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: Exact moment closure gives real linear frequencies for the isotropic circulating Hartree branch without establishing nonlinear stability or selecting particle indices.

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. Exact moment dynamics

Consider the single scalar equation i partial_t psi=(p^2+g x^T M x)psi, M=<xx^T>, g>0, with unit conserved norm and the flat, fixed-centroid assumptions of the circulating-orbital note. Define real matrices

Kab=⟨papb⟩,Cab=12⟨xapb+pbxa⟩. K_{ab}=\langle p_ap_b\rangle,\qquad C_{ab}=\tfrac12\langle x_ap_b+p_bx_a\rangle.

Quadratic operator commutators close exactly, even for a non-Gaussian state:

M˙=2(C+CT),C˙=2K−2gM2,K˙=−2g(MC+CTM). \dot M=2(C+C^T),\qquad \dot C=2K-2gM^2,\qquad \dot K=-2g(MC+C^TM).

There is no moment-factorization approximation here. In particular C-C^T, the angular-momentum tensor, is conserved because dot C is symmetric. These equations use the scalar Schrödinger-type functional, not the full Dirac master dynamics or a dissipative radiative model.

2. Linearize about an isotropic circulating stationary point

Let M=mI, K=g m^2 I and C=A, with A real antisymmetric. Write omega^2=g m. Let X,Y,Z be the symmetric perturbations of M,K,C. The antisymmetric perturbation of C is separately constant. Then

X˙=4Z,Z˙=2Y−4gmX,Y˙=−4gmZ−2g[X,A]. \dot X=4Z,\qquad \dot Z=2Y-4gmX,\qquad \dot Y=-4gmZ-2g[X,A].

Eliminating Y and Z gives

X(3)+24ω2X˙+16g[X,A]=0. X^{(3)}+24\omega^2\dot X+16g[X,A]=0.

The commutator map X-> [X,A] on symmetric matrices is skew-adjoint for the Frobenius inner product. Its complex eigenvalues are i a with real a, and |a| is at most twice the largest rotation rate of A. Positivity of the phase-space covariance gives ||A||_op<=m omega, hence |g a|<=2 omega^3. For example this follows from the Schur complement of the positive matrix [[mI,A],[-A,gm^2 I]]. The stronger quantum uncertainty inequality is not needed for this bound.

For a moment proportional to exp(-i Omega t), the frequency equation is

Ω3−24ω2Ω+16ga=0. \Omega^3-24\omega^2\Omega+16ga=0.

Its discriminant is bounded below by

4(24ω2)3−27(16ga)2≥27648ω6>0. 4(24\omega^2)^3-27(16ga)^2 \ge27648\omega^6>0.

Thus each commutator channel has three real frequencies. The a=0 channels include a zero frequency and plus/minus sqrt(24) omega. This proves absence of exponential growth in the linearized moment system, not nonlinear orbital stability or damping toward a unique state.

3. Relation to the complete scalar linearization

Write psi=exp(-i mu t)(phi+eta) around a harmonic-shell stationary state. The linearized field equation is

iη˙=(Hω−μ)η+g∑a,bδMab xaxbϕ. i\dot\eta=(H_\omega-\mu)\eta +g\sum_{a,b}\delta M_{ab}\,x_ax_b\phi.

The homogeneous operator is self-adjoint; the forcing depends only on the closed moment perturbations. A perturbation invisible to all delta M evolves under this homogeneous operator. A nonzero exponential instability could therefore neither hide in a zero-moment perturbation nor be generated by the bounded-frequency moment channels. Resonant forcing and neutral directions can still produce polynomial behavior; bounded norm for every linearized perturbation and nonlinear stability are not claimed.

For phi in shell N=2, every x_a x_b phi lies in shells 0,2,4. The entire nonlinear part of the Bogoliubov operator is therefore captured in their 148-dimensional scalar sum, doubled to 296 dimensions. All omitted shell directions have free real frequencies. With q_ab=x_a x_b phi, the blocks are

A=Hω−μ+g∑a,b∣qab⟩⟨qab∣,B=g∑a,bqabqabT,L=(AB−Bˉ−Aˉ). \mathcal A=H_\omega-\mu+g\sum_{a,b}|q_{ab}\rangle\langle q_{ab}|, \qquad \mathcal B=g\sum_{a,b}q_{ab}q_{ab}^T, \qquad \mathcal L=\begin{pmatrix}\mathcal A&\mathcal B\\-\bar{\mathcal B}&-\bar{\mathcal A}\end{pmatrix}.

The transpose in B is not an adjoint; the background is complex. Summing over a<=b instead requires a factor sqrt(2) in off-diagonal q_ab columns.

4. Explicit circulating N=2 result

For omega=1, g=6/5, m=5/6 and A=-J_0/3, the nonzero commutator rates are a=plus/minus 2/3. The additional cubic is Omega^3-24 Omega+64/5=0. The distinct positive field frequencies in the active block are approximately 0.5398903, 4, 4.6066714, 4.8989795 and 5.1465617, along with their negatives and neutral modes. The non-cubic sqrt(24) and free-shell frequency 4 are included in this list.

circulating_hartree_checks.py compares the analytic roots with all 296 numerical eigenvalues. Raw near-zero eigenvalues have imaginary roundoff around 4e-8; nonzero modes agree with real values to numerical precision. The proof uses the analytic closure, not a claim that floating-point zero eigenvalues are exactly diagonalizable.

There is a genuine current-carrying stationary branch with no exponential linear instability in this scalar model. It is not excluded by the earlier ground-state covariance theorem. But the argument applies broadly to isotropic harmonic-shell stationary states, so passing this test cannot select the observed particle indices. The missing cross-sector, radiative, spinor, and background-selection dynamics remain essential.