Part IV - Integrity: the geometry of preservation

Programme reconstruction notice — August 2026 This record forms part of a ground-up reconstruction of the Integrodynamics programme. Earlier versions are retained as part of the historical record, but the programme is being rebuilt from first principles so that each component is independently specified, formally bounded and connected to the others only through explicit structural interfaces. Part IV is the fourth reconstructed component, following Part I – Contradiction, Part II – Symmetry and Part III – Variance. It begins at the precise boundary left by Part III: contradiction determines whether binding commitments can coexist, symmetry determines whether required structural correspondences are preserved, and variance quantifies departure from those correspondences, but none of those theories alone determines whether the resulting structural requirements are jointly preserved. Part IV therefore develops integrity as that higher-order preservation property, while retaining the preceding structures as non-reducible components rather than collapsing them into a new scalar coordinate. Current status: preliminary formal architecture. This version presents the frozen formal architecture before literature positioning and publication-prose development. Definitions, propositions, theorems, proofs, canonical examples, realisability results and non-identifiability boundaries are included for architectural review. Citation apparatus and novelty claims are intentionally deferred. Abstract This paper develops a mathematical theory of structural integrity from the preceding theories of contradiction, symmetry and variance. Integrity is defined not as an additional independent quantity, but as a higher-order preservation property of an explicitly specified structural frame. Universal structural integrity holds exactly when the binding propositions of a finite epistemic state are jointly satisfiable and every response correspondence required by the governing symmetry structure is preserved. The framework distinguishes binary integrity status from the richer geometry that produces that status. Contradiction and symmetry remain non-reducible structural components, while faithful departure magnitude and variance quantify the geometry of symmetry-relative failure without creating an additional Boolean integrity condition. The theory distinguishes universal pointwise integrity from measured almost-everywhere integrity, identifies the information lost through domain restriction and null-set quotienting, and establishes conditions under which measured structure recovers universal preservation. For structurally separated domains, integrity composes conjunctively: complete product integrity holds if and only if every required component is intact. This yields an exact no-compensation result without requiring probabilistic, statistical or causal independence. The paper further develops typed failure witnesses, quantitative failure geometry, repair boundaries, structural completeness, recoverability and compression relations, and explicit non-identifiability boundaries. No canonical scalar integrity score, total ordering, common severity scale or global repair operator follows from the primitive structure. The resulting theory therefore characterises structural integrity as binary at its terminal preservation boundary while retaining an irreducibly typed and information-rich geometry underneath that classification.