[DennisKoshta]

Summary

This PR records

$$C_{1c} \ \ge\ \frac{134648548624804778248790016}{328144294688458468928334649} \ =\ 0.41033335274\ldots \ \ge\ 0.4103333,$$

which improves the listed $0.4103287$ ([R2026], exact value $21072342208353792/51354782902387969 = 0.41032871754\ldots$) by $4.6 \times 10^{-6}$. Equivalently, $\tau = C_{1c}^{-1/2} \le 1.56111$. It appends one row to the lower-bound table of constants/1c.md, adds the [Kos2026] reference and a contribution note, and updates the README cell for 1c and "Recent progress". Markdown only. The bound is printed rounded down. It carries the asterisk, being a self-archived computation, until someone replays it.

Stacked on #243 (which is itself stacked on #232). #243 edits the same row, README cell and Recent-progress block. This branch therefore carries #232's commit 6d83f98 and #243's commit 53b9999 underneath, so until they are merged the diff also shows their changes to 1a.md, 1b.md, 1d.md and the README. This PR's own change is the single commit 4a6c4b4 (constants/1c.md and README.md only). Once #232 and #243 are merged only that commit remains.

Construction and scoring

The certificate has the same form as [R2026]: $f = \sum_i w_i 1_{[i/K,(i+1)/K)}$ with $K = 24576$ and 24592 non-negative integer weights, the last 8 of them zero, so the support is $[0, 3073/3072)$. For equal steps, $t \mapsto \int f(x)f(x+t),dx$ is linear between consecutive multiples of $1/K$ ([K2021], proof of Thm 3.1). Hence

$$\min_{0\le t\le 1}\frac{\int f(x)f(x+t),dx}{(\int f)^2} ;=; \frac{K\min_{0\le k\le K}\sum_i w_iw_{i+k}}{(\sum_i w_i)^2},$$

a finite computation in integers. The minimum is attained at $k = 21264$.

How it was found (not needed for validity). We split each cell of the [R2026] certificate in two ($K = 12288$), maximised a soft-min of the $K+1$ normalised correlations with L-BFGS-B (FFT gradients), split again ($K = 24576$), and optimised for about 25 more minutes. The weights were then scaled to a maximum of $10^{12}$ and rounded. The shape is that of [R2026]: about 22% of the mass still sits in the interval $[1/6144, 2/6144)$, next to the left end (now four nearly equal cells), and the optimiser left it there. Re-optimising the [R2026] certificate at its own $K = 6144$ already gives $0.4103327$, so part of the gain is convergence and part is resolution.

Verification

• verify.py (Python 3 standard library only, about a second) computes all lags $A_k=\sum_i w_iw_{i+k}$ at once and exactly by Kronecker substitution: one big-integer product, with slots wide enough that no carry crosses a slot. It checks that the result equals the fraction stored in the certificate. --direct also re-sums every lag term by term. --selftest runs the controls: the ruler ${0,1,4,6}$ gives $3/8$; a claim $10^{-15}$ too high is refused; 200 random cases agree with direct sums; negative, fractional, boolean, string and all-zero weights are refused. On [R2026]'s certificate it prints that certificate's stated fraction.
• verify_independent.py, written by a separate Claude agent from the definition and the certificate format only, with no access to our code, reproduces the same fraction and reports that it beats [R2026]'s.
• [R2026]'s own verify.py and check_independent.py (commit 19f3971; the latter integrates block overlaps directly at the minimiser and at off-grid shifts) accept the file.
• PROOF.md in the artefact gives the argument, and the integer comparisons against [R2026] and against $0.4103333$.

How to reproduce

git clone https://github.com/DennisKoshta/c1c-autocorrelation-certificate c1c-cert && cd c1c-cert && git checkout a4ee94ef29ff
sha256sum -c SHA256SUMS
python3 verify.py K24576_0.4103333.json            # exact; prints the fraction and OK, about a second
python3 verify.py --selftest                       # controls
python3 verify.py K24576_0.4103333.json --direct   # term-by-term re-sum of every lag, 1-2 min
python3 verify_independent.py --beat 21072342208353792/51354782902387969 K24576_0.4103333.json   # VALID ... BEATS

Credits

• rozetyp [R2026], for the step-function certificate this one refines and its exact scoring.
• Kravitz [K2021], for the linearity argument.
• C. Toledo [T2026] / C1c, C1d: the two step functions independently replayed; asterisks removed, and the C1d bound printed rounded up #243, for the replay conventions followed here.
• T. Tao and contributors, for this table.

Disclosure (AI use)

This bound was found and prepared with Claude (Anthropic, Claude Opus 5.5), directed by Dennis Koshta, who takes responsibility for the submission. Claude did the numerical re-optimisation and wrote the certificate, the verifiers and this text. The independent check was done by a separate Claude agent with its own verifier, written from the definition and the certificate format only. No human mathematician has checked the argument.

🤖 Generated with Claude Code