A binomial compatibility law for actual spatial spinors

A family of circulating spinor patterns gives an exact orientation dependence for the declared local current interaction. The individual linear frequencies stay fixed under spatial rotation, while the interaction integral can become very small. Its binomial factor comes from integrating the actual spatial modes, rather than assigning a small coefficient to a particle sector.

The construction uses the conditional round four-sphere feature in ordinary six-dimensional space and the spinor parent already defined in A spatial spinor parent for the curvature frequency. It is one spatial family at depth six. It is not a photon model, a depth-five neutrality mechanism or a calculation of interaction rates in matter. The common surface, normal carrier and endpoint-current action remain assumptions.

Complete local spinor identity

In the Killing frame use five Hermitian ambient Clifford matrices Gamma_A on C^4. At a unit surface point x, the tangent matrices are gamma_a=-i Gamma_x Gamma_(P_x e_a), and C=Gamma_x is surface chirality. For every full complex spinor psi, direct Clifford algebra gives

$$\sum_{A=1}^5(\psi^\dagger\Gamma_A\psi)^2 =(\psi^\dagger\psi)^2. \tag{1}$$

The analogous local set of four tangent matrices and C also satisfies (1). Thus rho=psi^dagger psi, sigma=psi^dagger C psi and the actual tangent current obey |j|^2=rho^2-sigma^2. No spinor component or complex phase has been discarded in this identity.

Polarize (1) with two arbitrary complex spinors A and B, average their relative phase, and remove their separate self terms. The complete unit-endpoint mixed current factor becomes

$$\boxed{ j_A\cdot j_B+\sum_a|A^\dagger\gamma_aB|^2 =\rho_A\rho_B+|A^\dagger B|^2 -\sigma_A\sigma_B-|A^\dagger C B|^2.} \tag{2}$$

The chirality terms are physical polarization bilinears of this specified surface spinor. They are not electric charges or assigned neutral projectors. Equation (2) concerns rank-four surface spinors; its unrestricted extension to other spinor ranks has not been assumed.

An actual rotating spatial family

Write z_1=x_1+i x_2 and z_2=x_3+i x_4 on the unit four-sphere, and choose the unit spinor z_0=(1,0,0,0). It is annihilated by both Gamma_1+i Gamma_2 and Gamma_3+i Gamma_4. Every normalized field

$$\chi_{L,u}(x)=\frac{(u_1z_1+u_2z_2)^L}{\sqrt{V B(L)}}z_0, \quad |u_1|^2+|u_2|^2=1, \quad V=\frac{8\pi^2}{3}, \tag{3}$$

with B(L)=Gamma(L+1)Gamma(5/2)/Gamma(L+5/2), is a monogenic positive B_K eigenfield with eigenvalue L+3/2. Therefore its chain frequency depends on L and the common radius and gap, and does not depend on u.

The parameter u labels a complex line in the actual four ambient coordinates. These lines are a restricted family of oriented real planes; they do not exhaust all relative planes or polarizations in the full spinor multiplet. Parameter spaces and spinor components add no physical dimensions.

The full tangent current of z_0 is explicitly

$$j_0(x)=(-x_2,x_1,-x_4,x_3,0),\qquad |j_0|^2=1-x_5^2. \tag{4}$$

For (3), multiply (4) by the scalar density. Cross bilinears between two such spinors contain the same vector with the complete complex scalar product. Thus every tangent component of the interaction is retained, including its interference term.

Let e_L be the two real endpoint coefficients of the complete normalized positive chain state. Define

$$k_L=e_L^Te_L,\qquad k_{ab}=e_a^Te_b,\qquad A_{ab}=k_a k_b+k_{ab}^2. \tag{5}$$

The phase-averaged cross term from the same action is g times

$$\mathcal C_{ab}(u,v) =A_{ab}\int |Y_{a,u}|^2|Y_{b,v}|^2(1-x_5^2)\,dA. \tag{6}$$

The phase average applies to the two distinct rapid carrier frequencies. It does not assert that arbitrary coherent phases are irrelevant to the unaveraged field equation.

