Algebraic Phase Theory V: Boundary Calculus, Rigidity Islands, and Deformation Theory

Abstract: We develop a general boundary calculus for algebraic phases and use it to formulate an intrinsic and purely structural deformation theory. Structural boundaries are shown to be inevitable, finitely detectable, and canonically stratified by failure type and depth. For each boundary we construct a canonical boundary exact sequence and identify a unique maximal rigid subphase, called a \emph{rigidity island}, that persists beyond global boundary failure. Rigidity islands are classified by intrinsic invariants and serve as universal base points for deformation theory. All deformations are boundary-controlled, restrict trivially to rigidity islands, and are governed by boundary quotients, which act as universal obstruction objects. Infinitesimal and higher-order obstructions are finite, stratified by boundary depth, and terminate intrinsically. As a consequence, deformation directions are discrete, formal smoothness is equivalent to the vanishing of boundary data, and the moduli of algebraic phases form a stratified discrete groupoid indexed by rigidity islands. No analytic or continuous moduli parameters arise intrinsically.