TITLE:
A Finite-Response Master Equation with a Primitive Operator Spectrum Derived from M 2 ( ℂ )
AUTHORS:
Christoforos N. Panagis
KEYWORDS:
Finite-Dimensional -Algebras, Logarithmic Spectra, Nonlinear Response, Symmetric Hyperbolic Systems, Mathematical Physics
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
24,
2026
ABSTRACT: A finite-response operator model is developed in which a discrete logarithmic spectrum is derived from the minimal noncommutative complex operator cell rather than selected from empirical data. Under primitive-factor reduction, diagnostic minimality, typed input-output and source-probe-output roles, and complete coefficient retention, the primitive cell is
M
2
(
ℂ
)
. Its center, traceless self-adjoint subspace, and full self-adjoint part have real dimensions 2, 3, and 4. The primitive response arities are 2 for self-response and 3 for source-probe-output interaction. The resulting complete coefficient modules have real dimensions 4, 9, 16, and 27. Their identification with observable logarithmic windings is a separate conditional step, requiring coefficient-basis-neutral unitary transport and observation through the least nontrivial determinant character. Under that explicitly stated observation realization, a self-adjoint response-number operator generates the associated modes in logarithmic comparison space. Independently, an associative finite-load composition law yields a unique convex saturating response profile. This constitutive law is incorporated into a block master system coupling propagation, ordering, localisation, metric response, interaction, memory, and drift. Local well-posedness is obtained conditionally under explicit symmetric-hyperbolicity assumptions. A standard-library Python implementation verifies composition, covariance, monotonicity, stability, convergence, quotient invariance, and defect sensitivity, and supplies controlled reductions for galaxy dynamics and lensing, turbulent transport, ordered-regional transfer, interaction, merger memory, tensor propagation, and a regular spherical geometry. Empirical studies are used only as external tests of visibility and do not enter the derivation. The uniqueness claim is limited to the stated complex-operator and primitive-response assumptions.