Measurement, uncertainty and later bridges

Fix a force model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: a nondegenerate open time window $I_0\subseteq \mathbb{R}$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ for $N\ge 1$ particle labels, with the collision set removed when the model's force law is singular there (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator clause (a)), fixed masses $m_1,\dots ,m_N>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), and a force map $F:I_0\times Q\times ({\mathbb{R}}^3{)}^N⟶({\mathbb{R}}^3{)}^N$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate. Fix one inertial frame of the model with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames) and read all times and positions below in this frame; dimensions and units are those of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

Protocol. An operational observable model of this force model in the frame consists of the following data.

1. Parameters. A nonempty set $\mathrm{\Theta }$, the model parameters, with a family $(F_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ of force maps on $I_0\times Q\times ({\mathbb{R}}^3{)}^N$, each satisfying the standing hypotheses of the force branch, so that $(I_0,Q,(m_i),F_{\theta })$ is a force model of that branch for every $\theta \in \mathrm{\Theta }$.
2. Preparations. A nonempty set $Z$, the apparatus and preparation settings, and a map $\zeta :Z\to I_0\times Q\times ({\mathbb{R}}^3{)}^N$, written $\zeta (z)=(t_0(z),x_0(z),v_0(z))$, assigning to each setting an initial datum of the model.
3. Observation times. A finite strictly increasing family $t_1<\cdots <t_k$ in $I_0$ with $k\ge 1$, the observation times.
4. Definedness. For every $(\theta ,z)\in \mathrm{\Theta }\times Z$ the model with force map $F_{\theta }$ and initial datum $\zeta (z)$ has a unique maximal motion $x_{\theta ,z}:J_{\theta ,z}\to Q$, with $J_{\theta ,z}\subseteq I_0$ an open interval containing $t_0(z)$ (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder), and the protocol requires $[t_1,t_k]\subseteq J_{\theta ,z}$. Its state at $t_j$ is $s_{\theta ,z}(t_j):=(x_{\theta ,z}(t_j),w_{\theta ,z}(t_j))\in Q\times ({\mathbb{R}}^3{)}^N,$ where $w_{\theta ,z}:=x_{\theta ,z}^{\mathrm{\prime }}$ is the coordinate velocity.
5. Readout. An integer $d\ge 1$ and a map $R:\mathrm{\Theta }\times Z\times ({\mathbb{R}}^{3N}\times {\mathbb{R}}^{3N}{)}^k⟶{\mathbb{R}}^d,$ the readout, which takes $(\theta ,z)$ and an ordered $k$-tuple of states as inputs and returns the $d$ numerical readout coordinates; the prediction map is $\mathrm{\Phi }:\mathrm{\Theta }\times Z⟶{\mathbb{R}}^d,\mathrm{\Phi }(\theta ,z):=R(\theta ,z,s_{\theta ,z}(t_1),\dots ,s_{\theta ,z}(t_k))\mathrm{.}$ No regularity of $R$ in the parameters is assumed; continuity, differentiability or measurability in $\theta$ or in the states is an additional hypothesis, declared where it is used.
6. Units. For each $j\le d$ a dimension $[{\mathrm{\Phi }}_j]$ from Dimensions and SI unit conventions of the non-relativistic classical mechanics framework is declared, and the coordinate ${\mathrm{\Phi }}_j(\theta ,z)$ is read in the corresponding unit.
7. Error and observation. A probability space $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ (Probability measures and probability spaces), an open error domain $V\subseteq {\mathbb{R}}^d$ with $0\in V$, and a random element $\epsilon :\mathrm{\Omega }\to {\mathbb{R}}^d$ with $\epsilon (\omega )\in V$ for every $\omega$ (Random elements and real random variables). The observation at $(\theta ,z)$ is the random element $Y_{\theta ,z}:=\mathrm{\Phi }(\theta ,z)+\epsilon :\mathrm{\Omega }⟶{\mathbb{R}}^d,$ the sum being read coefficientwise in the declared units, so that the coordinate ${\epsilon }_j$ carries the dimension $[{\mathrm{\Phi }}_j]$ for every $j$ (rule (b) of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

Well-definedness and conventions

Well-definedness. The prediction map is well defined: for each $(\theta ,z)$ the maximal motion is unique by Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder, and clause 4 puts the observation times in its interval. For fixed $(\theta ,z)$ the observation is a random element in the sense of Random elements and real random variables: the map $\tau (u):=\mathrm{\Phi }(\theta ,z)+u$ is a translate of ${\mathbb{R}}^d$, hence continuous, its preimages of Borel sets are Borel, and $Y_{\theta ,z}^{-1}(B)={\epsilon }^{-1}({\tau }^{-1}(B))\in \mathfrak{F}$ by the measurability of $\epsilon$. This repeats the argument of Measurement models and additive-error observation models for the additive-error observation model with $h=\mathrm{\Phi }$; the mathematical interface used below is exactly that definition.

