Discrete Symmetry, Block Spectra, and 4D Axis Correspondence in a Fibonacci Modulo 9 Multiplication Matrix

This paper formalizes a finite-matrix framework built from the Fibonacci sequence modulo 9. By defining a 24x24 multiplication matrix Mᵢj = (Fᵢ x Fⱼ) mod 9, we reveal a highly structured, periodic, and symmetric algebraic object. This document establishes priority on four reproducible computational findings: 1. Exact Algebraic Partition: A three-class partition (A=1, 4, 7, B=2, 5, 8, N=0, 3, 6) that perfectly governs the matrix multiplication algebra. 2. Spectral Hierarchies: The spontaneous emergence of a three-tier positive eigenvalue spectrum from a specific family of 4x4 blocks, screened via a discrete Dirac-gamma anticommutation scoring rule. 3. 4D Geometric Correspondence: An exact algorithmic generation of the 60 unoriented axes of the 600-cell (H4 symmetry), demonstrating a high cosine alignment (>0. 95) with the K=4 spectral centroids of the 4x4 blocks. 4. Entropy Coarse-Graining: Localized Shannon entropy steps near the N-class (vacuum) boundaries under a width-3 sliding-window model. Crucially, this preprint serves as a defensible and conservative first disclosure. It isolates finite, reproducible mathematical facts from broader physical interpretations, explicitly deferring claims regarding empirical Standard Model mass ratios, global detailed-balance breaking, and biological (DNA-codon) isomorphisms to subsequent studies. The disclosed algorithms allow for full independent reproduction without requiring the immediate release of the original source code.