Finite Quantum Histories: Holonomy Spectra, Minimal Clocks, and Exact Clock-Change Covariance

Abstract

We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy $M=U_{L-1}\cdots U_0$. Its spectrum is $\lambda_{a,k}=1-\cos((2\pi k-\theta_a)/L)$, where $e^{i\theta_a}\in\mathrm{spec}(M)$. Thus the exact history sector is isomorphic to $\mathrm{Fix}(M)$, while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step $U$, the minimal number of clock events is the projective order of $U$. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by $U(r)\times\mathbb{Z}_L$ on a rank-$r$ history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.