math.NT — Number Theory

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

Contributed by Zeng Chunping

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 $D$ equals the narrow class number $h^{+}(D)$, i.e. the number of $\rho$-orbits of Zagier-reduced indefinite binary quadratic forms; on all 36 fundamental discriminants $5\le D\le 120$ the equality holds as an exact integer. (2) With the corrected counting (all non-square discriminants $D\equiv 0,1 \pmod 4$, primitive forms, $\rho$-orbits), the prime geodesic theorem has error exponent $1/2$: $N(X)/\mathrm{li}(e^X)=0.9963$ at $X=12$ with $|\mathrm{err}|/e^{X/2}$ bounded and decreasing overall. (3) The level-one Maass spectrum is long-range GOE and short-range non-Wigner: on 1110 even levels, $\Sigma_2(20)=1.203\ [1.02,1.26]$ against Poisson 20 and compatible with GOE 1.049, while the spacing distribution fails the Wigner surmise with a $17\times$ small-gap excess. This is the standard signature of arithmetic quantum chaos: rigidity without local Wigner repulsion. We also record the archimedean parity law $\Gamma^{a}_{2k}=2\Gamma^{a}_{2k+1}\sim r^{-2k}$, 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.