The Rytova-Keldysh interaction is a positive Gaussian mixture: kernel-checked admissibility for planar universal optimality and Keldysh-jellium Madelung energies of TMD Wigner lattices
- 1SocrateAI Lab
Abstract
The Rytova–Keldysh potential V(r) = (π/2r0)[H0 − Y0](r/r0) describes the in-plane interaction in a monolayer transition-metal dichalcogenide. We give a human proof (Tier C, not machine-checked) that, as a function of t = r2, it is a positive mixture of Gaussians, with weight w(a) = r0−1 erfcx(√a/r0). It is therefore completely monotone in the squared distance, which is the class covered by the openai/math theorem on universal optimality of the triangular lattice. Every claim is filed in an Elenchus claim ledger with tiers A (kernel-checked Lean), L (cited literature, quoted verbatim), C (human argument, not machine-checked) and X (floating-point exploration, which never supports a claim). Tier A (Lean 4.34.1, pinned openai/math and Mathlib; axioms within propext, Classical.choice, Quot.sound; planted sorry and planted-axiom controls caught): a generic lemma: positive exponential mixtures with finite moments are AdmissiblePotential; admissibility and strict positivity of the mixture-defined Keldysh potential, and of its Gaussian regularisation; the triangular-lattice energy inequalities obtained from upstream's universal_energy_minimum. These corollaries are conditional on the upstream theorem, which is AI-generated and not peer reviewed, and on the fidelity of its definitions. The unregularised inequality is expected to read ⊤ ≤ ⊤ (Tier C). Finiteness of the regularised energy is not proved. Tier C: the identification of the Lean mixture with the Struve form, given as a step-by-step human demonstration written for undergraduates. Its main route uses only positive integrands: DLMF 11.5.2, the subordination formula (used without a citation) and an elementary erfcx integral. Tier X: a new rusty-SUNDIALS module computes Keldysh-jellium Madelung energies of triangular and square Wigner lattices by an Ewald split in the mixture variable. It reproduces the Coulomb values at r0 = 0 and the Dedekind-eta log-gas gap at large r0, and both negative controls are caught. The square lattice lies above the triangular one at all 51 sampled points. Material curves for MoS2, MoSe2, WS2 and WSe2 in vacuum take r0 = 2πχ2D from Berkelbach et al. (Tier L). The HuggingFace dataset colabfit/JARVIS_C2DB (CC-BY-4.0) is used only to identify the 2H phases and to cross-check lattice constants; it contains no polarizabilities. All of this is static and classical, and compares two lattices only. No breakthrough is claimed: complete monotonicity follows quickly from textbook integrals. The ledger gate reports 0 blocking findings and 9 flags, one for each Tier A row whose statement adequacy no human has audited. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content.
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BibTeX
@misc{callens2026rytovakeldysh,
title = {{The Rytova-Keldysh interaction is a positive Gaussian mixture: kernel-checked admissibility for planar universal optimality and Keldysh-jellium Madelung energies of TMD Wigner lattices}},
author = {Xavier Callens},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23646106},
url = {https://doi.org/10.5281/zenodo.23646106},
note = {CC BY},
}
RIS
TY - UNPB
TI - The Rytova-Keldysh interaction is a positive Gaussian mixture: kernel-checked admissibility for planar universal optimality and Keldysh-jellium Madelung energies of TMD Wigner lattices
AU - Xavier Callens
PY - 2026
DA - 2026/10/25
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23646106
UR - https://doi.org/10.5281/zenodo.23646106
AB - The Rytova–Keldysh potential V(r) = (π/2r0)[H0 − Y0](r/r0) describes the in-plane interaction in a monolayer transition-metal dichalcogenide. We give a human proof (Tier C, not machine-checked) that, as a function of t = r2, it is a positive mixture of Gaussians, with weight w(a) = r0−1 erfcx(√a/r0). It is therefore completely monotone in the squared distance, which is the class covered by the openai/math theorem on universal optimality of the triangular lattice. Every claim is filed in an Elenchus claim ledger with tiers A (kernel-checked Lean), L (cited literature, quoted verbatim), C (human argument, not machine-checked) and X (floating-point exploration, which never supports a claim). Tier A (Lean 4.34.1, pinned openai/math and Mathlib; axioms within propext, Classical.choice, Quot.sound; planted sorry and planted-axiom controls caught): a generic lemma: positive exponential mixtures with finite moments are AdmissiblePotential; admissibility and strict positivity of the mixture-defined Keldysh potential, and of its Gaussian regularisation; the triangular-lattice energy inequalities obtained from upstream's universal_energy_minimum. These corollaries are conditional on the upstream theorem, which is AI-generated and not peer reviewed, and on the fidelity of its definitions. The unregularised inequality is expected to read ⊤ ≤ ⊤ (Tier C). Finiteness of the regularised energy is not proved. Tier C: the identification of the Lean mixture with the Struve form, given as a step-by-step human demonstration written for undergraduates. Its main route uses only positive integrands: DLMF 11.5.2, the subordination formula (used without a citation) and an elementary erfcx integral. Tier X: a new rusty-SUNDIALS module computes Keldysh-jellium Madelung energies of triangular and square Wigner lattices by an Ewald split in the mixture variable. It reproduces the Coulomb values at r0 = 0 and the Dedekind-eta log-gas gap at large r0, and both negative controls are caught. The square lattice lies above the triangular one at all 51 sampled points. Material curves for MoS2, MoSe2, WS2 and WSe2 in vacuum take r0 = 2πχ2D from Berkelbach et al. (Tier L). The HuggingFace dataset colabfit/JARVIS_C2DB (CC-BY-4.0) is used only to identify the 2H phases and to cross-check lattice constants; it contains no polarizabilities. All of this is static and classical, and compares two lattices only. No breakthrough is claimed: complete monotonicity follows quickly from textbook integrals. The ledger gate reports 0 blocking findings and 9 flags, one for each Tier A row whose statement adequacy no human has audited. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content.
ER -