Paul Erdős posed the finite partition problem in 1955. Given a partition of
1, …, 2n into two sets of size n, count how often each integer occurs as a
difference between opposite colors; then minimize the largest count. The exact
asymptotic constant remains unknown.
CodexProLong continued Hyra's public solution #2440 and submitted solution #2507. The frozen result is an explicit 3,584-step construction with recorded continuous objective bound
0.3808585748578583091444423330164409480469.
This is an upper-bound improvement, not a determination of the unknown
constant. The selected machine-readable certificate record is
artifacts/certificates/erdos-min-overlap-continuous.json,
and the campaign verification receipt is
artifacts/receipts/erdos-min-overlap.json.
The submitted payload contained n = 3,584 binary64 values. The certification
procedure parsed every value as binary64, converted it with
as_integer_ratio(), lifted the dyadic rationals to a shared denominator, and
normalized them exactly.
Write the resulting integer numerators as X_i, set T = Σ X_i, and let
m = n/2 = 1,792. Define f_i = m X_i / T on consecutive subintervals of
width 1/m in [0, 2], and let g = 1 − f on that interval and zero outside
it. The recorded checks establish exact integral 1, values in [0, 1], and
all 2n − 1 = 7,167 grid lags.
For grid lag k, if A_k is the sum of source numerators in the overlapping
range and B_k is the corresponding sum of products X_i X_j, the overlap is
the rational
(T A_k - m B_k) / T².
The unique recorded maximizing lag is −192.
Both f and its complement are constant on the same regular grid. Between
consecutive grid shifts, the set of intersecting interval pairs is fixed and
each intersection length is affine in the shift. Their weighted sum is
therefore affine there, so its maximum occurs at an endpoint. This reduces the
continuous maximum to the checked grid boundaries.
The receipt pins candidate SHA-256
43d6096c5ebd143a03f56e5c07de335e2c1b64bf3485336633df16d7f8257db6
and verifier SHA-256
7c0e78d9dc40f27584ee2de01348fddcc6ff4a540908ddc902a4c6ef920920b0.
The certificate preserves the exact fraction, normalization facts, maximizing
lag, and continuous-reduction statement.
Its certificate_source_sha256 pins the certification program from an earlier
public revision; that program is not included in the current release tree.
The candidate bytes are not redistributed in this release because the search continued a public Arena payload whose redistribution terms were not established. Consequently, this page documents the frozen certificate and its claim boundary; it does not claim that a fresh clone can recompute the certificate without obtaining the referenced payload through authorized means.
- Paul Erdős, “Some remarks on number theory” (1955).
- Jan Kristian Haugland, The minimum overlap problem revisited (2016).
- Ethan Patrick White, Erdős' minimum overlap problem (2022).
- Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, and Adam Zsolt Wagner, Mathematical exploration and discovery at scale (2025).
- Sungyoon Kim and Mert Pilanci, AI-Assisted Discovery of Convex Relaxations via Dual Agents (2026).
The approved claim is: CodexProLong recorded an improved upper-bound construction with an exact-arithmetic certificate record. It must not be shortened to “Codex solved the problem.” The exact constant remains open.