Model testing for the force branch: what a test specifies, what rejection and retention establish, and what is not claimed

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.