A Unified Maxwell Equation for Engineering Electromagnetics: Derivation from M3(C) Structure via Cognitional Mechanics

This is an expanded version of the Maxwell unification paper, prepared for arXiv submission. The original version (Version 1) is available at DOI: 10. 5281/zenodo. 18312668. This work presents a unified formulation of classical electromagnetism derived from Cognitional Mechanics (CM), an axiomatic framework that formalizes intelligence as a system of non-commutative, finite-depth operations. The core result is a Unified Maxwell Equation derived from CM: ( (1/c) * d/dt + div) F = J, where the electromagnetic field is embedded as an su (3) -valued matrix operator F in M₃ (C), and J is a conserved operational current. Despite its abstract origin, the equation reproduces all four Maxwell equations, energy conservation, and the Poynting theorem in standard SI units—requiring no modification to engineering practice, simulation tools, or pedagogical materials. The derivation demonstrates that the structure of electromagnetism (including c = 1/sqrt (epsilon₀ mu₀), gauge symmetry, and energy positivity) emerges as a stability condition of non-commutative algebraic dynamics under the CM axioms. Crucially, this approach does not alter classical electrodynamics; it explains why it takes the form it does. All physical quantities—including the fine-structure constant (alpha^-1 = 137. 036) —are derived as invariant ratios from the geometry of M₃ (C), with numerical agreement at the 0. 004% level. The framework is fully compatible with Griffiths- and Jackson-level electrodynamics while offering a structural foundation for the unification with gravity and particle physics within the CM-GUT program. This paper is intended for physicists, electrical engineers, and applied mathematicians seeking a deeper origin of classical field theory without sacrificing practical utility. Version 2 Changes: This is the second version of the Maxwell unification paper. Changes from Version 1 (DOI: 10. 5281/zenodo. 18312668) include: Added Section 1. 1 clarifying the distinction from Riemann-Silberstein formalism (1907). While Riemann-Silberstein uses E +/- iB as a notational convenience, CM-Maxwell embeds f = E + icB into traceless 3x3 anti-Hermitian matrices, making div B = 0 a structural consequence of the traceless condition rather than an independent postulate. Expanded Appendix A with four subsections: A. 1: Philosophical context (CM is a meta-theory, not a physical theory) A. 2: M₃ (C) uniqueness proof sketch (why n=3, why complex matrices) A. 3: Tier structure explanation (Tier 0 axioms, Tier 1 mathematics, Tier 2 physics, Tier 3 experiments) A. 4: Comparison table with string theory, loop quantum gravity, geometric algebra, and Riemann-Silberstein approaches Clarified Section 4. 5 (fine-structure constant derivation) with explicit note that this subsection demonstrates CM's broader predictive framework and may be skipped without loss of continuity for readers focused solely on electromagnetic results. Added footnote to Section 3 clarifying that the operator Dcm is not a Clifford algebra construction but a first-order differential operator in standard vector calculus. Submitted to arXiv (physics. class-ph) on January 31, 2026. arXiv submission ID: submit/7208040. Awaiting moderation and public release (expected within 2-3 business days). The core scientific content remains unchanged from Version 1. All additions are clarifications and contextualization responding to potential misunderstandings regarding historical precedents (Riemann-Silberstein) and theoretical foundations (Cognitional Mechanics). Related Publications: This work is part of the Cognitional Mechanics theoretical program establishing M₃ (C) as the unique minimal operational algebra. Related papers (all on Zenodo): Foundations of Cognitional Mechanics (DOI: 10. 5281/zenodo. 18005554) M₃ (C) Necessity in Cognitional Mechanics (DOI: 10. 5281/zenodo. 18285838) Einstein Equations from M₃ (C) (DOI: 10. 5281/zenodo. 18160010) Standard Model from M₃ (C) (DOI: 10. 5281/zenodo. 18163123)