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: An exact positive-square identity makes the circulating Hartree states global energy minima at fixed mean circulation.

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.

Model and conserved constraint

Use only the isolated flat six-dimensional scalar functional E = ⟨p²⟩ + (g/2) tr M², M = ⟨xxᵀ⟩, norm one, g > 0. It is invariant under SO(6), and its Hamiltonian evolution conserves the angular-momentum tensor. In particular fix

J=∑a=13La,a+3,⟨J⟩=−j,j≥0. \mathcal J=\sum_{a=1}^3L_{a,a+3},\qquad \langle\mathcal J\rangle=-j,\quad j\ge0.

j is an expectation constraint and need not be an integer. This is not a new quantization rule. The states φ_N constructed earlier have j = N.

Exact identity, without a Gaussian ansatz

For any positive oscillator scale ω, let N₊ count the circular oscillator quanta whose angular momentum has the opposite sign to the constraint. The oscillator identity is

p2+ω2r2+2ωJ=4ωN++6ω. p^2+\omega^2r^2+2\omega\mathcal J =4\omega N_++6\omega.

Choose ω³ = g(j+3)/6 and m = ω²/g. Completing the matrix square gives the exact expectation identity

E−32ω(j+3)=4ω⟨N+⟩+g2∥M−mI6∥F2≥0. E-\frac32\omega(j+3) =4\omega\langle N_+\rangle +\frac g2\|M-mI_6\|_F^2\ge0.

No restriction to a fixed oscillator shell, a Gaussian, or an isotropic trial density was made in deriving the bound. Finite kinetic energy and second moments suffice for these quadratic-form identities. Equality requires no opposite-circulation occupation and M = mI₆. For every integer N ≥ 2 the earlier pure state φ_N satisfies both conditions at j = N. It is therefore a global constrained energy minimum, despite being excited relative to the unconstrained scalar ground state.

What this controls dynamically

At fixed initial j, the chosen ω is fixed. Both E and j are conserved by the isolated scalar evolution, so the entire right-hand side is conserved. An initially small energy excess controls the opposite-circulation occupation and the Frobenius deviation of M from mI at all times for which the evolution exists. Since N₊ has integer spectrum with gap one above its kernel, the squared Hilbert-space norm outside that kernel is also bounded by ⟨N₊⟩. This is a nonlinear invariant bound for these observables.

It is not stability of a particular wavefunction modulo phase. Many states have the same constraints and second moments; their spatial evolution and higher moments need not stay near one representative φ_N. No unique orientation, selected degree, or radiative lifetime follows. The result complements rather than replaces the linear spectral analysis.

Cross-sector boundary

The fixed nuclear embedding used in the 3-in-6 moment model breaks the mixed rotations generated by this J. For a general quadratic mean-field potential xᵀ W x, the covariance equations give

ddt⟨L⟩=2[W,M]. \frac{d}{dt}\langle L\rangle=2[W,M].

Self-interaction W = gM makes this vanish. An added nuclear matrix need not commute with M, and the scalar constrained-minimum proof then cannot be carried across unchanged. Even initially isotropic M does not ensure that anisotropic W preserves the circulating stationary solution. Radiation and other channels can likewise exchange angular momentum.

Reproduction

circulation_energy_checks.py checks the symbolic positive-square identity, saturation by φ₂, and arbitrary complex finite-Fock-state examples. The proof is the displayed operator identity, not sampling those examples.