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Degeneracy and the level-one Maass spectrum: a measured verdict on arithmetic quantum chaos

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Contributed by Zeng Chunping ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We isolate the spectral axis of a seven-world program: the length spectrum of the modular surface, its degeneracy, and the statistics of the level-one Maass spectrum. (1) The true degeneracy of a closed geodesic of discriminant DD equals the narrow class number h+(D)h^{+}(D), i.e. the number of ρ\rho-orbits of Zagier-reduced indefinite binary quadratic forms; on all 36 fundamental discriminants 5≤D≤1205\le D\le 120 the equality holds as an exact integer. (2) With the corrected counting (all non-square discriminants D≡0,1(mod4)D\equiv 0,1 \pmod 4, primitive forms, ρ\rho-orbits), the prime geodesic theorem has error exponent 1/21/2: N(X)/li(eX)=0.9963N(X)/\mathrm{li}(e^X)=0.9963 at X=12X=12 with ∣err∣/eX/2|\mathrm{err}|/e^{X/2} bounded and decreasing overall. (3) The level-one Maass spectrum is long-range GOE and short-range non-Wigner: on 1110 even levels, Σ2(20)=1.203 [1.02,1.26]\Sigma_2(20)=1.203\ [1.02,1.26] against Poisson 20 and compatible with GOE 1.049, while the spacing distribution fails the Wigner surmise with a 17×17\times small-gap excess. This is the standard signature of arithmetic quantum chaos: rigidity without local Wigner repulsion. We also record the archimedean parity law Γ2ka=2Γ2k+1a∼r−2k\Gamma^{a}_{2k}=2\Gamma^{a}_{2k+1}\sim r^{-2k}, synthetic-spectrum pipeline self-certification, and the corrected upper-envelope saturation criterion. Every measured number carries a confidence interval; corrections applied are recorded in an appendix.

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References

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Version history

  1. v1 · 9 pagesSubmitted by Zeng ChunpingInitial depositCurrentNov 06, 2026