Experimental Interface

Adopt the SI conventions of SI units and coordinate conventions of the special-relativity framework, the ideal-clock and ideal-light assumptions of Adopted ideal-clock and ideal-vacuum-light interpretation, the ideal plane-wave and frequency-reading model of Adopted ideal plane-wave and frequency-reading model, the operational radar protocol of The operational radar protocol of an ideal observer, the massive-particle measurement identification of Adopted massive-particle momentum, observer measurement and isolated-collision conservation clause (ii) and the massless-particle measurement identification of Adopted momentum model for an ideal massless particle clause (ii). This definition introduces the framework's experimental-interface vocabulary; it selects no experiment, records no value and asserts no comparison.

(a) Run descriptions. A run description is a finite tuple $D=(d_1,\dots ,d_r)$ whose entries are framework objects of the following kinds, each declared with its domain and regularity as specified by the cited item, together with the frame, observer, event or unit convention required for a numerical reading where such a reading is used: a point of the affine Minkowski space $A$ (an event, Oriented time-oriented affine Minkowski space (the spacetime primitives)); a vector of $V$ of declared causal character (Timelike, null, spacelike and causal vectors of a Lorentz vector space); a future timelike worldline together with its proper-time function (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Proper time along a future timelike worldline); a future unit timelike vector or a smooth observer congruence (Charts, tetrads, observer congruences and adapted charts in affine Minkowski space); a future null wave covector with its affine phase (Affine phase functions and wave covectors on affine Minkowski space); a radar time or radar distance assigned by the operational radar protocol (The operational radar protocol of an ideal observer); an observer angular frequency (Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave); an observer energy, spatial momentum, kinetic energy or invariant mass (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum); and a real parameter vector in a declared subset of ${\mathbb{R}}^m$ with declared units. A run description is a description of a model configuration; the tuple asserts nothing about any apparatus or about nature.

(b) Operational model. An operational model is a tuple $M=(S,\mathrm{\Theta },\theta ,D,F)$ in which: $S$ is a nonempty set, the preparation index set; $\mathrm{\Theta }\subseteq {\mathbb{R}}^m$ is a nonempty parameter set and $\theta :S\to \mathrm{\Theta }$ is a surjection, the parameter map; $D$ assigns to each $s\in S$ a run description $D(s)$ of (a); and for some integer $d\ge 1$ the map $F:\mathrm{\Theta }\to {\mathbb{R}}^d$ is a prediction map for $D$, subject to the declared interpretation $(\ast )$: for every $s\in S$, each component of $F(\theta (s))$ is a number that the framework's definitions, postulates and proved consequences assign to the description $D(s)$, with every intermediate quantity, frame, event, sign and unit declared. An operational model is called an operational model for the quantity whose values $F$ computes. Condition $(\ast )$ is a declared interpretation that reviewers check when the model is used; it is not a consequence of the tuple structure, and no clause of this definition asserts that any apparatus realizes $M$ or that $F(\theta (s))$ is measured.

(c) Data records. A data record is a tuple $R=(N,z,\epsilon ,a)$ with an integer $N\ge 1$; $z=(z_1,\dots ,z_N)$ with $z_k\in {\mathbb{R}}^d$; $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_N)$ with ${\epsilon }_k\in [0,\mathrm{\infty }{)}^d$; and $a=(a_1,\dots ,a_N)$ a finite tuple of condition annotations, each a finite string identifying the declared run condition of the datum and, when the datum is taken from a report, the bibliographic identification of that report. The record declares: for each $k$, the value of the quantity of (b) under the condition annotated by $a_k$ lies in the closed box ${\prod }_{j=1}^d[z_{k,j}-{\epsilon }_{k,j},z_{k,j}+{\epsilon }_{k,j}]$, in the declared units. A data record contains no probability space, random variable, distribution, sampling law, independence assumption, likelihood or coverage statement; it declares nothing about how the values $z_k$ were produced and nothing about the correctness of any calibration.