No hypothesis on the error is part of the definition. Mean zero, square-integrability, boundedness, a stated law, independence across repetitions, stationarity and calibration are not part of the definition; each must be stated at its point of use, with its exact quantifiers, exactly as in Measurement models and additive-error observation models. In particular a deterministic error, for which $\epsilon$ is a constant random element, is admitted; the definition then records a modelled residual rather than a probabilistic error.

Frame relativity of the protocol. The observation times are times of the declared frame, and the prediction map is read in that frame's coordinates; a change of inertial frame re-expresses the coordinate times and states, and the readout must be transformed according to its declared law for the same physical protocol. Its numerical value need not change: the admitted constant readout $R\equiv 0$, for example, has $\mathrm{\Phi }\equiv 0$ in every frame. The framework therefore always names the frame in which a protocol is specified; no frame-independent reading of $\mathrm{\Phi }$ or $\epsilon$ is asserted.

Scope. The definition constructs an interface, not a measurement. It does not assert that any apparatus realizes the protocol, that any preparation can be reproduced, that the declared error domain or law is correct, or that $\mathrm{\Phi }$ agrees with any physical system. A statement that a physical procedure satisfies the protocol requires separate empirical evidence, and no such evidence is part of this item.

Fix the force branch and one inertial frame, and fix an operational observable protocol as in Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error, with prediction map $\mathrm{\Phi }:\mathrm{\Theta }\times Z\to {\mathbb{R}}^d$, nonempty $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$, $p\ge 1$, and its declared observation times, readout and numerical coordinates. Specify a nonempty set $C$ of apparatus settings and a map $\sigma :C\to Z$ selecting a preparation setting $z_c:=\sigma (c)$ for every $c\in C$. A calibration family is the family of preparation slices $({\mathrm{\Phi }}_c{)}_{c\in C},{\mathrm{\Phi }}_c:\mathrm{\Theta }⟶{\mathbb{R}}^d,{\mathrm{\Phi }}_c(\theta ):=\mathrm{\Phi }(\theta ,z_c)\mathrm{.}$ Thus the initial datum used in ${\mathrm{\Phi }}_c$ is $\zeta (z_c)$, and the readout uses $(\theta ,z_c)$ and its ordered model states. The preparation selection is part of the family data; fixing times and readout alone does not select it. The dimensions and units are those of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Coordinate ${\theta }_j$ is the numerical coefficient of a quantity in one fixed unit $u_j$ with declared dimension; different coordinates may use different units. Choose positive numerical scales $s_j$ and define $\mathrm{\parallel }v{\mathrm{\parallel }}_s=({\sum }_j(v_j\mathrm{/}{s}_j{)}^2{)}^{1\mathrm{/}2}$. This norm compares normalized numerical deviations, not sums of different physical dimensions.

A calibration datum for this family consists of:

