Papers

The $n = 11$ series:

Part I proves the project’s lower bounds.
Part II reviews Marc Leonard Overbeck’s certified bound $s (11) > 31 / 8$ (T-037).
Part III reviews Manasseh Ahmed’s machine-checked proof that Trump’s 1979 packing of eleven squares is optimal (T-060, V3/C3); the review record is pending.

How Record Square Packings Are Found follows the discovery process from a seed through search and local refinement to a certified upper bound, with figures drawn from published record packings.

New Lower Bounds for Square Packing for $n = 11$

September 5, 2026

How weighted points and 2-of-3 threshold atoms prove T-018, T-025 and T-026,

s (11) \geq 3.8264 \dots

, with interactive figures.

A Review of the Certified Lower Bound $s (11) > 31 / 8$ for 11 Squares

October 5, 2026

Explains Marc Leonard Overbeck’s proof that

s (11) > 31 / 8

(T-037): five-site k-of-m charges, k-of-m charges on shrunken parents with strict cores, and a reoptimized certificate over 12,028 angle rows.

A Review of the Optimality Proof of the Trump Packing of 11 Squares

September 30, 2026

Explains Manasseh Ahmed’s proof that Trump’s packing is optimal,

s (11) = 3.8770835 \dots

(T-060): construction, case exclusions, capture and local isolation.

Methods tutorial

How Record Square Packings Are Found

October 8, 2026

How seeds, search, local refinement and exact checks produce record upper bounds.