标准等价类的结构性能与归约动力学:兑换律守恒

摘要 矩阵乘法张量的等价分解类是代数复杂性理论的古典对象,但文献集中于最优算法的唯一性证明,等价类全谱的结构与几何从未被系统构造。本文以2×2×2(三元三阶)立体矩阵乘法张量为对象,给出其标准等价类 S* 的显式构造(朴素27项加三个零和模共39项),全部系数经 Brent 729 方程逐一整数精确核验。核心发现是对称性兑换律:S* 的稳定子群在648元注册群下阶为1,而朴素类为648——冗余由对称性全额支付;S* 携带8个零和模与6个独立可弃模,构成64种经认证的精确降级组合,且弃单砖12/12必坏(负结果入册)。归约动力学实验表明,归约流由势、约束与吸引子三要素构成,其收敛依赖操作集完备性:纯滑移引擎无降势通道(30/30负结果),与 Arai–Ichikawa–Hukushima 连通性定理互证。本文给出归约动力学与重整化群流的结构对应表,并讨论其在容错设计、可验证计算与复杂系统分析中的应用路径。本文进一步以模注入梯度实验判定兑换律的守恒形式:对称量呈阈值型坍缩(648→4→1),冗余量呈线性积累(ρ=3m),二元守恒函数不存在(负结果入册),据此提出兑换律的阈值-线性修正定律,并给出 SW 先验结构的存在性论证。

Abstract The equivalence classes of factorizations of the matrix multiplication tensor are a classical object of algebraic complexity theory, yet the literature concentrates on uniqueness proofs for optimal algorithms, and the structure and geometry of the full spectrum of equivalence classes have never been systematically constructed. Taking the 2×2×2 (ternary cubic) stereo matrix multiplication tensor as object, this paper gives an explicit construction of its standard equivalence class S* (the naive 27 terms plus three zero-sum modules, 39 terms in all), every coefficient verified integer-exactly through the Brent 729 equations. The core finding is a symmetry exchange law: under the 648-element registered group the stabilizer of S* has order 1, whereas that of the naive class is 648 — redundancy is paid in full by symmetry; S* carries 8 zero-sum modules and 6 independent dispensable modules, forming 64 certified exact degradation combinations, and discarding a single brick fails 12/12 (negative result registered). Reduction-dynamics experiments show that the reduction flow consists of three elements — potential, constraint and attractor — and that its convergence depends on the completeness of the operation set: a pure-slide engine has no potential-decreasing channel (30/30 negative result), corroborating the Arai–Ichikawa–Hukushima connectivity theorem. We give a structural correspondence table between reduction dynamics and renormalization-group flow, and discuss applications to fault-tolerant design, verifiable computation and complex-system analysis. We further determine the conservation form of the exchange law through module-injection gradient experiments: the symmetry quantity collapses threshold-wise (648→4→1), the redundancy quantity accumulates linearly (ρ=3m), and no binary conservation function exists (negative result registered), whence we propose the threshold–linear corrected exchange law and give an existence argument for the SW a priori structure.

本记录含中文版与英文版两个文件(Bilingual: Chinese + English DOCX)。