1. a finite family, with $n\ge 1$, ${\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}\in \mathrm{\Theta }$, $r=1,\dots ,n$, of nominal reference inputs assigned by a reference standard under stated realization conditions. Each coordinate uses its fixed unit $u_j$, and the nominal assignment is traced to declared realizations of the framework's primitive quantities, such as durations, lengths or masses (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)). The nominal value is the standard's assigned value, not an assertion that the actual realization equals it;
2. one description of the actual reference inputs for the whole family: either fixed values ${\theta }_{\mathrm{a}\mathrm{c}\mathrm{t}}^{(r)}\in \mathrm{\Theta }$ for all $r$, with declared bounds $\mathrm{\parallel }{\theta }_{\mathrm{a}\mathrm{c}\mathrm{t}}^{(r)}-{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}{\mathrm{\parallel }}_s\le {\delta }_r$, ${\delta }_r\ge 0$, or a Borel random deviation vector $\eta =({\eta }_1,\dots ,{\eta }_n):{\mathrm{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}\to ({\mathbb{R}}^p{)}^n$ on one probability space $({\mathrm{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\mathcal{F}}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\mathbb{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ with a stated joint law, where ${\mathrm{\Theta }}_{\mathrm{a}\mathrm{c}\mathrm{t}}^{(r)}:={\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}+{\eta }_r$ belongs to $\mathrm{\Theta }$ for every outcome and every $r$ (Random elements and real random variables). No independence is imposed. A mixture of fixed and random reference inputs uses this second branch: each fixed input is represented by its constant deviation component, retaining any declared bound on that component. In the random branch declare whether the law describes uncertainty about fixed realizations or physical variation across realizations; a model for one is not automatically a model for the other. Additions are coordinatewise in the fixed units, and deviations need not belong to $\mathrm{\Theta }$;
3. a set $Y\subseteq ({\mathbb{R}}^d{)}^n$ of admissible reference observations and a declared realized vector $y=(y_1,\dots ,y_n)\in Y$, together with a separate uncertainty specification for these readouts: declared bounds, a joint probability model, or an explicit statement that readout uncertainty is idealized as zero. If a random vector $Y_{\mathrm{r}\mathrm{e}\mathrm{f}}$ models the readings before realization, specify its joint law with the reference-input deviations, including any correlations; $y$ denotes its recorded realization, not the random vector itself;
4. a calibration rule, that is, a map $\mathrm{\Gamma }:D⟶C,(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})⟼c,D\subseteq Y\times {\mathrm{\Theta }}^n\mathrm{.}$ Here ${\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}=({\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(1)},\dots ,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(n)})\in {\mathrm{\Theta }}^n$ and $(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})\in D$; each observation coordinate is read in the fixed readout unit of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error.

Given the datum, the calibrated setting is $c:=\mathrm{\Gamma }(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and the calibrated prediction map is ${\mathrm{\Phi }}_c$. This is the physical instance of the calibration structure of Calibration data and calibration rules, with the framework's primitive quantities as reference realizations. If normalized coordinates ${\vartheta }_j={\theta }_j\mathrm{/}{s}_j$ are used, the deviation norm is exactly the Euclidean norm of that mathematical structure. A rule with fixed references is only a slice of this two-input rule, not a rule that discards reference values.

Well-definedness and conventions

The sets $\mathrm{\Theta }$, $C$, $Y$, the preparation selection $\sigma$ and the maps $\zeta$, $R$ of the underlying protocol, the reference inputs, the deviations and the rule $\mathrm{\Gamma }$ are data. Nothing in the definition requires $\mathrm{\Gamma }$ to be injective, continuous or measurable, or to return a setting whose prediction map is accurate; Calibration data and calibration rules records the same conventions for the mathematical structure. Clauses 2 and 3 specify reference-input and reference-readout uncertainty separately, and both must be carried into every quantitative conclusion about ${\mathrm{\Phi }}_c$; a calibration with those uncertainty terms dropped supports no quantified uncertainty conclusion. A random calibrated setting $\mathrm{\Gamma }(Y_{\mathrm{r}\mathrm{e}\mathrm{f}},{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})$ requires the additional hypothesis that this composite is measurable and is in the rule domain; the deterministic realized rule does not imply that hypothesis.

Calibration is the step at which the framework's units and a standard's values assign numerical values to model parameters. The framework by itself fixes no numerical value of any parameter (clause (e) of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), and the value ${\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}$ is the value attached by the standard, not a measured value of this item. A calibrated prediction map is a model object; a claim that an apparatus realizes it requires separate evidence.

Fix an operational observable model of the force branch (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) in a fixed inertial frame, with parameter set $\mathrm{\Theta }$, preparation settings $Z$, prediction map $\mathrm{\Phi }:\mathrm{\Theta }\times Z\to \mathbb{R}$, probability space $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$, error domain $V\subseteq \mathbb{R}$ and error random element $\epsilon$; the readout is scalar here ($d=1$), since the mathematical sampling interface used below is stated for real random variables, and vector readouts are treated coordinatewise where needed.

A repeated-observation protocol for this model consists of

1. a repetition count $n\ge 1$ and one error random element ${\epsilon }_k:\mathrm{\Omega }\to \mathbb{R}$ for each $k=1,\dots ,n$, with ${\epsilon }_k(\omega )\in V$ for every $\omega$, the error of the $k$-th repetition on the common probability space (Probability measures and probability spaces, Random elements and real random variables);
2. the observations $Y_k:=\mathrm{\Phi }(\theta ,z)+{\epsilon }_k$ at a fixed pair $(\theta ,z)\in \mathrm{\Theta }\times Z$: the same parameter value and the same preparation are declared for every repetition, so that the repetition changes only the error.

