Testing a Micro/Macro Dual-Scale Hypothesis on K3 x T2: Scale-Factor Duality, the Cohen-Kaplan-Nelson Bound, and the Dark-Energy Length in Lean 4

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22837832
Primary Topic
Cosmology and Gravitation Theories
Type
preprint
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preprint

Testing a Micro/Macro Dual-Scale Hypothesis on K3 x T2: Scale-Factor Duality, the Cohen-Kaplan-Nelson Bound, and the Dark-Energy Length in Lean 4

Xavier Callens, SocrateAI Scientific Agora Collaboration
Zenodo (CERN European Organization for Nuclear Research)
Cosmology and Gravitation Theories
preprint

Testing a Micro/Macro Dual-Scale Hypothesis on K3 x T2: Scale-Factor Duality, the Cohen-Kaplan-Nelson Bound, and the Dark-Energy Length in Lean 4

Xavier Callens, SocrateAI Scientific Agora Collaboration
preprint en

Abstract

We report DualScaleCosmology (Lean 4, toolchain v4.33.1, Mathlib), a companion library that formally tests one reading of a "micro/macro dual-scale" hypothesis motivating this research programme: that a microscopic length $\ell_{\mathrm{micro}}\sim\ell_P$ and a macroscopic length $\ell_{\mathrm{macro}}\sim H_0^{-1}$ are linked by non-perturbative duality. The hypothesis is not, and cannot be, a theorem; we instead formalize the two pieces of the literature that actually bear on it — string-cosmology scale-factor duality (Gasperini–Veneziano) and the Cohen–Kaplan–Nelson (CKN) ultraviolet/infrared bound — and test their numerical and structural consequences. The results are mixed and mostly negative for the literal hypothesis. Taken literally, $\ell_P$ as the ultraviolet cutoff at $H_0^{-1}$ fails the CKN bound by more than $10^{30}$ (Tier A arithmetic on Tier L constants). Read instead as a T-dual pair, the two scales saturate the CKN bound exactly at their self-dual length $s=\sqrt{\ell_P c/H_0}≈47 μ$m; we then prove, as an exact identity, that $s$ equals the observed dark-energy length $\Lambda^{-1/4}$ up to a fixed, computable factor $(8\pi/3\Omega_\Lambda)^{1/4}$ — so the numerical coincidence often quoted between such lengths is algebra, not independent evidence. As a string (Regge) scale $s$ is excluded by collider data by more than $10^{30}$ in $\alpha'$; as the radius of a single extra dimension it is disfavored only by an order-one factor, indistinguishable at this level of rigor from the "dark dimension" proposal of Montero–Vafa–Valenzuela. We also report, as a matter of methodological record, a citation error caught only by computing a numeric consequence of it, and an infrastructure misconfiguration that silently starved a local prover of its models for most of a work session. All Lean content is Tier A (44 declarations, 31 theorems, 0 sorry, standard axioms only); every physical identification is stated as Tier L (literature) or Tier C (this project's own reading), never conflated with the former. We close with a concrete list of open problems and proposed research directions for the community. Epistemic tiers. Claims are labelled Tier A (checked by the Lean 4 kernel; axioms propext, Classical.choice, Quot.sound only), Tier L (literature, cited) or Tier C (conjecture). Only Tier A statements are machine-checked; physical interpretation is not. Lean artifact: SocrateAI-Scientific-Agora-LeanMaster, tag v3.17.0 (Lean 4 v4.33.1, Mathlib). Release gates on this tag: all ten libraries build; 621 theorems pass #print axioms with standard axioms only; no sorry/admit. AI-assisted tooling (Anthropic Claude models) was used in preparing the formalization and the manuscript, under the direction of the author.

Zenodo (CERN European Organization for Nuclear Research)
Laboratoire de Mathématiques d'Orsay (FR)
Quality Education
Cosmology and Gravitation Theories
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