From multiscale quantum mechanics to Newton’s initial-value problem

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Result, 2026-10-27. For a smooth nonnegative potential on the line with bounded second and higher derivatives, we construct Euclidean \(0+1\) quantum mechanics at every supplied \(\hbar>0\) using time refinement and a normalized density at every dyadic scale. The reference fluctuations are Gaussian Brownian bridges, and the full interaction and large-field configurations are retained. Real-time evolution is the unitary group of the reconstructed self-adjoint Hamiltonian.

For every initial \((q_0,p_0)\), means of finite quantum time-slice products are a Cauchy family as both \(\hbar\) and the time mesh tend to zero. An explicit \(O(\hbar+|\pi|)\) estimate controls both scales simultaneously for the means, while the packet widths remain \(O(\sqrt \hbar)\). Their limit solves \(\dot q=p/m\), \(\dot p=-V'(q)\); it is the unique solution. An explicit bound compares any two positive values of \(\hbar\). The proof does not assume a classical trajectory in advance. Its stability hypothesis and Gronwall estimate are the same ingredients used by Cauchy–Lipschitz, so the construction is a different route, not a weakening of that theorem. For smooth confining potentials with locally Lipschitz force, including the positive quartic, a cutoff and energy conservation construct the classical trajectory from quantum families with cutoff potentials. This argument does not prove the joint approximation estimate for the uncut quartic quantum evolution.

The theorem concerns the displayed smooth scalar-potential class. Velocity-dependent forces, hard walls and singular/cusp selection are separate constructions, with their assumptions retained in the catalog.

This treatment assembles of standard Feynman–Kac, conditional expectation and unitary-conjugation ingredients; no priority claim is made. The normalized-density architecture is inspired by Dimock–Balaban, not an import of their four-dimensional Yang–Mills estimates. The construction assumes a positive action parameter \(\hbar\), and does not prove it necessary. A separate tangent-groupoid formulation follows only after this theorem.

Targeted advisor check (GPT-6.1 Sol, 2026-10-27). The finite-step identities, constants, joint Cauchy estimate, domain argument, quartic energy barrier and cusp counterexample were checked in writing. This is not an independent proof audit of the imported Wiener, Feynman–Kac, self-adjointness or spectral theorems. The centered \(O(\hbar)\) improvement below follows a subsequent user-suggested calculation and is not covered by that earlier check.

1. Hypotheses and the quantum theory to be constructed

The supplied action parameter is written \(\hbar\) throughout. It is distinct from Planck’s \(h_P=2\pi\hbar\), the time mesh and the dimensionless \(\epsilon=\hbar/S_0\) for a specified characteristic action \(S_0\).

Let \(m>0\) and let \(V:\mathbb R\to\mathbb R\) be \(C^\infty\), \(V\ge0\), with all derivatives of order at least two bounded. Set \[L=\sup_x|V''(x)|<\infty.\tag{1}\] Thus \(V'\) is globally Lipschitz with constant \(L\) and \(V\) grows at most quadratically. A nonlinear confining example is \[V(x)=\tfrac12m\omega_0^2x^2+\lambda[1-\cos(x/\ell)], \qquad\omega_0,\lambda,\ell>0.\tag{2}\] Its curvature is bounded, while the force is not linear.

For each \(\hbar>0\), set \[H_\hbar=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\tag{3}\] on \(L^2(\mathbb R)\). Its Friedrichs realization is nonnegative and self-adjoint; for this smooth bounded-below class the restriction to \(C_c^\infty(\mathbb R)\) is a core. Put \[U_\hbar(t)=e^{-itH_\hbar/\hbar}.\tag{4}\] The spectral theorem makes \(U_\hbar\) a strongly continuous unitary group. No classical ODE solution is used to define it. If \(V(x)\to\infty\) at infinity, compactness of the form embedding and positivity improvement additionally give the unique vacuum and positive spectral gap as in the preceding construction. They are not required for the initial-value theorem below.

Fix endpoints \(x,y\), duration \(T>0\), and a partition \(\pi=(0=t_0<\cdots<t_N=T)\). Write \(\delta_j=t_{j+1}-t_j\) and \[ S_\pi^E(q)=\sum_j \left\{\frac{m(q_{j+1}-q_j)^2}{2\delta_j} +\frac{\delta_j}{2}[V(q_j)+V(q_{j+1})]\right\}. \tag{5} \] The normalized finite-lattice measure has endpoints \(q_0=x,q_N=y\) and density proportional to \(e^{-S_\pi^E/\hbar}\) in the internal coordinates.

