Structural Interpretability - Unifying Deep Network Analysis via Piecewise Linear Path Algebras

The operational deployment of Deep Neural Networks (DNNs) in safety-critical domains remains limited by theirstructural opacity. Post-hoc Explainable Artificial Intelligence (XAI) tools provide useful but potentially diver-gent views of the same network decision. In this work, we introduce Unified Quiver Analysis of Neural Networks(UQANN), a framework that connects empirical XAI probes to structural properties of local path algebras. Build-ing on the continuous piecewise-linear (CPWL) structure of standard rectified architectures, we describe an input-specific computation through a local polyhedral linearity cell and an associated instantiated path algebra RQ|x .Under the stated masking and path-granularity assumptions, the combinatorial path support is invariant withinthe interior of a cell, while the local forward computation admits a decomposition over active path morphisms.UQANN provides a common reference in which several diagnostic procedures can be related: Feature Visualiza-tion is expressed through path-wise gradient directions; Network Dissection is implemented through a path-fluxbased Algebraic Coherence Index; Topological Data Analysis is applied through a spatial proxy for sampled path-support structure; and Integrated Gradients is related theoretically to a path-level completeness identity underexplicit assumptions, while the present implementation evaluates a coarser morphism-level approximation. Thetheoretical framework is stated at channel-level path granularity. However, several empirical probes use computa-tionally tractable approximations. In particular, the TDA, Integrated Gradients, and poly-semantic-unit analysesuse computationally tractable proxies, including thresholded filter-level spatial supports, macroscopic morphisms,and thresholded Integrated Gradients, as detailed in Appendix Appendix D. Accordingly, empirical results shouldbe interpreted as evidence concerning these operational proxies, not as exhaustive validation of the full channel-levelpath algebra. We evaluate the framework using a CNN2QAN conversion pipeline applied to a sequential VGG16architecture trained from scratch on Tiny ImageNet. The model provides a non-pretrained proof-of-concept baselinefor the present diagnostic study; detailed training and performance metrics are reported in Appendix Appendix C.2.Across a set of comparative visual cases, path-inspired operational routing proxies reveal differences beneath similarscalar activation profiles, while targeted channel ablations provide preliminary intervention evidence that selectedintermediate channels affect the logits of the examined classes. These results support the use of local path represen-tations as a structured coordinate system for comparing attribution, semantic alignment, topology, and interventionoutcomes. They do not establish a complete resolution of the disagreement problem. Finally, we discuss extensionsto residual architectures and propose RQ-regularization as a future direction for studying path complexity andsemantic alignment during training. All semantic conclusions are conditional on this proof-of-concept model andshould be interpreted as methodological demonstrations.