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The self-sufficiency contrast: where Newton’s axioms leave motion undefined, the theory with \(h>0\) defines it

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The thesis in STATE says that Newton’s mechanics is the large-action regime of a theory with \(h>0\), and that it needs \(h\) because its own axioms do not make a complete theory of motion. This note proves the first piece of evidence for that claim as one theorem with four parts, for Kepler’s force. The classical flow of \(V=-k/r\) is incomplete: every zero-angular-momentum datum that moves inward, or is bound, reaches the centre in finite time, with the explicit collision time \(t_c=\tfrac\pi2\sqrt{mr_0^3/2k}\) for fall from rest, and Newton’s axioms give no continuation there. The quantum Hamiltonian \(-\tfrac{h^2}{2m}\Delta-k/r\) is self-adjoint on the domain of the Laplacian and generates a global unitary group for every state and every \(h>0\), by Kato’s theorem, cited with its hypotheses. The same contrast holds for the quartic between the fixed-\(h\) unitary limit and the classical-first construction that needs a branch rule. And the classical trajectory is a derived object of the \(h>0\) theory: for quadratic forces the centroid of any state obeys Newton’s equation exactly at every \(h\), and for Kepler’s force a coherent state follows the orbit in the large-action regime for finite times away from the collision set, by Hepp’s theorem, cited with its hypotheses and with the one further localization hypothesis the singular potential needs. Parts (a), (c) and the quadratic half of (d) are proved here; (b) and the Kepler half of (d) rest on the cited theorems. Theorem 1 was refereed by GPT-6.1 Sol (Codex) in two bounded passes on 2026-10-01, REFINE then ACCEPT on its statement and proof as printed; the reports are in reviews/2026-10-01-theorem1-self-sufficiency-contrast-referee.md. Proposition 2 was assessed separately and not endorsed: its spin-zero threshold is the real-indicial one with the branch selection an additional premise, and its endpoints are not settled; the wording below records that. Proposition 2 then gives the gap the exercise is for, in two relativistic models: with finite \(c\), the theories of Kepler’s force that define motion without an added rule are those with \(h>k/(\alpha_cc)\), by the self-adjointness thresholds of the Dirac and Klein–Gordon Coulomb problems, cited; Newton’s theory at \(h=0\) and every theory with \(h\) below \(k/(\alpha_cc)\) are not self-sufficient. Written by Claude Fable, 2026-10-01; proof status recorded in the final section.

Notation: \(h\) is the reduced phase constant, \(m>0\) the mass, \(k>0\) the force constant, \(r=|x|\) in \(\mathbb R^3\); the planar case is a remark.

1. Statement

Theorem 1 (self-sufficiency contrast for Kepler’s force).

  1. Classical incompleteness. Let \(L=x\times p\) be the angular momentum. Every initial datum with \(L=0\) whose radial velocity is inward or zero, and every datum with \(L=0\) and negative energy, reaches \(r=0\) at a finite time. For a body released from rest at distance \(r_0\) the collision time is \[t_c=F(r_0):=\frac\pi2\sqrt{\frac{m\,r_0^3}{2k}}.\tag{1}\] For bound data, \(E<0\) with \(R=-k/E\): \(t_c\le F(R)\) if the radial velocity is inward or zero, and \(t_c\le2F(R)\) if it is outward; for inward data with \(E\ge0\), \(t_c\le\tfrac23\sqrt{m/2k}\,r_0^{3/2}\). The collision set, on the phase space \(\{(x,p):x\neq0\}\), \[\mathcal C=\{L=0\ \text{and}\ (\dot r\le0\ \text{or}\ E<0)\},\] is nonempty, invariant under the flow while solutions exist, and of measure zero. Along every solution in \(\mathcal C\) the force magnitude \(k/r^2\) and the momentum \(|p|\sim\sqrt{2mk/r}\) diverge as the centre is approached, so there is no continuation as a solution of the original vector field in phase space: the flow of Kepler’s force is not complete. Newton’s axioms, the Laws with the force law, assign no motion at the centre; any continuation is an added prescription (regularization), and the theorem establishes incompleteness, not that no classical completion is possible.

