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The central charge of an arithmetic system: a seven-world framework

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometrymath.DS — Dynamical Systemsmath.GT — Geometric Topologymath.RT — Representation Theory

Contributed by Zeng Chunping ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

To an arithmetic system (prime elements, norms, weights, LL-functions) we attach the central charge Cν=∑p(log⁡Np)ν/(Np(Npν−1))⋅(weight of Frobp)−1C_\nu=\sum_{\mathfrak p}(\log N\mathfrak p)^\nu/(N\mathfrak p(N\mathfrak p^\nu-1))\cdot(\text{weight of }\mathrm{Frob}_{\mathfrak p})^{-1}. Seven worlds share it: number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics. We record four universal laws with their exact status. (I) Index-set identity in all seven worlds (number fields to 101310^{13}, tropical skeletons to 103110^{31}). (II) Critical law Cν∼Φ⋅(ν−1)!/ΔνC_\nu\sim\Phi\cdot(\nu-1)!/\Delta^\nu, with Δ\Delta the distance to the controlling singularity: a pole (Tate) or the natural boundary (cuspidal); Φ\Phi measures reachability (≈0.7\approx 0.7 Tate, ≈0.07\approx 0.07 cusp). (III) Diagonal main-term theorem: LL entire iff cuspidal iff no diagonal main term iff vanishing connected/remainder ratio; item (v) (never-saturating spectral form factor) is at conjecture level, the bridge closed by a companion paper's recurrence-peak criterion. (IV) Law of forgetting in six formulations, unified: forgetting = increasing Δ\Delta = rising diagonal fraction = approaching the semicircle (free) = setting η24=1\eta^{24}=1. The core dichotomy (horizontal coordinate σ=1+κ\sigma=1+\kappa universal, amplitude fingerprinted) is checked on moments. The 24 network (Δ=qη24\Delta=q\eta^{24}, Yau--Zaslow, L(11a1,1)/Ω=1/5L(11a1,1)/\Omega=1/5) and the Neron wild tables are included; everything carries a status label and a corrections register. We prove neither abc, Szpiro, RH, nor BSD.

Tools used

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Version history

  1. v1 · 13 pagesSubmitted by Zeng ChunpingRevised after moderator-requested changesCurrentNov 12, 2026