R-8539DC: the third autocorrelation constant, and the functional it belongs to

Run 9698c990-e812-4890-94d8-f30a65b820af, 2026-08-23. Issue teal-sea/zeta-lab#123 (https://github.com/teal-sea/zeta-lab/issues/123). Reproduce with python3 hunts/r_8539dc/probe.py (standard library only).

Verdict in one line: the reporter was right about the artifacts he read, the authors fixed it in December 2025, and the improvement survives, but the two problems are still labelled in opposite senses by the paper and the colab, and the public problem page has not been updated.

1. The two functionals, and why the published abs() is a no-op

Write f for a step function on [-1/4, 1/4] with n equal steps of width h = 1/(2n) and heights a_0 … a_{n-1}. Then f*f is supported on [-1/2, 1/2], is piecewise linear with knots at multiples of h, and takes the value h·b_k at the k-th knot, where b = a ⋆ a. A piecewise-linear function attains its extrema at knots, so with ∫f = h·Σa:

The published verification cell computes abs(2n·max(conv)/sum²). That outer abs is a no-op, and not by accident: ∫_{-1/2}^{1/2} f*f = (∫f)² > 0 forces max_t f*f(t) > 0 for every admissible f. So the code computes A, cleanly and unambiguously, while the inequality printed above it defined B. Those are two different problems, exactly as Tao said.

Since A ≤ B always, a bound under B is also a bound under A, never the reverse. That asymmetry is what makes the mix-up load-bearing rather than cosmetic.

2. Both functionals on both published sequences, exact

Heights are published as ten-place decimals, so they are exact rationals with denominator 10^10; the whole computation below is integer arithmetic, no float anywhere in the chain. Thirty digits shown, truncated.

Read this table as follows.

• The n=400 construction reproduces 1.4557 only under A. Under B it gives 4.3340…, which is not an improvement on anything, it is three times the trivial π/2 bound. The convolution's extreme knot is negative (min_k b_k = −4.3340… at knot 215, against max_k b_k = +1.4556… at knot 176): the construction buys its A-score by pushing the autoconvolution deeply negative somewhere else, which is precisely the freedom that B forbids. This confirms the reporter's observation on the artifacts as published in November 2025.
• The n=150 construction gives the same number under both functionals, its max|b| is its max b. So it is a legitimate bound for both problems, and it is the one that has to be compared with Matolcsi–Vinuesa's 1.4993. It also reproduces 1.4688 exactly, which is the check that keeps this hunt's discretisation honest: had the knot argument or the 2n factor been wrong, this number would not have landed.

3. Which convention each prior bound belongs to

Answered by the corrected paper itself, not inferred here:

arXiv v1 cited 1.45810 to matolcsi-vinuesa; the correction moved that citation to vinuesageneralized. So the misattribution the problem page confesses to ("some results for one problem incorrectly attributed to another") is visible in the v1→v2 diff as a changed \cite key, not just as a changed formula.

4. Was it corrected? Yes, twice, in December 2025

The brief's premise that "no corrected table has appeared" is therefore wrong, and the reason it looked true is worth naming: the correction landed in the paper and in a different repository's notebook, while the issue that reported it and the problem page that hosts it were both left as they were. A reader who follows the report to its own thread still sees an unfixed problem.

height_sequence_3 was not changed by the correction, the fix moved the absolute-value bar in the statement to match the code, kept the number, and added a second construction for the other problem. That is the honest repair, not a quiet restatement: the n=400 object always was a witness for A.

5. Verdict

1. Which number is stated against which functional. 1.4557 and its predecessor 1.4581 are bounds on A = max f*f / (∫f)², the sign-unrestricted version of the first autocorrelation problem. 1.4993 and 1.4688 are bounds on B = max|f*f| / (∫f)², the Matolcsi–Vinuesa problem. As published in November 2025, 1.4557 sat under a statement that defined B; that is the defect the reporter found, and it was real.
2. Does the claimed improvement survive the definition the paper writes down? Under the corrected statements, yes, and with no number moving: 1.4557 < 1.4581 under A, 1.4688 < 1.4993 under B, both verified here in exact arithmetic. Under the original v1 statement, no: the n=400 witness scores 4.3340… on that functional. Nothing was withdrawn; a statement was repaired and a second experiment was reported.
3. What remains broken. The primed and unprimed labels are swapped between the two corrected artifacts. arXiv v2/v3 calls max|f*f| the constant C_3 (1.4993 → 1.4688) and |max f*f| the constant C_3' (1.4581 → 1.4557). The colab calls |max f*f| C_3 (1.4581 → 1.4557) and max|f*f| C_3' (1.4993 → 1.4688). Anyone citing "AlphaEvolve's C_3" without saying which artifact they read has a fifty-fifty chance of naming the other problem. The public problem page, which is what the paper's per-problem URL points at, has neither correction.

This hunt makes no new bound and takes no position on what should be said to whom upstream. Both quantities above are measured on the certainty ladder, in the strong sense that every arithmetic step is exact rational arithmetic on the published decimals; what is not exact is the published sequences themselves, which are ten-place truncations of whatever the search actually found. A truncated witness is still a witness, the bound is whatever the written-down step function gives, and that is what was computed.

Loose threads

• The n=400 witness is a very lopsided object under B. Its autoconvolution reaches −4.334 while its positive peak is +1.456, a ratio of about 2.98. Whether an A-optimal construction must be lopsided under B, or whether this one merely is, would say something about how far apart the two constants can be. First step: sweep A-optimising step functions at small n (say n ≤ 24, where exhaustive local optimisation is cheap) and record max|b| / max b for each optimum. If that ratio grows with n, the two problems separate structurally rather than numerically.
• 1.4581 is sourced to page 75 of a 2009 thesis, and this hunt did not open it. The convention attribution in §3 is taken from the corrected paper's own citation change, which is a strong but secondary source. First step: fetch https://www.icmat.es/Thesis/CVinuesa.pdf, read p. 75, and confirm the construction there is scored under max f*f and not max|f*f|. This matters because it is the one link in the corrected chain that was repaired by moving a citation rather than by rerunning a computation.
• The sibling problem has no public problem page. problems/4.html says a sibling will be added; nine months on there is none, so the repository's problem list is missing an entry that the paper references. First step: check whether problems/*.html numbering has a free slot or whether adding one renumbers the set, which would tell you whether the omission is editorial inertia or a structural cost.
• The lab has no instrument for "recompute a published constant from a published witness under a stated functional". This hunt hand-rolled 60 lines of exact rational convolution to do it. Reported as a Core candidate in HANDBACK.json, not built here.