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Optimal Packing of Eleven Unit Squares: The Formalized Voronoi-Cell Proof

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Manasseh Ahmed ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We explain the computer-assisted proof that eleven independently rotated unit squares require a square container of side at least T = 3.8770835900228141773 . . ., the side attained by the known construction attributed to Walter Trump. The proof reduces arbitrary packings to 2,184 patterns of occupied closed Voronoi cells. Exact geometric certificates exclude 2,180 patterns, a square-symmetry argument reduces the remaining four to one, and a complete subdivision of that case places every possible smaller packing inside an explicitly certified isolation neighborhood. We give the geometric definitions, prove the continuous soundness rules underlying the finite calculations, and explain the common-cone local isolation argument used in the Lean development. The large certificate instances are identified by pinned source references; they are not replaced by numerical experiments. The reported formal verification uses Lean’s kernel together with its native compiler for specified numerical certificates.

Provenance statement

The initial proof was generated by Astra, building off incremental progress from X. So far, the basic argument idea has been checked and seems reasonable, but the specific computations haven't yet been hand checked. A Lean formalization was also produced.

Tools used

OpenAI
ChatGPTVersion Astra

References

  1. Queuingtheorydotcom and contributors, 11SquaresFormalized , source snapshot cdc746ed907d258057c283aeb6d077cb2c27e349 . https://github.com/Queuingtheorydotcom/11SquaresFormalized/tree/cdc746ed907d258057c283aeb6d077cb2c27e349.link
  2. Queuingtheorydotcom and contributors, 11SquaresOptimal , source snapshot 4262f07c7723428c1725bb757ed06d0c4d28be1a , especially PROOF.md PROOF.md and its linked evidence sources. https://github.com/Queuingtheorydotcom/11SquaresOptimal/tree/4262f07c7723428c1725bb757ed06d0c4d28be1a.link
  3. Source review: why an unconditional v9 PASS is a continuum case exclusion , src/evidence/docs/V9_CONTINUUM_SCOPE_REVIEW.md source note in 11SquaresOptimal .
  4. A focused rectangle around the eleven-square candidate , src/evidence/research/global-math/FOCUSED-LOCAL-RECTANGLE.md source note in 11SquaresOptimal .
  5. Verification report, October 6, 2026 , docs/VERIFICATION_20261006.md report in 11SquaresFormalized .
  6. Acknowledgements and provenance , ACKNOWLEDGEMENTS.md acknowledgements and PROVENANCE.md provenance in 11SquaresFormalized ; THIRD_PARTY_NOTICES.md third-party notices in 11SquaresOptimal .

Version history

  1. v1 · 28 pagesSubmitted by Manasseh AhmedInitial depositCurrentNov 12, 2026