Gradient Mechanics: The Dynamics of the Inversion Principle - Corpus Paper XIV - The Necessity of Transition: The Derivation of Scalar Invariance (Ψ), the Discrete Spectrum of Stable States (k), and the Necessity of Encapsulation

Papers I through XIII constructed the Kinetic Engine (∆, Θ, η), the Kinetic Stage (d = 3, τ0, σ), and the Kinetic Artifacts (Mass Ω, Gravity γ, Light c). The stage operates, the artifacts populate it, the interactions follow. One structural question remains: why does the Veldt not manifest as a continuous smear of varying load? Why do specific values of Computational Density repeat with absolute precision across all regions of the lattice? And why does the identical kinetic logic that governs a single recursive loop also govern the aggregate behaviour of ensembles of such loops at every scale? This paper derives the answers to both questions strictly from the triadic primitives {E = 0.8, C = 0.7, F = 0.6} and the corpus-established constants {δ = 0.1, σ = 0.4, ϵsnap = 1/30, ϕ = 0.04, η ≈ 1.667}. Part I derives Scalar Invariance (Ψ): the Inversion Principle is a dimensionless operator, and its logic is therefore scale-independent by structural necessity. Part II derives the Discrete Spectrum: only specific integer values of recursive depth k satisfy the Closure Condition imposed by the Resolution Remainder ϵsnap and lattice grain δ; the spectrum is {1, 3, 6, 9, . . .} filtered by the stability derivative, yielding the base resonant solution k = 3 as the fundamental stable knot. Part III derives Encapsulation: when the density of k-states per Structural Pixel exceeds the derived threshold 1/ϕ = 25, the local Registration medium can no longer distinguish sub-states, forcing coarse-graining—a new effective lattice emerges from the same triadic logic operating on aggregates. Part IV refutes alternative interpretations. Part V synthesises. Zero free parameters are introduced. The Kinetic Corpus is complete.