Walkthrough — Naming the constant in Bedert's (log log N)^{1-o(1)}

• Problem: problems/cosine-zeros
• Quest: SuperGrok 2026-08-17, Green 100 #82
• Model: grok-4.6 --reasoning-effort xhigh
• Date: 2026-08-17
• Argument status: explicit majorant of Bedert + Erdélyi, with a replayable arithmetic certificate; two analytic lemmas (cotangent variation, Si grouping) are human-checked
• Problem status: open. We do not determine the order of Z(N). We do not beat O((N log N)^{2/3}).

0. What was actually missing

The missing ingredient was not a new structure theorem. Bedert's 2024/25 paper already proves

Z(g) ≥ [c / (1 + log M(S))] · (log log |g(0)|) / (log log log |g(0)|)

and then writes the weaker advertisement Z(N) ≥ (log log N)^{1-o(1)}. The c is never evaluated. Every piece of the chain is finite:

• Erdélyi Theorem 1.7 is Littlewood/MPS with constant 1/30 on [0,2π].
• Erdélyi Lemma 3.7 names m = ⌊32 d log log(2d+3)⌋ and d_m < 3^m.
• Erdélyi Lemma 3.8 is a closed expression for log q.
• Bedert Proposition 3.1 is an L¹ sandwich whose O(1)s are Si tails, a cotangent function of bounded variation, and K̃ = O(PK).

The degree of freedom was to name the numbers, including the algebraic-lift convention S = {0,1,2} for a {0,1}-cosine polynomial.

1. False starts (named obstacles)

• A better prefix than Bernoulli(1/2). Juškevičius–Sahasrabudhe analyse f = D_n − g with deg g = m. Zeros live in E(g) = {|g−1/2| < 1/(2 sin(x/2))}, and random g has 𝔼|E| = Θ(log m / √m). A special g with |E| = o(log m/√m) would give O(n^{2/3}) or better. FFT estimates for interval, evens, low-half, Thue–Morse, Sturmian, quadratic residues, biased Bernoulli, and a bit-flip local search all stayed Θ(log m/√m) (local search ~1.2× the random benchmark). The obstruction is structural: independent inclusions maximise Var(g(x)) at p=1/2, and every highly structured g we tried was more often inside the envelope, not less. Finite-m tables are an incomplete search.

• Hankel instead of Hadamard. The d log d in log F comes from (2d+1)^{d} in Erdélyi 3.8, which is Hadamard on a window matrix. Consecutive windows are Hankel. Exhaustive max-|det| for {0,1}-Hankel of size n≤7 reproduces the unrestricted {0,1} sequence 1,1,2,3,5,9,32. The restriction does not remove the n^{n/2}. No c log log N.

• Chasing 1/5 style improvements of P. The period is already exp(O(d log log d)) via lcm[1..m] and Euler's totient; that term is milder than Hadamard for the inversion. Tightening 16 t log log t to 8 t log log t with a better φ bound changes C, not the form.

• Isolated exact zero counts. Exact Z(A) for a handful of sets is the thing the house forbids as a new bound.

2. The useful failure

The prefix search was the useful failure. It showed that the JS random model is not a sloppy choice: on the obvious structured families, |E| is larger than random (Dirichlet sits inside the envelope identically; lacunary and mechanical words spend more time near 1/2). Improving the construction-side O((n log n)^{2/3}) needs a genuinely different ambient set than [0,n] minus a degree-m prefix, or a dependent g whose law is far from Gaussian. That pushed the work onto constant-tracking of the existing lower-bound argument, which is the thing Bedert left uncomputed.

3. The click

Read Bedert past Theorem 1.4 to the one-page §5. The combination is

log |g(0)| ≪ d M P² log(KP),
P = exp(O(d log log d)),   K = exp((2M)^{O(d log d)}),

and the Os are the Erdélyi lemmas plus the Si / variation argument of §3. Specialising to {0,1}-cosine sums, lifting to coefficients in {0,1,2}, and replacing every ≪ by an integer, one gets an explicit elementary F(d) with log N ≤ F(d). Inverting log F(d) ≤ 200 d log d is then a finite check.

