Gauge Theory on the Metric Bundle of a Spin 4-Manifold, I: Chimeric Spinors, Clifford Contraction, and Real Structure

We develop the mathematical foundations for a gauge-theoretic framework formulated on the 14-dimensional metric bundle Y^{14} = Met(X^4) over an oriented spin 4-manifold X, building on Weinstein's Geometric Unity program and Cox's subsequent formalization. We construct the chimeric bundle carrying Cl(0,14) spinors of dimension 2^7 = 128, define the inhomogeneous gauge group G = H ⋉ N, and give a precise construction of the Clifford contraction operator (the "Shiab operator" of Weinstein) using the real Clifford algebra, thereby resolving the complexification obstruction identified by Nguyen–Polya. We prove that the Shiab operator is unique (up to scale and exact terms) among local, frame-independent, G-covariant linear contractions, via classification of invariant tensors under the inhomogeneous group. We construct the augmented torsion, establish its transformation law under the full group G, and prove a projection-variation theorem ensuring consistency between 14-dimensional and 4-dimensional dynamics. All constructions are real (no complexification is required) and reduce to standard Einstein–Cartan geometry upon restriction to observation slices.