Specified-model comparison with calibrated bounds and reported uncertainty scales

Adopt Operational model, data, calibration and uncertainty vocabulary of the experiment interface. Comparison labels below range over explicitly specified prediction hypotheses, each with its own apparatus and calibration assumptions. When a label is an operational model of the SR framework, its prediction map must satisfy clause (b), condition $(\ast )$, of that vocabulary using the specified SR suppliers. A competing prediction hypothesis need not be an SR operational model and is not represented as satisfying $(\ast )$. Reported standard uncertainties are a separate record type, not the vocabulary's declared hard-bound records.

(a) General finite comparison. An apparatus/calibration nuisance parameter means an additional declared model input constrained together with the physical parameter; its domain is part of the admissible set below, and no probability law is implicit. Flatten a finite $d$-dimensional record with $N$ entries to $m=Nd$ real coordinates. For each label $\mathrm{\ell }$, let $A_{\mathrm{\ell }}$ be a declared set of admissible tuples $a=(\theta ,\eta ,b)$, where $\theta$ is a physical parameter, $\eta$ an apparatus or calibration nuisance parameter, and $b\in {\mathbb{R}}^m$ an additive calibration correction. Let $P_{\mathrm{\ell }}:A_{\mathrm{\ell }}\to {\mathbb{R}}^m$ be its corrected prediction, $P_{\mathrm{\ell }}(a)=F_{\mathrm{\ell }}(\theta ,\eta )+b$; every correction and admissibility constraint is explicitly stated. For a declared hard-bound record $(z,e)$ put $B(z,e)={\prod }_{j=1}^m[z_j-e_j,z_j+e_j]$ and $C_{\mathrm{\ell }}=\{a\in A_{\mathrm{\ell }}:P_{\mathrm{\ell }}(a)\in B(z,e)\}$. Then $C_{\mathrm{\ell }}\mathrm{\ne }\mathrm{\varnothing }$ if and only if $P_{\mathrm{\ell }}(A_{\mathrm{\ell }})\cap B(z,e)\mathrm{\ne }\mathrm{\varnothing }$. If the true calibrated vector satisfies the declared bounds and a candidate with its auxiliaries assigns that vector the value $P_{\mathrm{\ell }}(a)$ for some $a\in A_{\mathrm{\ell }}$, then $a\in C_{\mathrm{\ell }}$. Thus $C_{\mathrm{\ell }}=\mathrm{\varnothing }$ excludes that conjunction of candidate, parameter, calibration and hard-bound assumptions, rather than proving another model. If every prediction in $P_{\mathrm{\ell }}(A_{\mathrm{\ell }})$ has coordinate $j$ in $[L_j,U_j]$ disjoint from $[z_j-e_j,z_j+e_j]$, then $C_{\mathrm{\ell }}=\mathrm{\varnothing }$. This criterion is sufficient; overlap of marginal intervals alone does not establish joint feasibility.

(b) Exact finite uncertainty propagation. For positive frequencies ${\nu }_a,{\nu }_p,{\nu }_1,{\nu }_2$ define $Q=\sqrt{{\nu }_a{\nu }_p\mathrm{/}({\nu }_1{\nu }_2)}-1$. If separately declared hard bounds place these frequencies in positive closed intervals $[{\nu }_{a,-},{\nu }_{a,+}]$, $[{\nu }_{p,-},{\nu }_{p,+}]$, $[{\nu }_{1,-},{\nu }_{1,+}]$ and $[{\nu }_{2,-},{\nu }_{2,+}]$, then

$Q\in [\sqrt{\frac{{\nu }_{a,-}{\nu }_{p,-}}{{\nu }_{1,+}{\nu }_{2,+}}}-1,\sqrt{\frac{{\nu }_{a,+}{\nu }_{p,+}}{{\nu }_{1,-}{\nu }_{2,-}}}-1]\mathrm{.}$

Both endpoints are attained on this rectangular input domain. This is conditional on the hard-bound event; standard-uncertainty entries alone assert neither that event nor probabilistic independence.

(c) Reported-estimate comparison. A reported-estimate record $H=(\widehat{z},u,\alpha ,\mathcal{M})$ consists of a finite real estimate vector, strictly positive reported scale entries $u_j$, bibliographic and run annotations $\alpha$, and a source description $\mathcal{M}$ of estimation and uncertainty construction. It asserts no hard bound or coverage. Here $\mathrm{\mid }t\mathrm{\mid }=t$ for $t\ge 0$ and $\mathrm{\mid }t\mathrm{\mid }=-t$ for $t<0$; a nonempty finite maximum is obtained by recursively comparing pairs. For a declared multiplier $k\ge 0$ define

$D_{\mathrm{\ell }}(a)={\max }_{1\le j\le m}\frac{\mathrm{\mid }{\widehat{z}}_j-P_{\mathrm{\ell },j}(a)\mathrm{\mid }}{u_j},S_{\mathrm{\ell }}(k)=\{a\in A_{\mathrm{\ell }}:D_{\mathrm{\ell }}(a)\le k\}\mathrm{.}$

Then $S_{\mathrm{\ell }}(k)=P_{\mathrm{\ell }}^{-1}({\prod }_j[{\widehat{z}}_j-k{u}_j,{\widehat{z}}_j+k{u}_j])$. This is a deterministic comparison of estimates and reported scales. No probability, $p$-value, confidence level or rejection error rate follows without a separately justified sampling and error model.