The protocol satisfies the IID sampling hypothesis when the family $({\epsilon }_1,\dots ,{\epsilon }_n)$ — equivalently the family $(Y_1,\dots ,Y_n)$ — is independent and identically distributed in the sense of Identical distribution and IID families; it satisfies the second-moment hypothesis when $E\left[Y_1^2\right]<+\mathrm{\infty }$. Under both hypotheses the common mean and variance $\mu :=E[Y_1]\in \mathbb{R},{\sigma }^2:=\mathrm{Var}(Y_1)\in [0,+\mathrm{\infty })$ are defined by Moments, variance, and covariance on a probability space, the sample mean is $\bar{Y}:=\frac{1}{n}{\sum }_{k=1}^n{Y}_k,$ and the mathematical consequences of the two hypotheses are those of the sampling model Repeated-observation sampling models with IID and finite-variance hypotheses, namely the conclusions of Finite IID sample-mean variance and Chebyshev bound: $E\mid \bar{Y}-\mu {\mid }^2={\sigma }^2\mathrm{/}n$ and, for every $a>0$, $\mathbb{P}(\mid \bar{Y}-\mu \mid \ge a)\le \frac{{\sigma }^2}{n{a}^2}\mathrm{.}$

The equivalence uses the same constant $c:=\mathrm{\Phi }(\theta ,z)$ in every repetition. For any finite selection of indices and Borel sets $B_k$, $\{Y_k\in B_k\}=\{{\epsilon }_k\in B_k-c\}$. Since translation and its inverse preserve Borel sets, independence factorizes these intersection probabilities for the $Y_k$ exactly when it does for the ${\epsilon }_k$. Each observation law is the pushforward of its error law under $u\mapsto u+c$; applying the inverse translation shows that identical distribution is also equivalent. This argument requires the fixed common prediction; random or varying predictions need separate hypotheses.

Well-definedness and conventions

The IID and second-moment hypotheses are adopted model hypotheses about the repetitions, not consequences of the protocol and not properties of the data. Independence of the errors fails when successive repetitions share a systematic effect, when the apparatus drifts between repetitions or when the same recorded data are reused; identical distribution fails when the preparation or the conditions change between repetitions. A use of the displayed bounds must state both hypotheses, the declared law (if any) of $\epsilon$, and the conditions under which the repetitions are claimed to be comparable.

The sample mean $\bar{Y}$ is a random variable of the model. The definition attaches no observed value to $\mu$ or ${\sigma }^2$; in particular ${\sigma }^2$ is model data and not a measured uncertainty, as Scope of the sampling, propagation and inference interfaces records for the mathematical interface. A physical claim that a stated procedure satisfies the IID sampling hypothesis requires separate empirical evidence and is not part of this item.

A test of a null set is a rejection region $R$ inside a measurable data space $S$ for a specified family $({\mathbb{P}}_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ of probability measures and a specified null set ${\mathrm{\Theta }}_0\subseteq \mathrm{\Theta }$; its power function is $\beta (\theta )={\mathbb{P}}_{\theta }(R)$ and its size is ${\sup }_{\theta \in {\mathrm{\Theta }}_0}\beta (\theta )$ (Statistical tests, rejection regions, power functions and size). The size is computed from the specified data space, family, null set and rejection region; no alternative set and no prior distribution enter it. A test at level $\alpha$ rejects a true null with probability at most $\alpha$ under the specified family.

For the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, a family and a rejection region are obtained only after specifying: one inertial frame; an operational observable model of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error — parameters, preparations, observation times, readout and units — whose prediction map is computed from the maximal motions supplied by Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder; a calibration datum of Calibration of a mechanical apparatus against the primitive standards of the framework: reference inputs, deviations, rules and calibrated prediction maps with its reference deviations; a sampling model of Repeated-observation protocols of the force branch: adopted IID and second-moment sampling hypotheses and the sample mean with its adopted IID and second-moment hypotheses, or an explicitly different dependence model; the law of the error random element, or an explicit class of laws satisfying declared bounds with the robust size defined as a supremum over that class as well as over ${\mathrm{\Theta }}_0$; bounds alone do not select a probability family; a null set ${\mathrm{\Theta }}_0$ of parameter values; and a rejection region. Until these data are fixed, the phrase "a test of Newton's law" names no test, and the size and power of no test are defined.

