Cubic Unification Without Gauge Unification: Band-Edge Capacity, Peirce-Heat Reconstruction, and a Zero-Intersection Gauge Obstruction

Abstract

The result is not a conventional grand unified theory. In ordinary GUT model building, a larger symmetry that is said to be 'broken', incomplete, or unable to reproduce the observed low-energy theory is usually treated as a model-building problem to be repaired by a different breaking sector, vacuum alignment, threshold structure or representation content. The theorem developed here points in the opposite direction: the failure of gauge completion is tied to the same component geometry that preserves the successful matter-incidence invariant. Gauge non-unification is therefore treated as a geometric feature of the construction rather than merely an unfinished symmetry-breaking mechanism. The resulting theory is stratified: exceptional geometry organizes matter incidence, finite record geometry selects physical components and resolution, and local gauge dynamics acts through the matter bundles rather than through a faithful exceptional carrier symmetry. Absolute four-dimensional normalization, derivation of the magnetic product allocation from the underlying action, and selection of one global complex orientation remain open. A reduced Riemannian Einstein-Maxwell sector on a product geometry has an exact marginal point λℓ² = 49/16 with (a⁻²,r⁻²,s⁻²) = (7/64,7/4,7/24). Hence ℓ²/a² = 7/64 and the spatial capacity budget is exactly B ≡ λa² = (49/16)(64/7) = 28. Independently, complete finite record spaces R_K = ⊕_{j=0}^K E_j, with dim E_j = (j+1)² and C_rec|_{E_j}=j(j+2), admit a unique source-neutral, unitary-invariant and rank-barycentric capacity functional: Tr(C_rec)/dim R_K = 3K(K+3)/5. The budget B=28 therefore selects K=5 and dim R_5=91. The heat-resolution law s_K = 3/[4(⟨C_rec⟩_K+1)] then gives s_5=3/100. No gauge-coupling datum enters this selection. A sequence of exact algebraic, spectral and invariant-census audits gives a coherent answer. On the positive cone of the Albert algebra, the single record functional has Hessian and, on the determinant-one trace-free tangent, third variation . Canonical normalization fixes the cubic term to . A primitive record yields Peirce ranks ; the chiral spinor occupation operator yields ; and the selected pair of commuting complex structures gives the weak/colour split . The finite first-order calculus retains all 16 weak cubic monomials and rejects all 24 coloured monomials. For the exceptional Jordan algebra H₃(O), fully Spin(10)-equivariant symmetric dressing gives the projective flavour invariant 81, whereas component-oriented dressing gives the protected value 27/8. The relevant component projector has stabilizer SU(5)×U(1); its 20-dimensional Spin(10) orbit is therefore bookkeeping. The nontrivial result is the zero-intersection census: the first-order weak scalar source has no component in the 20-dimensional bridge required to complete the local algebra toward Spin(10), and the action-derived source rank into that bridge is zero. Thus local gauge completion requires directions that the physical scalar calculus does not source, while the component-resolved matter-incidence invariant survives on the restricted projector.