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Summary: The reciprocal tensor-response model admits uniformly precessing electron–response solutions whose rate obeys a cubic, with distinct branches and stability that must be checked rather than assumed.

Exact collective solution

This result uses the effective response channel in the linked note; it is not a derived microscopic nuclear current or a replacement for the actual-orbit target.

Let K generate a planar rotation in O, put a in (0,1), q=sqrt(1−a²), and use the following time-dependent mixed blocks:

A=aK,C=−aK,Q(t)=diag⁡(qR(ωt),1),R(θ)=eθK2. A=aK,\quad C=-aK,\quad Q(t)=\operatorname{diag}\big(qR(\omega t),1\big), \qquad R(\theta)=e^{\theta K_2}.

Take a response rotating at the same rate,

X(t)=diag⁡(xR(ωt),z),Y=MXX˙. X(t)=\operatorname{diag}\big(xR(\omega t),z\big),\qquad Y=M_X\dot X.

The oscillator equation and electron equation independently require

z=χg,x=χgq1−ω2/ΩX2,ωq=2gaIex. z=\chi g,\qquad x=\frac{\chi gq}{1-\omega^2/\Omega_X^2},\qquad \omega q=\frac{2ga}{\mathcal I_e}x.

With kappa=chi g², the common rate must therefore obey

ω(1−ω2ΩX2)=2κaIe. \boxed{ \omega\left(1-\frac{\omega^2}{\Omega_X^2}\right) =\frac{2\kappa a}{\mathcal I_e}. }

Unlike assigning X(t) as an external drive, this ansatz solves both reciprocal equations if the cubic holds. It constructs a family of self-consistent motions, not attraction from arbitrary initial conditions. The response has its own stored energy and momentum. The endpoint cases, zero coupling, and the response pole omega=plus/minus Omega_X are excluded from this nonzero-q derivation and require separate treatment.

Branch structure

Define omega_ad=2 kappa a/I_e > 0. For

ωad<2ΩX33 \omega_{\rm ad}<\frac{2\Omega_X}{3\sqrt3}

there are two positive sub-resonant roots and one negative root below −Omega_X. The two positive branches merge at Omega_X/sqrt(3). Beyond that threshold only the negative real branch remains. The negative branch has an out-of-phase planar response, not a negative susceptibility parameter. This root count follows from the extrema of the cubic.

For a fixed positive a and increasingly fast response, the low positive branch approaches omega_ad. This is the finite-response continuation of the static precession result. The relation determines the rate given (kappa,I_e,Omega_X,a); it does not determine those physical inputs or select an integer petal count.

Full tangent stability

The rotating-frame linearization includes all six CP3 tangent directions plus nine X and nine Y variables, not only the planar ansatz. Choose the dimensionless example chi=I_e=1, Omega_X=2, a=3/5, kappa=25/64. Its three self-consistent roots are

ω−=−1−614,ωlow=12,ωhigh=−1+614. \omega_- =\frac{-1-\sqrt{61}}4,\qquad \omega_{\rm low}=\frac12,\qquad \omega_{\rm high}=\frac{-1+\sqrt{61}}4.

For reproducible rational algebra use U=gX and Z=gY. In the frame rotating only the O plane at rate omega, the equations are

J˙=−[BU,J]−[diag⁡(ωK,0),J],U˙=Z/MX−ωKU,Z˙=−U+κQ−ωKZ. \dot J=-[B_U,J]-[\operatorname{diag}(\omega K,0),J],\qquad \dot U=Z/M_X-\omega KU,\qquad \dot Z=-U+\kappa Q-\omega KZ.

At the low root, Q=diag(4/5,4/5,1), U=diag(1/3,1/3,25/64), and Z=K/24. The exact 24-dimensional characteristic polynomial, up to its positive leading normalization, factors as

λ4(λ2+4)(4λ2+1)(4λ2+9)(4λ2+25)(144λ4+1324λ2+1825) p8(λ), \lambda^4(\lambda^2+4)(4\lambda^2+1)(4\lambda^2+9) (4\lambda^2+25)(144\lambda^4+1324\lambda^2+1825)\,p_8(\lambda),

where

p8(λ)=5435817984λ8+96096780288λ6+601412116420λ4+1573811889137λ2+1436009168896. p_8(\lambda)=5435817984\lambda^8+96096780288\lambda^6 +601412116420\lambda^4+1573811889137\lambda^2+1436009168896.

Exact real-root counting after setting t=lambda² shows that every nonzero factor has strictly negative real t roots. Thus the nonzero linear eigenvalues are imaginary for this example, without a numerical tolerance being used as the proof. Zero has algebraic multiplicity four, nullity three, and squared-matrix nullity four: one size-two Jordan block remains. The result is absence of exponential linear growth, not bounded linear evolution or nonlinear orbital stability.

The high positive branch has no numerically resolved exponential mode in the same full tangent test. This is a numerical result only; it is not covered by the exact low-branch polynomial proof.

The negative branch has a positive growth rate approximately .665240. A small constraint-preserving perturbation grows in an independent nonlinear rotating-frame integration. At amplitudes 1e-5 and 5e-6, the differences from linear evolution at time 3 are approximately 1.31e-10 and 3.56e-11. Thus this branch supplies a genuine unstable counterexample to treating every self-consistent rotation as stable.

Verification and boundaries

collective_precession_checks.py checks the three branches at the same coupling, the full equilibrium equations, tangent-space closure, nonlinear perturbations, exact characteristic factors and root counts, and the neutral Jordan structure. Its normal run completes 32 evaluations across 13 runtime conditions.

No autonomous capture into one branch, integer L selection, Coulomb orbital clock, physical current normalization, or petal readout has been derived. What is new is an explicit reciprocal model with self-consistent precession and a nontrivial tested stability distinction between branches.