math.NT — Number Theory

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

Contributed by Zeng Chunping

To an arithmetic system (prime elements, norms, weights, $L$-functions) we attach the central charge $C_\nu=\sum_{\mathfrak p}(\log N\mathfrak p)^\nu/(N\mathfrak p(N\mathfrak p^\nu-1))\cdot(\text{weight of }\mathrm{Frob}_{\mathfrak p})^{-1}$. 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 $10^{13}$, tropical skeletons to $10^{31}$). (II) Critical law $C_\nu\sim\Phi\cdot(\nu-1)!/\Delta^\nu$, with $\Delta$ the distance to the controlling singularity: a pole (Tate) or the natural boundary (cuspidal); $\Phi$ measures reachability ($\approx 0.7$ Tate, $\approx 0.07$ cusp). (III) Diagonal main-term theorem: $L$ 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 $\Delta$ = rising diagonal fraction = approaching the semicircle (free) = setting $\eta^{24}=1$. The core dichotomy (horizontal coordinate $\sigma=1+\kappa$ universal, amplitude fingerprinted) is checked on moments. The 24 network ($\Delta=q\eta^{24}$, Yau--Zaslow, $L(11a1,1)/\Omega=1/5$) and the Neron wild tables are included; everything carries a status label and a corrections register. We prove neither abc, Szpiro, RH, nor BSD.