Bilateral polarization coupling

Summary: Keeping both objects polarized gives covariant tensor pairings, distinct source and response zeros, and a reciprocal orientation model with exactly conserved relative overlap.

Both objects carry spatial polarization

Let the two particle patterns carry spatial tensors $J_a(x;\alpha)$ and $J_b(y;\beta)$, where $\alpha,\beta$ label their respective preparations. These parameters describe fields on actual nested macroscopic spatial submanifolds of one infinite manifold. They are not additional physical dimensions. Each object has its own polarization and can change in response to the other; a fixed-target calculation is a conditional snapshot.

The working physical hypothesis is that this reciprocal motion can improve or worsen the match between the spatial patterns. An encounter may therefore strengthen its subsequent coupling or move the patterns toward incompatibility. Changing compatibility is a proposed part of the mechanism, to be tested from the same field equations.

The calculation must follow both patterns during the encounter and determine which responses initiate and sustain that change. Initial overlap alone does not determine the outcome. Neither automatic alignment nor conserved relative overlap is a general requirement; the exactly conserved overlap in the benchmark below is a property of its specified restricted dynamics.

A specified reciprocal two-form term can be written

$$U[\alpha,\beta] =-g\operatorname{Re}\int\!\int \left\langle J_a(x;\alpha), \mathcal K_{ab}(x,y)J_b(y;\beta)\right\rangle d\mu_a(x)d\mu_b(y). \tag{1}$$

The kernel includes the actual spatial maps, frame transport, and any derivatives prescribed by the action. All tensors must be paired in compatible frames. Reciprocity uses the adjoint kernel for the reverse variation. Equation (1) is a conditional term of the one field's energy; the kernel and tensor observables must come from its action.

Under a common rotation of all physical geometric data, $J\mapsto RJR^T$ and $\mathcal K$ transforms correspondingly, leaving (1) unchanged. Rotating only the particles while keeping an anisotropic environment fixed is a different physical preparation and need not preserve the energy.

For a local isotropic benchmark, with two real tensors at one common point, the expression reduces to

$$U=-g\langle A,B\rangle, \qquad\langle A,B\rangle=-\frac12\operatorname{tr}(AB). \tag{2}$$

If $A=R_aA_0R_a^T$ and $B=R_bB_0R_b^T$, this depends on the relative rotation $R_a^TR_b$. Either tensor can be the lower or higher pattern. The formula does not privilege one as the only polarized object.

Plane alignment includes both principal directions

For oriented orthonormal pairs $u_1,u_2$ and $v_1,v_2$, put $A=u_1\wedge u_2$ and $B=v_1\wedge v_2$, with $u\wedge v=uv^T-vu^T$. Direct expansion gives

$$\boxed{ \langle A,B\rangle =(u_1\cdot v_1)(u_2\cdot v_2) -(u_1\cdot v_2)(u_2\cdot v_1) =\det(U^TV),} \tag{3}$$

where $U=(u_1,u_2)$ and $V=(v_1,v_2)$. The singular values of $U^TV$ are $\cos\theta_1,\cos\theta_2$, the two principal-angle cosines. Hence the squared pairing is

$$|\langle A,B\rangle|^2 =\cos^2\theta_1\cos^2\theta_2. \tag{4}$$

The sign of (3) also retains relative orientation. For several rotational planes, expand $A=\sum_i a_i A_i$ and $B=\sum_j b_j B_j$:

$$\langle A,B\rangle =\sum_{i,j}a_i b_j\det(U_i^TV_j). \tag{5}$$

Contributions to this same scalar pairing add before it is squared. Opposite orientations can cancel despite nonzero overlap of individual planes. Complex oscillating tensors additionally retain their relative phases through the Hermitian pairing. A temporal or ensemble average must be justified by the specified preparation.

A zero static source need not be a zero response

Let a photon amplitude $a$ couple to a target preparation $\beta$ through

$$U(a,\beta)=-g\,a\,s(\beta), \qquad s(\beta)=\langle B_\gamma,J_b(\beta)\rangle \tag{6}$$

for one real wave quadrature. The background can satisfy $s(\beta_0)=0$ while the mixed derivative is nonzero:

$$\boxed{\frac{\partial^2U}{\partial a\,\partial\beta} =-g\left\langle B_\gamma, \frac{\partial J_b}{\partial\beta}\right\rangle.} \tag{7}$$

For an exact example in three ordinary spatial directions, choose $B_\gamma=e_1\wedge e_2$ and

$$J_b(\theta) =\cos\theta\,(e_2\wedge e_3) +\sin\theta\,(e_1\wedge e_2).$$

This is a unit simple two-form for every $\theta$. Its static pairing at zero is zero, but $s'(0)=1$. An incoming wave can therefore drive the target's angular variation even when the target's unperturbed projection supplies no source in that wave direction.

