Planar universal optimality, tiered: kernel-checked consequences, a non-uniqueness theorem, and a CVODE N-body consistency check for the Yukawa interaction
- 1SocrateAI Lab
Abstract
The openai/math release contains a Lean 4 proof that the triangular lattice minimises the lower g-energy among all locally finite planar configurations of density one, for every smooth completely monotone g. Every claim this note makes about it is filed in a claim ledger with the tiers of the SocrateAI-Scientific-Elenchus method: A kernel-checked, L cited literature, C argument, and X floating-point exploration, which may steer a search but never supports a claim. Tier A, conditional on the fidelity of upstream's definitions and of our transcription: the rescaled triangular lattice is universally optimal at every density, in the sense of Cohn–Kumar–Miller–Radchenko–Viazovska's Definition 1.3, stated with upstream's lower energy (weaker than their notion, which also requires the energy limit to exist); universal optima are not unique among all configurations: the lattice with one vacancy is another, and it is not an isometric image of the lattice. This formalises the remark after CKMRV Theorem 1.4, also noted upstream. Tier L: CKMRV's history. Dimension 1 was settled before dimensions 8 and 24, and dimension 2 was then the lowest open case. Tier C: the 2D Yukawa interaction lies in the theorem's class. Tier X: a preregistered overdamped N-body relaxation with the CVODE solver of rusty-SUNDIALS, from initial conditions derived from the Hugging Face OC20 dataset (CC-BY-4.0). No relaxed periodic configuration fell below the triangular-lattice energy (0 of 100 relaxations at kappa = 2; 0 of 50 at kappa = 1). The secondary prediction, that annealing reaches the perfect lattice, held narrowly at kappa = 2 and failed at kappa = 1. It is reported as failed. The ledger gate reports 0 blocking findings and 6 flags (exit 1). Each flag marks a Tier A row whose statement adequacy no person has audited. The author reviewed the draft and chose to publish with those flags open, seeking that review from the community. The proofs, numerics, ledger, audits and text were produced by AI agents (Claude) under the author's direction; verification and audit runs were separate instances of the same model. The author is responsible for the content.
Related
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BibTeX
@misc{callens2026planar,
title = {{Planar universal optimality, tiered: kernel-checked consequences, a non-uniqueness theorem, and a CVODE N-body consistency check for the Yukawa interaction}},
author = {Xavier Callens},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23293111},
url = {https://doi.org/10.5281/zenodo.23293111},
note = {CC BY},
}
RIS
TY - UNPB
TI - Planar universal optimality, tiered: kernel-checked consequences, a non-uniqueness theorem, and a CVODE N-body consistency check for the Yukawa interaction
AU - Xavier Callens
PY - 2026
DA - 2026/10/11
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23293111
UR - https://doi.org/10.5281/zenodo.23293111
AB - The openai/math release contains a Lean 4 proof that the triangular lattice minimises the lower g-energy among all locally finite planar configurations of density one, for every smooth completely monotone g. Every claim this note makes about it is filed in a claim ledger with the tiers of the SocrateAI-Scientific-Elenchus method: A kernel-checked, L cited literature, C argument, and X floating-point exploration, which may steer a search but never supports a claim. Tier A, conditional on the fidelity of upstream's definitions and of our transcription: the rescaled triangular lattice is universally optimal at every density, in the sense of Cohn–Kumar–Miller–Radchenko–Viazovska's Definition 1.3, stated with upstream's lower energy (weaker than their notion, which also requires the energy limit to exist); universal optima are not unique among all configurations: the lattice with one vacancy is another, and it is not an isometric image of the lattice. This formalises the remark after CKMRV Theorem 1.4, also noted upstream. Tier L: CKMRV's history. Dimension 1 was settled before dimensions 8 and 24, and dimension 2 was then the lowest open case. Tier C: the 2D Yukawa interaction lies in the theorem's class. Tier X: a preregistered overdamped N-body relaxation with the CVODE solver of rusty-SUNDIALS, from initial conditions derived from the Hugging Face OC20 dataset (CC-BY-4.0). No relaxed periodic configuration fell below the triangular-lattice energy (0 of 100 relaxations at kappa = 2; 0 of 50 at kappa = 1). The secondary prediction, that annealing reaches the perfect lattice, held narrowly at kappa = 2 and failed at kappa = 1. It is reported as failed. The ledger gate reports 0 blocking findings and 6 flags (exit 1). Each flag marks a Tier A row whose statement adequacy no person has audited. The author reviewed the draft and chose to publish with those flags open, seeking that review from the community. The proofs, numerics, ledger, audits and text were produced by AI agents (Claude) under the author's direction; verification and audit runs were separate instances of the same model. The author is responsible for the content.
ER -