A rejection is a decision about the specified triple of null family, apparatus behaviour and sampling assumptions. It is evidence against that triple at the declared size; it is not a proof that a force law, a spacetime, or classical mechanics is false, and the realized discrepancy does not by itself identify which ingredient failed. A discrepancy may expose the force law, a wrong or incomplete calibration, a changed preparation, correlated errors violating the sampling hypothesis, or an analysis error; separating these requires further stated models and further data. Conversely, retaining the null is not proof of the null: a test with low power fails to reject alternatives it cannot distinguish, and nonrejection alone neither proves nor estimates an effect size. Under a specified family the nonrejection event has parameter-dependent probability $1-\beta (\theta )$ and can therefore inform comparisons of effect values; quantification requires that family and an explicit inferential construction.

The indices of this page attach no probability to a parameter value. The coverage of a confidence set $C:S\to 2^{\mathrm{\Theta }}$ at $\theta$ is $\gamma (\theta )={\mathbb{P}}_{\theta }\{s:\theta \in C(s)\}$, with each displayed set required to be measurable. For $\alpha \in [0,1]$, it has coverage level $1-\alpha$ when ${\inf }_{\theta \in \mathrm{\Theta }}\gamma (\theta )\ge 1-\alpha$; the declared level is a pre-data lower bound, and need not equal that infimum (Confidence sets and their coverage level); a likelihood value is the density of the observed datum at a parameter value, not the probability of the value (Likelihood functions for a dominated parametric family); and a size is a supremum over the null set of probabilities of a region. In particular a $p$-value computed from a specified family is not the probability that the tested hypothesis is true, as Scope of the sampling, propagation and inference interfaces records for this mathematical interface. The cited interpretive warnings of the sources state the same properties for statistical tests in general.

Agreement of a prediction with observations is evidence for the combined adopted assumptions — force model, calibration, preparation, sampling model and analysis — at the stated uncertainty; it is not a deductive proof of the model and it does not test "classical physics" as a whole, only the specified family. A finite sample can fail to exhibit a feature that the declared sampling model predicts with positive probability per trial; such a sample is not by itself a refutation of the model, and the elementary probability of that outcome under the sampling model is part of the specification, not an empirical result.

No datum, no rejection, no retention, no $p$-value, no effect size and no measured uncertainty is asserted anywhere in this item.

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.

The spacetime of the framework is the affine event space with its surjective absolute-time covector, its three-dimensional Euclidean spatial displacement space and its fixed orientation (Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure); it contains no inner product on the full displacement space, no light cones, no null displacements and no universal speed bound. The frames are the adapted affine charts of these data, and the allowed changes between them are the Galilean transformations $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Rx+vt+a$ with $R\in SO(3)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames); the resulting kinematic laws, the frame-independence of vanishing coordinate acceleration and the group structure of the changes are the content of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes. The force model transforms by its own declared covariance rule (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance), and the existence and uniqueness statements for maximal motions are stated in the chosen frame (Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder).

An operational observable model (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) is specified in one inertial frame: its observation times are times of that frame, its readout is a function of the coordinate states, and its error random element carries the units of the readout coordinates. Changing the frame shifts the coordinate times, transforms positions affinely and rotates and boosts coordinate velocities. The prediction map in the new frame must be computed using the transformed protocol and its declared readout law; it need not differ numerically from the original map. For example, the allowed constant dimensionless readout $R\equiv 0$ gives $\mathrm{\Phi }\equiv 0$ in both frames. A measurement model must therefore name its frame, and a claim that two frames "agree" is a claim about the declared transformation law of the model, not a separate empirical discovery.

The dimension group of the framework is generated by length, duration and mass (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (a)); the framework contains no quantity of dimension $\mathrm{\ell }{\tau }^{-1}$ that is a constant of nature. Consequently the framework cannot by itself form a dimensionless parameter that measures how close a system is to some other kinematic regime, and it states no numerical cutoff: every regime statement such as "the motion is adequately described by this model at the required accuracy" is an external, empirically qualified premise of a use of the model, not a theorem of it. A deductive approximation bound relative to a distinct physical model requires explicit hypotheses and a proved comparison estimate; a speed scale is needed only if that comparison uses one. Calibrated observations can instead bound discrepancies between specified predictions and readouts under declared apparatus, calibration and sampling assumptions, carrying their uncertainty and tested conditions. Such empirical bounds require neither another physical model nor a universal speed scale, and do not establish a universal validity boundary. No comparison theorem or empirical bound is asserted here.

