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AI-found construction improves the planar cube maximal constant

For the centered Hardy–Littlewood maximal operator over axis-parallel cubes in R², the optimal weak-type (1,1) constant c₂ is at least Φ=1.6855099933…, so c₂>1.685.

Sources for this record

What was already known

Aldaz’s earlier planar cube construction gave a lower bound of about 1.62119. Lin supplied a later seed near 1.66856.

AI involvement AI-centered

Yongxi Lin set the target and supplied a seed. Claude Fable 5.1 found the weighted periodic configuration and exact certificate; Claude Opus 5 built the Lean proof.

The human designed the objective and supplied a seed; Claude's search found the retained weighted lattice and exact Φ certificate. The proof was subsequently formalized by an agent.

Importance Notable

Improves a longstanding lower bound for the sharp planar cube constant.

Limitations

The result is only a lower bound; the separate cube and ball upper bounds have different origins. The exploratory search files and research brief are unavailable, and no external human proof review is documented.

Technical verification Formal (Lean 4/Mathlib kernel, Comparator, NanoDa and con-ron)

Palomar v3 reports Comparator matching the exact lower-bound declaration and numerical bounds to the Mathlib-only challenge, with Lean, NanoDa and con-ron accepting the exported proof. The six AI referees and numerical exploration are separate checks and are not counted as human checking.

Formal checking

Palomar v3 reports Comparator matching the exact lower-bound declaration and numerical bounds to the Mathlib-only challenge, with Lean, NanoDa and con-ron accepting the exported proof. The six AI referees and numerical exploration are separate checks and are not counted as human checking. This review did not rerun Lean.

Checker
Palomar Comparator, Lean kernel, NanoDa and con-ron
Scope
CenteredMaximal.ofReal_phi_le_weakTypeConstant_two together with CenteredMaximal.lt_phi and phi_lt at registration v3
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

The detailed proof account is present, but the documented research and proof writing were agent work. Lin's brief and oversight do not by themselves document a human explanation of the retained weighted-lattice mechanism; Few is the missing-evidence default.

Correction and publication history