  2. Quantum completeness. For every \(h>0\), \(m>0\) and real \(k\), the operator \[H_h=-\frac{h^2}{2m}\Delta-\frac kr\qquad\text{on }L^2(\mathbb R^3)\tag{2}\] is self-adjoint on the domain \(H^2(\mathbb R^3)\) of the Laplacian and bounded below, and \(U_h(t)=e^{-itH_h/h}\) is a strongly continuous unitary group defined for every \(t\in\mathbb R\) and every initial state in \(L^2(\mathbb R^3)\). This is Kato’s theorem [@Kato1951; metadata], whose hypothesis, that the potential is the sum of an \(L^2(\mathbb R^3)\) and a bounded function, is verified in §3.

  3. The quartic contrast. For \(V=\lambda q^4\) on the line, \(\lambda>0\), Theorem 2(b) of the 1998-conjecture note gives, for every \(h>0\), the strong convergence of the time-sliced operators to the unitary evolution \(e^{-iTH_h/h}\) along every sequence of partitions, with no extra datum; Theorem 1 there shows that the unrestricted halved classical-first construction at zero resolution has no limit on a two-cell partition with three stationary paths, and Theorem 2(a) there exhibits one repair, a branch selection chosen by hand. The contrast is between the fixed-\(h\) construction, which exists for all data, and that specific classical-first prescription, which needs a datum the action does not supply; it says nothing against the quartic Newtonian initial-value problem itself, which is globally well posed.

  4. The classical trajectory as a derived object. (d1) For \(H=p^2/2m+V\) with \(\deg V\le2\), the expectations \(\langle q\rangle_t\), \(\langle p\rangle_t\) of every state with finite second moments obey Newton’s equations exactly for every \(h>0\), \[\frac{d}{dt}\langle q\rangle_t=\frac{\langle p\rangle_t}m,\qquad \frac{d}{dt}\langle p\rangle_t=-V'(\langle q\rangle_t),\tag{3}\] and a coherent state stays a Gaussian whose centroid is the classical trajectory. (d2) For Kepler’s force, let \(z_0=(x_0,p_0)\) lie on a classical orbit whose distance from the centre stays at least \(\rho_{\min}>0\) on \([0,T]\). Under Hepp’s theorem, stated in §5 with its hypotheses, applied to a smooth potential \(V_\delta\) equal to \(-k/r\) for \(r\ge\delta\) with \(0<\delta<\rho_{\min}\), the coherent state \(\varphi^h_{z_0}\) with \(\sigma_h^2\propto h\) evolves under \(H^{(\delta)}_h\) into a Gaussian centred on the Kepler orbit \(z_t\), with the phase fixed as in (4), up to an \(L^2\) error tending to zero as \(h\to0\) at fixed \(T\) and \(\delta\), uniformly on \([0,T]\); and under the localization hypothesis (L) of §5 the same holds for the Coulomb evolution \(U_h(t)\) itself. Here “large action” means this fixed-scale limit \(h\to0\) at fixed orbit, and nothing uniform in \(\delta\) or in \(T\) is claimed.

Conclusion. The theory with \(h>0\) defines motion for all data and all times, by (b), and yields Newton’s trajectories where they exist, exactly for quadratic forces and, under the cited theorem and (L), for Kepler’s force away from collisions, by (d), while Newton’s axioms leave motion undefined on the collision set, by (a), and the unrestricted classical-first construction at zero needs a datum the action does not supply, by (c).

2. Proof of (a)

Conservation of \(L\) under a central force is Proposition I of Book I; for \(L=0\) the motion is on a line through the centre and \(E=\tfrac m2\dot r^2-k/r\) is conserved. Released from rest at \(r_0\), \(E=-k/r_0\) and \[\dot r^2=\frac{2k}m\Big(\frac1r-\frac1{r_0}\Big)=\frac{2k}{m}\,\frac{r_0-r}{r\,r_0},\] with \(\dot r<0\) for \(t>0\) because the force points inward. Hence \[t_c=\int_0^{r_0}\frac{dr}{|\dot r|} =\sqrt{\frac{m r_0}{2k}}\int_0^{r_0}\sqrt{\frac{r}{r_0-r}}\,dr.\] With \(r=r_0\sin^2\theta\), \(dr=2r_0\sin\theta\cos\theta\,d\theta\) and \(\sqrt{r/(r_0-r)}=\tan\theta\), the integral is \(2r_0\int_0^{\pi/2}\sin^2\theta\,d\theta=\pi r_0/2\), which gives (1).

