Boundary Calculus, Rigidity Islands, and Deformation Theory in Algebraic Phase Structures

We develop a general boundary calculus for algebraic phases and use it to formulate an intrinsic structural framework for deformation and obstruction phenomena. Structural boundaries are shown to be finitely detectable and canonically stratified by failure type and depth. For each boundary we construct a canonical boundary exact sequence and identify a maximal rigid subphase, called a rigidity island, that persists beyond global boundary failure. Rigidity islands are organised by intrinsic invariants and serve as canonical base points for deformation theory. Deformation behaviour within the standing admissibility framework is governed by boundary quotients, while rigidity islands remain stable under admissible deformation. Boundary quotients act as obstruction objects whose associated strata organise higher-depth deformation behaviour. As a consequence, deformation behaviour is naturally stratified by boundary depth and failure type, while formal smoothness is associated with the vanishing of boundary data. The resulting moduli behaviour is organised by rigidity islands together with their associated obstruction patterns, without requiring intrinsic analytic or continuous deformation parameters.