The Chrysene Formalism Revisited: A Twelve-Axiom Foundation for Molecular Characterization, Tensorial Noncommutative Geometry, and Operator Response

Abstract

We present a revised twelve-axiom formulation of the Chrysene Formalism that rigorously separates its molecular, tensorial, noncommutative, spectral, and physical layers. The molecular reduct is restricted to a finite four-ring polyhex comparison class whose canonical reference point group is conjugate in O(3) to the standard C2h realization and whose two primary attachment directions are intrinsically inequivalent. The 3,6,9,12 <-> 2,5,8,11 dual cipher is realized as a uniform one-step perimeter displacement.

A local structural tensor Psi-C generates a Z3-indexed tensor network. Its quasi-local observable algebra and lattice-translation action produce the crossed-product sequence algebra A-C = Q-C semidirect product Z3, whose noncommutativity follows from translation of localized operations rather than from geometric orthogonality. The dual cipher defines a second structural tensor, a native Z2 x Z2 four-state alphabet, and a commutant representation of the primary sequence algebra.

The sequence algebra is equipped with the spectral triple (A-C-circ, H-omega, D-C). An explicit faithful tracial representation and finite-filtration Dirac operator give bounded commutators and compact resolvent on the infinite Z3 architecture without compactifying the lattice or requiring translation invariance of D-C. Physical transfer, polarization response, and admissibility geometry are introduced as distinct subsequent structures.

An exhaustive enumeration of the seven free four-cell polyhexes selects a unique canonical carbon framework with the required C2h point-group type, and its graph is independently identified with chrysene. An explicit extension of the standard numbered chrysene reference node satisfies all twelve axioms, proving non-vacuity. Consequently, every full model has canonical carbon-framework reduct polyhex-congruent to chrysene, while the standard 3,6,9,12 <-> 2,5,8,11 locant frame is retained on the reference and standard-locant realization class rather than derived uniquely from the axioms.

The Ny = Nx + 2 relation, a structurally induced GF(4) field law, directional transfer and optical anisotropy, spectral--path metric reconstruction, real spectral structure, spectral dimension, higher-algebraic braiding, emergent assembly chirality, and dynamical scission remain theorem-level problems.