Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Auxiliary model only: A local Dirac spin-orbit term supplies mixed generators, but a single simple bivector retains a twofold spinor degeneracy.

Interpretation correction

This calculation concerns an auxiliary field or local spin model, not the actual electron orbit constructed in the 6D demo. Its stationary states, angular expectations, and stability results have not been mapped to that trajectory or its observer readout. They must not be used as a replacement for the existing orbit-shape derivation. See Actual orbit research correction.

Source and approximation

The scalar ground-state moment calculation has no spontaneous mixed source. A next candidate is the derivative structure already present in a Dirac Hamiltonian, rather than an inserted phenomenological spin bilinear.

This calculation assumes a local flat six-spatial-dimensional Hamiltonian H=beta m+O+V, O=alpha_A p_A, p_A=-i partial_A, with constant m>0, Hermitian Clifford generators, beta anticommutes with alpha_A, and a smooth static time-component vector potential V times the identity. Units are hbar=c=1. V is not a Lorentz-scalar mass potential beta S; that has a different reduction. Curvature, magnetic fields, time dependence, and spatially varying mass are excluded here. The nonrelativistic expansion needs small momenta and potential energies relative to m and controlled derivatives.

The leading Foldy–Wouthuysen expansion uses

H+=m+p22m+V−p48m3−18m2[O,[O,V]]++⋯ . H_+=m+\frac{p^2}{2m}+V-\frac{p^4}{8m^3} -\frac1{8m^2}[O,[O,V]]_++\cdots.

The commutator coefficient follows from the stationary expansion in Silenko, Eq. (7), specializing its leading terms. The six-dimensional Clifford computation below is performed here; the cited paper is not evidence that V2 realizes this local Hamiltonian.

1. The derivative vertex retains the mixed generators

Define Sigma_AB=-i[alpha_A,alpha_B]/2. Direct operator differentiation gives

[O,[O,V]]=−ΔV−2∑A<BΣAB((∂AV)pB−(∂BV)pA). [O,[O,V]]=-\Delta V-2\sum_{A<B}\Sigma_{AB} \left((\partial_A V)p_B-(\partial_B V)p_A\right).

Consequently the leading spin-dependent term is

Hso=14m2∑A<BΣABBAB,BAB=(∂AV)pB−(∂BV)pA, H_{\rm so}=\frac1{4m^2}\sum_{A<B}\Sigma_{AB}B_{AB}, \qquad B_{AB}=(\partial_A V)p_B-(\partial_B V)p_A,

along with the scalar Darwin term Delta V/(8m^2). Symmetry of the Hessian makes each antisymmetrized coefficient Hermitian. Unlike the contracted extrinsic Levi–Civita term, this operator does contain mixed Sigma_ai.

For f=grad V and a frozen commuting momentum p, its mixed 3-by-3 block is

BOH=fOpHT−pOfHT. B_{OH}=f_Op_H^T-p_Of_H^T.

This can have rank two. For a potential supported only on O, f_H=0 and the frozen mixed block has rank at most one. Neither fact alone classifies the complete spinor operator, which includes B_OO and B_HH as well.

2. A simple full bivector does not select a unique spinor ray

At each frozen phase-space point, B=f wedge p is a simple bivector. Let

S(B)=∑A<BBABΣAB \mathcal S(B)=\sum_{A<B}B_{AB}\Sigma_{AB}

act on a chiral four-dimensional Spin(6) space. The Clifford relations give

S(B)2=(∣f∣2∣p∣2−(f⋅p)2)I4,tr⁡S(B)=0. \mathcal S(B)^2= \left(|f|^2|p|^2-(f\cdot p)^2\right)I_4, \qquad \operatorname{tr}\mathcal S(B)=0.

For nonzero area A, the spectrum is {-A,-A,A,A}. When the area vanishes the operator is zero. The statement is about the frozen matrix symbol, not the global differential-operator spectrum.

3. Explicit full-operator versus mixed-block counterexample

Take f=(1,0,0;1,0,0) and p=(0,1,0;0,2,0). Then

BOH=(020−100000),B12=1,B45=2. B_{OH}=\begin{pmatrix}0&2&0\\-1&0&0\\0&0&0\end{pmatrix}, \qquad B_{12}=1,\qquad B_{45}=2.

The mixed tensor has rank two. Its isolated spinor contraction has eigenvalues {-3,-1,1,3}, apparently selecting a unique ground ray. Restoring the same-sector terms dictated by the same Dirac reduction gives {-sqrt(10),-sqrt(10),sqrt(10),sqrt(10)}.

The Procrustes rank-two theorem remains correct for the specified isolated mixed operator. What fails is dropping other terms of the same derived operator without a dynamical reason. A nonzero mixed block is not enough.

4. A supplied two-plane source can select a locked ray

The non-simple bivector

B=2e1∧e4+e2∧e5 B=2e_1\wedge e_4+e_2\wedge e_5

has full rank four and purely mixed rank two. Its chiral spinor operator has eigenvalues {-3,-1,1,3}. In the Clifford chirality convention used by the check, its ground ray induces A=C=0 in the complex-structure blocks, with Q=diag(-1,-1,1) in SO(3). It genuinely selects an oriented locked ray within this supplied-source model. Choosing the opposite spinor chirality changes the orientation convention, not the nondegeneracy result.

This source is a sum of independent simple planes, not a single f wedge p. An orbital projection, multiple fields, or a non-simple magnetic curvature could supply such a sum, but none is derived by displaying it. Rank four is a useful sufficient example; arbitrary rank-six fields need a separate chiral spectral analysis and can have special degeneracies.

5. Next calculation and limits

Project the complete differential operator onto specified normalized orbital states before identifying an effective polarization Hamiltonian. A sum or expectation of simple bivectors need not itself be simple. Conversely, an orbital-state ansatz that already contains an oriented correlation cannot count as deriving that orientation. The full orbital/spin dynamics or a controlled supplied-source response must decide this.

The C4 spinor used here is a positive mass-grading block of a local Dirac model. Its identification with the homogeneous base coordinate Z in V2 is not supplied by this calculation. It also does not compute P_3 D_infinity P_6 or the rate of a physical lock.

Verification

spin_orbit_source_checks.py tests the double commutator on a nonconstant polynomial spinor, the Clifford square and characteristic polynomial, the full/mixed-only counterexample, and the complex structure of the supplied two-plane ground ray using exact matrices.