Let \(B\) be the Brownian bridge from \(x\) to \(y\) with variance parameter \(\hbar/m\) and mean \(b(t)=x+(y-x)t/T\). Its covariance is \[\operatorname{Cov}[B(s),B(t)] =\frac hm\left(\min(s,t)-\frac{st}{T}\right).\tag{6}\] With \(p_h(t;u,v)=\sqrt{m/(2\pi ht)} \exp[-m(v-u)^2/(2ht)]\), the normalized kinetic factor \(\prod_jp_\hbar(\delta_j;q_j,q_{j+1})/p_\hbar(T;x,y)\) is exactly the joint density of \(B(t_j)\). Define \[W_\pi(B)=\sum_j\frac{\delta_j}{2} [V(B(t_j))+V(B(t_{j+1}))],\qquad Z_\pi=\mathbb E_0e^{-W_\pi(B)/\hbar}.\tag{7}\] The interpolated lattice measure is the pushforward of \(Z_\pi^{-1}e^{-W_\pi/\hbar}d\mathbb P_0\) by polygonal interpolation.

Brownian continuity gives uniform convergence of this interpolation to \(B\), and \(W_\pi\to\int_0^T V(B(t))dt\) almost surely, for every sequence of partitions with \(|\pi|\to0\). The weights lie in \([0,1]\). Consequently dominated convergence constructs the limit \[d\nu_\hbar=Z_\hbar^{-1} e^{-\hbar^{-1}\int_0^T V(B(t))dt}d\mathbb P_0,\qquad Z_\hbar=\mathbb E_0e^{-\hbar^{-1}\int_0^T V(B(t))dt}>0.\tag{8}\] It proves weak convergence of the polygonal measures on \(C([0,T])\) and \(L^1(\mathbb P_0)\) convergence of their lifted densities. The polygonal measures themselves are not asserted to converge in total variation to the non-polygonal path measure.

The normalization has an explicit finite-partition lower bound. Since \(V''\le L\), Taylor’s inequality gives \(\mathbb EV(B(t))\le V(b(t))+\frac L2\operatorname{Var}B(t)\). Writing \(M_{xy}=\max_{z\in[x,y]}V(z)\), using \(\operatorname{Var}B(t)\le hT/(4m)\) and Jensen’s inequality yields \[Z_\pi\ge \exp\!\left[-\frac{TM_{xy}}\hbar-\frac{LT^2}{8m}\right] \quad\text{for every partition.}\tag{9}\] Thus normalization and large-field paths are included at every mesh. The bound is finite for each \(\hbar>0\) but is not uniform as \(\hbar\to0\).

Projective Gaussian preparation and raw kinetic action

The free reference has a useful distinction between consistency and scale selection. More generally give a partition’s centred bridge nodes covariance \(\kappa_\pi G(s,t)/m\), where \(G(s,t)=\min(s,t)-st/T\) and \(\kappa_\pi\ge0\). A refinement retaining one interior time \(s\) has the same marginal only if \(\kappa_{\pi'}s(T-s)/(mT)=\kappa_\pi s(T-s)/(mT)\), hence the parameters agree. Any two nontrivial partitions have a common refinement; conversely fixed \(\kappa\) gives exactly the required Gaussian covariance submatrices. The endpoint-only partition fixes no scale. The quantum Euclidean reference above chooses \(\kappa=\hbar\); a classical zero-width reference is also projectively consistent.

For \(d=N-1\) internal nodes, the free polygon’s excess kinetic action over the endpoint chord obeys \(2D_\pi/\kappa\sim\chi_d^2\) at \(\kappa>0\). This follows by whitening its positive Dirichlet quadratic form under the normalized Gaussian density. Therefore

\[\mathbb ED_\pi=d\kappa/2,\qquad \operatorname{Var}D_\pi=d\kappa^2/2,\qquad \Pr(D_\pi<d\kappa/4)\le8/d.\]

The unnormalized action diverges in probability as nodes are added, whereas \(\widehat\kappa=2D_\pi/d\) has mean \(\kappa\) and variance \(2\kappa^2/d\). Its mean-square convergence estimates the supplied parameter; it is not an additive surviving action. A triangular family with \((N-1)\kappa_N\to\ell>0\) changes retained interior variances toward zero and is not restriction of one fixed-width experiment.

