Structural Depth Theory III: Spectral Accessibility, Structural Depth, and Sheaf-Theoretic Constraint Geometry

Structural Depth Theory III establishes a formal bridge between sheaf-theoretic invariant structures and spectral accessibility. Starting from a Structural Constraint Site, this work constructs an Invariant Relational Residue, a Containment Poset, an Accessibility Nerve, a Relational Skeleton Graph, a Structural Laplacian, and an Accessibility Spectrum. The resulting framework provides a complete structural determination chain linking local admissibility conditions to observable accessibility structures. The central result, the Spectral Accessibility Construction Theorem, demonstrates how accessibility can emerge from invariant relational geometry without assuming a predefined state space. Accessibility is treated as a derived structural property rather than a primitive feature of reality. This paper extends the Structural Depth Theory research program developed by Halil Ibrahim Guven and directly builds upon: • Structural Depth Theory I: Foundations of Structural Constraint Geometry via Grothendieck Sites • Structural Depth Theory II: Structural Determination and Sheaf-Theoretic Coherence The work forms part of the broader Market Alchemy Framework, an interdisciplinary research program investigating structural constraints, accessibility, emergence, complex systems, markets, cognition, and organizational dynamics through relational geometry. Keywords:Structural Depth Theory, Structural Depth, Structural Determination, Structural Constraints, Accessibility Spectrum, Spectral Accessibility, Category Theory, Sheaf Theory, Grothendieck Topology, Spectral Graph Theory, Relational Geometry, Complex Systems, Emergence, Constraint Geometry, Market Alchemy Framework. Author:Halil Ibrahim Guven ORCID:0009-0008-4942-5495 Research Portal:https://marketalchemy.io Author Page:https://amazon.com/author/higuven