Reciprocal evolution within degenerate frequency families

The complete source calculation identifies resonant forcing of spatial patterns outside the initial two modes. The appropriate leading weak interaction equation therefore retains each entire degenerate family in the stated spatial symmetry class. It follows by averaging the declared quartic action over the relative particle oscillation phase.

This is an interaction calculation on overlapping spherical features. The feature, common gap and endpoint interaction are inverse assumptions. It does not describe separated trajectories or derive their sustaining geometry. The endpoint bilinear is also distinct from the kinetic norm flux, as derived in The kinetic current and the chosen endpoint interaction.

Average the full action, including both objects' currents

Write the field at leading order as

$$\Psi(t,x)=e^{-i\omega_a t}A(s,x)+e^{-i\omega_b t}B(s,x), \qquad s=g t. \tag{1}$$

Here A and B are arbitrary complex vectors in their respective positive parent-frequency eigenspaces. The unit-radius notation absorbs the spatial units of g. Each eigenspace includes both signed B polarizations; neither is a particle-antiparticle assignment. The amplitudes describe the evolving spatial pattern and canonical norm. Mass is still the selected particle frequency, not any one of these amplitudes.

Using the equal-endpoint bilinear, define

$$j_A=\sum_eA_e^\dagger\gamma A_e,\quad j_B=\sum_eB_e^\dagger\gamma B_e,\quad z=\sum_eA_e^\dagger\gamma B_e. \tag{2}$$

The full current is j_A+j_B+exp[-i(omega_b-omega_a)t]z plus its complex conjugate interference term. Integrating its square over one relative phase gives the exact averaged energy

$$\boxed{\mathcal U_{\rm av} =\frac12\int|j_A+j_B|^2dA+\int|z|^2dA.} \tag{3}$$

The currents of A and B need not vanish during the evolution. Let P_a and P_b be the full orthogonal spectral projections onto the retained frequency families. Varying (3) gives both equations:

$$\begin{aligned} i\partial_s A&=P_a\big[(j_A+j_B)\cdot\gamma A +z^*\cdot\gamma B\big],\\ i\partial_s B&=P_b\big[(j_A+j_B)\cdot\gamma B +z\cdot\gamma A\big]. \end{aligned} \tag{4}$$

This is a Hamiltonian evolution. Independent phase symmetry in each family conserves Q_A and Q_B separately, while (3) and the common rotation momentum are conserved. It need not decrease an overlap monotonically. Energy can move between the individual currents and their mutual interference, so coupling can weaken and subsequently strengthen without an imposed alignment rule.

For the O(m-1)-invariant basic spatial class, a degree L family contains q=-L,-L+2,...,L and both Hodge signs. This gives 2(L+1) complex canonical coefficients. The depth-six degree-14/36 pair therefore has 104 coefficients, with every allowed relative phase and both signed form polarizations. They are evaluated as actual spatial forms using the normalized Jacobi basis, not as an isotropic density trace.

What makes this an approximation

Equation (3) is an exact phase average of the declared interaction. Using (4) as dynamics is a weak-interaction reduction. Off-resonant frequencies remain in the complete parent and have the explicit first response in Complete form response to the current interaction. Their forcing divided by their detuning must be small for the average to be adequate. A large intermediate gap alone is insufficient, because additional low form families exist.

No nonzero cross-frequency resonance was found in the initial cubic source at the assigned controls, beyond the two complete original frequency families. This finite source calculation is not a nonlinear stability theorem or a classification of every disturbance. The evolving family approximation and the full unrestricted spatial equation remain different levels of calculation.

Numerical validation is bounded in time

The executable calculation checks (3) by direct averaging of the complete spatial current-square energy. Fourier coefficients of its full source independently verify both equations (4). Arbitrary complex variations check the Hamiltonian derivative, each family norm identity and common rotation covariance. Radial and angular refinement check the projections.

An exploratory evolution through s=50 conserved energy and norm well but failed independent trajectory refinement in the depth-ten degree-2/24 control. Its discrepancy was about 4e-8 at s=15, 7e-7 at s=20, and 0.58 at s=50. Conservation alone does not validate that late trajectory. The full report and its source versions are preserved. The degree-two branch is a mathematical control, not an assigned photon.

The refined report spatial-form-resonance-20260914T004927_072850Z.json passes 52 checks through s=15, using tolerances 2e-11 and 2e-12 and separate radial and angular resolutions. The largest final trajectory difference is 2.63e-10. Over this verified interval the interference integral in (3) has the following ranges:

In the degree-2/36 depth-six control, for example, the current interference falls substantially and recovers while the density overlap changes little. The evolving polarization matters independently of simple spatial density overlap. These rows use the stated normalized time and interaction; they are not absolute comparisons of observed coupling strengths across depths.

A separate calculation now resolves the complete complex tangent equation and two actual finite nonlinear differences through s=20. It measures the longer-time sensitivity without adopting the earlier unrefined s=50 trajectory.

spatial_form_resonance.py contains (4) and its complete real-linear derivative. spatial_form_variations.py expands the cubic force exactly:

$$\frac{F(\psi+\epsilon\delta)-F(\psi)}{\epsilon} =DF(\psi)[\delta]+\epsilon F_2(\psi,\delta) +\epsilon^2F(\delta). \tag{5}$$

This avoids subtracting nearly equal trajectories to measure a very small physical disturbance. It retains conjugate variations and all complex polarizations in the chosen spatial class. The exact difference equation also permits an independent comparison of tangent growth with finite nonlinear perturbations.

Verified finite-time shape sensitivity

The report spatial-form-variations-20260914T015614_934737Z.json passes 13 checks. The depth-ten degree-2/24 control uses a unit reflection-even initial disturbance, orthogonal to each band's norm and overall phase. Both spatial refinements evolve the base state, the complete conjugate tangent and the exact differences at epsilon equal to 1e-6 and 5e-7. The latter are actual nearby solutions through the exact cubic identity (5), not two independently subtracted floating-point trajectories.

The refined tangent differs by less than 3e-9 relative to its size at s=20. While the finite disturbance remains small, halving epsilon halves its deviation from the tangent prediction, as required. At s=20 the actual epsilon=1e-6 field difference is approximately 0.0109699; its departure from the linear tangent is then appreciable.

The independent report spatial-form-shape-variation-20260914T020034_305133Z.json passes 20 checks. It removes norm and phase directions in each band, common azimuthal rotation and, separately, the direction along the trajectory. The large remaining norm in the last column shows that the growth is not explained by those symmetry motions.

The same tangent changes actual spatial observables. At s=20 the derivative of the integrated interference with respect to the unit initial disturbance is approximately 60.78694; the density-overlap derivative is -259.75481. Direct symmetric spatial variations verify these derivatives, and the independent trajectory refinement agrees. These are response derivatives, not the observable values themselves.

This demonstrates strong finite-time sensitivity of shape and coupling compatibility in the stated averaged control. It does not establish an asymptotic Lyapunov exponent, complete spatial stability, a physical depth-ten particle encounter or the validity of the earlier late trajectory. The degree-two pattern remains a mathematical control.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.