Numerical error in a prediction pipeline: certified velocity-Verlet bounds, additive per-step error, the analytic long-time branch, and the separation from model error

The prediction map of an operational observable model (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) is defined from the exact maximal motions of the force model. A numerical pipeline replaces those motions by a computed sequence, and at least five distinct sources of discrepancy must be kept apart. A model-to-system discrepancy can be defined only after an application supplies a reconstructed physical trajectory $\chi :J\to Q$ in the same frame and a metric $d$ on $Q$: for a model motion $x:J\to Q$, one possible pointwise discrepancy is $t\mapsto d(x(t),\chi (t))$. This framework supplies neither $\chi$ nor $d$ and proves no bound on that discrepancy. The discretization error is the difference between a computed sequence and the exact flow of the same model; the implementation error is the per-step residual of the arithmetic actually performed; the observation error is $\epsilon$ of the protocol; and the sampling error is the fluctuation of an estimator such as $\bar{Y}$ around its model mean in Repeated-observation protocols of the force branch: adopted IID and second-moment sampling hypotheses and the sample mean, under its adopted dependence and moment hypotheses. Only the discretization and implementation errors admit the certified statements below, and those statements are always about the model.

Fix a splitting $H=A+B$ of a mechanical Hamiltonian and the velocity-Verlet map $V_h=K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}$ of The splitting Hamiltonian $H=A+B$, the drift and kick maps, and the velocity-Verlet map, and let ${\mathrm{\Phi }}_t$ be the exact flow of the system (The local flow of an autonomous first-order system). Under the hypotheses of Finite-time global error of the velocity-Verlet method: $\parallel z_n-{\mathrm{\Phi }}_{nh}(z_0)\parallel \le C(e^{LT}-1)h^2\mathrm{/}L$ — a bounded convex tube with the local truncation constants $h_1,C_1$ of The velocity-Verlet step matches the exact flow through order two: local truncation error $O(h^3)$ on a convex tube, a compact set $K_1$ inside the tube, a step $0<\mid h\mid \le h_1$ and $n\mid h\mid \le T$, with the computed iterates $z_j=V_h^j(z_0)$ and the exact flow points $y_j={\mathrm{\Phi }}_{jh}(z_0)$ both retained in $K_1$ — the finite-time bound $\parallel z_n-y_n\parallel \le C_1{h}^2\frac{e^{LT}-1}{L}(L>0),\parallel z_n-y_n\parallel \le C_{1}T{h}^2(L=0)$ holds with $L$ the sup-norm bound on the derivative of the Hamiltonian field over the tube. The hypothesis that the iterates remain in the compact set is part of the statement; no first-exit claim is made here.

