Re-Phase Vol.1.8: Compatibility Conditions and Enrichment Axes

This volume gives a foundation map for the Re-Phase framework. Vol. 1. 8 clarifies that the primitive object of Re-Phase is not necessarily an observation map, but a compatibility condition relating configurations and given conditions. A given condition oᵢ ∈ Oᵢ determines a compatible set Kᵢ (oᵢ) ⊂ X, and a context S ⊆ I determines an admissible set AS = ⋂₈∈ₒ Kᵢ (oᵢ). The Level 0 core includes compatibility relations, compatible sets, admissible intersections, context order, recoverability regions, and context-relative indistinguishability quotients. In particular, context posets and quotients X/≡S are treated as core set-relation structures rather than later enrichments. The volume distinguishes the Level 0 core from enrichment axes that require additional assumptions: functional/map-fiber, metric/robust, linear/algebraic, differential/tangent, inner-product/spectral, probabilistic/Bayesian, temporal/continuation, and computational/search axes. Some axes are composite enrichments, such as differential/tangent structures and inner-product/spectral structures. This volume also states core well-definedness principles: compatibility truth values, compatible sets, admissible intersections, order reversal, recoverability regions, and context-relative quotients are well-defined at the set-relation level once the relevant data are specified. Additional assumptions are required only when moving to enrichment axes. Vol. 1. 8 is intended as an integrated foundation map for the preceding and following Re-Phase volumes. It clarifies what belongs to the minimal compatibility core, what is well-defined at that level, and what requires additional structure in later volumes.