Reciprocal shape response in the spinor family

The same endpoint-current action has a complete resonant evolution within a specified polynomial spinor family. Both shapes change, and their interaction integral can change even though rigid canonical rotation would preserve its orientation overlap. The exchange is fixed by the same self-interaction and cross-interaction energy.

This is a weak-interaction average on an overlapping round field feature. It is not a separated-particle encounter, a solution for the normal carrier or the complete spatial field equation at arbitrary coupling. All complex coefficients in the stated resonant family are retained; its symmetry and spectral restriction are described below.

The complete compatible resonant space

Use the actual depth-six surface spinors in A binomial compatibility law for actual spatial spinors. Replace the single coherent polynomial (u_1 z_1+u_2 z_2)^L by every homogeneous holomorphic polynomial of degree L, multiplying the same vacuum spinor z_0. The full normalized basis is

$$\Phi_{L,k}(x)= \sqrt{\frac{\Gamma(L+5/2)}{V\Gamma(5/2)(L-k)!k!}} \,z_1^{L-k}z_2^k z_0,\qquad k=0,\ldots,L. \tag{1}$$

Its coefficient vector has L+1 arbitrary complex entries. Both the canonical norm and all three spatial SU(2) angular generators are retained. The spinor and orbital basis labels add no spatial dimensions.

This family is complete within its stated resonant symmetry class. Under the common rotation of the two complex planes, the base vacuum spinor's character fixes the other spinor components' required orbital charges to L+1, L+1 and L+2. At scalar degree L these are impossible: an orbital charge cannot exceed its harmonic degree. The surviving component has charge L and is precisely a holomorphic degree-L polynomial. Every such polynomial is monogenic and has B_K eigenvalue L+3/2. Thus no other spinor or signed-B component at the same degree has been arbitrarily removed from this rotation class.

The opposite parent frequency and higher scalar degrees are not resonant at the original positive frequency. They do receive forcing in the full equation. Their response must be controlled before using the average at a finite physical coupling. The nonlinear fixed-point calculation already demonstrates this complementary forcing for the single circulating branches.

Multiply the actual spatial polynomials

For coefficient vectors A in degree a and B in degree b, define

$$[M_{ab}(A,B)]_s =\sum_{i+j=s} \sqrt{\frac{\binom ai\binom bj}{\binom{a+b}s}}\,A_iB_j. \tag{2}$$

This is the maximal total angular-momentum projection. Vandermonde's identity gives unit norm to each row of its matrix. Therefore the linear map on the product space is a coisometry and

$$\|M_{ab}(A,B)\|^2\le\|A\|^2\|B\|^2. \tag{3}$$

It has rank a+b+1 on a product space of dimension (a+1)(b+1). It is not a globally rank-one interaction operator. Its rank-one blocks at fixed total angular weight follow from the actual spatial integral, rather than from an assumed sector coupling matrix.

Let

$$I_{ab}=\frac{B(a+b)}{V B(a)B(b)} \left(1-\frac1{2(a+b)+5}\right), \tag{4}$$

with the moment B(q) defined in the geometric coupling note. The full ordinary spatial integral is exactly integral |Phi_A|^2 |Phi_B|^2(1-x_5^2)dA =I_ab ||M_ab(A,B)||^2. Thus (2) is a compact evaluation of actual surface spinors, not a substitute for their spatial overlap.

One Hamiltonian supplies both responses

Use the complete canonical endpoint coefficients k_a, k_b and k_ab. Set

$$\kappa_a=k_a^2 I_{aa},\qquad \kappa_b=k_b^2 I_{bb},\qquad \kappa_{ab}=(k_a k_b+k_{ab}^2)I_{ab}. \tag{5}$$

They are matrix elements of the same action, not separately fitted pair constants. At R=1 remove the two rapid linear carrier phases and use slow time s=gt. The averaged Hamiltonian is

$$\boxed{ H=\frac{\kappa_a}{2}\|M_{aa}(A,A)\|^2 +\frac{\kappa_b}{2}\|M_{bb}(B,B)\|^2 +\kappa_{ab}\|M_{ab}(A,B)\|^2.} \tag{6}$$

Variation with respect to both complex conjugate vectors gives i dA/ds=partial H/partial A^* and i dB/ds=partial H/partial B^*. For example, the cross source in the first equation is kappa_ab M_ab(B)^dagger M_ab(A,B), with the corresponding adjoint map in the second equation. The complete self terms remain in both. The real-linear response also retains conjugate perturbations.