For a bound datum with \(L=0\), \(E<0\) and \(R=-k/E\): \(r\le R\) throughout, \(|\dot r|^2=\tfrac{2k}m(1/r-1/R)\), and the time from \(r_0\) to the centre along the inward branch is \(I(r_0)=\sqrt{m/2k}\int_0^{r_0}\sqrt{rR/(R-r)}\,dr\le I(R)=F(R)\). If the radial velocity is outward, the body first rises to \(R\) in the time \(F(R)-I(r_0)\) and then falls in the time \(F(R)\), so its collision time is \(2F(R)-I(r_0)\le2F(R)\), and it exceeds \(F(R)\): one fall time does not bound it. For inward data with \(E\ge0\), \(|\dot r|\ge\sqrt{2k/(mr)}\) gives the time bound \(\int_0^{r_0}\sqrt{mr/2k}\,dr=\tfrac23\sqrt{m/2k}\,r_0^{3/2}\). Invariance of \(\mathcal C\): \(L\) and \(E\) are conserved; for \(E<0\) the condition is then automatic, and for \(E\ge0\) the radial velocity cannot vanish, since \(\tfrac m2\dot r^2=E+k/r>0\), so its inward sign is preserved. The condition \(L=0\) is that \(p\) be parallel to \(x\), which for \(x\neq0\) defines a four-dimensional subset of the six-dimensional phase space, hence of Liouville measure zero; it is nonempty. As \(r\to0\) along a solution in \(\mathcal C\), the force magnitude \(k/r^2\) is unbounded and \(\tfrac m2\dot r^2=E+k/r\) gives \(|p|\sim\sqrt{2mk/r}\), so the solution leaves every compact subset of phase space in finite time and admits no continuation as a solution of the vector field. A continuation through the centre (Levi-Civita or Kustaanheimo–Stiefel regularization) is an added prescription with an enlarged notion of solution, in the same position as the branch rule of (c). \(\square\)

3. Proof of (b), with Kato’s hypothesis verified

Kato’s theorem [@Kato1951; metadata; the statement used is the one standardly quoted from it, the primary text was not read in this session]: let \(V=V_1+V_2\) be real with \(V_1\in L^2(\mathbb R^3)\) and \(V_2\in L^\infty(\mathbb R^3)\). Then for every \(a>0\) the operator \(-a\Delta+V\) is self-adjoint on \(H^2(\mathbb R^3)\) and bounded below. The mechanism is the Sobolev bound \(\|u\|_\infty\le\epsilon\|\Delta u\|+C_\epsilon\|u\|\) for \(u\in H^2(\mathbb R^3)\), which makes \(V_1\) relatively bounded with respect to \(\Delta\) with relative bound zero, so that the Kato–Rellich theorem applies for every value of \(a=h^2/2m\).

Verification for \(V=-k/r\): fix a cutoff radius \(R>0\) and put \(V_1=-\tfrac kr\mathbf 1_{r<R}\), \(V_2=-\tfrac kr\mathbf 1_{r\ge R}\). Then \(|V_2|\le|k|/R\), and \[\int_{\mathbb R^3}|V_1|^2d^3x=k^2\int_0^R\frac{4\pi r^2}{r^2}\,dr=4\pi k^2R<\infty.\] So the hypothesis holds for every real \(k\), and \(H_h\) in (2) is self-adjoint on \(H^2(\mathbb R^3)\) for every \(h>0\), \(m>0\). Stone’s theorem then gives the unitary group \(U_h(t)\) for all \(t\in\mathbb R\), strongly continuous, defined on every \(\psi\in L^2\), with \(U_h(t)\psi\in H^2\) for \(\psi\in H^2\); states outside the operator domain have the mild unitary evolution, those inside satisfy the Schrödinger equation. No initial datum and no time is excluded; the states supported near \(r=0\) evolve like all others. \(\square\)

Planar remark. In \(\mathbb R^2\) the function \(1/r\) is not square integrable near the origin, and the operator-sense theorem does not apply; the planar Coulomb Hamiltonian is defined in the form sense by the KLMN theorem, since \(1/r\in L^p_{\rm loc}(\mathbb R^2)\) for \(p<2\) is form-bounded relative to \(-\Delta\) with relative bound zero. The planar Kepler problem used elsewhere in this repository therefore also has a global unitary dynamics, by that standard route, which is not reproved here.

