Coupled circulating branch

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Summary: Auxiliary model only: The coupled scalar 3-in-6 model admits excited circulating stationary states with an exact oscillatory linear moment spectrum along the equal-coupling family.

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. Same functional, different branch

Use the positive rank-one, fixed-embedding 3-in-6 scalar functional from the ground-state moment test, with both norms and kinetic coefficients one. The mean-field quadratic matrices are

$$W_3=v_3^2M_3+v_3v_6PM_6P^T,\qquad W_6=v_6^2M_6+v_3v_6P^TM_3P, \qquad P=(I_3,0).$$

The unique ground-state covariance has no mixed current. That result did not exclude excited stationary solutions. The following construction includes the nuclear-sector response instead of assuming it is absent.

2. Elliptical circular modes

Put r = √(v₃/v₆). Choose an integer N ≥ 2 with N > 3r/2, and set

$$\alpha^2=\frac12-\frac{3r}{4N},\qquad \beta^2=\frac12+\frac{3r}{4N},\qquad \omega^3=v_6^2\left(\frac{N+3}{6}+\frac r4\right).$$

Use three orthonormal complex oscillator modes z_a = αe_a − iβe_(a+3), and the pure shell state φ = 3^(-1/2) Σ_a |N quanta in z_a⟩. For α ≠ β its coordinate polynomial is not simply a harmonic power: it is defined by applying the oscillator creation operator N times. It remains in the oscillator shell N, but need not have a single angular degree ℓ.

Take the nuclear field to be the oscillator ground state with frequency ω₃ = rω. The moments are

$$M_3=aI_3,\quad a=\frac1{2r\omega},\qquad M_6=\operatorname{diag}(m_OI_3,m_HI_3),$$

$$m_O=\frac{1/2+N\alpha^2/3}{\omega},\qquad m_H=\frac{1/2+N\beta^2/3}{\omega}.$$

Substitution gives exactly W₃ = r²ω²I₃ and W₆ = ω²I₆. Increased unobserved width compensates the additional observed-plane nuclear term. Both fields are consequently eigenfunctions of their own self-consistent operators, with μ₃ = 3rω and μ₆ = (2N+6)ω. These are phase eigenvalues, not total energies or physical rest masses.

The mixed position covariance vanishes, but the angular expectation is

$$\langle L\rangle=-\frac{2N\alpha\beta}{3}J_0.$$

It has full rank for the strict threshold N > 3r/2. At equality α = 0, the circulation disappears. This is a family of current-carrying excited states, not an orientation selected by minimizing the unrestricted energy.

3. Exact linear moment spectrum at one point

For N = 2 and v₃ = v₆ = √(12/13), both frequencies are one: α² = 1/8, β² = 7/8, M₃ = I₃/2, M₆ = diag(7I₃/12,13I₃/12), C₆ = −√7 J₀/12, C₃ = 0. Here C_ab = ⟨{x_a,p_b}⟩/2 and K_d = ⟨p_dp_dᵀ⟩ = M_d.

For each sector the exact quadratic moment equations are

$$\dot M_d=2(C_d+C_d^T),\quad \dot C_d=2K_d-2M_dW_d,\quad \dot K_d=-2(W_dC_d+C_d^TW_d).$$

Their full real linearization has 99 variables. Six independent symmetric 3-by-3 spatial components each carry an identical 13-dimensional block; three antisymmetric components each carry a 7-dimensional block. The characteristic polynomials for the growth exponent λ are

$$p_S(\lambda)=\frac{\lambda^3(\lambda^2+16)(\lambda^2+24)}{169} (169\lambda^6+7800\lambda^4+89856\lambda^2+7168),$$

$$p_A(\lambda)=\frac{\lambda^5(13\lambda^2+288)}{13}.$$

For the remaining squared frequencies u = −λ², the cubic 169u³ − 7800u² + 89856u − 7168 has one root in each of (0,1), (20,22), and (24,25), by endpoint sign changes. These exhaust its roots and are positive. Thus every nonzero moment exponent is purely imaginary.

As in the single-sector analysis, the full field linearization is a pair of self-adjoint oscillator evolutions forced only by these closed moment variables. No nonzero exponential instability can hide entirely outside the moments. Neutral modes, resonant polynomial growth, and nonlinear stability still need separate analysis.

The symbolic calculation extends to every N ≥ 2 on the equal-coupling branch, choosing units with ω = 1 and g = v₃² = v₆² = 12/(2N+9). Here g denotes the rescaled coefficient: for an original fixed physical coupling g₀ it is g₀/ω³. This rescaling is fixed by self-consistency and does not retune g₀ separately for each state. Physical growth exponents and frequencies are restored by multiplying by ω. Put D = 2N+9. The symmetric block keeps the factors λ³(λ²+16)(λ²+24); its remaining squared-frequency polynomial is

$$F_N(u)=D^2u^3-24D(4N+17)u^2+1152(N+4)Du -1024(4N^2-9).$$

Its signs at u = 0,16,24,32 are respectively negative, positive, negative, positive for every N > 3/2. Thus it has three distinct positive roots, one in each intervening interval. The antisymmetric block has the sole nonzero squared frequency 48(N+4)/D. Hence the equal-coupling circulating family has no exponential linear moment instability for every integer N ≥ 2, not just the displayed N = 2 example. Unequal couplings and nonlinear stability remain outside this result.

4. Significance and remaining boundary

The scalar cross interaction does not eliminate all mixed circulation: self-consistent excited widths can restore the frequency degeneracy it needs. This gives an explicit cross-sector source candidate within the imported scalar model only as an angular expectation. No derivation connects it to the actual electron-orbit generator in the 6D demo; calling it an established cross-sector orbit source would be incorrect. The isolated fixed-circulation energy bound is not a proof of nonlinear stability for this coupled branch, since that mixed circulation is no longer separately conserved.

Reproduction

coupled_circulation_checks.py verifies four stationary parameter choices and both exact characteristic polynomials, including the symbolic equal-coupling family. coupled_circulation_bdg_checks.py independently constructs the two-field Bogoliubov matrix at N = 2 from actual Fock amplitudes and quadratic interaction columns. The nonlinear coupling is entirely captured by nuclear shells 0,2 and electron shells 0,2,4, giving 155 complex amplitudes and a 310-dimensional doubled matrix. Its nonzero frequencies agree with the exact moment polynomials; omitted shell directions have the free self-adjoint oscillator evolution. Tiny near-zero imaginary roundoff is not interpreted as an instability or as proof that the zero eigenspace is diagonalizable. All couplings, embedding, field types, and normalization assumptions are the ones stated above; no observed constants were fitted.

Related

• Spin-orbit source audit — conditional derivative vertex needed before orbital current becomes a spin source.
• Self-consistent moment lock test — same coupled functional, with the distinct unrestricted ground-state result.
• Circulating isotropic orbitals — isolated equal-width construction generalized here.
• Fixed circulation energy bound — explains why the isolated energetic argument cannot establish coupled stability.
• Research status — current proof boundaries and correction routes.