Relative Obstructions and Spectral Diagnostics for Sheaves on Cell Complexes

TL;DR

This work reframes structural inconsistency as a relative obstruction between a model sheaf ${\mathcal F}$ and a grounding ${\mathcal W}$ via a cochain map $\epsilon:{\mathcal F}\to{\mathcal W}$, and diagnoses it with the mapping cone cohomology $H^{*}(K;{\mathcal F},{\mathcal W})$. It introduces spectral witnesses from both regular and mapping-cone Laplacians that quantify the magnitude, robustness, and localization of obstructions, enabling comparison of distinct inconsistency mechanisms without altering the underlying representation. A geometric and algebraic cone correspondence links relative cohomology to observable spectral signals, delivering a practical, evaluative toolkit for assessing feasibility in fixed systems. The framework is domain-agnostic and supports a nuanced, scale-aware view of feasibility through a filtration-based, interleaving-stable suite of global and local indicators. Together, these contributions advance a principled, computable approach to grounded consistency in sheaf-theoretic models, with potential applications in data fusion, constraint reasoning, and counterfactual feasibility analysis.

Abstract

Many structured systems admit locally consistent descriptions that nevertheless fail to globalize when constrained by an ambient reference or feasibility condition. Diagnosing such failures is naturally an evaluative problem: given a fixed model and a grounding, can one determine whether they are structurally compatible, and if not, identify the nature and localization of the obstruction? In this work, we introduce a sheaf-theoretic and spectral framework for evaluating structural inconsistency as a \emph{relative} phenomenon. A model is represented by a cellular sheaf $\mathcal F$ on a cell complex, together with a morphism into a grounding sheaf $\mathcal W$ encoding admissible global behavior. Failure of compatibility is captured by the mapping cone of this morphism, whose cohomology computes the relative groups $H^*(K;\mathcal F,\mathcal W)$ and separates intrinsic obstructions from inconsistencies induced by the grounding. Beyond exact cohomological classification, we develop \emph{spectral witnesses} derived from regular and mapping-cone Laplacians. The spectra of these operators provide computable, quantitative indicators of inconsistency, encoding both robustness and spatial localization through spectral gaps, integrated energies, and eigenmode support. These witnesses enable comparison of distinct inconsistency mechanisms in fixed systems without learning, optimization, or modification of the underlying representation. The proposed framework is domain-agnostic and applies to a broad class of structured models where feasibility is enforced locally but evaluated globally.

Figures (2)

• Figure 1: Local spectral witnesses for hidden twist (top row) and noisy trivial assignment (bottom row). Each row visualizes local spectral witnesses $\mathcal{W}_{j,\epsilon}$ on the same cycle complex, comparing intrinsic (base) Laplacian channels with grounded (mapping-cone) diagnostics. Shown are heatmaps induced by low-energy eigenmodes of the intrinsic $j=0$ and $j=1$ Laplacians, together with the relative cone channel corresponding to the intrinsic $j=1$ structure. Although the hidden twist and noisy trivial constructions exhibit comparable global spectral gaps, the hidden twist produces sharply localized low-energy modes, while the noisy trivial case displays diffuse energy. This localization behavior distinguishes structured logical defects from unstructured noise at the level of local spectral witnesses.
• Figure 2: 1D intersection $\mathcal{F}(u)\cap\mathcal{F}(v)$, each of which is spanned by two 3-dimensional vectors.

Theorems & Definitions (40)

• Remark 1: Degree-dependent interpretation of obstruction
• Definition 1: Ambient grounding
• Proposition 1: Geometric cone realizes the translated mapping cone
• proof
• Definition 2: Incidence-level defect
• Remark 2: Two regimes
• Lemma 1: The induced long exact sequence.
• proof
• Remark 3
• Remark 4: Relation to Morse-theoretic viewpoints.