(d) Calibration declarations. Let $Z$ be a nonempty set of declared instrument-output values. A calibration declaration is a tuple $C=(Z,c,\delta ,h)$ with $c:Z\to {\mathbb{R}}^d$, $\delta :Z\to [0,\mathrm{\infty }{)}^d$ and $h$ a condition annotation; it declares that for every $z\in Z$ obtained under the condition annotated by $h$, the value of the declared quantity lies in the closed box ${\prod }_j[c_j(z)-{\delta }_j(z),c_j(z)+{\delta }_j(z)]$ in the declared units. A calibration record is a data record of (c) whose declared quantity is the value of the calibration map's output. The declaration asserts no accuracy beyond the bound $\delta$ and no statistical property of $c$ or $\delta$.

(e) Uncertainty vocabulary. For a data record of (c), the declared uncertainty of the $k$-th datum is the vector ${\epsilon }_k$; a source label is an element of the finite declared set $\{\text{calibration},\text{ systematic},\text{ sampling-variability},\text{ rounding},\text{ other}\}$ attached to a bound; and a total declared bound for a value computed from finitely many recorded values by a declared finite arithmetic expression is any $B\in [0,\mathrm{\infty })$ for which the declaration states: whenever each recorded value lies in its declared closed box, the computed value lies in $[\widehat{y}-B,\widehat{y}+B]$ for the declared number $\widehat{y}$ (for instance, for a sum of recorded scalars the sum of the recorded bounds is such a bound). Every uncertainty entry is a declared constraint of the form in (c) and (d); no probability law, coverage probability, confidence level, p-value or posterior is defined by this vocabulary.

(f) Comparison claims are separate. A claim that a data record $R$ and an operational model $M$ are consistent, or that $R$ supports, refutes or fails to distinguish $M$, is not part of (a)-(e): it requires its own statement, an explicitly declared comparison rule and adequate primary-report and analysis prerequisites. For the current stored-lithium-ion optical Doppler report, that research prerequisite, labelled M20 in the source design, is supplied by exp-sr-stored-lithium-ion-doppler-time-dilation-2014 and pthm-sr-specified-model-comparison. The comparison theorem distinguishes hard-bound records of (c) from separately defined source-reported estimates and standard-uncertainty scales; the latter are not hard bounds or coverage guarantees. It permits comparison with specified alternative prediction hypotheses, which are not thereby operational models of the SR framework and are not asserted to satisfy condition $(\ast )$ in (b). Other primary-report families remain subject to their own retrieval and analysis obligations. These are nonlogical status pointers to the separate empirical record and comparison claim; no clause of this vocabulary licenses a conclusion about the truth of a postulate or of the framework.

(g) Well-definedness and scope. All components of (a)-(e) are finite tuples of sets, maps and real numbers with declared domains and units; the constructions use no choice principle and no measurability, integrability, completeness or differentiability beyond the cited items, and clause (b) evaluates a prediction only at parameter values for which the cited items supply its ingredients. The vocabulary asserts no measurement, no apparatus behaviour, no empirical support or refutation, no value of the constant $c$ or of any other physical parameter, and no statistical inference.

Scope-and-queue record for the empirical layer of the flat special-relativity framework, stated in the vocabulary of Operational model, data, calibration and uncertainty vocabulary of the experiment interface.

(i) The actual primary Botermann et al. 2014 stored-lithium-ion Ives-Stilwell report has now been retrieved and read in full. The experiment item records its apparatus, calibration, finite sampling, frequency estimates and $1\sigma$ budget; the comparison theorem supplies explicit conditional bound and reported-scale analysis. No raw dataset or new coverage claim is supplied.