The framework's kinematic data distinguish no inertial frame (Status audit and model limits of the framework page: primitive data, adopted assumptions, derived objects and what is not claimed); "at rest" is a frame-relative description, not an absolute state. No measured speed, no numerical validity boundary, no comparison with any other kinematic framework, and no claim that a physical system is exactly Galilean is made. The Galilean transformations relate the adopted inertial frames and give the velocity and acceleration laws of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes. Covariance of the equation of motion also uses the transformed force map on the transferred domain prescribed by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (vi). A symmetry of a particular force model means a transformation carrying its solution curves to solutions of the same prescribed force law on the same domain; this requires the force law and its domain to be invariant under that transformation and does not follow from the kinematic theorem alone. Whether a physical situation is adequately represented by the model is an empirical question that this item leaves open.

The particle side of the framework is complete on its own terms: labels with $C^2$ worldlines parametrized by frame time and fixed positive masses (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses); a force branch in which the force map is declared model input with standing continuity and state-local-Lipschitz hypotheses (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance); motions, unique maximal motions and the continuation criterion (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder, Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain); the mechanical quantities and their balance theorems, conditional on the declared interaction structure (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy, Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries); and the operational observable model through which predictions at observation times are read (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error). The framework contains no charge, no field and no coupling law.

A charged-particle coupling is a new physical assumption, not a consequence of the framework, and it must supply at least:

1. Charge attributes. To each label an attribute with a declared dimension and unit. The dimension group of the framework is generated by length, duration and mass alone (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (a)); the dimension and unit of charge, and of the components of any field used, must therefore be declared by the coupling and cannot be inferred from this framework.
2. Field maps. For an electric/magnetic vector-field presentation, specify maps $\mathbf{E},\mathbf{B}:{\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}\times I_0\to {\mathbb{R}}^3$ in the declared inertial frame, with ${\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}\subseteq {\mathbb{R}}^3$ open and $x_i\in {\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}$ for every configuration and label on the coupling domain. At fixed time these are spatial vector fields; their component dimensions, units and regularity are additional declared data. The fields may be prescribed external maps without a model of their sources or any boundary or radiation conditions. If instead they are determined by field equations, the independent field model must supply its source data and whatever initial, boundary or radiation conditions its formulation requires. In either case, the total particle force must satisfy the standing regularity hypotheses below on the chosen domain; singular source loci must be excluded where those hypotheses fail, and fields requiring a distributional force need a separate solution notion. A presentation by potentials likewise requires their exact maps and the rule constructing the fields or particle force from them.
3. A coupling law. An added force map $F^{\mathrm{E}\mathrm{M}}:I_0\times Q\times ({\mathbb{R}}^3{)}^N\to ({\mathbb{R}}^3{)}^N$ with components $F_i^{\mathrm{E}\mathrm{M}}$ of force dimension $\mu \mathrm{\ell }{\tau }^{-2}$, read in newtons, for instance a declared function of the fields, the charge and the label's velocity in the chosen frame, as in the Lorentz force of the cited electromagnetic treatment. The coupling law is adopted: the framework neither derives it nor tests it.
4. Field-side bookkeeping. If statements about the energy or momentum of the combined system are wanted, the field contributions and their balance with the particles must be declared as separate objects; this framework contains no field energy or field momentum.

The particle ODE results apply once the coupling supplies the closed time-state force map $F^{\mathrm{E}\mathrm{M}}$ of item 3 on the same domain as $F$, and $F+F^{\mathrm{E}\mathrm{M}}$ satisfies the standing hypotheses of the force branch (continuity and state-local Lipschitz, locally uniformly in time). Then Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder and Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain give local existence, uniqueness, maximal motions and the continuation criterion for that particle model. If the fields are sourced by evolving labels, their values along a coupled solution are not automatically prescribed time-space maps or functions of the instantaneous particle state. A closed force map must be supplied by a justified reduction; otherwise the independent field equations and particle equations require a separate coupled solution theory, to which these particle ODE theorems alone do not apply.

The balance laws recorded in Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries retain their individual hypotheses. In particular, a coupling force can be added wholly to the external contribution of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i) only when $F_i^{\mathrm{E}\mathrm{M}}(t,x,v)=H_i(t,x_i,v_i)$ for declared single-label maps $H_i$ on open domains containing all these arguments, with the standing regularity. Prescribed fields independent of the evolving particle labels, with a local coupling law depending only on $(t,x_i,v_i)$ and fixed charge attributes, have this form if the resulting $H_i$ satisfies that regularity. Then replace $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ by $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}+H_i$ in an existing external-plus-pair presentation. More generally, reuse of that presentation requires a declared identity $F_i^{\mathrm{E}\mathrm{M}}(t,x,v)=H_i(t,x_i,v_i)+{\sum }_{j\mathrm{\ne }i}{K}_{ij}(t,x_i,x_j,v_i,v_j),$ with the component domains and regularity of that clause; add $K_{ij}$ to the corresponding pair map. Antisymmetry, centrality and any potential or residual-force presentation required by a particular balance must be checked for the resulting total force. Regularity alone implies none of these restrictions. The work-energy identity for the total force retains its force-model hypotheses without requiring an external-plus-pair presentation.

