Microscopic Closure of Collective Aether-Cell Recruitment in Neutron-Star Matter:Singular Phase Response and Conductance--Flux Equilibrium

This paper completes the microscopic recruitment mechanism developed in the Aether Metric Coordinates (AMC) series for dense neutron matter. The preceding collective construction established a pair-preserving hierarchy of coherent Aether cells, $M=2K$, but left two questions unresolved: why a saturated fixed-$M$ branch must recruit an additional two-seat cell, and why the enlarged branch enters at the specific coherence $q_{\rm entry}$ required to preserve the matter--space boundary. The local QMU boundary is expressed through the reciprocal conductance and magnetic-flux identities $$\mathrm{cond}=\frac{\mathrm{chrg}}{\mathrm{angm}},\qquad\mathrm{mflx}=\frac{1}{\mathrm{cond}}.$$ The collective phase state is described by the maximum-entropy relation $$q=\frac{I_1(\kappa)}{I_0(\kappa)},$$ with exact susceptibility $$\chi_q=\frac{dq}{d\kappa}=1-q^2-\frac{q}{\kappa}.$$ As a fixed-$M$ branch approaches full coherence, $q\rightarrow1$, the required coupling satisfies $\kappa\rightarrow\infty$ while $\chi_q\rightarrow0$. In particular, $$\kappa(1-q)\rightarrow\frac{1}{2}.$$ Because the collective deformation $D(M,q)$ retains a finite nonzero slope at $q=1$, this phase-response singularity propagates directly into the conductance boundary and neutron-density coordinates. Exact saturation therefore represents an infinitely stiff fixed-branch limit rather than a regular state attainable at finite coupling. Opening the next pair-preserving branch, $M\rightarrow M+2$, provides the regular continuation. Once the enlarged branch is admitted, its entry state is selected by minimizing reciprocal QMU conductance--magnetic-flux mismatch. Defining $$r(q)=\frac{D(M+2,q)}{D(M,1)},$$ a class of positive reciprocal boundary actions has the common stable equilibrium $r=1$. The resulting condition is $$D(M+2,q_{\rm entry})=D(M,1).$$ Thus the continuity relation used geometrically in AMC II acquires an independent variational interpretation: the newly recruited branch enters at the unique coherence that preserves both the conductance boundary state and its reciprocal magnetic-flux state. The invariant physical result is the matched boundary equilibrium rather than the choice of a particular algebraic mismatch action. The resulting recruitment sequence is reconstructed without using stored entry coherences, density thresholds, or equation-of-state data. The first transitions remain $$n_n=0.722972969267\ {\rm fm}^{-3}\qquad(M=2\rightarrow4),$$ and $$n_n=1.104423378641\ {\rm fm}^{-3}\qquad(M=4\rightarrow6).$$ The SLY4 endpoint remains within the regularized $M=4$ branch at finite coherence and finite coupling. Equation-of-state information enters only after construction of the recruitment hierarchy and therefore serves as an external validation layer rather than an input to the microscopic mechanism. A logical-independence audit verifies the complete ordering $$\text{QMU reciprocity}\rightarrow\text{two-seat collective geometry}\rightarrow\text{fixed-}M\text{ response singularity}\rightarrow\text{branch opening}\rightarrow\text{variational boundary matching}\rightarrowq_{\rm entry}\rightarrowD_{\rm entry}\rightarrown_{\rm entry}.$$ The paper therefore closes the principal microscopic questions left by the first two AMC papers: fixed-branch saturation supplies the recruitment trigger, while reciprocal conductance--magnetic-flux equilibrium selects the entry coherence of the enlarged branch. The resulting mechanism is parameter-free and preserves the pair structure and continuous matter--space boundary of the preceding collective construction. Beyond dense-matter applications, the derived conductance--magnetic-flux--coherence chain identifies a prospective constitutive route for testing whether a local Aether deformation state can be manipulated experimentally. This provides a bridge from compact-object physics toward future investigations of controlled Aether boundary states and space-engineering applications.