The Coherence Field: A Canonical Reversible Operator Arising from Curvature

This paper introduces and rigorously defines a canonical Coherence Field associated with any system admitting a monotone functional with a well-defined second variation. While prior work within the Tier-0 framework established a unique dissipative generator (the Λ-field) governing irreversible dynamics, the present work identifies the complementary reversible structure implied by the same curvature data. The coherence field is constructed operator-theoretically from the Hessian of an admissible monotone at equilibrium, yielding a bounded, selfadjoint operator whose unitary flow captures phase-stable, record-free, and reversible dynamics. This construction is universal, functorial, and independent of microscopic details, relying only on curvature properties rather than model-specific assumptions. The paper develops the full mathematical framework of the coherence field, including its spectral definition, coherence propagator, determinant, and associated coherence budget. A coherence monotonicity law is established, demonstrating stability and invariance under admissible embeddings. The resulting structure provides a law-level characterization of reversible dynamics that complements irreversible Λ-field behavior without introducing new forces or modifying existing physical theory. Physical realizations are discussed across quantum statistical mechanics, relativistic field theory, renormalization group flows, and geometric settings. In particular, relativistic null propagation is identified as a canonical realization of coherence: a regime in which phase and correlation are transported without proper-time accumulation or record formation. Together with the Λ-field, the coherence field completes a second-variation description of admissible dynamics within the Tier-0 framework, unifying dissipative and reversible structures as complementary consequences of curvature-driven lawfulness. Companion Work: • The Canonical Λ-Field: Uniqueness, Spectral Determinants, and Dissipative Generators (https://doi.org/10.5281/zenodo.18091880)establishes a rigidity theorem for the dissipative (Δ-sector) component of the framework, proving that once irreversible dynamics is present, the associated scalar spectral invariant (the Λ-field) is uniquely forced. This result supplies the definitive collapse-sector anchor of the universal closure architecture. • λ–Profiles and the Universal Spectral Budget: Lawful Structure of Dissipative Generators (https://doi.org/10.5281/zenodo.18092618) develops the law-level consequences of the Λ-field rigidity theorem by showing how the uniquely forced dissipative scalar organizes universal behavior across systems. Rather than re-deriving Λ, the paper studies its profiles, normalization, stability, and cross-domain comparability, establishing a universal spectral budget for irreversible dynamics. This work elevates the Λ-field from a collapse-sector invariant to a lawful organizing quantity, enabling meaningful comparison of dissipative systems across physics, geometry, computation, and data without model-dependent assumptions. The Tier-0 Framework and the Everything Equation: A Universal Recursion Law for Physics, Mathematics, and Information (https://doi.org/10.5281/zenodo.17813117), where the global admissibility principle for lawhood is introduced abstractly. The present work provides a definitive, sector-level resolution of the Lambda field within that framework. • The Everything Equation: A Universal Closure Principle for Law Structure (https://doi.org/10.5281/zenodo.18081205) provides the mathematical foundation for the universal closure recursion at the core of this framework, proving its inevitability and uniqueness within an abstract admissible setting. • The Everything Equation in Physics: A Universal Closure Principle for Physical Law (https://doi.org/10.5281/zenodo.18080442) demonstrates the physical instantiation of this abstract structure in standard field theories and renormalisation group universality, showing how physical laws arise as fixed points of the same closure recursion.