Derive the binomial factor

Common spatial SU(2) rotations allow u=(1,0) and v=(cos(theta),sin(theta)). Put x=|u^dagger v|^2=cos^2(theta). The squared radii (|z_1|^2,|z_2|^2,x_5^2) have surface integration weights corresponding to parameters (1,1,1/2). Expanding the two ordinary azimuths therefore gives, exactly,

$$\boxed{ \frac{\mathcal C_{ab}(u,v)}{\mathcal C_{ab}(u,u)} =F_{ab}(x)=\frac{\displaystyle\sum_{j=0}^{\min(a,b)} \binom aj\binom bj x^j}{\binom{a+b}{a}}.} \tag{7}$$

One explicit derivation first expands the degree-b field, producing sum_k binom(b,k) binom(a+k,k) x^k(1-x)^(b-k), divided by binom(a+b,a). Use binom(a+k,k)=sum_j binom(a,j)binom(k,j). The binomial average of binom(k,j) is binom(b,j)x^j, giving (7). Vandermonde's identity then supplies

$$F_{ab}(1)=1,\quad F_{ab}(0)=\binom{a+b}{a}^{-1},\quad F'_{ab}(1)=\frac{ab}{a+b}. \tag{8}$$

The complete aligned integral, including the spatial current and canonical endpoint weights, is

$$\boxed{\mathcal C_{ab}(u,u) =A_{ab}\frac{B(a+b)}{V B(a)B(b)} \left(1-\frac1{2(a+b)+5}\right).} \tag{9}$$

At radius R the integral gains the common factor R^(-4). It has not been divided by either particle's frequency to make responses appear similar.

For the assigned degree-14/36 patterns at R=1 and gap G=10000, the aligned integral is approximately 2.088613623. At theta=0.7 radians it is 0.012767877. For perpendicular planes it is 2.227033423e-12, with the exact relative factor 1/binom(50,14)=1.066273532e-12. This suppression is conditional on that spatial preparation; it is not a depth-five prediction.

A low-degree mathematical probe also provides a useful comparison. The aligned degree-2/14 and degree-2/36 integrals are 0.268861396 and 0.303574875 under the same normalization. Their difference is modest compared with the electron-to-muon frequency ratio. These degree-two fields belong to this depth-six surface family and must not be identified with photons.

Both rigid orientations follow the canonical kinetic term

Pull the complete normalized field kinetic term back to (3). The spinor z_0 is invariant under the spatial SU(2) subgroup in question, so the orbital Berry curvature is that of spin L/2. Direct surface integration gives -L sin(vartheta)/2 in polar-angle coordinates, with the sign fixed by the canonical time convention.

For unit Bloch vectors n_a,n_b representing the two complex lines, x=(1+n_a dot n_b)/2. Hamiltonian H=g C_ab(1) F_ab(x) gives

$$\dot n_a=\frac{H'(x)}a(n_b\times n_a),\qquad \dot n_b=-\frac{H'(x)}b(n_b\times n_a). \tag{10}$$

Both responses come from the same action. Their total angular momentum a n_a/2+b n_b/2 and their relative x are conserved. They precess around that constant total vector. A potential gradient alone therefore does not justify imposing an alignment rule on the rigid family. Changing compatibility requires a spatial deformation, another retained mode or a separately derived physical process.

spinor-geometric-overlap-20260914T021339_299491Z.json passes 88 checks: the exact full-complex quartic identity, phase averaging, actual spatial integration at two resolutions, all binomial factors, the canonical field curvature and both reciprocal orientations. An independent exact precession verifies the latter. Its earlier 021023_940799 report is preserved; its looser trajectory tolerance failed three small conservation checks. The current run tightens that tolerance and adds the exact trajectory comparison.

The complete compatible polynomial shape space, used to go beyond these rigid orientations, is developed in the companion resonant spinor calculation. The physical carrier and full self-supported particle construction remain open.

Related research

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