4. Proof of (c)

The statements are Theorems 1, 2(a) and 2(b) of the 1998-conjecture note, accepted on refereeing there (status line of that note). Theorem 2(b) proves the product formula for \(H_h=-\tfrac{h^2}{2m}d^2/dq^2+\lambda q^4\) at every \(h>0\): essential self-adjointness, the strong limit of the symmetric time slicing along any partition sequence, and the composition \(\|U_h(T)\|=1\). Theorem 1 proves that the halved classical-first functional on the two-cell partition has, as the resolution tends to zero, the oscillating value (4) there with no limit, because the three stationary paths interfere; Theorem 2(a) restores a limit only by restricting to a neighbourhood of one path, a datum the action does not contain. Hence for the quartic the \(h>0\) theory exists for all data without extra rules and the construction at zero does not. \(\square\)

5. Proof of (d)

(d1), quadratic forces. By Theorem A(a) of the fifth-postulate note, the Heisenberg equations are Newton’s second law as operator identities for every real \(h\): \(\dot q=p/m\) and \(\dot p=-V'(q)\). Taking expectations in a state with finite second moments gives \(\tfrac{d}{dt}\langle q\rangle=\langle p\rangle/m\) and \(\tfrac{d}{dt}\langle p\rangle=-\langle V'(q)\rangle\); for \(\deg V\le2\) the function \(V'\) is affine, so \(\langle V'(q)\rangle=V'(\langle q\rangle)\), which is (3). The centroid therefore follows the classical trajectory through \((\langle q\rangle_0, \langle p\rangle_0)\) for every \(h>0\) and every such state, with no approximation. For a coherent state the Wigner function is the Gaussian transported by the affine flow (proof of Theorem 5 of the reachability note), so the state stays Gaussian with the classical trajectory as its centroid and covariance \(M_t\Sigma_hM_t^{\sf T}\). The width at time \(t\) depends on the flow: for free motion, scaling the mass by \(\lambda\) at fixed \(h\) and widths sends \(M_{12}=t/m\to0\) and the position width to the initial \(\sigma\), while for Hooke’s force at fixed frequency the width \(s_t^2=\sigma^2\cos^2\omega t+h^2\sin^2\omega t/(4m^2\omega^2\sigma^2)\) tends to \(\sigma^2\cos^2\omega t\); the centroid transport is exact in every case. \(\square\)

(d2), Kepler’s force. Hepp’s theorem [@Hepp1974; metadata; the statement used is the one standardly quoted from it, the primary text was not read in this session], in the form used here: let \(W\) be a real \(C^\infty\) potential on \(\mathbb R^3\) with bounded derivatives of every order \(\ge2\), so that \(-\tfrac{h^2}{2m}\Delta+W\) is self-adjoint and the classical flow \(\Phi_t\) of \(p^2/2m+W\) is complete; let \(\varphi^h_{z}\) be the coherent state at \(z=(x,p)\) with position width \(\sigma_h\), \(\sigma_h^2=hL_*/P_*\) for fixed \(L_*,P_*\); and let \(\varphi^h_{z_t,M_t}\) be the Gaussian obtained by propagating the centred initial Gaussian with the continuously lifted metaplectic representation of the linearized flow \(M_t=D\Phi_t(z_0)\), which fixes its phase as well as its covariance, and then translating it in phase space to \(z_t=\Phi_t(z_0)\). Then for every \(T>0\), \[\sup_{|t|\le T}\big\|e^{-itH^{W}_h/h}\varphi^h_{z_0} -e^{iS_t/h}\varphi^h_{z_t,M_t}\big\|_{L^2}\longrightarrow0 \qquad(h\to0),\tag{4}\] with \(S_t=\int_0^t(p\cdot\dot x-H_{\rm cl})\,ds\) the classical action along the trajectory in the translation convention matching the metaplectic lift. The phase convention is part of the statement: the covariance alone leaves an order-one phase undetermined, and a free centred Gaussian already acquires the dispersive phase \(-\tfrac32\arctan(ht/2m\sigma_h^2)\) while its centroid action vanishes. The constants in (4) may depend on \(\delta\) and \(T\). Apply this with \(W=V_\delta\), a \(C^\infty\) function equal to \(-k/r\) for \(r\ge\delta\) and bounded with bounded derivatives inside the ball: for \(r\ge\delta\) all derivatives of \(-k/r\) of order \(\ge1\) are bounded by constants times \(\delta^{-2},\delta^{-3},\ldots\), so \(V_\delta\) satisfies the hypothesis. Since the Kepler orbit through \(z_0\) stays at distance \(\ge\rho_{\min}>\delta\) on \([0,T]\), it is also the orbit of \(V_\delta\), and (4) gives the first claim of (d2): under \(H^{(\delta)}_h\) the coherent state follows the Kepler orbit.

