Calibration of a mechanical apparatus against the primitive standards of the framework: reference inputs, deviations, rules and calibrated prediction maps

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.