The Octavian Coherence Gate: The Four Irreducible Integers of the E8(1) ⊃G2(1) ×F4(1) Conformal Embedding

We prove a constraint theorem for the octonionic unification program: any construction that fixes its dimensionless data (conformal weights, quantum dimensions, or coupling ratios) with zero continuous parameters is forced to use a single algebraic engine, and we identify it completely. Three requirements define the class of theories under study: (R1) the underlying lattice satisfies five closure axioms (integrality, evenness, unimodularity, positive-definiteness, and minimality of rank); (R2) both factors of its conformal sub-algebra are automorphism groups of octonionic structures; (R3) the emergent Z3 of the broken SU(3) closes through a dimension-one current extension. Under R1–R3, the lattice is E8, the sub-algebra is G2(1) ×F4(1), and the derivation chain uniquely determines four irreducible integers (n7,n26,d10,d11) = (7,26,2,3). Three faces of the conformal-embedding identity c(G2,1) + c(F4,1) = c(E8,1) = 8 lock them together, and every formal single-integer perturbation violates at least one constraint. An exhaustion theorem shows that all five alternative conformal embeddings of E8(1) fail independently. The result is a constraint theorem on an existing research program: any octonionic unification that fixes its data with zero continuous parameters must use the engine E8(1) ⊃G2(1) ×F4(1) and inherit the dictionary of rational constants assembled here. A single quarantined identification hypothesis is tested against the charged-lepton spectrum: the exported Koide ratio Q0 = 2/3 and Brannen phase θ= 2/9 each require mτ = 1776.97 MeV, consistent with the PDG 2024 value at the 1σ level.