(d) Primary-report instantiation. Under the qualified conditions of exp-sr-stored-lithium-ion-doppler-time-dilation-2014, additionally stipulate that resonance identifies each laser frequency in the ions' instantaneous rest frame with its stated rest transition frequency, that those transition frequencies are unchanged under active boosts, and that the ideal vacuum phase and frequency law applies in the interaction section after the report's beam, Gouy-phase and line-shape corrections. These atomic, optical and apparatus premises are not derived by Doppler shift and aberration for an ideal plane wave between inertial descriptions. At each selected reception event, additionally represent the ion by an inertial rest observer with velocity $c\beta e_1$, $\mathrm{\mid }\beta \mathrm{\mid }<1$, in a laboratory inertial frame $(o;e_0,e_1,e_2,e_3)$. This is a stipulated reception idealization in the interaction section, not an application of the supplier to an accelerating storage-ring worldline. For oppositely propagating laboratory waves, that theorem gives

${\nu }_1=\gamma (1-\beta ){\nu }_p,{\nu }_2=\gamma (1+\beta ){\nu }_a,\gamma =(1-{\beta }^2{)}^{-1\mathrm{/}2}\mathrm{.}$

Hence $Q_{\mathrm{S}\mathrm{R}}=0$. To make that interpretation explicit, fix arbitrary positive model transition frequencies ${\nu }_1,{\nu }_2$ in hertz. For each $\beta \in (-1,1)$ take the chart $x_{\beta }=B(\beta e_1)x$, its rest vector $n_{\beta }=B(-\beta e_1)e_0$, and laboratory frequencies ${\nu }_p={\nu }_1\mathrm{/}[\gamma (1-\beta )]$, ${\nu }_a={\nu }_2\mathrm{/}[\gamma (1+\beta )]$. Take covectors with laboratory components $k_p=(2\pi {\nu }_p\mathrm{/}c)(-1,1,0,0)$ and $k_a=(2\pi {\nu }_a\mathrm{/}c)(-1,-1,0,0)$ and phases $p\mapsto k_p(p-o)$, $p\mapsto k_a(p-o)$. Let $D(\beta )$ contain these waves, $n_{\beta }$, reception event $o$ and the declared scalar parameters, with the two frames attached to the readings. Then $S=\mathrm{\Theta }=(-1,1)$, parameter map $\mathrm{i}\mathrm{d}$, and prediction $F(\beta )=Q_{\mathrm{S}\mathrm{R}}=0$ give an operational model for $Q$ under the stipulated resonance identification. The transition frequencies here are model parameters; this construction does not assert exact knowledge of the report's uncertain frequency values. Define the alternative prediction hypothesis by replacing the common factor $\gamma$ in those resonance equations by $\mathrm{\Gamma }(\beta )=1$, retaining the same directional factors and apparatus assumptions. It predicts $Q_1(\beta )=(1-{\beta }^2{)}^{-1\mathrm{/}2}-1$. This deliberately alternative hypothesis is not an SR operational model and makes no claim to satisfy $(\ast )$; it is not identified with all classical physics.

For the source-reported dimensionless estimate $\widehat{z}=1.5\times {10}^{-9}$ and scale $u=2.3\times {10}^{-9}$, the declared $k=1$ criterion gives $D_{\mathrm{S}\mathrm{R}}=15\mathrm{/}23<1$, with descriptive scale band $[-0.8,3.8]\times {10}^{-9}$. At the report's nominal $\beta =0.338$, one has $0.062<Q_1(0.338)<0.063$, hence $D_1>{10}^7$: that nominal alternative prediction fails this scale criterion. This is not a statistical significance claim. The speed is nominal, no calibrated speed uncertainty is supplied for this evaluation, and no distribution or coverage is inferred. The SR product prediction is independent of $\beta$; the alternative prediction depends on its declared value. Agreement of SR with the descriptive scale band tests the physical and apparatus conjunction within the report's uncertainty treatment; it neither proves SR nor uniquely identifies its auxiliary assumptions. All conclusions inherit the primary report's conditions, calibration dependence and statistical and systematic limitations.