Part III - Variance: the geometry of difference

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 III is the third reconstructed component, following Part I – Contradiction and Part II – Symmetry. It begins at the precise boundary left by Part II: symmetry can determine whether a required structural correspondence is preserved, but binary breach alone cannot determine the magnitude of departure from that correspondence. 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 Part III develops a structural theory of variance as quantitative departure from a previously specified symmetry reference. Part II determines whether an eligible transformation preserves the required response correspondence. Part III refines that binary result by introducing a faithful non-negative departure field whose zero set recovers the complete symmetry-breach structure while positive values retain magnitude information unavailable to binary symmetry alone. A measured action domain then supplies the weighting required to induce a departure distribution. Mean departure, mean-square departure and variance are shown to encode distinct information, with the exact decomposition Qμ = Mμ² + Vμ. Using the symmetry quotient as a conditioning geometry, total departure variance further decomposes as Vμ = VμW + VμB, separating dispersion within structural equivalence classes from dispersion of mean departure between them. The resulting three-part second-order structure is Qμ = Mμ² + VμB + VμW. The paper establishes composition rules, realisability results, recoverability relations and a canonical measured departure structure from which the scalar variance geometry is derived. It also fixes the corresponding inferential boundary: variance determines the measured magnitude and dispersion of departure relative to a specified reference, geometry and measure, but does not determine cause, correctness, agency, importance or normative judgement.