(ii) The source design declares four primary-report search destinations for future experiment items of this framework. Each is named here as a destination, that is, a family of published reports to retrieve and read in full text, and no destination is evidence for any postulate: (a) historical ether and light-propagation experiments (interferometric or resonance tests of the constancy of the coordinate speed of light between inertial charts, and Doppler-shift tests of moving emitters); (b) unstable-particle lifetime measurements (beam or storage-ring measurements of a particle's lifetime at a measured speed compared with its lifetime at rest); (c) accelerator kinematics of charged particles at high Lorentz factor (measurements of momentum, energy or speed relating a particle's measured quantities to its velocity); (d) clock-transport and clock-comparison experiments (aircraft-, satellite- or laboratory-borne atomic clocks comparing the elapsed times of differently moving or differently situated ideal clocks). The secondary narratives consulted for this page mention instances of these families, among them the historical ether-and-Maxwell context (Crowell pp. 13-15), the 1938 Ives-Stilwell Doppler test (Crowell p. 57), storage-ring muon-lifetime measurements (Crowell pp. 28-29), the 1971 Hafele-Keating aircraft atomic-clock comparison (Crowell pp. 15, 26, 28; Tong pp. 24-25) and a 2010 low-speed atomic-clock measurement (Crowell p. 28); those mentions are contextual motivation only and are not documentation of reported measurements here.

(iii) Before any experiment item can be authored on a destination, the actual primary report must be retrieved and read, and the record must carry the apparatus and preparation, operational definitions of the measured quantities, the calibration model, the sampling and error model, the treatment of systematic errors, the measured values with their reported uncertainties, and the source's limitations. Textbook and review narratives are not substitutes for the primary reports.

(iv) The comparison of model predictions with recorded data is a separate promised physics-theorem obligation which the source design declares to require the primary-report and analysis layer (research label M20); that layer is now supplied for the specified 2014 stored-lithium-ion report by the experiment and comparison theorem, while the other primary-report families remain research queues.

(v) Qualifications any future comparison must carry: observations test the combined model and apparatus assumptions; finite-sample agreement does not establish a postulate or a model; a p-value would not be the probability that a theory is false and a confidence level would not be a posterior probability; and a finite bounded record does not by itself determine a unique parameter or adjudicate between distinct models.

(vi) This remark is a queue/status pointer; measured estimates and their qualifications are recorded only in the experiment item. It supplies no independent empirical conclusion or certainty.

The four-family search queue remains intact: this single optical Doppler primary report supplies one branch of family (a); historical ether/interferometer reports and the lifetime, accelerator and transported-clock families still require actual retrieval and reading. Source-cited RMS/SME analyses remain separate queue obligations.

Current nonlogical status pointers: exp-sr-stored-lithium-ion-doppler-time-dilation-2014 records the primary observations; pthm-sr-specified-model-comparison gives the separate conditional comparison. These pointers add no logical prerequisite.

Quantities and apparatus notation. The laboratory speed parameter is $\beta =v\mathrm{/}c$, with $v$ the selected ion-class speed in an approximately inertial laboratory description; its quoted value is nominal. Frequencies ${\nu }_1,{\nu }_2,{\nu }_p,{\nu }_a>0$ are scalar estimates in hertz of the two rest transitions and the parallel/antiparallel laser readings, respectively; the ideal-model convention is $\nu =\omega \mathrm{/}(2\pi )$ and $\lambda =c\mathrm{/}\nu$ from Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave, while the real apparatus has finite-width spectra. Here $\mathrm{M}\mathrm{H}\mathrm{z}={10}^6{\mathrm{s}}^{-1}$, $\mathrm{n}\mathrm{m}={10}^{-9}\mathrm{m}$, and $\mu \mathrm{r}\mathrm{a}\mathrm{d}={10}^{-6}$ radians. For a single-peaked nonnegative profile, its full width at half maximum (FWHM) is the separation of the two frequencies (or speeds) at half its peak height, when those crossings exist; the reported $\delta v\mathrm{/}v$ uses the speed width divided by the selected speed. Beam divergence and alignment are angles between ion trajectories and the specified beam axis. A Lambda system here means three source-labelled atomic levels, two lower and one common upper, with the two listed driven transitions; the symbols $3S_1$, $3P_2$ and $F$ distinguish the report's atomic levels, rather than supplying a quantum-state model. Metastable lifetime means the reported characteristic persistence time of the prepared level; no survival distribution is inferred. The Penning source, ESR (Experimental Storage Ring), electron cooling, lasers and optics below are the report's named preparation and control apparatus; their performance is reported empirically, not derived from SR. The energy label $\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{u}$ is energy per atomic-mass unit: $1\mathrm{M}\mathrm{e}\mathrm{V}={10}^6$ electronvolts, with one electronvolt the energy acquired by elementary charge across one volt; it is not used here to deduce the nominal speed.

