Universal Pullbacks from the Tautological Connection: A Universal Carrier for Connection Data

We prove that principal-bundle connection data admit a canonical universal pullback carrier from which analytic and spectral structures are determined once completion is fixed. For any smooth principal G-bundle p: P → M, there is a smooth affine bundle of connections δ: Conn (P) → M whose smooth sections are exactly principal connections on P, and the pullback bundle δP → Conn (P) carries a unique tautological connection D₀ characterized by D = χD₀ for every section χ. We prove gauge equivariance, base-change functoriality for pullback connections, and the affine curvature-translation law. In the abelian case the tautological curvature defines a canonical closed 𝔤-valued two-form on Conn (P) ; in the rank-one abelian U (1) sector, this yields the distinguished symplectic realization of the theory, with Lagrangian fibers. After specifying the completion lock Lock—that is, the completion, normalization, and regularization data fixed in advance—a section χ, and a twist direction ϑ, the same universal carrier determines an analytic holonomy/twist family, a Kato-analytic family of Laplace-type operators Δ_θ, and a zeta-regularized generating functional Z (θ; Lock) on regular parameter domains. We prove the connection-geometric and analytic hypotheses needed for the resulting locked spectral datum, including kernel stability, zeta-regularity, joint meromorphy and parameter-analyticity of zeta data, and the corresponding zeta-variation formulas with explicit kernel conventions. Determinant and zeta-type structures are thus not introduced as independent analytic inputs, but are supported canonically by analytic descent from the tautological connection once completion is fixed. The tautological connection D₀ therefore serves as a universal carrier for connection data together with their locked analytic realization. Beyond the underlying bundle-theoretic data, only the completion lock and a choice of section and twist direction are required; no prior Laplacian, Hamiltonian, metric background, or independently specified spectral operator enters as input. The same locked datum canonically determines primitive holonomy and spectral channels, through which branch projections and invariant quantities factor once the corresponding fixed data are specified. The theorems therefore determine a single-carrier factorization hierarchy in which connection data are carried by D₀, analytic and spectral structures descend once completion is fixed, and invariant branches factor through common locked channels. License note: Distributed under CC BY-NC-ND 4. 0.