有限量子历史：和乐谱、最小时钟与精确时钟变换协变性
Finite Quantum Histories: Holonomy Spectra, Minimal Clocks, and Exact Clock-Change Covariance
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中文总结 AI 辅助
该研究求解含时幺正步的循环有限维量子历史，推导其和乐谱等闭式结果，定义锐有限时钟的预测商，给出有限误差定理，分类时钟变换并验证核心结论。
中文摘要 AI 辅助
我们求解任意含时幺正步的循环有限维量子历史，不假设某一步具有有限阶。传播哈密顿量是环上的幺正联络拉普拉斯算子，其完整规范不变量为单值性M=U_{L-1}…U_0。其谱为λ_{a,k}=1−cos((2πk−θ_a)/L)，其中e^{iθ_a}∈spec(M)。因此，精确历史区同构于Fix(M)，而 frustration（挫败）、非空零能区之上的能隙、行列式及有限温度迹均以闭式形式得到。普通谱数据可恢复单值性相位余弦的多重集，但无法恢复相位取向；低能特性可证明其与精确关系历史的接近程度。随后，我们定义了相对于可达算子系统的锐有限时钟的预测商，即保留历史区上所有条件统计的唯一最粗事件字母表。有限误差定理表明，当不等价事件通道的最小钻石间距超过估计误差的4倍时，阈值聚类可恢复该商，并证明了最优记录计数界。在全矩阵访问和齐次步U的情况下，时钟事件的最小数量为U的射影阶。我们区分时钟代数的正规化子与保持相干历史码的变换，并在秩为r的历史区上按U(r)×Z_L对取向精确锐时钟变换进行分类；无取向时，循环因子变为二面体。全信息时钟纤维的可逆变换必为幺正变换，因此不可逆粗粒化并非精确时钟协变性。完整历史格拉姆核的最小实现是唯一幺正等价的，具有有限数据的普罗克拉斯提斯界。独立有限矩阵码验证了主要结果。
英文摘要
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 $λ_{a,k}=1-\cos((2πk-θ_a)/L)$, where $e^{iθ_a}\in\mathrm{spec}(M)$. 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.