No field-momentum theorem is imported. With the electromagnetic force represented by single-label external maps as above, the particle mechanical momentum alone need not be conserved; the balance laws record the exchange through the external force, and the open-system boundary of Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope applies. A conservation statement for matter plus field requires the field contributions of item 4 above as additional declared data, and is not a consequence of this framework.

No field equation, no value or dimension of any charge or coupling constant, no derivation or validation of a coupling law, no prediction of any electromagnetic phenomenon and no measured electromagnetic quantity is asserted. The observable models of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error acquire an electromagnetic readout only after a coupling has been declared; until then this framework is silent about electric and magnetic effects.

This framework describes a physical system by labels with $C^2$ worldlines parametrized by frame time, fixed masses (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses) and a declared force map whose maximal motions are deterministic objects (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder). The point-particle idealization itself is an adopted description: the cited treatment records that it neglects the quantum-mechanical uncertainties of the position. A quantum framework is a different model whose state space, measurement structure and probability rule must be specified in that model; this item chooses none of them. The probability space of an operational observable model (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) is attached to the declared error random element of a coordinate readout; it is not a state, not an observable algebra, and not a probability rule for a physical system, as Scope of the sampling, propagation and inference interfaces already records for the mathematical interface.

One common type of relation, a semiclassical correspondence theorem, requires at least two separately specified parts:

1. a reconstruction or quantization construction: a declared map from data of the mechanical model to the objects of the quantum framework, with its domain, its algebraic consistency conditions and its treatment of ordering ambiguities. The cited treatment presents canonical quantization as exactly such a prescription between Poisson brackets and commutators; it is an additional rule, not a theorem of this framework, and the replacement of a classical phase-space variable by an operator is not a deduction inside it.
2. limit theorems with explicit hypotheses: statements that specified quantum predictions converge, in a stated topology and under stated regularity and small-parameter conditions, to specified classical predictions, with an error estimate. For the cited one-dimensional model, let $J\subseteq \mathbb{R}$ be an open position interval, let $I\subseteq \mathbb{R}$ be an open time interval, let $m>0$ and $\mathrm{\hslash }>0$ be constants, and let $V:J\to \mathbb{R}$ be a continuous potential in energy units. The wavefunction $\psi :J\times I\to \mathbb{C}$ satisfies the Schrödinger equation $i\mathrm{\hslash }{\mathrm{\partial }}_t\psi =-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_q^2\psi +V(q)\psi \mathrm{.}$ On an open subregion where $\psi \mathrm{\ne }0$ and a phase is chosen, assume $\psi ,A,W$ have one continuous time derivative and two continuous position derivatives, and write $\psi (q,t)=A(q,t)e^{iW(q,t)\mathrm{/}\mathrm{\hslash }}$ with real amplitude $A$ and phase $W$. Here $[W]=[\mathrm{\hslash }]$ is action and $[q]=\mathrm{\ell }$. Where ${\mathrm{\partial }}_{q}W\mathrm{\ne }0$, define the dimensionless ratio $r(q,t):=\frac{\mathrm{\hslash }\mid {\mathrm{\partial }}_q^{2}W(q,t)\mid }{\mid {\mathrm{\partial }}_{q}W(q,t){\mid }^2}\mathrm{.}$ This ratio records the phase-curvature comparison, but alone does not control the amplitude derivatives. With $A\mathrm{\ne }0$, the real part of the substituted equation gives the exact identity ${\mathrm{\partial }}_{t}W+\frac{1}{2m}({\mathrm{\partial }}_{q}W{)}^2+V(q)=\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\mathrm{\partial }}_q^{2}A}{A}\mathrm{.}$ Indeed, differentiating twice gives $e^{-iW\mathrm{/}\mathrm{\hslash }}{\mathrm{\partial }}_q^2\psi ={\mathrm{\partial }}_q^{2}A+\frac{2i}{\mathrm{\hslash }}({\mathrm{\partial }}_{q}A)({\mathrm{\partial }}_{q}W)+\frac{iA}{\mathrm{\hslash }}{\mathrm{\partial }}_q^{2}W-\frac{A}{{\mathrm{\hslash }}^2}({\mathrm{\partial }}_{q}W{)}^2,$ while $e^{-iW\mathrm{/}\mathrm{\hslash }}{\mathrm{\partial }}_t\psi ={\mathrm{\partial }}_{t}A+iA{\mathrm{\partial }}_{t}W\mathrm{/}\mathrm{\hslash }$; taking real parts yields the identity. Thus, on a region where ${\mathrm{\partial }}_{q}W\mathrm{\ne }0$, an additional explicit bound $\eta (q,t):=\frac{{\mathrm{\hslash }}^2\mathrm{\mid }{\mathrm{\partial }}_q^{2}A\mathrm{/}A\mathrm{\mid }}{\mathrm{\mid }{\mathrm{\partial }}_{q}W{\mathrm{\mid }}^2}\le ϵ,0\le ϵ<1,$ makes the Hamilton-Jacobi residual at most $ϵ\mathrm{\mid }{\mathrm{\partial }}_{q}W{\mathrm{\mid }}^2\mathrm{/}(2m)$ in absolute value. Neglecting that bounded residual gives the local Hamilton-Jacobi approximation; convergence of solutions requires a separate theorem and is not established by this comparison. Both $r$ and $\eta$ are dimensionless. The Classical Dynamics PDF's equation (4.204) prints a comparison without the square on $\mathrm{\mid }{\mathrm{\partial }}_{q}W\mathrm{\mid }$; since $[W]$ is action and $[q]$ is length, that comparison is dimensionally inconsistent. The squared phase-curvature comparison appears in Advanced Quantum Mechanics, equation (6.13), in its stationary exponential ansatz. It does not by itself supply the amplitude bound for the real phase-amplitude representation used here.

