Energy in Extended Classical Mechanics (ECM): Mathematical Support — Part - 2

This paper provides the formal mathematical support for the Extended Classical Mechanics (ECM) framework introduced in Part 1. Building on the tripartite state progression of existence — the latent state, the phase state, and the manifest state — we derive explicit functional forms for the latent force limit F_latent = g(Δx) that satisfy the structural constraints of ECM: a bounded displacement domain within the pre-Planck interval (ℏℏₚ/360²) · (1/360) ≤ Δx ≤ (ℏₚ/360)·(359/360), asymptotic divergence as Δx → 0 while remaining finite at every discrete non-zero state, and consistency with the potential-gradient relation F_latent = −dPE_ECM,latent/dΔx. Two candidate forms are developed: (1) an inverse-power potential gradient exhibiting power-law hyper-stiffness, and (2) a phase-modulated exponential barrier incorporating 360-degree geometrical scaling. We evaluate the definite transition integrals of the energy-accounting scheme, demonstrating that transferred potential energy ΔPE_ECM between any two non-zero displacement endpoints is finite and maps directly onto kinetic energy ΔKE_ECM under the ECM positive-magnitude convention. The explicit velocity scaling of the emergent manifestation is derived, and the partition of the latent reservoir into residual effective potential energy (PE_ECM)^eff and transferred energy ΔPE_ECM is formalized. Four mathematical distinctions are preserved throughout: latent-state divergence versus finite states; the displacement interval versus the transition threshold ℏₚ/360; potential versus transferred energy at vanishing displacement; and phase velocity versus propagation velocity.