The actual spinor parent admits a manageable spatial symmetry class for testing a circulating nonlinear branch. Fix the two rotation quantum numbers of a highest-weight spinor pattern, then retain every spinor component and every compatible scalar harmonic. This supplies a basis for a fixed-point calculation; it does not establish the fixed point or stability against perturbations in other rotation classes.

Use the unit four-sphere feature in ordinary six-dimensional space, with the spinor operator B_K in A spatial spinor parent for the curvature frequency. The underlying four-sphere sits in five ambient directions. Its second ordinary normal direction and its physical sustaining field are still outside this surface calculation.

Actual coordinates and canonical measure

At zero azimuths a representative point is

x=(t,0,1−t1−u2,0,1−t u),0<t<1,−1<u<1.(1) x=(\sqrt t,0,\sqrt{1-t}\sqrt{1-u^2},0,\sqrt{1-t}\,u), \quad 0<t<1,\quad -1<u<1. \tag{1}

The first and second planes also have azimuths phi_1 and phi_2. Their integration gives the measure

dAreduced=2π2(1−t)1/2 dt du,∫dA=8π23.(2) dA_{\rm reduced}=2\pi^2(1-t)^{1/2}\,dt\,du, \qquad \int dA=\frac{8\pi^2}{3}. \tag{2}

There is no replacement of physical coordinates by oscillator labels. The azimuth dependence is known exactly from the declared rotation quantum numbers and is retained in every derivative.

Every compatible spinor component

Choose the standard ambient Clifford frame with i Gamma_1 Gamma_2=-Z tensor I and i Gamma_3 Gamma_4=-I tensor Z. Let L_0 be the circulating base degree. For spinor components 0,1,2,3 the scalar azimuthal charges are

(q1,q2)=(L0,0), (L0,1), (L0+1,0), (L0+1,1).(3) (q_1,q_2)=(L_0,0),\ (L_0,1),\ (L_0+1,0),\ (L_0+1,1). \tag{3}

Together with the spin rotation, all four have the same total rotation labels. The component-zero polynomial (x_1+i x_2)^(L_0) is a monogenic positive B_K mode. In the parity class of this initial field, all compatible scalar degrees are K=L_0+2s, s nonnegative.

For one component and fixed K, choose a rest-sphere degree l satisfying K=q_1+l+2j, j nonnegative, l at least q_2. The complete scalar basis is

FK,l(t,u,ϕ1,ϕ2)=C tq1/2(1−t)l/2Pj(l+1/2,q1)(2t−1)Plq2(u)ei(q1ϕ1+q2ϕ2).(4) F_{K,l}(t,u,\phi_1,\phi_2) =C\,t^{q_1/2}(1-t)^{l/2} P_j^{(l+1/2,q_1)}(2t-1)P_l^{q_2}(u) e^{i(q_1\phi_1+q_2\phi_2)}. \tag{4}

Here the second P is an associated Legendre polynomial, and C normalizes (4) using (2). Jacobi and associated-Legendre orthogonality give the full canonical norm; different spinor components are also orthogonal. Up to s=J the basis has (J+1)(2J+1) scalar-spinor modes. Each coefficient may be complex.

All four physical tangent derivatives are used. In particular the azimuthal gradients are i q_1/sqrt(t) and i q_2/sqrt((1-t)(1-u^2)); omitting them would remove the circulating motion and change the spinor operator.

Complete first-order closure

Apply the actual ambient formula B_K=3/2-Gamma_x Gamma_grad to (4) and integrate its full spinor matrix elements. Each K block closes exactly and satisfies

BK†=BK,BK2=(K+3/2)2I.(5) B_K^\dagger=B_K,\qquad B_K^2=(K+3/2)^2I. \tag{5}

At order s its dimension is 4s+1. Its positive and negative B_K multiplicities are respectively 2s+1 and 2s. The first block therefore contains the single monogenic positive mode; the other spinor branches enter at higher compatible spatial degree. This is a consequence of the rotation labels, not an imposed removal of their components.

The same low-frequency chain supplies both positive and negative parent frequencies on each block. Both must be retained in a nonlinear fixed-point and response calculation. An efficient canonical embedding can use the two exact B projections in each block, avoiding a discretionary choice of basis inside degenerate eigenspaces.

What has been verified

spinor-torus-basis-20260914T012425_701894Z.json passes 153 checks. It includes complete canonical Gram matrices, direct full spatial Dirac closure, both branch multiplicities, arbitrary complex field projection and the current in every tangent direction. Independent radial and polar refinement verifies the current integral. The largest relative full spatial closure residual is below 4.1e-14 in the tested base degrees 2, 14 and 36 with J=8.

The next use is the same declared local endpoint current action at one common radius, gap and coupling. Its complete stationary frequency, spatial harmonics and both chain signs must be solved together, with canonical norm fixed independently. No nonlinear solution or physical mass-operator derivation is asserted by these basis checks alone.