Relativistic Kepler: a singular-core angular-action threshold
Regular bound motion in the fixed potential \(-k/r\) exists exactly for \(|L|>k/c\). Replacing it by \(-k/\sqrt{r^2+a^2}\), with any fixed \(a>0\), admits regular bound circles for every \(|L|>0\). Thus the positive infimum uses the singular core as well as finite speed and binding.
Object and physical premises
Use planar canonical coordinates \((r,\theta,p_r,L)\), \(r>0\), and \(H=\sqrt{m^2c^4+c^2(p_r^2+L^2/r^2)}+V(r)\), with fixed \(m,c,k>0\). Energy includes rest energy. A regular bound orbit is a complete Hamiltonian trajectory whose radius remains in a compact subset of \((0,\infty)\). The angular action is \(I_\theta=|L|\): the angular canonical cycle has \((2\pi)^{-1}|\int_0^{2\pi}L\,d\theta|=|L|\). This cycle normalization does not require a closed spatial rosette. Define \(g_\theta=\inf\{|L|:\text{a regular bound orbit exists}\}\). Its units are action, since \([k]=\text{energy}\times\text{length}\). This is an admissibility threshold, not a spacing of adjacent action values.
The potential is external and static; recoil, radiation and field dynamics are excluded from this one-body model. Hamilton’s velocity always has magnitude less than \(c\) at finite momentum.
Singular potential: domain and proof
Set \(\ell=|L|\) and \(x=1/r\). The turning-point energy is \(F_\ell(x)=\sqrt{m^2c^4+c^2\ell^2x^2}-kx\). For \(\ell>0\) its second derivative is strictly positive, its derivative at zero is \(-k\), and its derivative at infinity tends to \(c\ell-k\). Consequently a finite minimum exists exactly for \(\ell>k/c\), with
\[ E_{\min}=mc^2\sqrt{1-k^2/(c^2\ell^2)},\qquad r_0=\frac{\ell\sqrt{c^2\ell^2-k^2}}{mkc}. \]
For each such \(\ell\), the complete bound domain has \(E_{\min}\le E<mc^2\). Equality gives the circle; strict inequality gives two turning radii. Indeed \(F_\ell(0)=mc^2\) and \(F_\ell(x)\to\infty\) at large \(x\), so the allowed interval \(F_\ell(x)\le E\) is compact and separated from zero. Energy bounds the momentum there; the smooth Hamiltonian vector field therefore extends for all time. For \(E\ge mc^2\) the outer radius is unbounded. For \(E<E_{\min}\) there are no states.
At \(\ell=k/c\), \(F_\ell(x)\) decreases to zero without attaining it. For \(\ell<k/c\) it decreases to minus infinity (also directly for \(\ell=0\)). There is no radial well in either case. For any allowed orbit with \(E<mc^2\), the single outer turning point leads to a plunge. The energy identity and inward radial velocity are
\[ c^2p_r^2=E^2-m^2c^4+\frac{2Ek}{r} +\frac{k^2-c^2\ell^2}{r^2},\qquad \dot r=\frac{c^2p_r}{E+k/r}. \]
For \(\ell<k/c\), \(|\dot r|\to c\sqrt{k^2-c^2\ell^2}/k>0\) at the centre. At equality the allowed energy satisfies \(E>0\), and \(|\dot r|\sim c\sqrt{2E/k}\sqrt r\). Both give finite arrival time by integrating \(dr/|\dot r|\). A collision continuation is outside the domain. Hence \(g_\theta=k/c\), with the infimum excluded and all larger angular actions admitted continuously. At fixed \(k\) it closes as \(c\to\infty\); at fixed \(c\) it closes as \(k\downarrow0\). These are limits of thresholds for positive couplings; at \(k=0\) there are no nonstationary bound orbits. At fixed \(\ell,k,m\), \(r_0\to\ell^2/(mk)\) and \(E_{\min}-mc^2\to-mk^2/(2\ell^2)\) in the Newtonian limit.
Boyer’s equations (19)–(20), (37)–(42), mapped by \(\alpha=k\), supply the established classification; B27 records the passage audit. The compactness and endpoint-time arguments above state the precise domain used here.
Softened core: the threshold closes at fixed coupling
Fix \(a>0\) and take \(V_a(r)=-k/\sqrt{r^2+a^2}\). For every \(\ell>0\) the turning-point energy
\[ U_{a,\ell}(r)=\sqrt{m^2c^4+c^2\ell^2/r^2} -\frac{k}{\sqrt{r^2+a^2}} \]
tends to infinity as \(r\downarrow0\). At large radius, \(U_{a,\ell}(r)=mc^2-k/r+\ell^2/(2mr^2)+O(r^{-3})\), so it takes values below its limit \(mc^2\). It therefore attains a global minimum at a finite positive radius. At that minimum \(p_r=0\) and \(U'_{a,\ell}=0\), giving a complete circular Hamiltonian trajectory. Every energy strictly between the minimum and \(mc^2\) also has compact allowed radial components. These give regular bound trajectories by the same continuation argument. Thus \(g_\theta(a)=0\) for every \(a>0\) even at fixed \(m,k,c\). This existence argument needs neither uniqueness of the minimum nor a small-velocity approximation.
For each fixed subcritical \(0<\ell<k/c\), minimizing radii approach zero as \(a\downarrow0\). To see this, write \(r=ay\) and choose a fixed large \(y\) with \(c\ell/y<k/\sqrt{1+y^2}\). Then \(aU_{a,\ell}(ay)\) tends to a negative constant, so the minimum tends to minus infinity. On \(r\ge\delta>0\), \(U_{a,\ell}\ge mc^2-k/\delta\) uniformly in \(a\), forcing the minimizer below any such \(\delta\). These circles leave the regular domain of the singular model in the limit. In particular \(\lim_{a\downarrow0}g_\theta(a)=0\), whereas the singular model has \(g_\theta=k/c\).
Next observable test
A15 should compare orbital action with a finite-window fluctuation in a specified bound preparation. M07 identifies which singular-core premise supports a positive angular threshold; it supplies no universal action value or attraction law for a physical-time field. Proof and literature status are recorded separately in the claim ledger and the follow-up audit.