Physical Laws as Top-Down Projections of a Minimal Algebraic Core: A Metatheoretical Unification Theorem from Cognitional Mechanics and Noology

This paper presents the Noological Unification Theorem: a formal proof that all physical laws are the surjective projection of a single minimal algebraic structure, the algebra M₃(ℂ) of 3×3 complex matrices, onto the category of physical laws L_phys. The composition ρ∘Π : M₃(ℂ) → L_phys is shown to be surjective while its inverse is structurally undefined. Surjectivity is established constructively via explicit identification of nineteen Standard Model parameters as coordinate components of ρ∘Π, each derived without free parameters in the Cognitional Mechanics (CM) corpus. Non-invertibility follows axiomatically from Axiom 1 (Non-Commutativity) alone, independently of any empirical input. These two proof paths are logically independent and establish complementary claims: surjectivity asserts that every physical law has an algebraic preimage; kernel non-triviality asserts that the projection discards non-trivial information, making inversion structurally impossible. Together they yield a categorical isomorphism C₁/ker(ρ) ≅ C_phys — the category of physical laws is identical to the algebraic quotient category with the unobservable kernel factored out. The generating mechanism is unique: since M₃(ℂ) is the sole algebra satisfying the four axioms of any system capable of internal distinction, and the structure map Π admits no free parameters, no alternative algebra can generate L_phys. Surjectivity is therefore not merely constructive but exclusive. Four corollaries follow: the top-down direction of physical determination is structurally necessary; the Wigner problem dissolves completely; fine-tuning and hierarchy problems dissolve as projection artefacts; and the exclusivity of the generating mechanism is formally established. The paper additionally formalises the Noological partial category C_N, which accommodates both law formation (ρ∘Π) and reality selection (A_formal∘Π) as orthogonal operations in disjoint output spaces. The morphism A_formal : S → {p*} exists within C_N but its transition rule is placed outside the language of C_N by axiomatic silence — formalising Arbitrium as the structural implementation of human final authority (IASER Norm 4). This design is shown to be free of Gödelian incompleteness: the transition rule is not an undecidable proposition but a relationship excluded from the language of C_N by construction. The result recasts the epistemology of physics: experiment is not discovery but decoding — the act of reading out coordinate values of a fixed algebraic structure that M₃(ℂ) determines independently of any measurement. Physical constants are not adjustable parameters but unique coordinate evaluations of the quotient structure, making the question "why this value rather than another?" incoherent: there is no "another." Changes from the 1st Edition (DOI: 10.5281/zenodo.19965570, May 2, 2026): The structural content of the 1st Edition — definitions, theorems, and proofs — is unchanged. The 2nd Edition adds Corollary 4.4 (Exclusivity of the generating mechanism), which establishes formally that surjectivity and uniqueness of the generating algebra are equivalent claims given the M₃(ℂ) Necessity Theorem, closing the logical gap between constructive surjectivity and exclusive generation. A corresponding item is added to Section 7 (Structural Closure of Apparent Gaps). The Abstract is supplemented to reflect this addition.