The velocity-Verlet step of the model's mechanical splitting is a canonical map of the presentation: exact flows of the drift and kick, unit determinant, and the oscillator check

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, time window $\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$, a time-independent $C^3$ potential $U$ and no residual forces (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions); by Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian the branch Hamiltonian is $H(x,p)=\mid p{\mid }^2\mathrm{/}(2m)+U(x)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions). Consider the mechanical splitting $H=A+B$ with $A(x,p)=\mid p{\mid }^2\mathrm{/}(2m)$ and $B(x,p)=U(x)$, the drift and kick maps

$D_h(x,p)=(x+h{m}^{-1}p,p),K_h(x,p)=(x,p-h\mathrm{\nabla }U(x)),$

and the velocity-Verlet map $V_h=K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}$, defined wherever the composition exists (The splitting Hamiltonian $H=A+B$, the drift and kick maps, and the velocity-Verlet map with the mass matrix $\mathsf{M}=m{I}_3$ and $V=U$). In this physical presentation $h$ is a time increment with $[h]=\tau$.

For the numerical-error assertion in claim 4, choose reference scales ${\mathrm{\ell }}_{\ast }>0$, $p_{\ast }>0$, ${\tau }_{\ast }>0$ and use dimensionless coordinates $\widehat{x}=x\mathrm{/}{\mathrm{\ell }}_{\ast }$, $\widehat{p}=p\mathrm{/}{p}_{\ast }$, $\widehat{t}=t\mathrm{/}{\tau }_{\ast }$, canonical for the normalized form $\widehat{\omega }:=\omega \mathrm{/}({\mathrm{\ell }}_{\ast }{p}_{\ast })={\sum }_{i}d{\widehat{x}}_i\wedge d{\widehat{p}}_i$, and the normalized Hamiltonian $\widehat{H}={\tau }_{\ast }H\mathrm{/}({\mathrm{\ell }}_{\ast }{p}_{\ast })$, and $\widehat{h}=h\mathrm{/}{\tau }_{\ast }$. The product ${\mathrm{\ell }}_{\ast }{p}_{\ast }$ has action dimension, so $\widehat{H}$ is dimensionless. All error norms and derivative bounds in that claim are those of the dimensionless phase vector $\widehat{z}=(\widehat{x},\widehat{p})$ with its ordinary Euclidean norm. In these coordinates $\widehat{H}=\frac{1}{2}{\widehat{p}}^{\mathsf{T}}{\widehat{M}}^{-1}\widehat{p}+\widehat{U}(\widehat{x})$, where $\widehat{M}=({\mathrm{\ell }}_{\ast }m\mathrm{/}({\tau }_{\ast }{p}_{\ast }))I_3$ is positive and $\widehat{U}(\widehat{x})={\tau }_{\ast }U({\mathrm{\ell }}_{\ast }\widehat{x})\mathrm{/}({\mathrm{\ell }}_{\ast }{p}_{\ast })$ is $C^3$. For a physical horizon $T_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, write $\widehat{T}=T_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\tau }_{\ast }$. Write $\widehat{\mathrm{\Phi }}$ and ${\widehat{V}}_{\widehat{h}}$ for the exact flow and Verlet map of $\widehat{H}$ in these normalized variables. Then:

1. $D_h$ is the exact time-$h$ flow of the Hamiltonian field of $A$, whose components are $X_A=(m^{-1}p,0)$; and $K_h$ is the exact time-$h$ flow of the field of $B$, $X_B=(0,-\mathrm{\nabla }U(x))$, because along the horizontal trajectory $x$ is constant and the momentum moves at the constant rate $-\mathrm{\nabla }U(x)$;
2. consequently $V_h$ is a canonical map of the presentation in exact arithmetic: $D{V}_h^{\mathsf{T}}JD{V}_h=J$ and $V_h^{\ast }\omega =\omega$ wherever $V_h$ is defined and $C^1$ (The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$, Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, A $C^1$ map between phase domains preserves the standard two-form iff $D{C}^{\mathsf{T}}JDC=J$, and a canonical diffeomorphism carries $X_{H\circ C}$ to $X_H$); the separate drift and kick maps are canonical as well, and claim 1 identifies them as exact time-$h$ flows of their respective split-component fields;
3. explicit check on the isotropic oscillator instance $U(x)=\frac{1}{2}k\mid x{\mid }^2$ with $k>0$ and stiffness dimension $[k]=\mu {\tau }^{-2}$: on each coordinate pair,