The implementation error is not analysed as floating-point arithmetic. Instead, fix a metric state set $(X,d)$, for example a subset of phase space with the Euclidean distance, on which every Verlet stage is defined and $V_h:X\to X$. Require a positive step $h>0$, a separately established map bound $d(V_h(x),V_h(y))\le (1+L_{\mathrm{m}\mathrm{a}\mathrm{p}}h)d(x,y)$ for all $x,y\in X$, $L_{\mathrm{m}\mathrm{a}\mathrm{p}}\ge 0$, and computed points $x_j\in X$ with $x_0=z_0$ and $d(x_{j+1},V_h(x_j))\le \eta$. The exact scheme iterates are $z_{j+1}=V_h(z_j)$. Additive per-step errors in a Lipschitzian one-step scheme accumulate geometrically: $\eta ((1+Lh{)}^n-1)\mathrm{/}(Lh)$ then gives $d(x_n,z_n)\le \eta \frac{(1+L_{\mathrm{m}\mathrm{a}\mathrm{p}}h{)}^n-1}{L_{\mathrm{m}\mathrm{a}\mathrm{p}}h}\le \eta e^{L_{\mathrm{m}\mathrm{a}\mathrm{p}}T}\frac{T}{h}(nh\le T,L_{\mathrm{m}\mathrm{a}\mathrm{p}}>0),$ with $d(x_n,z_n)\le n\eta \le \eta T\mathrm{/}h$ if $L_{\mathrm{m}\mathrm{a}\mathrm{p}}=0$. For a readout $F:X\to \mathbb{R}$ with $0\le \mathrm{\Lambda }<\mathrm{\infty }$ and $\mathrm{\mid }F(x)-F(y)\mathrm{\mid }\le \mathrm{\Lambda }d(x,y)$, multiply these bounds by $\mathrm{\Lambda }$. The constant $L_{\mathrm{m}\mathrm{a}\mathrm{p}}$ is a bound for the numerical map, not the exact-field derivative constant $L$ of the preceding section; that flow bound does not establish the map hypothesis. To combine discretization and implementation bounds, take $X$ to be a subset of the declared numerical phase domain and additionally require $y_j\in X$ for $0\le j\le n$. Require either that $d$ is the Euclidean distance used for discretization, or a finite constant $c_d\ge 0$ with $d(u,v)\le c_d\parallel u-v\parallel$ for all $u,v\in X$; in the Euclidean case set $c_d=1$. Let $D_n$ and $I_n$ denote the displayed discretization and implementation upper bounds, respectively, with all their hypotheses and a common positive step satisfied. Then the triangle inequality in $(X,d)$ gives $d(x_n,y_n)\le d(x_n,z_n)+d(z_n,y_n)\le I_n+c_d{D}_n$. For the readout $F$ above, evaluated in these same numerical state coordinates, $\mathrm{\mid }F(x_n)-F(y_n)\mathrm{\mid }\le \mathrm{\Lambda }(I_n+c_d{D}_n)$. A multi-time readout requires a Lipschitz bound in a declared product-state metric and comparison bounds in that metric for the ordered state tuples. A common underlying set alone supplies no comparison between metrics. The number $\eta$ is a hypothesis on the per-step residual, not a measurement of an arithmetic unit, and the lemma says nothing about the distribution of rounding.

For analytic splitting data on a complex neighbourhood, with the domain radius, step-size restriction $0<h\le h_0$ and compact retention of the iterates through the stated time window, there are constants $C,c>0$ with $\mathrm{\mid }{H}_h(z_n)-H_h(z_0)\mathrm{\mid }\le Cnh{e}^{-c\mathrm{/}h},\mathrm{\mid }H(z_n)-H(z_0)\mathrm{\mid }\le C{h}^2\text{when }nh\le e^{c\mathrm{/}(2h)},$ where $H_h$ is the finite optimally truncated modified Hamiltonian of Analytic Verlet retains order-two energy accuracy on exponential times. Adding the implementation error, Bounded per-step state perturbations add a separate linear energy term gives $\mathrm{\mid }H(x_n)-H(x_0)\mathrm{\mid }\le 2C_E{h}^2+Cnh{e}^{-c\mathrm{/}h}+C_{G}n\eta$ for computed points with per-step residual at most $\eta$. Both statements keep every hypothesis of the cited items, including the small-step bound, the holomorphy domain and the retention of iterates in the compact set; the first is a discrete-time energy estimate, not an estimate of the original trajectory on exponentially long times.

The bounds above compare the computed sequence with the exact motion of the same model. They are silent about the difference between the model and a physical system, and they do not certify any apparatus. Exact symplecticity of the Verlet map is a statement about exact arithmetic (Scope of the certified velocity-Verlet estimates: exact arithmetic, compact tubes, roundoff, and absent all-time claims); the additive residual hypothesis is the only interface through which the arithmetic enters, and rounded symplecticity is not claimed. Consequently a numerically computed trajectory, fit or error budget is a computation on the model under the stated hypotheses, not an experiment, not a measurement, and not evidence for the model. The constants $C_1,L,L_{\mathrm{m}\mathrm{a}\mathrm{p}},h_1,C,C_E,C_G,c$ are data of the cited statements; no numerical value of any of them for a physical system is asserted here.