● Covered by 1 source · 1 reportMedium impact1 neutral
AI-assisted proof verifies optimal packing for 11 squares using Lean 4.34.1
Key points
- Optimal packing proof for 11 squares verified.
- Verification used AI assistance and native numerical certificates.
- Proof passed EvolvingPrograms verification with zero admissions.
- Project pins Lean 4.34.1 and Mathlib revision d13f23b723b8a846827a245b89c10fc7d3f11612.
Proof Verification Details
The complete optimality proof for packing 11 squares has successfully passed verification. This process involved the use of native numerical certificates and was audited by EvolvingPrograms, which accepted all 7,920 local Lean modules with zero admissions. The verification report provides evidence and scope of this achievement.
Methodology and Trust Model
Selected expensive, exact numerical certificate checks utilized native_decide, while geometry, checker soundness, and proof assembly relied on ordinary Lean proofs. The final theorem trusts Lean's kernel and native compiler, indicating a verification claim that extends beyond kernel-only verification. Approved numerical declarations and their exact source hashes are recorded in verification/native-certificates.json.
Optimal Side Length and Construction
The optimal side length is defined by the formula T = (6u+4)/(1+2u-u^2), where u is the unique root in (9/25, 37/100) of the equation 5u^8-10u^7-2u^6+14u^5+12u^4-6u^3+2u^2+2u-1=0. This construction achieves an approximate value of 3.8770835900228141773. The model accommodates arbitrary orientations, legal boundary contact, and disjoint open interiors for the squares.
Technical Environment
The project pins Lean 4.34.1 and Mathlib revision d13f23b723b8a846827a245b89c10fc7d3f11612. The public statements in ElevenSquare/Optimality.lean and the complete T03 source tree remain unchanged from the repository's previous main branch. Verification can be run on Linux with Python 3, Git, curl, and tar, or on macOS after installing the elan launcher.
Verification Process
The verification script checks every local module and performs a final source, receipt, dependency, and axiom audit. Successful completion requires OPTIMALITY_PROVED_WITH_NATIVE_CERTIFICATES, zero admissions, and a trust_model of lean_kernel_and_native_compiler in the final result. Reaching 100% of compiled modules alone is not sufficient for full verification.
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Primary sources
GitHub cmc_internal/api GitHub github/collect GitHub Queuingtheorydotcom/11SquaresFormalized GitHub _private/browser GitHub get-started/accessibility GitHub open-source/sponsorsReporting from
An AI-assisted proof has been verified for the optimal packing of 11 squares, passing verification with native numerical certificates. This achievement demonstrates the increasing capability of AI in complex mathematical proofs, particularly in geometry and optimization problems.