Unified Geometry: Relational Parallel Axiom and Relatively-Symmetric Quantum Gravity

We formulate a tensor-principal version of Unified Geometry (UG). The entire physical content is carried by a single axiom---the relational parallel-operator axiom: the parallel postulate is no longer a binary proposition but a spectral problem for a normal operator P(γ₁,γ₂) on a Hilbert fibre, whose spectrum contains an intersecting mode $+1$, a non-intersecting mode $0$, and a coherent mode in C R. Relational symmetry is not an independent meta-axiom but the domain-and-covariance clause of this axiom: P is defined only on source-matched morphism pairs and transforms covariantly under local frame changes. From this single axiom we strictly derive, where mathematically permitted, the path-groupoid and fibre-bundle structure; the complex fibre C³⊕ C²⊕ C¹ with automorphism group SU(3)× SU(2)× U(1); wave functions as coherent groupoid sections; spin as irreducible fibre representations; noncommutative holonomy as the origin of commutation relations and uncertainty; the Born rule as the Gleason measure on the projection lattice; the Einstein--Cartan tensor equation as the classical limit of curvature-sector decoherence; and the constants of nature as holonomy spectral gaps. We give both differential-form and index-form total actions, and list falsifiable predictions with explicit death conditions. The paper is a conditionally rigorous programme, not a completed theory: measure positivity, the full classical limit, Bell correlations, three generations, chiral representations, and all numerical predictions remain open and are mapped explicitly.