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: Pure excited orbitals can have isotropic position moments while carrying a full-rank circulation that supplies a conditional singlet spin source.

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.

1. Position moments do not exhaust orbital data

The positive scalar ground-state result remains intact. The following is an excited-state construction, not another ground-state minimum. It shows why isotropic second moments do not imply vanishing orbital current.

Use a flat six-dimensional oscillator H_omega=p^2+omega^2 r^2, with unit norm, and split its real basis into O=(e1,e2,e3) and H=(e4,e5,e6). Define orthonormal circular modes z_a=(e_a-i e_(a+3))/sqrt(2), a=1,2,3. In the oscillator shell of total degree N>=2 take

ϕN=13∑a=13∣N quanta in za⟩. \phi_N=\frac1{\sqrt3}\sum_{a=1}^3|N\hbox{ quanta in }z_a\rangle.

This is a pure superposition, not a mixture. The coordinate polynomials (x_a-i x_(a+3))^N are harmonic, so the state lies in ell=N as well as shell N. Its one-body cross matrix elements vanish because a one-body operator cannot transfer all N>=2 quanta between two orthogonal modes.

With rho the trace-N one-body density matrix,

ρ=N3P−,P−=12(I+iJ0),J0=(0I3−I30). \rho=\frac N3 P_-,\qquad P_-=\frac12(I+iJ_0),\qquad J_0=\begin{pmatrix}0&I_3\\-I_3&0\end{pmatrix}.

Thus Re rho=N I/6, while Im rho=N J_0/6 is nonzero. For a fixed shell, the anomalous expectations of two creation or two annihilation operators vanish, giving

M=⟨xxT⟩=N+36ωI6,⟨L⟩=−2Im⁡ρ=−N3J0. M=\langle xx^T\rangle=\frac{N+3}{6\omega}I_6, \qquad \langle L\rangle=-2\operatorname{Im}\rho=-\frac N3J_0.

The position covariance is isotropic and the circulation has full rank six. The density itself need not be isotropic: the quadratic scalar kernel sees only its second moments, not its higher angular moments.

2. These are scalar self-consistent stationary solutions

For the single-sector functional E=T+(g/2)tr M^2 with g>0, variation gives the mean-field potential g x^T M x. It equals omega^2 r^2 precisely when

ωN3=g(N+3)6. \omega_N^3=\frac{g(N+3)}6.

At that width, phi_N is an exact stationary scalar Hartree solution, with phase eigenvalue mu=(2N+6)omega_N. Its energy is E=(3/2)(N+3)omega_N, not mu. For N=2, choosing omega=1 gives g=6/5, M=5I/6, and mu=10.

The scalar problem supplies an entire symmetry-related family, not a chosen absolute J_0 or a particle-index selector. The state is above the unconstrained scalar ground state. Its linear stability is a separate calculation.

3. Conditional spin source and the circulation sign

If this orbital is used in the local vector-potential Dirac reduction of the spin-orbit audit, its angular expectation produces

S=−N3∑a=13Σa,a+3. \mathcal S=-\frac N3\sum_{a=1}^3\Sigma_{a,a+3}.

In the stated chiral convention this has eigenvalues {-N,N/3,N/3,N/3}. For a positive spin-orbit coefficient the unique lowest ray lies in the locked orbit, with complex structure J_0. Reversing the orbital circulation negates the matrix and gives a triply degenerate lowest eigenvalue instead. Thus the current sign relative to chirality and the interaction coefficient must be retained; a magnitude-only source calculation misses the distinction.

The scalar Hartree functional does not prove that its potential is the time-component vector potential of the Dirac model. These bridges remain open.

4. A shared-well normalization constraint

With fixed g and unit norm, these isolated self-bound solutions have N-dependent omega_N. They cannot all be treated as excitations of one fixed harmonic well while each is simultaneously taken to source that well alone. For charge Q the relation becomes omega_N^3=g Q(N+3)/6. Keeping omega fixed would require Q proportional to 1/(N+3), or a separate dominant background that fixes the well. Neither is supplied by the one-quantum normalization.

This does not disprove a fixed-background cumulative-band construction. It specifies what must distinguish that construction from isolated self-binding. N here is a flat oscillator degree, not an independently derived assignment to an electron, a chemistry orbital, or a V2 projective band.

Verification

circulating_hartree_checks.py verifies the N=2 state's norm, moment matrix, self-consistent width, one-body current, and both signed chiral spectra. Its further linearized checks are documented separately. The general N>=2 statements follow from the one-body selection rule and the displayed moment identities.