This paper specifies the representation geometry used to measure residual distortion in finite obstruction systems. It provides a formal mapping from combinatorial ℓ¹ cochains to spectral trace-class operators, enabling calculation of the Representative Distortion Metric Ξ. Crucially, it establishes the Combinatorial Primary Doctrine: the discrete topology of the obstruction field is the primary authority, while the spectral representation is a derived, lossy, and tertiary refinement. The paper formalizes the Lossy Collision Principle, which states that combinatorial structure determines representation, but the representation layer cannot uniquely reconstruct the source combinatorial configuration. This asymmetry is intentionally enforced to prevent the collapse of the verification framework into ungrounded representation metaphysics. By defining the Density Lift Fden and the resulting trace-distance metrics, this work provides the final tie-breaking mechanism used in the guarded descent evaluator, allowing high-resolution diagnostic comparisons that complement the primary topological obstruction metrics Φ₁ and Γ₂. Key technical contributions: 1. Formalization of the Combinatorial Primary Doctrine and the strict subordination of spectral metrics. 2. Definition of the Linear Trace-Class Embedding Fₗin and its norm-preservation properties. 3. Specification of the Density Lift Fden and the Trace Metric Ξ for representative comparison. 4. Proof of the Lossy Collision Principle and the resulting reconstruction obstruction. 5. Establishment of the Hilbert-Lift Quarantine, explicitly separating representation metrics from physical or quantum mechanical claims.
JEREMY H. CARROLL (Fri,) studied this question.