The experiment used ${}^7{\mathrm{L}\mathrm{i}}^{+}$ ions from a Penning ion gauge source, accelerated to $58.6\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{u}$ and stored in the ESR at nominal $\beta =0.338$. Electron cooling reduced the reported longitudinal velocity spread to $\delta v\mathrm{/}v\approx 9\times {10}^{-6}$ (FWHM) and transverse divergence to less than $100\mu \mathrm{r}\mathrm{a}\mathrm{d}$. About ${10}^7$ to ${10}^8$ ions were stored, with an estimated metastable fraction below $1\mathrm{\%}$. The metastable $3S_1$ lifetime was $15$–$30\mathrm{s}$ in the ring, compared with $50\mathrm{s}$ in vacuum. The Lambda system used the transitions $3S_1(F=3\mathrm{/}2)\to 3P_2(F=5\mathrm{/}2)$ and $3S_1(F=5\mathrm{/}2)\to 3P_2(F=5\mathrm{/}2)$. Parallel and antiparallel beams near $386\mathrm{n}\mathrm{m}$ and $780\mathrm{n}\mathrm{m}$ addressed its two legs; an additional $780\mathrm{n}\mathrm{m}$ beam repumped ions into the system.

Count and fit objects. A spectrum in this account is a finite list of frequency settings $f_i\in \mathbb{R}$ with detected pulse counts $n_{i,h}\in {\mathbb{N}}_0$ in each switching configuration $h\in \{1,2,3,4\}$ and window. Fluorescence denotes the emitted light counted by the photomultipliers; a dark count is a registered pulse without applied excitation. Subtracted or averaged spectra are real-valued lists, not nonnegative integer counts or independent trials. A Lorentzian peak plus linear background means a fitted function $L:\mathbb{R}\to \mathbb{R}$ of the form $L(f)=A\mathrm{/}[1+4(f-f_0{)}^2\mathrm{/}{w}^2]+b_0+b_1(f-f_0)$ with $A>0$, $w>0$; $f_0,w$ have frequency units, and the peak term has half height at $f_0\pm w\mathrm{/}2$. Residuals are the differences of processed values and the fitted values at the recorded settings. A weighted average of fitted centers has form ${\sum }_i{w}_i{f}_{0,i}\mathrm{/}{\sum }_i{w}_i$ for nonnegative weights with positive sum; the report does not give the weights or their construction. Frequency locking means controlling a laser reading relative to a specified reference or offset; second-harmonic generation denotes conversion to twice the optical frequency. Optical-optical double resonance is the selected condition that both listed transitions are resonant for the same ion velocity class; it is an apparatus selection model, not a measured probability law.

The parallel beam was generated by doubling a $772\mathrm{n}\mathrm{m}$ Ti:Sa laser locked to an iodine marker previously calibrated with a frequency comb against a rubidium clock. The antiparallel diode laser was offset-locked to a rubidium reference, with a $600\mathrm{M}\mathrm{H}\mathrm{z}$ tuning range. Polarization-maintaining fibers transported the beams. Motorized stages controlled angles and positions to $20\mu \mathrm{r}\mathrm{a}\mathrm{d}$ and $50\mu \mathrm{m}$; scrapers established laser/ion overlap within $80\mu \mathrm{r}\mathrm{a}\mathrm{d}$. Photomultipliers counted fluorescence in four successive $100\mu \mathrm{s}$ configurations: both excitation beams, antiparallel only, parallel only, and neither. Dark counts were subtracted, then the separate-beam background was subtracted from the both-beam spectrum. Optical-optical double resonance selected ions for which both Lambda legs were resonant at the same velocity. To compensate metastable depletion, an $81$-point scan was paused after $69$ points for injection and cooling. Figure 2 shows an example averaged over $44$ scans, with $49\mathrm{M}\mathrm{H}\mathrm{z}$ laboratory linewidth and $35\mathrm{M}\mathrm{H}\mathrm{z}$ rest-frame linewidth. The authors fitted $40$ spectra with Lorentzians on linear backgrounds and formed a weighted average. Raw count tables, fit weights, a covariance matrix and a complete likelihood are not published in this report.

