Directed Layered Bases: A Fiber-Theoretic Approach to Structural Memory

This paper introduces Directed Layered Bases (DLB) as a sub-program of the SHA-DNS framework, addressing systems where basis layers are statically orthogonal but dynamically connected through threshold-activated transitions. The central construction is a path-based formulation of structural residual and its interface with hardening degeneracy. We define directed layered basis systems on a graph G = (V, E), formalize Carry and Reset operators as state transitions, and introduce the Jump Residual R_{i→j} as the componentwise discrepancy between a direct transition and a stepwise reference path, projected onto skipped layers. The deeper structural observation is that degeneracy is compositional: successive projections Path(i,j) --P--> S --h_core--> H produce successive fibers, and the composite fiber decomposes as |Φ^{-1}(H)| = Σ_{S ∈ h_core^{-1}(H)} D_path(S; i, j). Residuals record what survives the projection; memory requirements record what must be retained to reverse it. We distinguish state memory from path memory, showing that these yield different information lower bounds. This paper is a sub-program of the SHA-DNS framework. It does not attempt to complete the theory of dynamically connected layered bases. It defines the core structure, establishes the Fiber Decomposition Identity and Factorization Bound, and invites further work. Eight open problems define the immediate research boundary. Version: v1.5 (September 2026) Series: Directed Layered Bases / SHA-DNS Sub-Program