Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error

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.