Table I gives the frequency estimates, in MHz, ${\nu }_1=546455143.0$, ${\nu }_2=546474960.7$, ${\nu }_a=384225534.98$ and ${\nu }_p=777210326.98$. The rest-frame inputs ${\nu }_1,{\nu }_2$ come from earlier measurements cited as references 21–22; those papers were not retrieved or read here. Equation (4) reports $ϵ=\sqrt{{\nu }_a{\nu }_p\mathrm{/}({\nu }_1{\nu }_2)}-1=(1.5\pm 2.3)\times {10}^{-9}$ at nominal $\beta =0.338$. These are estimates with the source-labelled $1\sigma$ uncertainty budget, not exact values or guaranteed bounds. The rounded Table I centers yield approximately $1.4744849565\times {10}^{-9}$, consistent with the reported rounded estimate, without reconstructing unrounded data.

The Gouy phase is the additional spatial phase of a focused beam relative to a plane-wave phase; for a $C^1$ phase correction $\psi :J\to \mathbb{R}$ on a proper-time interval $J\subseteq \mathbb{R}$, the phase-reading convention assigns a frequency correction $-(2\pi {)}^{-1}d\psi \mathrm{/}d\tau$ under the stipulated phase-reading interpretation. No beam profile $\psi$ is reconstructed here. The $1\sigma$ notation below is the source's label for an uncertainty scale; without its full estimation model it specifies neither a guaranteed interval nor a sampling distribution.

Table I labels its uncertainties $1\sigma$. In MHz, the aggregate entries are $0.43$ for each rest frequency, $1.6$ for the antiparallel frequency and $1.25$ for the parallel frequency. Antiparallel contributions are: weighted-fit statistical uncertainty $0.6$, rubidium reference $0.05$, diode stabilization $0.64$, background linearization $0.5$, Gouy phase $1.0$, angular alignment $0.6$, and ion divergence $0.4$. Parallel contributions are: iodine reference $0.06$, Ti:Sa stabilization $0.12$, Gouy phase $1.0$, alignment $0.6$, and divergence $0.4$. The inferred Gouy correction for each laser was $0.44\pm 1.0\mathrm{M}\mathrm{H}\mathrm{z}$, from experimentally validated numerical simulation using measured beam parameters. The authors reported no significant residual asymmetry; simulated background linearization was assigned $0.5\mathrm{M}\mathrm{H}\mathrm{z}$. The excess linewidth, about five times the expected width, was attributed mainly to drift-induced fluorescence. The $0.6\mathrm{M}\mathrm{H}\mathrm{z}$ statistical entry comes from the weighted analysis of forty fitted spectra, not forty independent raw clock readings. These reported statistical and systematic scales are preserved without inventing independence, a Gaussian total sampling law, fit weights, covariance or coverage.

The measured observable tests the combined Doppler/time-dilation, active atomic-transition and vacuum-wave invariance assumptions together with resonance selection, calibration, beam alignment and correction models. The source reports no resolved deviation at its uncertainty level and quotes $\mathrm{\mid }\alpha \mathrm{\mid }\le 2.0\times {10}^{-8}$ for $\alpha =ϵ\mathrm{/}{\beta }^2$. That quotation is recorded as the source's test-theory sensitivity statement, rather than a newly certified hard bound or confidence interval. RMS and SME parameter translations and future higher-boost prospects remain contextual research topics; they are not prerequisites or claims of the comparison theorem.

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.