At a new cut after duration \(a\) in a cell of duration \(a+b\), the endpoint interpolation is \(\bar y=(bx+az)/(a+b)\) and the exact splitting identity is

\[\frac m2\left[\frac{(y-x)^2}a+\frac{(z-y)^2}b -\frac{(z-x)^2}{a+b}\right] =\frac{m(a+b)}{2ab}(y-\bar y)^2.\]

The fixed-\(\kappa\) conditional variance is \(\kappa ab/[m(a+b)]\), so the expected added kinetic cost is \(\kappa/2\) at every split; it is zero for deterministic interpolation. This nonnegative kinetic addition is a different comparison from moving a sampled point onto a forced classical curve, whose action includes potential and whose chord defect decreases under refinement. Source coverage: Pitman–Yor 2018, printed pp. 6, 10, 12–13, read for the bridge/restriction framework in B12; the independently retained regulator manuscript gives the finite quadratic-form law. No hard-speed law or positivity selection follows.

3. The exact all-scale retained density

For the dyadic partition let \(\mathcal F_n\) be the sigma-algebra of the \(2^n+1\) vertex values, and let \[\rho=\frac{d\nu_\hbar}{d\mathbb P_0},\qquad \rho_n=\mathbb E_0[\rho\mid\mathcal F_n].\tag{10}\] These are densities relative to the normalized free bridge. Conditional expectation gives, exactly, \[\rho_n=\mathbb E_0[\rho_{n+1}\mid\mathcal F_n],\quad \mathbb E_0\rho_n=1,\quad 0<\rho_n\le Z_\hbar^{-1}.\tag{11}\] At level \(n+1\), conditional on level \(n\), the reference midpoint of a cell of length \(\delta=T/2^n\) has mean \((a+b)/2\) and variance \(\hbar\delta/(4m)\); reference midpoints in different cells are independent. The interacting conditional law is the product of these Gaussian reference laws multiplied by \(\rho_{n+1}/\rho_n\). The latter density must be retained; the interacting midpoint variables are not assumed independent.

Dyadic evaluations generate the Borel sigma-algebra of continuous path space. Hence the bounded martingale theorem gives \[\rho_n\longrightarrow\rho\quad\text{in }L^1(\mathbb P_0) \text{ and almost surely.}\tag{12}\] Equations (11)–(12) are an exact normalized blocking tower at all scales. They use an already constructed Wiener measure as reference; they do not claim a lattice-only construction or a four-dimensional polymer estimate. In particular the \(Z_\hbar^{-1}\) bound does not give control uniform in \(\hbar\downarrow0\).

The distinction between a finite lattice step and the complete effective density can be seen explicitly. Before any blocking, integrate the middle vertex \(z\) of two cells of length \(s\), with retained endpoints \(a,b\) and \(\mu=(a+b)/2\). Its conditional potential is \[F_\mu(z)=\frac m s(z-\mu)^2+sV(z).\tag{13}\] For \(2m/s-sL>0\), put \(\kappa_s=2m/s-sL\). Then \(F_\mu''\ge\kappa_s\); there is a unique minimizer \(z_*\) and the normalized conditional law satisfies \[\mathbb P(|z-z_*|\ge r\mid a,b) \le 2\exp[-\kappa_s r^2/(2\hbar)].\tag{14}\] Indeed, integration by parts gives \(\mathbb E(z-z_*)^{2n}\le(2n-1)!!(\hbar/\kappa_s)^n\); expand the hyperbolic cosine and optimize the exponential Markov bound. Conditioned on all retained vertices, the eliminated vertices are independent for this original nearest-neighbour lattice. After a blocking step the generated action must be kept: (13)–(14) cannot be iterated as though the original potential were unchanged.

This is the precise \(0+1\) version of the source lesson: retain the entire normalized effective density, not merely its background minimizer or an unnormalized large-field action cost. Compare Dimock I, sections 2.1, 2.4 and 4.5, and Dimock II, introduction and sections 3.13–3.14 (selected passages read there). The latter papers treat a three-dimensional scalar theory, not (3).

4. From the Euclidean limit to real-time quantum mechanics

The Feynman–Kac theorem identifies the kernel of \(e^{-TH_\hbar/\hbar}\) as \(K_\hbar(T;x,y)=p_\hbar(T;x,y)Z_\hbar(x,y)\); (8) is the corresponding normalized bridge, not the kernel itself. It also makes the semigroup reflection positive in the ground-state construction when a normalizable ground state exists. Real time is defined by the self-adjoint operator through (4), not by treating the Euclidean probability law as a real-time probability law.

