From Algebra to Geometry: A Canonical Finite Construction of Geometry from Algebraic Invariants

Whether a canonical geometry can be derived from purely algebraic information, without external metric or topological assumptions, is a fundamental question in emergent spacetime. This paper answers it affirmatively for the canonical 64-node TAGC assembly established in LT1 and characterised algebraically in LT2. Starting from seven relational axioms, we prove that the assembly admits a unique canonical minimal geometry determined by the orbital structure of the wreath-product action S4 wr S16. We introduce the Canonical Geometric Induction Criterion (CIG) and show that the sign trichotomy of the inter-meta-group adjacency operator is the unique LT2 invariant satisfying it. This invariant canonically induces a pseudometric of signature (1,15) on the quotient space R^64/V_48, providing the first enriched geometric structure of the programme. A canonical relational representation is constructed on the quotient of the sixteen intrinsic meta-groups, yielding an injective embedding into the pseudometric space. We prove a Structural Identification Theorem establishing that the quotient geometry and its image are canonically isomorphic within the category of L2 invariants. The 64-node assembly is recovered as a canonical K4-fibration over this quotient, identifying the two-scale structure analysed in L3. These results complete the deductive development of TAGC through the stage of canonical structural identification.