Analytic Detection of Singularities in Categorified Spectral Geometry: Heat Kernels, Spectral Flow, and Obstruction Theory

The Categorified Spectral Duality (CSD) framework provides a geometric and cohomological description of operator-semantic systems through the spectral stack. A fundamental gap, however, remains between this structural description and the analytic extraction of quantitative invariants. This paper closes this gap by developing an analytic theory for categorified spectral spaces. Our first main result constructs a canonical analytic enhancement of the categorified spectrum, equipped with a Hilbert-type structure, a closed operator, a compatible connection, and a trace measure. We prove that this enhancement is functorial, Morita-invariant, and compatible with effective hyperdescent. Our second main result establishes a Rees-type comparison theorem giving a canonical filtered equivalence between the algebraic nilpotent filtration of an admissible realization and the geometric singular filtration of the categorified spectrum. The equivalence is mediated by the analytic enhancement and induces an isomorphism of the corresponding Rees modules. Under an additional detection-faithfulness hypothesis, the comparison identifies maximal nilpotent depth with singular thickness. Our third main result shows that the singular filtration is analytically detectable. Localized heat coefficients recover obstruction characteristic classes through a bigraded pairing between the Chern character of the linearized obstruction object and universal local heat-kernel densities. Under spectral-faithfulness hypotheses, the resulting localized heat signature is injective on obstruction cohomology. The associated deformation classes further admit an index-theoretic realization through spectral flow and analytic boundary pairing. These correspondences culminate in an analytic detection theorem identifying three a priori distinct quantities: the maximal nilpotent depth of the algebraic realization, the singular thickness of the truncated categorified spectrum, and the maximal infinitesimal layer detectable by localized heat asymptotics. Thus, in a precise analytic sense, infinitesimal algebraic thickness can be “heard” from spectral data. Together, these results establish a local-to-global analytic mechanism connecting non-semisimple algebraic structure, singular spectral geometry, heat-kernel asymptotics, obstruction theory, and index theory. As an application, we show that the CSD nilpotent invariant detects the classical reducibility locus for principal series of GL2(L).