Geometric Derivation of the Scaling Law Eₙ = xⁿ from D = 3 Euclidean Geometry

This work introduces a dynamical scalar field n ∈ (0, 4) that quantifies the complexity of the interaction network among Planck units in three-dimensional physical space (D = 3). The field evolves together with the volume, generates discrete phase transitions (d = 0 → 1 → 2 → 3), and carries an energy ledger Eₙ = xⁿ (x = 2) derived from the geometric measure. The framework equips n with two geometry-derived functions — a kinetic function K(n) and a potential U(n) — thereby establishing a non-singular, closed bounce cycle.Dark energy is interpreted as the unfolding energy released as form gains dimensionality. With the structural bound w ≥ −1, the framework yields the prediction w₀ ≈ −0.88 ± 0.02 (numerical; slow-roll check −0.86). Because both x and the energy form are derived from geometry, this is an uncalibrated prediction (the only calibrated parameter is n_today ≈ 3.3), and it is structurally consistent with the dynamical-dark-energy direction of DESI DR2 (w₀ > −1, w_a < 0). The non-singular bounce is realized at the two ends by two independent physical agents: a gravity-triggered turnaround at the matter end (n → 4⁻), and a classical bang established by Planck-density closure at the Planck end (n → 0⁺). The same equations describe both cosmological and local black-hole collapse; this self-similarity predicts every black hole to be a cycle core.A geometric motivation for the matter–energy–force hierarchy is offered; however, it should not be read as a derivation of the gauge structure or of Standard-Model phenomenology (see Section 1, Scope of the Work).