Topological Confinement Analogues, Discrete Path Propagation, and Chiral-Gap Proxies in a Fibonacci Mod-9 Algebra

Abstract This paper isolates a finite algebraic subsystem from a Fibonacci mod-9 construction and develops it as an independently reproducible discrete model. Based on the 24-step Pisano cycle, the model introduces a 24 24 multiplication table, a class-valued addition table, and a phase-valued propagator matrix. Core Algebraic Theorems (Layer A/B) The fundamental mathematical statements are exact. The state space partitions perfectly into three residue classes (denoted 147, 258, and 369), which form a closed algebra under both addition and multiplication. Crucially, the six distinguished "NIL" positions generate an absorbing 6 6 multiplication block, and the class-averaged phase sum over the 369 sector vanishes identically. Physical Interpretations and Analogues (Layer C) These exact results motivate a physically interpreted discrete analogue of non-perturbative structures from gauge theory. Confinement & Wilson Loops: The absorbing NIL block and phase cancellations act as a topological analogue to color confinement and Wilson-loop phase behavior. Discrete Path Integral: The 24 24 finite propagator matrix allows for the algorithmic computation of aggregate amplitudes and representative channel ratios (e. g. , e^+e^- ^+^-) without ultraviolet divergence. Chiral-Gap Proxy: The idempotent structure of Z₉ (multiplicative fixed points 0 and 1) provides a minimal discrete analogue of chiral condensate formation (mass gap). Methodology and Claim Boundary The construction is not presented as a continuum proof of Yang-Mills confinement or the mass gap problem. It provides a finite, explicit algebraic toy model. To ensure maximum reproducibility without disclosing the proprietary source code, a complete algorithmic protocol is explicitly stated.