The Algebra of Being: Finitude, Evolution, Comprehension

We reconstruct the algebraic structure of reality from a single epistemic principle: I think, therefore the Universe is. Interpreted operationally rather than metaphysically, the existence of experience requires that primitive reality be internally complete—and hence finite in representation—and timeful. Finitude forces an attribute-free set of primitive states that can be known only through reversible enumeration, yielding a homogeneous finite cycle. Timefulness requires a temporal polarity distinct from spatial reversal; at symmetry-complete prime stages its minimal finite-field support is the quadratic extension Fp2 equipped with the Frobenius involution x→ xp. The coexistence of spatial and temporal involutions generates Klein symmetry, forcing innovation to occur in irreducible four-element packets and yielding the minimal shell growth law qt+1 = qt + 4 under minimal innovation. We prove a Uniqueness Theorem showing that these consequences determine the Finite Ring Continuum (FRC) uniquely within this class: any finite, attribute-free, timeful universe with minimal innovation is isomorphic to its shell hierarchy.