K-Theory of the Sequence Algebra, Spectral Dimension Three, and Path-Calibrated Spectral Metrics

Abstract

We study three operator-algebraic constructions associated with the sequence framework: crossed-product $K$-theory, a confining spectral operator, and a calibrated finite-graph metric. For the quasi-local UHF algebra of type $d^\infty$, the product trace proves that lattice shifts act trivially on $K_0\cong\mathbb Z[1/d]$. The first crossed product has $K_0\cong K_1\cong\mathbb Z[1/d]$. For all three crossed-product stages, the inclusion-induced localization theorem for noncommutative Bernoulli shifts yields the exterior-power groups, trace decomposition and unconditional Morita obstructions. This replaces an insufficient elementary Pimsner--Voiculescu induction with an explicit analytic theorem dependency. Independently, a finite-fibre graded confining operator satisfies $N_D(\Lambda)=32\pi d\Lambda^3/(3\kappa_x\kappa_y\kappa_z)+O(\Lambda^2)$ and has spectral dimension three. For every finite weighted graph, an edge representation augmented by isolated-vertex blocks is faithful and its spectral distance equals the extended shortest-path distance. The fifteen-state source example has 21 finite and 204 infinite ordered-pair distances. A graded product retains both this finite-graph metric and the confining counting exponent. Cut-set degeneration, rather than deletion of an arbitrary single edge, gives a sufficient condition for distance divergence. The mathematical constructions are separated from molecular realization, and companion verification and Lean dossiers certify finite consequences within their stated scope.