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

Joe Gildea

TL;DR

This work develops a boundary calculus for terminating algebraic phases, showing that all structural variation is governed by intrinsic boundaries rather than analytic data. Boundaries induce a maximal rigid subphase, the rigidity island, and a universal obstruction encoded in the boundary quotient $\mathcal{B}(\mathcal{P})$, with deformations fully boundary-controlled and finite. Rigidity islands are classified by intrinsic invariants (defect rank $d$, termination length $L$, and interaction signature $\Sigma$), and moduli form a finite stratified groupoid indexed by these islands. The boundary quotient decomposes into finitely many obstruction components $\mathcal{B}_i(\mathcal{P})$, corresponding to distinct boundary failure types, and every deformation factors through this finite, discrete obstruction data, yielding a structurally closed deformation theory for algebraic phases.

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.

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

TL;DR

This work develops a boundary calculus for terminating algebraic phases, showing that all structural variation is governed by intrinsic boundaries rather than analytic data. Boundaries induce a maximal rigid subphase, the rigidity island, and a universal obstruction encoded in the boundary quotient , with deformations fully boundary-controlled and finite. Rigidity islands are classified by intrinsic invariants (defect rank , termination length , and interaction signature ), and moduli form a finite stratified groupoid indexed by these islands. The boundary quotient decomposes into finitely many obstruction components , corresponding to distinct boundary failure types, and every deformation factors through this finite, discrete obstruction data, yielding a structurally closed deformation theory for algebraic phases.

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.
Paper Structure (28 sections, 38 theorems, 100 equations)

This paper contains 28 sections, 38 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Standing Framework and Structural Assumptions
  3. Structural Boundaries Revisited
  4. Boundary Failure and Detection
  5. Types of Boundary Failure
  6. Boundary Detection Criteria
  7. Boundary Exact Sequences, Rigidity Islands, Classification, and Obstruction
  8. Boundary Exact Sequences
  9. Rigidity Islands
  10. Classification of Rigidity Islands
  11. Boundary Quotients and Obstruction Objects
  12. Predictive Boundary Calculus and Deformation Theory
  13. Predictive Power of the Boundary Calculus
  14. Deformation Theory in Algebraic Phase Theory
  15. Boundary Structure in Representative Phase Families
  16. ...and 13 more sections

Key Result

Proposition 3.2

Let $\mathcal{P}$ be a terminating algebraic phase with defect rank $d$ and boundary depth $k$. Then necessarily Moreover, if $k>d$, one may write where $k_{\mathrm{ext}}$ is the weak extension depth: the number of filtration steps beyond defect generation for which canonical propagation of higher interaction constraints remains functorial before the first failure.

Theorems & Definitions (110)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Proposition 3.7
  • ...and 100 more