Papers

These are the project’s papers, each written to be read on its own. The first three are one series on $n = 11$ , read in order: Part I proves the project’s own lower bounds, Part II reviews Kleddamag’s certified bound $s (11) > 31 / 8$ , T-037, and Part III reviews the proof that settles the case. Trump’s 1979 packing of eleven squares has been proved optimal by Queuingtheorydotcom’s proof, T-060, which is machine-checked and reviewed here with its review record pending (V3/C3). T-060 settles the case; Part III explains it.

Part INew lower bounds for square packing for

n = 11

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.Part IIA review of the certified lower bound

s (11) > 31 / 8

for 11 squaresExplains Kleddamag’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.Part IIIA review of the optimality proof of the Trump packing of 11 squaresExplains Queuingtheorydotcom’s proof that Trump’s packing is optimal,

s (11) = 3.8770835 \dots

(T-060): construction, case exclusions, capture and local isolation.TutorialSquare packing from first principlesAn introduction for anyone new to the problem: what the objects are, why the approach is shaped the way it is, and what the research has and has not established. Each outside idea it uses, from linear programming to algebraic number fields, is introduced where it is first needed.