Project website: https://www.integrodynamics.org/ This paper establishes the axiomatic foundations of Integrodynamics: the unified dynamical theory governing contradiction, symmetry, adaptation, and integrity across epistemic, procedural, and institutional systems. Building on the preceding six papersof the Integrity series, this work consolidates their implicit assumptions into a minimal and complete axiom set from which the full field theory follows. Integrity is treated not as a moral property but as a conserved structural quantity governed by free-energy dissipation, symmetry constraints, evidential collapse, and bounded adaptation. Eight axioms are stated, defining contradiction observability,free-energy monotonicity, conservation of evidential structure, symmetry-stabilised equilibria, legitimacy as invariance, bounded adaptation, statistical irreversibility, and deferred intent. From these axioms, the integrodynamic law is derived, yielding a unified architecture in which contradiction manifests as curvature, symmetry as stability, adaptation as feasible flow, and evidential collapse as irreversible directional structure. The paper records the resulting theorems – including the Lyapunov structure of free-energy, symmetry-stability correspondence, legitimacy-invariance equivalence, and the conditions under which evidential ignition is unavoidable. This axiomatic closure renders Integrodynamics a complete, falsifiable physical theory of integrity, specifying clear empirical failure modes under which the system collapses. All subsequent applications derive their validity exclusively through thesefoundations. [References in progress] Keywords: integrodynamics; axiomatic systems; integrity theory; free-energy formalism; structural integrity; contradiction curvature; symmetry constraints; procedural symmetry; bounded adaptation; integrity entropy; evidential collapse; statistical irreversibility; legitimacy invariance; Lyapunov structure; phase structure; field-theoretic integrity; structural falsifiability; integrity diagnostics; epistemic entropy; institutional dynamics.