FDS–H2 develops a formal bridge architecture between finite causal-screen recovery geometry and independently registered premetric response obstructions. The paper begins from the finite-screen recovery framework of FDS–H1 and asks when regional recovery data can support a physically identifiable response obstruction. It separates several logically distinct structures that must not be conflated: raw regional descent; physical quotient descent; strict liftability and its non-Abelian lifting obstruction; mixed compatibility between regional gluing and parameter transport; recoverable-support curvature; task-defined physical gauge reduction; and response visibility. The principal geometric construction is a relative Kato connection on a smooth constant-rank recoverable-support bundle inside a registered ambient Hermitian bundle. This construction is deliberately support-limited: recovery channels with the same registered support projector have the same Kato connection and curvature even when their spectra, weights, fidelities, probabilities, or channel actions differ. H2 therefore distinguishes a completed recoverable-support branch from an open full recovery-channel geometry. The response side is divided into two independent bridge branches. Operational-covector branch. A directly registered G1 response one-form ωth is related to the physical recovery curvature through a pre-registered parallel flat-coefficient readout: dωth = ℓ(Fphys). This branch is tested through held-out Stokes loop–surface comparisons. A global covector realization additionally requires the vector-valued de Rham class [ℓ(Fphys)]dR to vanish. Closed-cycle periods provide an operational test of the global obstruction. Response-bundle branch. An independently registered response connection is related to the physical recovery bundle by a parallel bundle morphism. The existence of such a morphism is equivalent to a base-point holonomy-intertwining condition. This branch has its own held-out-loop validation protocol, curvature-naturality relation, and Hom-bundle defect for approximate bridges. The paper also establishes visibility limits. General parallel flat-coefficient readouts are constrained by the actual registered holonomy group, whereas character-based one-dimensional responses see only the Lie-algebra Abelianization of the identity component. Discrete characters may additionally detect global flat monodromy. Observable families are blind to curvature sectors lying in the kernel of their registered representation. A three-qutrit calculation is included only as a code-seeded candidate-support audit. The correctability-preserving product-local control is flat in the registered test, while a tested nonzero-support-curvature deformation violates the Knill–Laflamme condition and is therefore ineligible for physical-quotient analysis. The existence or impossibility of a smooth, fully correctable code manifold with nonzero physical support curvature is retained as an open problem. The release package includes the manuscript and deterministic companion materials for the qutrit audit, including the Python script, expected JSON output, dependency specification, reproduction instructions, and checksums. Scope. H2 establishes a formal recoverable-support bridge architecture, branch-specific existence criteria, visibility bounds, finite-cost non-absorption rules, and validation protocols. It does not yet provide a completed nontrivial physical bridge, a universal full-channel recovery geometry, a curved quantum-error-correcting-code realization, general relativity, the G1 residual tensor, or the optical M3/4 branch. Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory