The Compatibility Bifurcation of the Aether Fabric: Reciprocal-Cycle Dynamics, Nonlinear Port Softening, and the Rank-Change Obstruction

This fourth foundational paper in the Aether-fabric series investigates whether the established reciprocal-cycle geometry, Gforce contact work, nonlinear packing functional, and six-port transport structure are sufficient to produce a mathematically defined transition from free phtn propagation to a locked compatibility state. The reciprocal geometry gives the exact parameter-free line element $$\mathrm{d}s_q^2=2,\mathrm{d}q^2,$$ while the natural QMU second-order scale is $$\dfrac{W_A}{{F_q}^2}=m_a{\lambda_C}^2.$$ The corresponding kinetic normalization remains undetermined by a single dimensionless factor $\mu_q>0$. Because the contact work depends only on reciprocal differences, a spatially uniform trajectory $q_A(t)$ is an exact zero mode. The balanced value $q_A=0$ is therefore an exchange-symmetric reference rather than an equilibrium selected by the contact law. For two otherwise registered neighboring Aether Units, the exact reciprocal contact work is $$V_q(\Delta q)=\dfrac{W_A}{6}\tanh^2\left(\dfrac{\Delta q}{2}\right).$$ The small-amplitude reciprocal stiffness is $$\kappa_{q,0}=\dfrac{W_A}{12},$$ which equals the previously derived binormal rigid-frame stiffness. The restoring gradient reaches its parameter-free maximum at $$|\Delta q|_*=\ln(2+\sqrt{3}),$$ where $$\dfrac{V_q(\Delta q_*)}{W_A}=\dfrac{1}{18}$$ and $$\dfrac{1}{W_A}\max\left|\dfrac{\partial V_q}{\partial\Delta q}\right|=\dfrac{1}{9\sqrt{3}}.$$ Beyond this point, the contact force remains restoring, but its local stiffness softens. When the reciprocal interaction is extended to the six-port Aether fabric, the collective reciprocal-strain mode is gapless because uniform reciprocal deformation remains an exact zero mode. Its leading continuum behavior is isotropic, while the first directional correction appears at fourth order and preserves the cubic directionality of the underlying six-port geometry. This mode is distinct from both the free propagation branch $v_{\mathrm{free}}=c$ and the locked branch $v_{\mathrm{locked}}=c/\sqrt{3}$. The central result is a conditional no-go theorem. The free transport operator acts on the twelve-dimensional complex space $$\mathbb{C}_{\mathrm{port}}^6\otimes\mathbb{C}_T^2,$$ whereas the exact locked operator is a constrained six-dimensional representation. Within the established smooth packing functional, fixed-rank port-temporal state space, and positive kinetic closure, smooth changes in reciprocal strain, frame mismatch, profile distance, packing participation, inertia, damping, or additional smooth profile modes cannot dynamically produce this persistent reduction of the accessible representation. The smooth packing functional has one interior relative-compatibility minimum and supplies no pitchfork, saddle-node, discontinuous transition, spectral gap, or other branch-changing mechanism. An exact free-to-locked conversion therefore requires an additional physical structure, such as a projection, quotient, singular constraint, gap-opening mechanism, or explicit QMU-closed constraint-strength variable. The result closes the nonlinear relative-compatibility dynamics as far as the established geometry permits. It also identifies two remaining foundational questions: the physical law governing the absolute unoccupied trajectory $q_A(t)$ and the mechanism that converts the independent temporal fiber into the locked port-temporal representation. No fitted transition parameter is introduced. The accompanying derivation package provides symbolic calculations, six-port numerical verification, coefficient-free diagnostic values, claims ledgers, validation reports, numerical tables, figures, source code, and SHA-256 manifests.