Part II - Symmetry: the geometry of fairness

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 II is the second reconstructed component, following Part I – Contradiction: the geometry of structural conflict. It develops symmetry as a distinct structural primitive rather than treating fairness, bias, discrimination or normative judgement as primitive inputs. 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 structural theory of symmetry as preservation under transformations that leave specified relevant structure unchanged. A resolved state space is equipped with a relevance map, and reversible structure-preserving transformations form an eligible symmetry group whose action partitions the state space into orbits representing structurally interchangeable positions. A response is symmetric when it is equivariant under a coherent corresponding action on the output space, with invariance as the fixed-output special case. This yields an intrinsic structural fairness result: an invariant response cannot distinguish between different representatives of the same symmetry class. Symmetry breaking is represented by a binary breach function over eligible transformation–state pairs and its associated breach set; global equivariance holds exactly when that set is empty. The resulting geometry identifies transformation-specific and state-level breach support, broken orbits and the unique maximal subgroup of eligible transformations under which the response remains equivariant. In the invariant case, broken symmetry is equivalently the splitting of a structural symmetry orbit by the response partition. Structurally separated symmetry domains compose through product actions and quotient structures. For a fixed eligible transformation architecture and output action, the breach function is complete for binary symmetry behaviour, while arbitrary equivalence-class geometries are realisable. The theory remains deliberately non-causal and non-normative: symmetry structure alone does not identify response magnitude, correctness, cause, agency, motive, bias, discrimination or legal or moral fairness.