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These corollaries are conditional on the upstream theorem, which is AI-generated and not peer reviewed, and on the fidelity of its definitions.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},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23646107},
url = {https://zenodo.org/records/23646107},
note = {CC BY 4.0},
}
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
DO - 10.5281/zenodo.23646107
UR - https://zenodo.org/records/23646107
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 -