For finite real-time steps use the kick–drift product \[T_{\hbar,s}=D_{\hbar,s}K_{\hbar,s},\quad K_{\hbar,s}=e^{-isV(Q)/\hbar},\quad D_{\hbar,s}=e^{-isP^2/(2mh)}, \quad P=-ih\,\partial_x.\tag{15}\] The factors are unitary. Its products converge strongly to \(U_\hbar(T)\) under arbitrary positive partitions at each fixed \(\hbar>0\). Here is the needed arbitrary-partition argument.

Write \(c_\hbar=\hbar^2/(2m)\), and take \(f\in C_c^\infty\). Integration by parts gives \[ \|H_hf\|^2=c_\hbar^2\|f''\|^2+\|Vf\|^2 +2c_\hbar\int V|f'|^2-c_\hbar\int V''|f|^2 \ge c_\hbar^2\|f''\|^2+\|Vf\|^2-c_hL\|f\|^2.\tag{15a} \] Since \(C_c^\infty\) is an operator core, closure identifies \(D(H_\hbar)=D(P^2)\cap D(V)\) and controls the two component graph norms. For \(f\) in this domain, \((T_{\hbar,s}-I)f/s\to-iH_hf/\hbar\) strongly, and these difference quotients are uniformly bounded from the \(H_\hbar\) graph domain to \(L^2\). The orbit \(\{U_\hbar(t)f:0\le t\le T\}\) is graph-norm compact. Thus the local first-order error is uniform on the orbit, and a telescoping sum over the cells proves the limit; density and unitarity extend it to all \(L^2\) vectors. This is the argument used for the symmetric product in Rivero 1998 test, Theorem 2(b), now written for kick–drift. It is a strong operator statement, not a finite complex measure on path space.

The Euclidean stationary equation is \(m\ddot q=+V'(q)\), whereas real-time Newton motion satisfies \(m\ddot q=-V'(q)\). Euclidean concentration cannot be used as a Newton trajectory by silently changing that sign.

5. A joint quantum and time-mesh construction of Newton motion

Fix \(\omega>0\), initial \((q_0,p_0)\), and the coherent packet \[ \psi_\hbar(x)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\!\left[-\frac{m\omega(x-q_0)^2}{2\hbar} +\frac{i p_0(x-q_0)}\hbar\right].\tag{16} \] Its position and momentum means are \(q_0,p_0\), with standard deviations \(\sqrt{\hbar/(2m\omega)}\) and \(\sqrt{hm\omega/2}\).

For a partition \(\pi\), apply the unitary kick and drift (15) successively. Let \(q_j,p_j\) be the resulting means at \(t_j\), and put \[W_j=\operatorname{sd}(Q)_j+ \frac{\operatorname{sd}(P)_j}{m\omega},\quad C=\sqrt{\frac2{m\omega}},\quad \beta=\omega+\frac{L}{m\omega},\quad a=\max\!\left(\omega,\frac L{m\omega}\right).\tag{17}\] No Gaussian assumption is made about the state after the first step.

The exact conjugation identities are \[K_{\hbar,s}^*QK_{\hbar,s}=Q,\quad K_{\hbar,s}^*PK_{\hbar,s}=P-sV'(Q),\quad D_{\hbar,s}^*QD_{\hbar,s}=Q+sP/m,\quad D_{\hbar,s}^*PD_{\hbar,s}=P.\tag{18}\] Both finite factors preserve Schwartz space under the stated derivative bounds, so (18) is valid on each propagated Gaussian. For the kick, the triangle inequality in \(L^2\) and \(\operatorname{sd}[V'(Q)]\le L\,\operatorname{sd}(Q)\) give \(\operatorname{sd}(P)_{\rm after}\le \operatorname{sd}(P)_{\rm before}+sL\,\operatorname{sd}(Q)\). For the drift the analogous bound adds \(s\,\operatorname{sd}(P)/m\) to the position deviation. Therefore \[W_{j+1}\le (1+s\omega)\left(1+\frac{sL}{m\omega}\right)W_j, \qquad W_j\le C\sqrt \hbar\,e^{\beta T}.\tag{19}\] These are finite-step statements for the full quantum state.

