TDDA Quantum Chaos and a Novel Universality Class of Random MatricesTDDA 量子混沌与一类全新随机矩阵普适类

Abstract

摘要 本文基于三维耗散代数(TDDA)两体态空间 ,提出并论证了一种全新随机矩阵普适类,命名为TDDA 强普适类。依托累计超 40000 组数值样本开展系统性模拟实验,结果表明:定义于正度规物理子空间的 TDDA 左乘算子系综,能级近邻间距比统计特征显著区别于经典高斯正交系综(GOE)与高斯酉系综(GUE);其间距比均值 ,与 GOE 理论特征值 0.536 相差约 13 倍标准误,统计差异高度显著。 该新型普适类的本质成因源于 TDDA 非结合乘法对矩阵元施加的强代数约束:9 阶左乘算子共计 81 个矩阵分量,仅由代数元 9 个独立随机自由度完全决定,整体谱统计行为受非结合乘法规则严格调控。数值实验同步验证,非结合纠缠度 与能级系统的随机程度存在单调对应关系。 本文独立完成前 10000 个黎曼 ζ 非平凡零点间距统计核验,得到间距比均值 ;同时完整记录一次零点向 TDDA 代数元映射的失败尝试,并系统剖析映射失效的底层原因。全部数值模拟基于 Python 与 NumPy 工具实现,核心代数恒等式通过 SymPy 符号计算完成严格校验。 关键词:非结合代数;三维耗散代数;随机矩阵;量子混沌;黎曼零点;谱统计 Abstract Based on the two-body state space of the three-dimensional dissipative algebra (TDDA), this paper proposes and verifies a new universality class of random matrices, named the TDDA-strong universality class. Systematic numerical simulations with over 40,000 samples demonstrate that the spectral statistics of TDDA left-multiplication operator ensembles restricted to the positive-metric physical subspace differ significantly from the classical Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE). The mean nearest-neighbour spacing ratio reads , with a deviation of approximately 13 standard errors from the GOE theoretical value 0.536, indicating highly statistically significant discrepancy. The intrinsic origin of this novel universality class lies in the strong algebraic constraints imposed by the non-associative multiplication of TDDA on matrix elements. The 9×9 left-multiplication operator contains 81 matrix entries fully determined by only 9 independent random degrees of freedom of the algebraic element, and its overall spectral behaviour is strictly governed by non-associative multiplication rules. Numerical experiments further verify a monotonic correspondence between the non-associative entanglement degree and the randomness of energy levels. We independently perform spectral statistics verification on the first 10,000 non-trivial zeros of the Riemann zeta function, yielding a mean spacing ratio . An unsuccessful mapping attempt from Riemann zeros to TDDA algebraic elements is fully documented, with thorough analysis of underlying failure mechanisms. All numerical simulations are implemented via Python and NumPy, and core algebraic identities are symbolically validated using SymPy. Keywords: non-associative algebra; three-dimensional dissipative algebra; random matrix; quantum chaos; Riemann zeros; spectral statistics