Phase Mathematics: Definitions, Operators, and Axiomatic Foundations

A typed relational framework for phase, coherence, closure, kernel reconstruction, and scale–phase geometry This monograph develops Phase Mathematics as a typed framework for phase-bearing states, covariant comparison, coherence measures, closure sectors, reclosure dynamics, relational kernels, and scale-extended geometry. It begins from a methodological correction: phase, coherence, closure, curvature, stress, resonance, and conservation are not interchangeable concepts. Each must be assigned a carrier, transformation law, codomain, regularity class, and epistemic status before it can enter a theorem or physical interpretation. The formal core uses Hermitian bundles, unitary transport, connection and holonomy, Hilbert and graph models, positive scalar and operator-valued kernels, gradient systems, variational source tensors, and extended position–scale connections. The manuscript proves a minimal foundational package: gauge invariance of transported coherence; existence of nontrivial holonomic models; exponential convergence of projection reclosure; local Abelian connection reconstruction from a gauge-covariant kernel; global patching from a valid kernel atlas; positive semidefinite metric extraction from a normalized positive kernel; local non-Abelian connection reconstruction from an operator-valued kernel; ordered-kernel convergence to path-ordered transport; endpoint-transitivity flatness; and exact graph local-to-global closure. Several disciplinary boundaries are made explicit. For nonzero complex scalars, the modulus of normalized overlap is identically one and cannot measure relative-phase alignment. A smooth second-order kernel Hessian generates a Riemannian rather than genuinely Finslerian infinitesimal geometry. A relational Hessian is not automatically a stress tensor. Mixed scale–phase curvature is mathematically ordinary curvature on an extended domain until scale receives an independent operational interpretation. NFNs are presently structured spectral signatures, not a completed new arithmetic. Coherence-related quantities may be exactly conserved, exchanged, topological, spectral, or monotonic according to their mathematical provenance. The resulting framework is neither a completed physical theory nor a claim that standard geometry, gauge theory, or number systems have already been replaced. It is a formally audited foundation and research program. Its leading proper-extension candidates are singular kernel-reconstruction invariants, reconstruction-residue classification, canonical operator-kernel metric selection, operational mixed scale–phase holonomy, and stable dynamic relational geometry. Keywords Phase mathematics; relational comparison; coherence; closure; reclosure; gauge connection; holonomy; positive kernels; operator-valued kernels; Finsler geometry; scale–phase curvature; NFN spectral signatures; Noether identities; Ward identities; dynamic relational geometry.