The two canonical norms, total H and all three components of the common spatial angular momentum are conserved. No imposed alignment, loss or damping is present.

Deformation changes the compatibility

At unit norm in each band define

$$P_a=\|M_{aa}(A,A)\|^2,\quad P_b=\|M_{bb}(B,B)\|^2,\quad C=\|M_{ab}(A,B)\|^2. \tag{7}$$

Each quantity is a spatial projection norm, and each is at most one. The absolute interaction integral is kappa_ab C. It is not divided by a particle mass. A coherent single-plane pattern has P_a=1 or P_b=1. For two initially coherent patterns, conservation of (6) gives

$$\boxed{\kappa_{ab}[C(s)-C(0)] =\frac{\kappa_a}{2}[1-P_a(s)] +\frac{\kappa_b}{2}[1-P_b(s)].} \tag{8}$$

Thus this positive-current action cannot lower C below its initial value for that particular preparation. It can increase C as the initially concentrated shapes redistribute their self energy. This is a consequence of the action and the starting shapes, not an imposed alignment rule.

More general initial shapes do not have P_a=P_b=1. Equation (8) then contains their actual initial P values, and compatibility can increase or decrease. The real coefficient Hamiltonian also permits the exact reversed trajectory A(T-s)^, B(T-s)^. Forward integration from that deformed state reverses the energy exchange under the same action. This mathematical operation has no antiparticle interpretation.

Rigid coherent orientations alone miss this response. Projecting the initial full velocity onto each coherent shape's phase, norm and rotation tangents leaves a nonzero complement at intermediate relative angles. For the assigned degrees at angle 0.7 radians its norm is approximately 0.0665421 in slow-time units. The initially coherent submanifold is therefore not invariant under the complete projected equations.

Verified examples and their scope

The report spinor-resonant-pair-20260914T022209_533999Z.json passes 123 checks. It uses unit canonical norm for each pattern, one radius, gap and interaction action, and all L+1 complex coefficients in each specified frequency family. The following interaction integrals are from independently refined evolution through slow time s=20:

The degree-two rows are mathematical probes within the same depth-six surface construction; they are not photons. The strongly misoriented last pair remains weakly engaged over this interval. The intermediate pair deforms substantially and becomes more compatible.

For the degree-14/36 angle-0.7 example, the electron-side mean-spin length falls to about 77.4 percent of its coherent maximum. That quantity measures shape deformation in this family, not a change of physical spatial dimension. Both mean angular vectors respond and their complete vector sum is conserved.

The largest difference between the two complete complex trajectories in the reported controls is below 2e-10. Canonical norms, energy, all three angular momenta and (8) pass independent checks. Actual ordinary-coordinate integration of arbitrary complex polynomials verifies the full projected force and quartic energy at two spatial resolutions. Complex variations, the conjugate Hessian, spatial covariance, initial overlap acceleration and reversed evolution are also checked.

The selected isolated-particle frequencies remain those of the specified parent plus its nonlinear shifts. These pair trajectories describe a slow interaction response and do not separate mass from frequency. The later Controlling the full source of the reciprocal spinor pair retains and bounds the complete complementary source. It also gives a finite-slow-time weak-coupling argument in the complete surface equation. The endpoint action, normal carrier and sustaining geometry remain physical premises; a useful finite physical coupling and complete inertia are not supplied by that asymptotic argument.

Related research

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