A Generative Field Equation for Cyclic Matter Dynamics: The Schrödinger–REK–RGB Functional

We introduce a variational energy functional E[Ψ_H] for a complex scalar field Ψ_H — the holistic field — governing the cyclic dynamics of generation, manifestation, and structural return within the G→X→Q→N cycle of the Kinematic Energy PHramework (KEPH). The functional extends the Gross-Pitaevskii–Ginzburg-Landau class through three original terms: a triaxial symmetry-breaking potential V_RGB encoding polar differentiation of space into contractive (R) and expansive (B) axes around a neutral generative axis (G); a state-dependent aggregation coupling g(Ψ_H) encoding centripetal densification; and a non-local memory functional ε_REK[Ψ_H] with oscillatory doubling kernel derived from the self-composition law (N∘X)·A·U = A²·U. Variation yields the Schrödinger–REK–RGB equation — a non-linear non-local integro-differential generalization of the Schrödinger equation. Standard formulations are recovered as limiting cases: Schrödinger (g=0, λ_REK=0), Gross-Pitaevskii (λ_REK=0, g=g₀), and free-field G limit. A linear stability analysis of the stationary Q solution yields a non-Markovian dispersion relation whose REK-modified Bogoliubov limit is derived analytically. The critical crystallization threshold λ_c = g₀(1+ω²_B τ²₀)²/(4ρ_Q) is established as the stability boundary above which the REK memory term overcomes the aggregation restoring force — structurally identical to the threshold derived independently in the traumatic memory application. The crystallization frequency ω_REK is shown to be structurally determined by the B-axis natural frequency ω_B = √(2α/m), reducing the parameter space by one degree of freedom. The temporal coordinate is derived from spatial differentiation by vectorial logic, consistent with the Page-Wootters mechanism. Four physical instantiations are identified: the non-thought state (G proximal, α→0, ℓ_z→∞), REM sleep (Q proximal, single-axis B), the vertebrate optical system (complete G→X→Q→N at optical scale), and the cellular mitotic cycle.