The means obey \[p_{j+1}=p_j-s_j\langle V'(Q)\rangle_j,\qquad q_{j+1}=q_j+\frac{s_jp_{j+1}}m.\tag{20}\] Fix \(\hbar_0>0\), restrict \(0<\hbar\le \hbar_0\), and use the phase-space norm \(\|(q,p)\|_\omega=|q|+|p|/(m\omega)\). Define \[\begin{gathered} b=\frac{|V'(0)|+LC\sqrt{\hbar_0}e^{\beta T}}{m\omega},\\ R_T=e^{\beta T}\big(\|(q_0,p_0)\|_\omega+Tb\big),\qquad F_T=|V'(0)|+L(R_T+C\sqrt{\hbar_0}e^{\beta T}),\\ C_\delta=\frac{F_T+LR_T}{m},\qquad C_\hbar=\frac{\|V'''\|_\infty C^2e^{2\beta T}}{2m\omega},\qquad G_T=\frac{e^{aT}-1}{a}. \tag{21} \end{gathered}\] Equation (19) and (20) imply recursively that every vertex mean has \(\|(q_j,p_j)\|_\omega\le R_T\) and \(|\langle V'(Q)\rangle_j|\le F_T\), independently of \(\hbar\) and \(\pi\). For example, the recurrence for the norm is bounded by \(R_{j+1}\le e^{\beta s_j}R_j+s_jbe^{\omega s_j}\).

Linearly interpolate the mean vertices, writing \(z_{\hbar,\pi}(t)=(q_{\hbar,\pi}(t),p_{\hbar,\pi}(t))\). Put \(\mathcal F(q,p)=(p/m,-V'(q))\). On every cell the two components of the residual are bounded by \[ |\dot q_{\hbar,\pi}-p_{\hbar,\pi}/m| \le |\pi|F_T/m,\quad |\dot p_{\hbar,\pi}+V'(q_{\hbar,\pi})| \le \tfrac12\|V'''\|_\infty C^2\hbar\,e^{2\beta T} +L|\pi|\,\omega R_T.\tag{22} \] Consequently \[\|\dot z_{\hbar,\pi}-\mathcal F(z_{\hbar,\pi})\|_\omega \le C_\delta|\pi|+C_\hbar \hbar.\tag{23}\] The first residual is the difference between the interpolated momentum and its post-kick value; the second is the sum of the packet’s force bias and the motion of the interpolated position. For the bias, Taylor expand \(V'\) around the packet mean \(q_j=\langle Q\rangle_j\). Its linear term has zero expectation, so \[ |\langle V'(Q)\rangle_j-V'(q_j)| \le\tfrac12\|V'''\|_\infty\operatorname{Var}_j(Q) \le\tfrac12\|V'''\|_\infty C^2\hbar\,e^{2\beta T}.\tag{23a} \] The bounded third derivative is part of the standing hypotheses. For quadratic potentials \(V'''=0\) and the force-bias term vanishes exactly. This improves the mean error; (19) still gives widths of order \(\sqrt \hbar\).

The vector field \(\mathcal F\) has Lipschitz constant at most \(a\) in this norm. Integrating the residuals of two products with common initial data, and applying Gronwall, gives \[ \sup_{t\le T}\|z_{\hbar,\pi}(t)-z_{k,\sigma}(t)\|_\omega \le G_T\!\left[ C_\delta(|\pi|+|\sigma|) +C_\hbar(\hbar+k)\right].\tag{24} \] Thus the means are Cauchy in \(C([0,T],\mathbb R^2)\) as the two scales \(\hbar\) and \(|\pi|\) tend to zero in any joint order. Let \(z=(q,p)\) be their limit. Equation (23) passes to the integral equation \[q(t)=q_0+\int_0^t\frac{p(s)}m\,ds,\qquad p(t)=p_0-\int_0^tV'(q(s))\,ds.\tag{25}\] It is continuously differentiable and solves Newton’s initial-value problem. Uniqueness follows from the same Gronwall inequality with zero residual. The construction extends to negative time using inverse unitary steps, or by replacing \(p\) by \(-p\) and reversing time.

Letting \((k,|\sigma|)\to(0,0)\) in (24) yields the explicit joint bound \[ \sup_{t\le T}\|z_{\hbar,\pi}(t)-z(t)\|_\omega \le G_T\big(C_\delta|\pi|+C_\hbar \hbar\big).\tag{26} \] The constants are independent of the number of cells and of \(\hbar\) in \((0,\hbar_0]\). Equation (26) is an observable-level statement about the packet means; it is not an operator-norm estimate on real-time propagators and does not assert a real-time positive path measure.

