1 (q = 0,…,n)
with W_j = w_j (j<n) and 1 (j≥n);
a = m ∑_r λ_r ‖p^{(r)}‖²
b = 1 + 2( (1/m) ∑_r λ_r ‖w^{(r)}‖² − L )
and if b>0 then F(N) ≤ √N + √(ab) N^{1/4} + O(1). The O(1) may depend on the finite certificate (hence on L) but not on N. For L=6 one needs N ≥ 12H with H ≍ N^{3/4}, which holds for large N.
5. Computer search
figures/published_C.png— published C against year.figures/table1_replay.png— independent QP vs R.figures/L_lift.png— γ against L for the published mix; the L=4 certificate sits above the L≥6 plateau.compute/search_results.jsonl— every floating experiment.compute/bose_greedy.json,compute/singer_greedy.json— extras = 0 (Singer always; Bose for p≥7).compute/certs/hz_kernels_L6.json— the exact L=6 certificate.- grid1 aborted on an infeasible covering row from a degenerate seed.
- Multi-scale (different m_r) and coordinate descent on the eight histograms were still at the L=6 plateau when this was written.
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
6. What is proved vs still open
Proved here. For all sufficiently large N,
F(N) ≤ N^{1/2} + 0.94349251 N^{1/4} + O(1),
with an explicit rational certificate whose a is Hou–Zhao’s a and whose √(ab) is 8.22×10^{-8} below their γ0. Independently, Hou–Zhao’s own L=4 certificate is valid, so C < 0.9435 was already in the literature.
Still open (2026-08-17). Erdős–Turán (do not claim it). A lower bound F(N) ≥ √N+ω(1) for infinitely many N. Any constant at the 0.9434 level or a new method. The $1000 form h(N)=N^{1/2}+O_ε(N^ε).
The L=6 lift is a strict numerical improvement of a published certificate. It is not a new proof idea. If a later reader wants only improvements that change the four-decimal statement 0.9435, the 17 August search is incomplete.
7. 2026-08-27: the six-mode cage
Hou–Zhao’s Table 1, rows R=4 through R=8, add one six-mode cosine profile at a time. The 17 August coordinate descent and Powell searches stayed inside that cage, or never returned. The 27 August search used L-BFGS on the eight kernels as free symmetric histograms (16 half-bin logits, no mode cutoff) at L=6.
The floating γ dropped by 2.5×10^{-4}, from 0.94349251 to 0.94324253. Adding R=9..12 twelve-mode kernels on top of the published mix only reached 0.94326. The degree of freedom was the shape class of the existing eight kernels, not a ninth kernel and not a longer boundary.
Rounded, η-shifted, and checked over Q: √(ab) = 0.9432425309706136, so F(N) ≤ √N + 0.94325 N^{1/4} + O(1). That does change the four-decimal statement 0.9435. Same lemma. New histograms.
Still open (after §7). Erdős–Turán. A lower bound F(N) ≥ √N+ω(1) infinitely often. The $1000 form. A constant at the 0.9431 level, or a method that is not Lemma 2.1.
8. 2026-08-27 later: the 32-bin cage
q1 leftover refine / dropped-symmetry never finished. An L-lift of the q1 kernels saturates (). Block descent and extra free histograms at only saved . The eight (then eleven) 32-bin shapes were already near a local floor.
The unused handle was the grid of the optimized mix, not of the published mix. q1’s phase started from Hou–Zhao’s histograms and never ran. Resampling the q1-grown mix to 48 bins first made worse (). Two block cycles on that finer grid then dropped it by , to .
Rounded, η-shifted, and checked over Q: √(ab) = 0.943006169985179, so F(N) ≤ √N + 0.94301 N^{1/4} + O(1). That is below the folder 0.94325 and below Hou–Zhao 0.9435. Same lemma. Finer free histograms.
Still open. Erdős–Turán. A lower bound F(N) ≥ √N+ω(1) infinitely often. The $1000 form. A constant at the 0.942 level, or a method that is not Lemma 2.1.