Unified Phase-Kernel Field - Axiomatic Foundation: Extended Classical Mechanics (ECM)

This paper presents the axiomatic foundation of Extended Classical Mechanics (ECM) through the unified phase-kernel field formalism. The ECM Cosmic-Quantum Invariant is established as the foundational postulate, declaring that all physical observables--energy, mass, frequency, and motion--are projection states of a single conserved phase-kernel frequency structure. A complete hierarchy of axioms is derived: kernel decoupling governs the transition from bound to free radiative modes; phase-mass conversion regulates the emergence of matter from kernel curvature; frequency-space and phase-drift-time emergence replace pre-existing geometric containers with integrated kernel dynamics; and spacetime consistency ensures mutual constraint of emergent geometry and temporal flow. Gravitational emergence is formulated as kernel instability relaxation, with curvature arising from second-order frequency non-uniformity. Cosmological boundaries, entropic drift, cyclic recurrence, and inter-cycle memory encoding are derived as natural consequences of kernel energy density evolution. The framework culminates in Axiom 3.0, which unifies quantum and cosmological regimes as scale-dependent projections of one invariant 0-D phase-kernel field, eliminating the fundamental distinction between quantum mechanics and cosmology.