For the Coulomb evolution itself, Duhamel’s formula on the common domain gives \[U_h(t)\varphi-U^{(\delta)}_h(t)\varphi =-\frac ih\int_0^tU_h(t-s)\,(V-V_\delta)\,U^{(\delta)}_h(s)\varphi\,ds,\] so that \(\|U_h(t)\varphi-U^{(\delta)}_h(t)\varphi\|\le\tfrac Th\sup_{0\le s\le T} \|(V-V_\delta)U^{(\delta)}_h(s)\varphi\|\) on the common domain \(H^2\), where \(V-V_\delta\) is supported in the ball \(r<\delta\). Hypothesis (L): \(\sup_{0\le s\le T}\|(V-V_\delta)U^{(\delta)}_h(s)\varphi^h_{z_0}\|=o(h)\) as \(h\to0\). Under (L) the Coulomb and regularized evolutions of the coherent state coincide in the limit and (4) transfers to \(U_h\). The hypothesis says that the evolved state carries negligible mass and energy inside the ball the orbit never visits; for the Gaussian \(\varphi^h_{z_t,M_t}\) alone it holds with an exponentially small bound, by a smooth cutoff to the ball followed by Hardy’s inequality \(\|f/r\|\le2\|\nabla f\|\) in \(\mathbb R^3\) and the Gaussian tails, but the \(o(1)\) remainder in (4) is an \(L^2\) statement, and (L) requires its energy-norm counterpart, which is not proved here. \(\square\)

5b. Proposition 2: with finite \(c\), self-sufficiency has a smallest \(h\)

The point of the exercise is a gap in the admissible values of \(h\), the elementary analogue of a spectrum \(\{0\}\cup[m,\infty)\). Theorem 1 alone gives none: by (b) every \(h>0\) makes the non-relativistic Coulomb theory complete, and the ground-state energy \(-mk^2/2h^2\) tends to \(-\infty\) as \(h\to0\) without any threshold. The threshold appears when the speed of light is finite, through the known self-adjointness limits of the relativistic Coulomb problem.

Proposition 2 (smallest \(h\) at finite \(c\)). Let \(c<\infty\) and \(\nu=k/(hc)\), the Coulomb coupling in units of \(hc\). (i) Spin one half. The Dirac–Coulomb operator \(c\,\alpha\cdot p+\beta mc^2-k/r\) on \(L^2(\mathbb R^3)^4\) has a distinguished self-adjoint extension, selected by finiteness of the potential energy on its domain, for \(\nu<1\) [@Schmincke1972; @Wuest1975; @Nenciu1976; metadata], and is essentially self-adjoint on \(C_c^\infty(\mathbb R^3\setminus\{0\})^4\) for \(\nu\le\sqrt3/2\); for \(\nu>1\) every self-adjoint realization requires a boundary condition at the centre, an added rule in the sense of Theorem 1(a) and (c). (ii) Spin zero. For the Klein–Gordon equation with the Coulomb potential the reduced \(s\)-wave radial function \(u=rR\) behaves at the origin like \(r^{1/2\pm\sqrt{1/4-\nu^2}}\) [@Case1950; metadata]: for \(\nu<\tfrac12\) the indicial exponents are real and distinct, at \(\nu=\tfrac12\) the second solution carries a logarithm, and for \(\nu>\tfrac12\) the exponents are complex and the solutions \(r^{1/2}e^{\pm i\beta\log r}\) oscillate in phase without limit at the centre. Below \(\nu=\tfrac12\) both branches of \(u\) are finite at the origin, so finiteness alone selects neither; the choice of the larger-exponent branch is the usual selection in this limit-circle regime and is an additional premise, not supplied by the equation. The threshold \(\nu=\tfrac12\) is therefore the real-indicial threshold, and the spin-zero line of (5) holds under that selection premise. (iii) Hence, within these two relativistic models and with the stated selection criteria (a distinguished extension for Dirac, the larger-exponent branch for Klein–Gordon), the theories of Kepler’s force at finite \(c\) that define the motion without an added rule are those with \[h\ >\ h_{\min}=\frac{k}{\alpha_c\,c},\qquad \alpha_c=1\ (\text{spin }\tfrac12),\quad\alpha_c=\tfrac12\ (\text{spin }0),\tag{5}\] and none with \(0<h<h_{\min}\); the endpoint \(h=h_{\min}\) needs its own extension criterion in the spin-one-half case (\(\nu=1\) is not settled by the cited subcritical results) and its own selection in the spin-zero case, so \(h_{\min}\) is the infimum of the self-sufficient values, attained or not. Together with Theorem 1(a), the set of values of \(h\) at which a theory of Kepler’s force is self-sufficient, in this sense, is an interval with left endpoint \(h_{\min}>0\): Newton’s theory at \(h=0\) is incomplete, no theory with \(h\) below \(h_{\min}\) is self-sufficient, and every theory above it is. This is the gap, in the two models considered; whether it is universal over all admissible relativistic completions requires further model and selection premises that are not supplied here. In the non-relativistic limit \(c\to\infty\), \(h_{\min}\to0\) and the gap closes, in agreement with Theorem 1(b).

