Fixed force ceiling: small bound circles and an excitation floor
One fixed smooth confining potential with a globally bounded force admits stable circular orbits with positive angular actions tending to zero. A lower bound on circular speed, together with the force ceiling, instead gives a positive action bound. This identifies the missing excitation premise in A16’s fixed-force follow-up. A17, C058–C059; written proof and B31 review.
1. Fixed Hamiltonian and exact circles
Fix \(m,c,K,b>0\) and use Cartesian configuration space \(\mathbb R^2\) with
\[ H(q,p)=\sqrt{m^2c^4+c^2|p|^2}+V(|q|),\qquad V(r)=Kb^2\left(\sqrt{1+r^2/b^2}-1\right). \]
The potential is smooth even at the origin, tends to infinity, and has inward force magnitude \(f(r)=Kr/\sqrt{1+r^2/b^2}<Kb\). Thus \(Kb\le F_{\max}\) gives a prescribed global force ceiling without changing the potential across the orbit family. Finite energy bounds both position and momentum, giving complete trajectories. Particle velocities are strictly below \(c\); this external-potential model specifies no propagating mediator.
For each radius \(R>0\), set \(s=Rf(R)\) and
\[ \gamma_R=\frac{s/(mc^2)+\sqrt{[s/(mc^2)]^2+4}}2, \quad v_R=c\sqrt{1-\gamma_R^{-2}},\quad P_R=m\gamma_Rv_R,\quad \ell_R=RP_R. \]
Circular force balance is \(P_Rv_R/R=f(R)\), equivalent to \(mc^2(\gamma_R-\gamma_R^{-1})=s\). The positive solution above therefore gives an exact complete circle with \(\omega_R=v_R/R\). The normalized canonical orbit integral is \(\ell_R\) in action units. Uniform circular phase gives A15’s \(\mathcal A_{\rm cov}=\ell_R\).
At fixed \(\ell=\ell_R\), the radial effective energy is \(U_\ell(r)=\sqrt{m^2c^4+c^2\ell^2/r^2}+V(r)\). Differentiation at the circle gives
\[ U_\ell''(R)=f'(R)+(3-v_R^2/c^2)\frac{f(R)}R>0. \]
Indeed the kinetic contribution is \(P_R^2/(m\gamma_R^3R^2)+2P_Rv_R/R^2\); use force balance and \(\gamma_R^{-2}=1-v_R^2/c^2\). Here \(f'(R)=K(1+R^2/b^2)^{-3/2}>0\). The circle is a strict radial minimum and is stable against sufficiently small radial perturbations at fixed angular momentum. This is the stability notion used here, rather than asymptotic attraction of the Hamiltonian flow.
2. The small-orbit limit keeps every upper ceiling
As \(R\downarrow0\) at fixed \(m,c,K,b\),
\[ s=KR^2+O(R^4),\quad v_R\sim\sqrt{K/m}\,R,\quad \ell_R\sim\sqrt{mK}\,R^2,\quad \omega_R\longrightarrow\sqrt{K/m}. \]
These follow by expanding \(\gamma_R=1+s/(2mc^2)+O(s^2)\). Kinetic energy above rest is \(mc^2(\gamma_R-1)\sim KR^2/2\) and \(V(R)\sim KR^2/2\), hence \(H-mc^2\sim KR^2\). Force and coordinate acceleration satisfy \(f(R)\sim KR\) and \(v_R^2/R\sim KR/m\), both tending to zero. Any strictly positive prescribed speed and acceleration upper bounds hold on all sufficiently small members, alongside the global force ceiling. The period tends to \(2\pi\sqrt{m/K}\): rapid cycling is unnecessary.
Every circle is nonstationary, regular and bound, with positive energy above the minimum and positive covariance action. Their infimum is nevertheless zero in this single fixed Hamiltonian. The limiting state is its smooth central equilibrium. Quantitative excitation restrictions, rather than merely positivity of excitation, would exclude this family.
3. A positive conditional bound
For any relativistic circular orbit under inward force \(0<f(R)\le F_{\max}\), force balance gives the exact identity
\[ \ell=RP=\frac{P^2v}{f(R)}. \]
If an independently specified speed floor \(v\ge v_*>0\), \(v_*<c\), is imposed, then the increasing function \(m^2v^3/(1-v^2/c^2)\) yields
\[ \ell\ge\frac{m^2v_*^3}{F_{\max}(1-v_*^2/c^2)}>0. \]
The dimensions are action. This is a lower bound on circular orbital action, and on A15’s estimator under uniform phase. The speed floor is a lower kinetic excitation premise; the ceiling alone gives Section 2’s zero-infimum family. The bound is parameter-dependent and does not establish a universal value or physical-time attraction. For Newtonian circular motion the same calculation uses \(P=mv\) and gives \(m^2v_*^3/F_{\max}\).
4. Next selection question
A18 should test the positive bound beyond circles, using total turning of a closed regular trajectory and the force ceiling. Its source audit should check the relevant total-curvature theorem before importing it.
The useful positive mechanism combines an upper force scale with a lower excitation scale. The next task should identify a physical source for persistent excitation in a closed mechanical model, with its energy allocation and any attraction claim explicit. Reuse A07’s prescribed-endpoint turn cost and A02’s prepared reservoir as comparison mechanisms. Another arbitrary upper ceiling would leave the small-circle family intact.