Law as Forms of Cognition: The Necessary Algebra of Distinction and Its Projection into Physics

What are physical laws? The dominant answer since the seventeenth century has been inductive: laws are regularities extracted from observation. The observer contacts the external world and derives structure from it. Cognitional Mechanics (CM) reverses this direction entirely. The starting point is not an observation but a logical question: what structure must any system capable of internal distinction necessarily possess? The answer, derived without assuming anything about the physical world, is a unique algebraic object. Since the physical world is itself a system of distinctions, it must exhibit the invariants of this structure. Physical laws are not discovered; they are projected. This paper develops the philosophical foundations of this inversion. We derive the four axioms that any distinguishing system must satisfy and demonstrate that they uniquely force M₃(ℂ), the algebra of three-by-three complex matrices, as their minimal solution. We then situate this result within the history of philosophy of science. CM agrees with Kant that the conditions of possible experience are a priori, but extends them beyond human cognition to any distinguishing system whatsoever. It agrees with Ontic Structural Realism that structure is ontologically fundamental, but arrives at this conclusion by downward derivation from logical necessity rather than upward inference from the success of physical theories. It agrees with Wallace that mathematical structure should come first, but treats the relevant structure as logically necessary rather than pragmatically adopted. Three consequences follow: the fine-tuning problem dissolves because the physical constants admit no alternative values; the role of experiment is redefined from discovery to decoding; and the IASER operational framework finds its philosophical foundation in the structural separation between the generation of possibilities and the selection among them. Published on May 5, 2026 DOI: 10.5281/zenodo.20032194