$V_h(q,p)=(q+\frac{h}{m}p-\frac{h^{2}k}{2m}q,p-hkq-\frac{h^{2}k}{2m}p+\frac{h^3{k}^2}{4m}q),D{V}_h=\left(\begin{array}{cc}1-\frac{h^{2}k}{2m}& \frac{h}{m}\\ -hk+\frac{h^3{k}^2}{4m}& 1-\frac{h^{2}k}{2m}\end{array}\right),$

and $\det D{V}_h=(1-\frac{h^{2}k}{2m}{)}^2-\frac{h}{m}(-hk+\frac{h^3{k}^2}{4m})=1$; in dimension two, $\det D{V}_h=1$ is equivalent to canonicity (Determinant tests on explicit linear maps: $\det A=1$ is equivalent to canonicity in dimension two, and strictly weaker in dimension four (the volume-preserving shear)). In grouped coordinates $(x,p)$ the full derivative has the corresponding $3\times 3$ scalar blocks; under the permutation to interleaved order it is $\mathrm{diag}(D{V}_h,D{V}_h,D{V}_h)$ and the canonical matrix is $\mathrm{diag}(J_2,J_2,J_2)$. Thus the full matrix identity and determinant-one claim hold on the six-dimensional instance; 4. accuracy and scope. Assume that in the normalized coordinates there are an open bounded convex tube $\widehat{W}$ in the phase domain, a nonempty compact set $\widehat{K}\subseteq \widehat{W}$ and $r>0$ with $\widehat{K}+{\bar{B}}_r(0)\subseteq \widehat{W}$, and finite bounds $M_0={\sup }_{\widehat{W}}\mathrm{\parallel }{X}_{\widehat{H}}{\mathrm{\parallel }}_2$, $L_1={\sup }_{\widehat{W}}\mathrm{\parallel }D{X}_{\widehat{H}}{\mathrm{\parallel }}_2$ and $L_2={\sup }_{\widehat{W}}\mathrm{\parallel }{D}^2{X}_{\widehat{H}}{\mathrm{\parallel }}_2$. The local-error lemma then supplies constants ${\widehat{h}}_1>0$ and $C_1<\mathrm{\infty }$. For $\widehat{T}>0$, $0<\mathrm{\mid }\widehat{h}\mathrm{\mid }\le {\widehat{h}}_1$, and $n\in \mathbb{N}$ with $n\mathrm{\mid }\widehat{h}\mathrm{\mid }\le \widehat{T}$, suppose both exact points ${\widehat{y}}_j={\widehat{\mathrm{\Phi }}}_{j\widehat{h}}({\widehat{z}}_0)$ and numerical points ${\widehat{z}}_j={\widehat{V}}_{\widehat{h}}^j({\widehat{z}}_0)$ remain in $\widehat{K}$ for every $0\le j\le n$. Then the suppliers give $\mathrm{\parallel }{\widehat{V}}_{\widehat{h}}(\widehat{z})-{\widehat{\mathrm{\Phi }}}_{\widehat{h}}(\widehat{z}){\mathrm{\parallel }}_2\le C_1\mathrm{\mid }\widehat{h}{\mathrm{\mid }}^3$ for $\widehat{z}\in \widehat{K}$, and $\mathrm{\parallel }{\widehat{z}}_n-{\widehat{y}}_n{\mathrm{\parallel }}_2\le C_1{\widehat{h}}^2(e^{\widehat{L}\widehat{T}}-1)\mathrm{/}\widehat{L}$ when $\widehat{L}=L_1>0$, or $C_1\widehat{T}{\widehat{h}}^2$ when $\widehat{L}=0$ (The velocity-Verlet step matches the exact flow through order two: local truncation error $O(h^3)$ on a convex tube, Finite-time global error of the velocity-Verlet method: $\parallel z_n-{\mathrm{\Phi }}_{nh}(z_0)\parallel \le C(e^{LT}-1)h^2\mathrm{/}L$). The constants depend on the declared normalized tube bounds; the retention and small-step hypotheses are assumptions. Roundoff, all-time energy claims and the holomorphic-domain hypotheses of the analytic estimates are outside this example and are governed by Scope of the certified velocity-Verlet estimates: exact arithmetic, compact tubes, roundoff, and absent all-time claims.

All statements are about exact arithmetic on the declared idealization; no measurement or floating-point claim is made.