Re-Phase Vol.3.1: Linear and Algebraic Admissible Structures

This volume develops the linear and algebraic axis of the Re-Phase framework. Vol. 3. 1 starts from the Level 0 compatibility core established in the preceding foundation volumes: compatibility relations, compatible sets, admissible intersections, context order, recoverability regions, and context-relative indistinguishability quotients. It then studies what happens when this core is enriched by linear or algebraic structure. When compatibility conditions are induced by linear maps (Lᵢ: X Yᵢ), a context (S) determines a block map (LS: X ₈ ₒYᵢ), and the admissible set becomes a fiber: AS=LS^-1 (oS). If this fiber is nonempty, it is an affine coset: AS=x₀+ LS. Thus, in the linear setting, admissible ambiguity becomes kernel freedom. Context-relative indistinguishability becomes a kernel quotient (X/ LS), and structural recoverability is evaluated over fibers or cosets. For a linear feature map (h: X Z), the feature image of an admissible coset satisfiesh (AS) =h (x₀) +h (LS). This separates target recoverability from unique feature recoverability. Target recoverability requires (h (AS) B), while unique feature recoverability requires (h (AS) =h (x₀) ), equivalently (LS h). The volume also identifies this latter condition as the linear descent condition: the feature map (h) descends to the quotient vector space (X/ LS) exactly when it is constant on cosets of (LS). Vol. 3. 1 does not develop coding theory, delayed-closure symbolic sequences, spectral weak modes, Fisher geometry, robust compatibility, probabilistic inference, temporal continuation, or computational search algorithms. Those structures are deferred to later volumes. The purpose of this volume is to provide the linear and algebraic foundation on which those later structures can be safely built.