立体矩阵的内蕴结构:胞腔复形与等价类全谱

摘要 本文剖析立体矩阵(SM)的内蕴结构,回答系列首篇之后的核心问题:SM对象的内部地形是什么形态?四个层次的结果。其一,胞腔结构:⟨3,3,3⟩方案空间含648座滑移四面体,被162个滑移交换腔整除瓜分(162×4=648,无重叠无遗漏),162腔构成单一群轨道,腔间面耦合网1,701条棱、每腔恒22度;体数公式 b(n)=12n³(n−1) 对 n≤7 机器吻合。其二,等价类全谱:秩27邻域的等价类谱系精确为深度0–4各类数1、1、58、2,939、140,931,合计143,930类;翻转图单连通、直径≤8、环秩424,777、最短改写回路4步。其三,成本律与规约地貌:加法成本等于3倍翻转深度,全谱143,930类逐一核验无一例外;标准等价类 S*(39个秩1项)为汇点,贪婪规约对全谱均匀平均6.98步、最坏7步抵达。其四,四族度量:A秩1幂零族48/48同构、B腿值族48/48区分、C耦合族谱半径7、D族19-正则谱半径22——四族度量互不替代,构成SM对象的多维坐标。本文另收录三阶过渡态理论:N阶向N−1阶的退化经冻结中介过渡态完成,是PDE冻结变量法的非交换推广;对素数三体,冻结整除锚后的核心为乘法矩阵,其对角面恰为平方剩余界面(|QR|=(p−1)/2,10素数全验)。

Abstract This paper dissects the intrinsic structure of the Stereo Matrix (SM), answering the core question that follows the first paper of the series: what shape does the internal terrain of an SM object take? Results at four levels. First, cell structure: the ⟨3,3,3⟩ scheme space contains 648 gliding tetrahedra, partitioned exactly by 162 gliding exchange chambers (162×4=648, no overlap, no omission); the 162 chambers form a single group orbit, with an inter-chamber face-coupling network of 1,701 edges and a constant degree of 22 per chamber; the cell-count formula b(n)=12n³(n−1) matches machine enumeration for n≤7. Second, the full spectrum of equivalence classes: in the rank-27 neighbourhood the class genealogy is exactly 1, 1, 58, 2,939, 140,931 at depths 0–4, totalling 143,930 classes; the flip graph is connected, of diameter ≤8, with cycle rank 424,777 and shortest rewriting cycle 4. Third, cost law and reduction landscape: additive cost equals 3× flip depth, verified class by class over all 143,930 classes without exception; the standard equivalence class S* (39 rank-one terms) is a sink, reached by greedy reduction in an average of 6.98 steps and at worst 7. Fourth, four families of metrics: family A (rank-one nilpotent) 48/48 isomorphic, family B (leg-value) 48/48 distinguishing, family C (coupling) spectral radius 7, family D 19-regular spectral radius 22 — the four are mutually non-substitutable, forming multidimensional coordinates for SM objects. The paper also records the theory of third-order transition states: degeneration from order N to N−1 proceeds via a frozen intermediary transition state, a non-commutative generalization of the PDE frozen-variable method; for prime three-bodies, the core after freezing the divisibility anchor is a multiplication matrix whose diagonal face is exactly the quadratic-residue interface (|QR|=(p−1)/2, verified for 10 primes).

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