Reciprocal Contact With Deforming Spatial Patterns

The scale-covariant contact action admits an exact two-grade invariant subsystem. Evolving it tests whether reciprocal contact survives when both patterns can deform. These are separate same-depth experiments in ordinary spatial dimensions three and five; they do not model a cross-depth encounter.

Equations and Conserved Quantities

Let $v_j$ be the odd parent component, $c_j$ the sign-reversed child, $M_j=\mu q_j$, $g_j=\gamma M_j^{(5-d)/2}$ and $w_j=M_j^{(3-d)/2}$. The canonical two-grade potential is

$$V=\sum_j\left[M_j^2|v_j|^2+4M_j^2|c_j|^2-g_j\operatorname{Re}(c_j^*v_j^2) +\frac{g_j^2}{4M_j^2}|v_j|^4\right] +\lambda\gamma^2 S^2,\qquad S=\sum_jw_j|v_j|^2.$$

Variation gives

$$\begin{aligned} \partial_t^2v_j&=\Delta v_j-M_j^2v_j+g_jc_jv_j^* -\frac{g_j^2}{2M_j^2}|v_j|^2v_j-2\lambda\gamma^2w_jSv_j,\\ \partial_t^2c_j&=\Delta c_j-4M_j^2c_j+\frac{g_j}{2}v_j^2. \end{aligned}$$

Each grade separately conserves its primitive phase charge

$$Q_j=-2\int\operatorname{Im}(v_j^*\dot v_j+2c_j^*\dot c_j).$$

The restricted action has independent global phase symmetries. Global relative phases cannot change its density histories. Spatial phase gradients and the internal parent-child phase remain dynamical. This symmetry claim is for the invariant subsystem, not the full graph.

Put $\rho_j=|v_j|^2$, $\beta=2\lambda\gamma^2w_1w_2$, and $U=\beta\int\rho_1\rho_2$. Let $H_j$ include each grade's kinetic, gradient and complete self energy, including its own density quartic. Then

$$\dot H_j=-\beta\int\rho_k\dot\rho_j,\qquad \frac{d}{dt}(H_1+H_2+U)=0,$$ $$\mathbf P_j=-2\operatorname{Re}\int(\dot v_j^*\nabla v_j+\dot c_j^*\nabla c_j), \qquad \dot{\mathbf P}_j=-\beta\int\rho_j\nabla\rho_k=-\dot{\mathbf P}_k.$$

For exact isolated profiles initially placed together with their stationary carrier velocities, $\dot\rho_j(0)=0$ and $\ddot\rho_j(0)=-2\beta\rho_1\rho_2$. Therefore

$$\boxed{\dot U(0)=0,\qquad \ddot U(0)=-2\beta^2\int\rho_1\rho_2(\rho_1+\rho_2)<0.}$$

Contact initially weakens through local deformation before rigid separation can explain the change. The lost interaction energy becomes self energy. This is conservative reversible evolution, not a relaxation theorem.

Primitive-charge centroids have zero initial acceleration in this preparation even though the component momenta have nonzero derivatives. They cannot be inserted into a rigid $F/E$ acceleration formula. The total energy centroid, including interaction energy, obeys $E\dot{\mathbf X}_E=\mathbf P_{\mathrm{total}}$.

Controlled Spatial Evolution

The pair uses counts 3 and 10, $\lambda=1/4$, $\kappa=1$, $\mu=\gamma=1$, initial separation 2 and evolution time 2. Each depth uses its inherited dimension-specific isolated profile, mapped to the amended density action. These are representative profiles, not selected observed particles or a common derived cross-depth normalization.

Axisymmetric evolution retains axial position and transverse radius with the full depth-dependent radial measure. The fine spacing is 0.04 and coarse spacing 0.08, axial half-length 12 and transverse radius 10. The timestep is 0.002; a depth-three control halves it. Velocity Verlet evolves a volume-weighted discrete Laplacian whose quadratic form is the reported gradient energy. Axial boundaries are periodic, the outer radial ghost is zero, and the axis is regular.

