Part I - Contradiction: the geometry of conflict

Programme reconstruction notice - August 2026 This version marks the beginning of a ground-up reconstruction of the Integrodynamics research programme. Earlier versions of Part I remain preserved in this record as the historical development of the previous framework. They have not been withdrawn. However, the present version introduces a substantially rebuilt formal architecture and should not be assumed to be structurally or terminologically compatible with earlier versions. Parts II onward are being rebuilt progressively from this new foundation. Readers primarily interested in the completed revised programme may therefore prefer to treat the earlier papers as historical versions until their reconstruction is complete. Current status: preliminary formal architecture. Literature positioning, citation integration, publication-prose development and novelty claims are intentionally deferred. Abstract This paper develops a finite structural theory of contradiction. It begins with resolved invariant propositions rather than raw representations, separating contradiction-relevant content from differences of language, notation and description. For a finite epistemic state, consistency is defined by the non-emptiness of the joint intersection of the admissible models of its binding propositions; contradiction is the binary failure of that intersection. Inclusion-minimal contradictory subsets form the hyperedges of a conflict hypergraph. The minimum number of binding commitments that must cease to bind in order to restore a common model is exactly the transversal number of that hypergraph. The resulting geometry yields common-core and component-additivity results, distinguishes repair burden from minimum-repair multiplicity, and identifies the complete family of irreducible repairs as the blocker of the conflict hypergraph. Because the conflict family is a finite clutter, blocker involution establishes an exact duality between minimal contradiction and irreducible repair. The minimal conflicts also uniquely determine the binary contradiction status of every subset of the finite epistemic state, establishing contradiction-structural completeness without claiming semantic completeness. Finally, the paper records a conditional additive-to-exponential bridge for later investigation, while expressly not claiming that contradiction derives Euler's number or supplies the additional normalisation such a claim would require.