The Gforce Contact Geometry of the Aether Fabric: Quantum-Area Force Closure, Four-State Packing, and Nonlinear Interface Stiffness

This paper develops the absolute contact geometry of the primitive, unoccupied Aether fabric within the Aether Physics Model and Quantum Measurement Units framework. It completes the relational and dynamical foundations established in the preceding Aether-fabric papers by deriving the nearest-neighbor spacing, registered packing participation, nonlinear interface work, and absolute stiffness of adjacent Aether Units. Two independent derivations establish the primitive center-to-center spacing $$d_A=\lambda_C.$$ The first follows from the balanced two-dimensional temporal metric: one complete joint temporal winding has physical transfer length $\lambda_C$. The second follows from magnetic-charge saturation: the maximum magnetic interaction between adjacent Aether Units equals Gforce only when their separation is $\lambda_C$. The shared relational face and compatibility-cell volume therefore satisfy $$A_{\mathrm{face}}={\lambda_C}^2$$ and $$V_{\mathrm{cell}}={\lambda_C}^3=\mathrm{volm}.$$ The corrected Singularity ledgers yield the common force-area invariant $$\mathrm{Gforce}\,{\lambda_C}^2=G{m_a}^2=A_u{e_a}^2=16\pi^2k_C{e_a}^2.$$ Gravity, electrostatic force, and magnetic force consequently share the geometric kernel $${\lambda_C}^2/r^2$$ while retaining their distinct mass, electrostatic-charge, and magnetic-charge participation factors. The corresponding one-quantum-length boundary work is $$W_A=\mathrm{Gforce}\,\lambda_C=m_ac^2.$$ The internal Aether state is represented by four forward-time loxodromic channels obtained by crossing the two electrostatic spheres with the two temporal-torque handednesses: $$L^-\rightarrow e^-,$$ $$L^+\rightarrow p^+,$$ $$R^-\rightarrow \bar p,$$ and $$R^+\rightarrow e^+.$$ These particle labels identify the occupied geometries associated with the four latent Aether channels; they do not imply four simultaneously occupied particles within an unoccupied Aether Unit. The electron and antiproton share the negative electrostatic sphere, while the proton and positron share the positive electrostatic sphere. The electron and proton occupy the two left-temporal-torque channels, and the positron and antiproton occupy the two right-temporal-torque channels. For electrostatic-sphere compatibility $C_E$ and temporal-torque compatibility $C_T$, coherent four-state registration gives the exact rigid overlap $$\mathcal R_4=\sqrt{C_EC_T}.$$ The earlier affine registration score is recovered as $$\mathcal R_{\mathrm{aff}}=(C_E+C_T)/2=1-\Phi/4,$$ so the affine law is the arithmetic-mean upper bound of the exact geometric-mean overlap. Both agree through quadratic order and coincide exactly on the balanced-mismatch subspace $C_E=C_T$. The local expansion recovers the interface anisotropy $$\omega_n^2+2\omega_b^2+\omega_c^2,$$ giving the stiffness ratio $1:2:1$. The registered higher-order packing functional is $$\Pi_{jk}^{(\alpha)}=\Pi_0\,\operatorname{sech}^2\left(\frac{\Delta q}{2}\right)\left|\frac14\sum_{\sigma=\pm1}\sum_{\tau=\pm1}e^{i(\sigma\delta_E+\tau\delta_T)}e^{-\mathcal D_{jk,\sigma,\tau}^{(\alpha)}/2}\right|.$$ Here $\Delta q$ is the reciprocal-cycle mismatch, $\delta_E$ and $\delta_T$ are the electrostatic-sphere and temporal-torque phase mismatches, and $\mathcal D_{jk,\sigma,\tau}^{(\alpha)}$ is the transported squared-logarithmic loxodromic-profile distance. The functional is bounded, reciprocal, six-port normalized, and reduces exactly to $\mathcal R_4$ in the rigid limit. The reciprocal temporal radii are represented by $$R(q)=\frac{\lambda_C}{2\pi}e^q$$ and $$r(q)=\frac{\lambda_C}{2\pi}e^{-q},$$ with invariant area $$4\pi^2R(q)r(q)={\lambda_C}^2.$$ The Aether quantities $$r_{a0}=\frac{\lambda_C}{2\pi a}$$ and $$r_a=\frac{\lambda_Ca}{2\pi}$$ are treated as the extreme reciprocal-radius limits permitted by the Aether fine structure $a$, not as fixed dimensions of the actual unoccupied Aether geometry. Expansion of the first loxodromic deformation mode yields $$\mathcal D_\varepsilon=C_2(\Delta\varepsilon)^2+C_4(\Delta\varepsilon)^4+C_6(\Delta\varepsilon)^6+O((\Delta\varepsilon)^8),$$ with $$C_2=3.256033146312710,$$ $$C_4=2.965847150120771,$$ and $$C_6=5.526339844977451.$$ The first phase-carrier quartic coupling is $$C_2\delta_E\delta_T(BM+EH),$$ The absolute nonlinear packing work is $$\mathcal E_{\mathrm{pack},jk}=W_A\Gamma_0(1-\mathcal P_{jk}),$$ where $\Gamma_0=1/6$ and $\mathcal P_{jk}=\Pi_{jk}/\Pi_0$. Its quadratic limit gives $$\kappa_{n,0}=\kappa_{c,0}=W_A/24$$ and $$\kappa_{b,0}=W_A/12.$$ The paper distinguishes derived results, results obtained under explicit minimal composition axioms, inherited QMU and APM identities, and remaining open dynamical questions. The time-dependent reciprocal trajectory $q_A(t)$, higher loxodromic deformation modes, occupied mass-string selection, and the free-to-locked compatibility bifurcation remain subjects for subsequent investigation.