Contact endpoints changed by 0.222% and 0.069% under spatial refinement. Fine-grid component-momentum imbalance was 0.084% and 0.075%; halving spacing reduced that defect by factors 3.99 and 3.98. This supports convergence toward reciprocal continuum momentum transfer.

Maximum relative total-energy drift was $2.55\times10^{-7}$ and charge drift $4.45\times10^{-16}$. Energy error was below 0.038% of the interaction-energy decrease. Halving the timestep changed the depth-three contact endpoint by 0.0327% and momenta by at most 0.0243%.

Increasing the coarse domain to axial half-length 16 and transverse radius 14 changed final contact energy by less than $3\times10^{-16}$ relatively and momenta by at most $2.34\times10^{-12}$ over this short interval. These are domain-sensitivity checks, not infinite-time guarantees. Independent initial radial/angular quadrature differed from the fine PDE grid by 0.0462% and 0.1480%, with approximately fourfold larger coarse-grid errors.

Isolated controls retain each grade's own density quartic and remove only cross contact. Field changes exceed those controls, although the depth-five isolated profile error is appreciable. Contact and momentum are the better-controlled observables. The force is outward at every recorded sample, but its magnitude and the contact energy rebound. Constant initial force would overestimate the actual momentum transfer.

Concentration and Response Time

For a fixed lighter profile and separation, the heavy profile has density proportional to $q_h^{d-1}F^2(q_hx)$. Changing variables in the exact overlap and curvature integrals yields

$$U(0)\propto q_h^{(1-d)/2},\qquad \ddot U(0)\propto-q_h,\qquad \tau_{\mathrm{curv}}:=\sqrt{\frac{2U(0)}{-\ddot U(0)}} \propto q_h^{-(d+1)/4}.$$

Thus weak integrated contact can coexist with fast initial local deformation. This Taylor-curvature timescale is neither a decay time nor an eigenfrequency. The short-time approximation is nonuniform at large count. Thirty-six initial-integral cases cover depths 2, 3, 4, 5, 6 and 10 with heavy counts 10, 35 and 126 at separations 2 and 8. At count 126 and separation 8, leading asymptotes differ from direct integrals by less than 0.27%. That comparison does not establish spatial stability; the inherited depth-six profile remains unbound.

What This Establishes

The declared action gives nonzero reciprocal momentum transfer while both field patterns change shape. A fixed-overlap interaction misses a substantial part of that response. The calculations do not establish completed scattering, nonlinear stability, polarization acceptance or an observed coupling rate. Exterior graph components remain zero in the invariant preparation; general pair perturbations have not been evolved.

The separate conserved charges and the continuous isolated-profile family remain. Energy exchange alone does not choose their values or a unique mass-frequency. Radiation might change a localized remnant while preserving total charge in its grade, but no long-time selection result is established here. The count-selection analysis explains why that remains a distinct requirement.

Reproducing the Calculation

Download the solver text, frozen profile inputs and run histories and comparison records. Save the solver as reciprocal-density-evolve.py beside reciprocal-density-inputs.json. It requires Python, NumPy and SciPy. For example:

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 reciprocal-density-evolve.py --d 3 --h .04 --dt .002 --end 2 --R 2 --zmax 12 --rmax 10 --cross 1 --tag reproduced-d3

Use --h .08 for the coarse control, --d 5 for the depth-five pair, and --cross 0 for isolated controls retaining the self interaction. The time control uses --dt .001; larger coarse domains use --zmax 16 --rmax 14. Run cases sequentially on a memory-limited machine.

This publication edition replaces binary-profile loading and local paths with portable JSON inputs. The evolution and diagnostic formulas are unchanged. Recorded hashes distinguish the executed original solver from this packaging revision; the archived numerical results are not represented as fresh executions of the revised file. Profile generation and infinite-domain error certification are separate from reproducing these frozen-input evolutions.