Finally fix \(\hbar>0\). At a fixed \(t\), choose partitions containing \(t\). Section 4 gives strong \(L^2\) convergence of the product state to \(U_\hbar(t)\psi_\hbar\). Equations (19)–(21) bound its position and momentum second moments uniformly. Strong \(L^2\) convergence gives convergence of bounded position and momentum cutoffs, and the uniform second moments make the first-moment tails at most \(O(R^{-1})\). The first moments therefore converge to those of \(U_\hbar(t)\psi_\hbar\). For a general refining partition, use the product truncated at the last vertex before \(t\); the same graph-norm telescoping argument is uniform on compact time intervals. The bounds on vertex momenta and forces make the interpolated means equi-Lipschitz. Pointwise identification therefore upgrades to uniform convergence on \([0,T]\). Passing to the limit in (26) gives \[ \sup_{t\le T} \left\|(\langle Q\rangle_t,\langle P\rangle_t)-z(t)\right\|_\omega \le G_T C_\hbar \hbar.\tag{27} \] This uses no differentiation of an unbounded quantum observable. It also proves the claimed consistency of the two limiting orders for these means. The construction assumes a semiclassical family \(\hbar\downarrow0\); a single fixed-positive-\(\hbar\) theory does not have simultaneously sharp position and momentum.

The Cauchy estimate and its domain argument are the present assembly. The broader relation between coherent quantum packets and classical orbits is established prior work, including Hepp, 1974 (introduction and theorem statements). For the time-splitting aspect, Borns–Weil and Fang, Theorem 2 and §3.2 (relevant passages read), establish semiclassical uniform observable error bounds for smooth symbol classes; their operator comparison and the present coherent-state estimate have different hypotheses and conclusions. No claim of literature novelty is made for the semiclassical mechanism or uniform observable accuracy as such.

6. Quartic extension and the limits of the implication

Let \(V(q)=\lambda q^4/4\) with \(\lambda>0\). Its Hessian is unbounded, so (26) does not apply globally with a single \(L\). For fixed initial data choose a smooth nonnegative \(V_R\) with bounded derivatives of order at least two, agreeing with \(V\) on \([-R,R]\). Apply the construction to \(V_R\). As long as its trajectory is inside that interval, \[E=\frac{p^2}{2m}+\frac{\lambda q^4}{4} =\frac{p_0^2}{2m}+\frac{\lambda q_0^4}{4}\tag{28}\] by the chain rule. Choosing \(R>(4E/\lambda)^{1/4}\) rules out a first exit, so the constructed trajectory solves the original quartic equation for all finite times. Local Lipschitz uniqueness identifies different choices of cutoff. The same argument applies to smooth confining potentials whose derivative is locally Lipschitz, with an appropriate compact energy sublevel. The cutoff quantum means are not claimed here to be those of the uncut quartic Hamiltonian.

Quantum existence does not alone imply classical uniqueness. The bounded \(C^1\) potential in the cusp note, with \(V(q)=-\frac23|q|^{3/2}\) near zero, has a bounded-perturbation self-adjoint quantum Hamiltonian; adding a constant makes \(V\ge0\) without changing the force. Yet, while remaining in that neighbourhood, Newton’s equation with \(m=1\) admits both rest and \[q(t)=\begin{cases} 0,&t\le t_0,\\ (t-t_0)^4/144,&t\ge t_0, \end{cases}\tag{29}\] for any waiting time \(t_0\ge0\). Thus extra force regularity, or some independent selection rule, is necessary for the uniqueness conclusion.

7. Comparison with Cauchy–Lipschitz

The direct theorem works with the integral operator on phase-space paths; Lipschitz continuity gives a contraction locally, then global continuation under the global hypothesis. The earlier polygon lesson gives a comparable refinement/Cauchy proof.

The route above first constructs a quantum family and then proves that its means form a Cauchy family as \(\hbar\downarrow0\). Its quantitative inputs are normalized multiscale Euclidean control, self-adjoint real-time evolution, a packet variance estimate, and the same Lipschitz/Gronwall stability that makes Cauchy–Lipschitz work. It gives the explicit approximation (26) and places the classical path beside quantum dynamics, but it invokes substantially more machinery and does not establish an ODE theorem beyond Cauchy–Lipschitz.

8. Consequence for STATE

The user-directed first document gives a complete scoped existence and classical-limit construction. The subsequent tangent-groupoid formulation places its observables and dynamics in a continuous field, keeping the deformation parameter, time mesh and coarse-graining distinct while relating their observable-state limits along chosen joint schedules.