Contemporary nuclear physics, molecular biology, and evolutionary theory each address their domains through independent frameworks with empirical parameters and no shared derivational foundation. This paper derives the origin of life and all three core conditions of Darwinian evolution — self-replication, mutation, and selection pressure — as necessary consequences of a single algebraic structure: M₃ (ℂ), the unique minimal non-commutative algebra of Cognitional Mechanics (CM). No free parameters are introduced at any stage. The derivation proceeds in two stages. First, the spectral position κ* = 1/6 is derived internally from CM Axioms 1–4 alone: the minimum occupied-state count generating non-commutativity is n = 2 (Axioms 1 and 4), and the unique Ad-invariant positive-definite metric scalar is A = 12 (Axioms 2 and 4), yielding κ* = 2/12 = 1/6 with no empirical input. This value determines the four-dimensional bonding space VC on which all subsequent structure is built. Second, life (Condition P) and Darwinian evolution (Condition Q) are shown to be algebraically distinct. A system satisfies Condition P if it sustains idempotent self-referential computation — the algebraic definition of life, substrate-independent in principle and realised at minimum cost by carbon. Condition Q additionally requires mutation, derived as the structural action of off-diagonal root operators forced by Axioms 1 and 3, and positive selection pressure, derived as N²-scaled relative entropy with the parameter-free coefficient 25/1296. A κ-space scan confirms κ* = 1/6 as the minimum-cost simultaneous realisation of both conditions within Z = 1–36. Biogenesis and Darwinian evolution are thermodynamic projections of M₃ (ℂ) spectral structure; molecular biology is their Tier-2 image.
T.O. (Thu,) studied this question.