Proof. (i) and (ii) are the cited theorems, read at the metadata level and in their standard statements; the thresholds are stated as they are standardly quoted, with the essential self-adjointness bound \(\sqrt3/2\) due to Weidmann, not separately cited here. (iii) is the translation \(\nu<\alpha_c\iff h>k/(\alpha_cc)\) in each model, with the endpoint assigned by the model’s selection criterion, together with Theorem 1(a) at \(h=0\). \(\square\)

The unit \(k/c\) is the one the necessity-unit note isolated as the unique mass-independent action built from \(k_e\) and \(c\), and the relativistic Kepler note found the same threshold classically as \(\ell>k/c\) for regular bound motion. Proposition 2 places it where the thesis needs it: as the lower edge of the set of self-sufficient theories. What it does not do is fix \(h\): any \(h\ge h_{\min}\) is admissible, and the physical value exceeds \(h_{\min}\) by the inverse fine-structure ratio, which is the separate calibration obligation.

6. What the theorem establishes and what it does not

The contrast is between two theories of the same force law. Newton’s, the Laws with \(F=-kx/r^3\), assigns no motion to the collision set, and for the quartic its unrestricted halved classical-first construction has no limit at zero resolution without a datum the action does not supply. The theory with \(h>0\) assigns a motion to every state for all time, by self-adjointness, and recovers Newton’s trajectories as the centroids of localized states, exactly for quadratic forces and, in the fixed-scale limit \(h\to0\) away from collisions, for Kepler’s force under the cited theorem and hypothesis (L). The thesis in STATE reads this as the sense in which Newton’s mechanics needs \(h\): its only completion is the large-action regime of a theory that contains \(h\).

Not established here: the energy-norm localization (L) for the Coulomb evolution; the planar operator beyond the standard form construction; any statement that no completion of Newton’s mechanics without \(h\) exists, which is the open obligation (ii) of STATE, and in particular nothing against the quartic Newtonian initial-value problem, which is globally well posed; the universality of Proposition 2’s gap beyond its two models; and anything about long times or uniformity in \(\delta\), where the Ehrenfest-type breakdown of (4) is a separate matter. The theorem is evidence for the thesis, in the form the thesis now has, and no more.

7. Consequence for STATE

Proposition 2 supplies the gap in its two models: the self-sufficient theories of Kepler’s force at finite \(c\) have \(h>k/(\alpha_cc)\), with Newton at \(h=0\) outside; it rests on cited thresholds, is not shown universal, and fixes no value of \(h\). Theorem 1 supplies the self-sufficiency contrast required by STATE’s obligation (iii): classical incompleteness with the explicit collision time and collision set (proved), quantum completeness by Kato’s theorem (cited, hypothesis verified), the quartic contrast (proved in the 1998-conjecture note), and the classical trajectory as the derived centroid, exact for quadratic forces (proved) and for Kepler’s force under Hepp’s theorem and hypothesis (L) (cited). Next: prove (L) for the Coulomb evolution of a coherent state, or replace Hepp’s \(L^2\) statement by an energy-norm one; then the open obligation (ii), that no completion without \(h\) exists. Proof status: written by Claude Fable, 2026-10-01; parts (a), (c), (d1) proved, (b) and (d2) cited with hypotheses; Theorem 1 refereed ACCEPT (second pass, GPT-6.1 Sol via Codex), Proposition 2 assessed separately with its selection premises recorded and not endorsed; the referee also notes that (4) with (L) gives norm approximation and bounded-observable localization, so the centroid language for Kepler’s force is shorthand pending moment bounds.