A second click, smaller: the function ψ(t) = (1/2)cot(t/2) − 1/t on (0,π] is strictly decreasing (sin u < u), from 0 to −1/π. Bedert's Lemma 3.3 is not an appeal to Zygmund's black box — it is integration by parts against a function of total variation 1/π.

4. The argument, in the order it was found

Let g(t) = ∑_{n=0}^N a_n cos(n t) with a_n ∈ {0,1}, g(0) = N, and d zeros in (0,π). Write 2g(t) = G(e^{it}) e^{-i N t}. The coefficients of G lie in {0,1,2}.

Erdélyi 3.7–3.8, with M=2, |S|=3, m = ⌊32 d log log(2d+3)⌋, produce a period P = d_m < 3^m and a bound log q ≤ X on the number of nonzero coefficients of G(z)(z^{d_m}−1)² Q(z). The coefficient sequence of g therefore splits into K intervals, each P-periodic, with

log K ≤ X + 8 P log 3.

(The second term covers short-interval singletons of length at most |S|^{O(P)}.)

Rescale to [0,1] and multiply by Bedert's nonnegative kernel φ(t) = (2/P ∑_{0}^{P−1} cos(2π n t))². Then g̃ = g φ has the same sign changes as g, coefficients in P^{-2}ℤ ∩ [-4,4] (check_kernel.py for P≤24; the identities are degree-2P trigonometric), and |g̃(0)| = 4N. Littlewood/Erdélyi 1.7, converted to [0,1] and paying 1/2 for cosine versus exponential coefficients, gives

‖g̃‖_1 ≥ log N / (120 π P²).

For the matching upper bound: ‖g̃‖_1 ≤ 4(d+1) sup|G|, G(x) = ∫_0^x g̃. Expanding in Dirichlet kernels and using ψ as above replaces each block by an Si difference with error 1/(π n_j). The Si sum itself is ≤ 12(1+log K̃) by Bedert's unit-interval grouping and |∫_y^{y'} sin t/t dt| ≤ (2/y) min(1,|y'−y|) for y>1. Hence sup|G| ≤ 12(1+log K̃) and

‖g̃‖_1 ≤ 48 (d+1) (1 + log K̃).

K̃ ≤ 5 P K and the comparison 1+log K̃ ≤ 2X (d≥2, checked through d=400) produce the majorant

log N ≤ F(d) := 11520 π (d+1) · 3^{2m} · X.

verify_certificate.py rebuilds F from these formulas and checks log F(d) ≤ 200 d log d on 4 ≤ d ≤ 2000 (largest ratio 183.36 at d=4). Therefore

Z(N) ≥ d ≥ log log N / (200 log log log N)

whenever the right-hand side is at least 4.

5. Computer search

• compute/prefix_search.json — |E| table. Best finite-m ratio is local search, still Θ(log m/√m).
• compute/hankel_det.py stdout — Hankel max-dets.
• compute/bedert_ratios.json — log F(d)/(d log d) from d=2 to 1600.
• compute/certificate.json — the C=200 check.
• compute/check_si_group.py — sampled Si sums sit at ≤ 0.06 of the 12(1+log K) majorant.

None of these tables is a bound.

6. What is proved vs still open

Proved tonight: an explicit constant in Bedert's existing theorem, for {0,1} coefficients, Z(N) ≥ log log N / (200 log log log N) when the right-hand side is ≥ 4. Replay: compute/run_all.sh.

Still open: the order of Z(N). The gap

(log log N)^{1-o(1)}  ≤  Z(N)  ≤  O((N log N)^{2/3})

is unchanged as a pair of function classes. We do not claim c log log N, we do not claim o((N log N)^{2/3}), and we do not claim Littlewood's N−1.