Paper 20: Dynamical Closure of the Scale-Space Framework: Complex Scale Potential, Newtonian Recovery, and the Hidden Configurational Sector

Papers 10–11 of the Scale-Space programme established the corrected five-dimensional metric gtt = −(1 +2/L)c^2 and  identified a configurational energy deficit that required a dynamical source. This paper provides that source through a systematic dynamical closure and proposes a candidate 5D covariant completion.

We first identify the correct primary variable (gAB) and derive the effective scale potential ΦR = c^2/L from the Paper 10 clock factor, making L a derived parameter. The Paper 11 deficit is then rewritten as a nonlinear potential-density relation for ΦR. We show that a purely real closure leads to an unscreened logarithmic correction to Newtonian gravity that is empirically dangerous, and that this is resolved by extending ΦR to a complex scale potential Ψs = ΦR + iΦI .
    The action for Ψs is:
    LΨ = − α/2  ∇A Ψ^∗s ∇^A Ψs − αγ/4 (|Ψs|^2 − Ψ^2 0)^2 + 4πGα ρm Re(Ψs).

The Euler-Lagrange equation is derived exactly and its polar decomposition yields a phase-current equation sourced by ordinary matter. Integrating over a spherical source establishes that mass sources exterior phase flux universally, and the real projection of the exterior solution recovers Newtonian gravity ΦR ≈ −GM/r at leading order, with corrections O(r^−4). The imaginary component ΦI ≈ Ψ0 − G^2 M^2 / (2Ψ0 r^2) constitutes a hidden configurational sector that carries the nonlinear scale-sector stress without appearing directly in observable gravity. All four validation tests (real-limit recovery, logarithm screening, hidden-sector regularity, observable-projection) are satisfied.

We further show that superposition of phase fluxes from multiple sources recovers standard Newtonian multi-body gravity exactly. We then address the structural question of whether Ψs alone can supply the Paper 11 anisotropic background deficit. It cannot: a complex scalar yields an isotropic cosmological-constant-like stress, not the required pure clock-sector form. We resolve this by deriving a constrained configurational action with a clock-ordering covector nA, which supplies T^conf AB = E(ΦR)nA nB exactly, with E = −3 / (κ5L^3). This reproduces the Paper 11 deficit with no unwanted spatial or scale components. A candidate 5D covariant form of the full action is stated.

The remaining open items are: the value of Ψ0 from a background constraint calculation; and the full 5D Bianchi and nonlinear closure (including the nonlinear [KAB] = 0 verification), which require GR specialist input. Interior matching at the stellar surface (both phase and amplitude sectors) is established unconditionally within the leading-order sourced-phase ansatz (§5, Task 21.C v2).