For two varying target patterns and a fixed kernel, the mixed variation of (1) contains

$$\delta_a\delta_b U =-g\operatorname{Re}\int\!\int \langle\delta J_a,\mathcal K_{ab}\delta J_b\rangle d\mu_a d\mu_b. \tag{8}$$

When the kernel, embeddings, or measures also change, include those derivatives. A claim of no linear coupling must test all allowed response tangents and the other action terms. The value of (1) at one preparation, by itself, is insufficient.

A complete mode-fibre pair action provides a conditional deformation benchmark: two initially coherent patterns leave the coherent subset, and their changing density contact is constrained by energy and rotation moments. This positive-density component model retains shape responses beyond two ray angles; its auxiliary mode-space dynamics still require a physical spatial and Dirac-current reduction.

A solvable reciprocal orientation benchmark

For the finite isotropic energy (2), give $A$ and $B$ fixed coadjoint orbits of $SO(D)$ and positive symplectic weights $\ell_a,\ell_b$. Specify the Lie–Poisson convention

$$\ell_a\dot A=[A,\nabla_AU], \qquad \ell_b\dot B=[B,\nabla_BU].$$

The equations are

$$\ell_a\dot A=-g[A,B], \qquad \ell_b\dot B=g[A,B]. \tag{9}$$

Both polarizations respond. Their weighted sum $S=\ell_aA+\ell_bB$ is constant, as are their individual spectra, the overlap $\langle A,B\rangle$, and the energy. Since

$$\dot A=\frac{g}{\ell_a\ell_b}[S,A], \qquad \dot B=\frac{g}{\ell_a\ell_b}[S,B],$$

the exact solution is

$$\boxed{ R(t)=\exp\left(\frac{gt}{\ell_a\ell_b}S\right), \qquad A(t)=R(t)A(0)R(t)^T, \qquad B(t)=R(t)B(0)R(t)^T.} \tag{10}$$

The weights are coefficients of a specified first-order orientation action, not values inferred from spatial depth. These orbit dynamics preserve tensor spectra and rotate both objects together. They produce precession, with no attraction toward a better matched relative angle. Testing the proposed improvement or loss of compatibility during an encounter therefore requires derived spatial motion, anisotropy, or response structure beyond this benchmark. The direction of that change must follow from the equations, rather than a rule that always favors alignment.

Equation (9) assumes the displayed orbit variations are admissible. Fixed sector embeddings or geometric constraints can restrict the allowed generators and require reaction terms. The benchmark does not establish that arbitrary rotations preserve every global nested embedding.

Following the full field beyond its mean rotation tensor

The coadjoint benchmark follows the two-form observables. They need not specify the complete polarized field at a higher spatial depth. Equal-current spatial polarizations derives a reciprocal six-inside-ten example from a rotation-moment energy and both full spatial kinetic terms. Its mean tensors remain fixed while the higher field can rotate into an orthogonal polarization. Thus even a two-form interaction can require more field information to describe its response than the mean two-forms alone retain.

Relation to the photon and higher-sector calculations

The tensor $J_d$ in photon plane coupling already specifies one target polarization, while $E,k,c$ specify the photon's polarization and propagation. Its projector calculation freezes the target at that preparation. Equations (7)–(8) supply the additional test when both can respond.

The descending source starts from the higher pattern; its derivative gives the reverse response. The Dirac source derivative computes that derivative explicitly for one written nonlinear term. These descriptions can coexist: the higher geometry may determine the source while the two objects' relative polarizations determine which response is engaged.

Verification

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Related

• Descending photon source — separates the source value from its reciprocal derivative.
• Photon plane coupling — fixes both spatial preparations for one instantaneous overlap.
• Dirac source linearization — retains the changes of the higher field and its nonlinear source.
• Canonical free-pair operator — gives an earlier two-electron example whose invariants depend on both complex structures.

Polarized response spectrum weights the spatially allowed responses by their frequencies and finite exposure time.

Coherent spatial overlap evaluates both objects' shapes and phases in exact Gaussian and opposing-lobe examples.