No quantization construction, no semiclassical limit theorem, no correspondence principle, no derivation of quantum mechanics from this framework and no derivation of this framework from a quantum one is asserted. No quantum phenomenon is predicted, explained or excluded here; this classical item assigns no trajectory between preparation and detection in a quantum experiment and makes no universal claim about trajectories in other quantum models; and the error random element of an observable model is not identified with a quantum measurement outcome. A bridge between the frameworks would be a separate development stating its own construction, hypotheses and error control, and would not change the status of any statement on this page.

A later experiment item for this framework is a report of an actual measurement, and must be assembled from an actually retrieved primary source and record, separately:

1. Source. A primary report with a resolvable reference, read in full: an apparatus paper, an instrument report or a data release, not a textbook exposition of a law and not a secondary anecdote.
2. Setup. The prepared system in the vocabulary of the model: labels, declared parameters, configurations and the initial-data region (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error); the apparatus and its settings; and the calibration with its reference inputs and their declared deviations (Calibration of a mechanical apparatus against the primitive standards of the framework: reference inputs, deviations, rules and calibrated prediction maps).
3. Procedure. The measurement protocol in a declared inertial frame: observation times, readout, units; the sampling model with its adopted IID and second-moment hypotheses, or an explicit statement of the dependence or drift that replaces them (Repeated-observation protocols of the force branch: adopted IID and second-moment sampling hypotheses and the sample mean); and the selection of the reported data.
4. Observations. The values actually reported by the source, with their units and their conditions, quoted or transcribed faithfully. If the source records no quantitative detail, only a correspondingly limited qualitative account is supported; no precision, count or value may be supplied from elsewhere.
5. Uncertainty. The statistical and systematic uncertainty treatment of the source and of the calibration: Type A evaluation from repeated observations, Type B evaluation from declared bounds and reference data, combination and expansion, and the deviations of the reference standards. Measured inputs of the comparison model carry their uncertainties and conditions into the conclusion (Scope of the sampling, propagation and inference interfaces).
6. Interpretation. A specified comparison between the reported observations and a specified prediction of the model, including the idealizations the prediction uses; a specified discrepancy measure; and a specified test — data space, model family, null set and rejection region with its size (Statistical tests, rejection regions, power functions and size) — or a specified estimation or confidence construction (Confidence sets and their coverage level, Likelihood functions for a dominated parametric family). Agreement is evidence for the joint assumptions of model, apparatus and analysis, and a discrepancy may expose any of them.

The treatments cited by this framework are model exposition; they are not primary reports of a measurement of the framework's parameters under a controlled preparation, and the citations of this page therefore close no experiment obligation. No experiment item, no observation, no measured value, no uncertainty, no $p$-value and no effect size is asserted by this page. The worked computations on the companion page are proposals: they compute what a protocol would imply under declared hypothetical data and uncertainties. Turning such a proposal into a reported experiment requires all six items above, in particular an actually retrieved primary source.