Model configurations, interface readouts, operational models, data records and the analysis vocabulary of the experiment interface

Adopt the unit conventions of SI unit and coordinate conventions of the classical gravitational framework, the spacetime postulate of Adopted spacetime model and classical domain of the gravitational framework and the clock, free-fall and light-ray postulates of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. This definition fixes the vocabulary in which the framework's empirical layer is stated: it names model configurations, the readouts assigned to them, operational models, data records, calibration declarations and declared uncertainty bounds. It selects no experiment, records no value and asserts no comparison.

(a) Model configurations. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). A model configuration is a finite tuple $D=(d_1,\dots ,d_r)$ whose entries are objects of the following kinds, each carrying the domain, parametrization and regularity required by the cited item:

• a worldline entry: a $C^1$ future timelike worldline $h:I\to M$ on an interval $I$ with nonempty interior, parametrized by proper time, so that its four-velocity $U:=h^{\mathrm{\prime }}$ satisfies $g(U,U)=-c^2$ (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent), or the restriction of such a worldline to a subinterval;
• an observer entry: an observer at a point, that is a future unit timelike vector $n$ with $g(n,n)=-1$, or a smooth observer congruence (Observers, rest spaces and the spatial projection at an event, Observer congruences, their observer curves and adapted charts);
• an isometric-identification entry: declared points $p,q\in M$ and a linear isomorphism $\mathrm{\Phi }:T_{p}M\to T_{q}M$ satisfying $g_q(\mathrm{\Phi }v,\mathrm{\Phi }w)=g_p(v,w)$ for every $v,w\in T_{p}M$; this map is supplied comparison data, not an identification selected by the metric alone, and any required relation to observer entries is specified by the readout;
• a ray entry: a smooth curve $\eta :J\to M$, where $J\subseteq \mathbb{R}$ is an interval with nonempty interior, that is a future-directed null geodesic of $\mathrm{\nabla }$; thus it is affinely parametrized and satisfies ${\mathrm{\nabla }}_{{\eta }^{\mathrm{\prime }}}{\eta }^{\mathrm{\prime }}=0$, $g({\eta }^{\mathrm{\prime }},{\eta }^{\mathrm{\prime }})=0$ and ${\eta }^{\mathrm{\prime }}$ is future-directed, with the one-sided convention at included endpoints (Geodesic of an affine connection, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework clause (iii));
• a chart entry: a smooth chart $(U,x)$ of $M$ (Smooth manifolds and their smooth charts);
• an azimuth entry: a continuous real function $\phi :K\to \mathbb{R}$ on a compact parameter interval $K=[t_1,t_2]$ of a worldline entry, with values in radians, together with the local azimuth chart designations and compatibility conditions specified in (b)4; this is supplied data, not an assertion that such a function exists;
• a detector entry: the data of the ideal two-body detector of the linearized-wave regime, namely a TT linearized-wave datum, a reference point $x_0$, a separation vector $\delta \mathrm{\ne }0$ with ${\delta }^0=0$, the arm direction $e_{\delta }$, and a parameter value $s$, as in The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path clauses (a)-(c);
• a parameter entry: a finite tuple of real numbers in a declared subset of ${\mathbb{R}}^m$, each component with declared units;
• an annotation entry: a finite string recording declared conditions of the configuration and, where a datum is taken from a report, the bibliographic identification of that report.

A model configuration is a description of a hypothetical model situation. The tuple asserts nothing about any apparatus and nothing about nature, and no clause of this definition asserts that a configuration with prescribed entries exists: existence claims belong to the items that construct the entries.

(b) Readouts. Let $D$ be a model configuration. The following readouts are defined whenever $D$ contains the entries named in the clause together with the stated incidence conditions; each is a real number or a finite tuple of real numbers determined by the entries of $D$.

1. Frequency ratio and redshift. Suppose $D$ contains a ray entry $\eta :J\to M$ with tangent $W:={\eta }^{\mathrm{\prime }}$, worldline entries $h_E:I_E\to M$ and $h_R:I_R\to M$ with four-velocities $U_E,U_R$, and parameter values $s_E<s_R$ in $J$, $t_E\in I_E$, $t_R\in I_R$ with $\eta (s_E)=h_E(t_E)$ and $\eta (s_R)=h_R(t_R)$. The observer energies of the ray at the two events relative to the unit observers $U_E\mathrm{/}c$ and $U_R\mathrm{/}c$ are

   $E_E:=-c{g}_{\eta (s_E)}(W(s_E),U_E(t_E)\mathrm{/}c),E_R:=-c{g}_{\eta (s_R)}(W(s_R),U_R(t_R)\mathrm{/}c),$

   the values of clause (a) of Observer-relative energy, spatial momentum and rest mass of a tangent vector for those observers; both are strictly positive by claim 1 of Pairing of an observer with future causal vectors, because $W$ is nonzero future null and $U_E\mathrm{/}c$, $U_R\mathrm{/}c$ are future unit timelike. The frequency-ratio readout and the redshift readout are

   $\mathcal{F}:=\frac{E_R}{E_E}=\frac{g_{\eta (s_R)}(W(s_R),U_R(t_R))}{g_{\eta (s_E)}(W(s_E),U_E(t_E))},z:=\frac{E_E}{E_R}-1=\frac{1}{\mathcal{F}}-1.$

2. Radar round-trip proper time. Suppose $D$ contains worldline entries $h_R:I_R\to M$ (the receiver) and $h_C:I_C\to M$ (the reflector), parameter values $t_1<t_2$ in $I_R$ and $s_C\in I_C$, and ray entries ${\eta }_1,{\eta }_2$ whose parameter intervals contain values exhibiting ${\eta }_1(s_1)=h_R(t_1)$, ${\eta }_1(s_1^{\mathrm{\prime }})=h_C(s_C)$ with $s_1<s_1^{\mathrm{\prime }}$, and ${\eta }_2(s_2)=h_C(s_C)$, ${\eta }_2(s_2^{\mathrm{\prime }})=h_R(t_2)$ with $s_2<s_2^{\mathrm{\prime }}$. The radar readout is the receiver proper-time increment $\mathrm{\Delta }\tau :=t_2-t_1$ between emission and reception. It is independent of the affine parametrizations of ${\eta }_1,{\eta }_2$, which enter only through the incidence statements (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold clause (d), Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).

3. Deflection angle. Suppose $D$ contains a ray entry $\eta :J\to M$ with tangent $W$, parameter values $s_1<s_2$ in $J$, observer entries $n_1,n_2$ at $p_1:=\eta (s_1)$ and $p_2:=\eta (s_2)$, and an isometric-identification entry $\mathrm{\Phi }:T_{p_1}M\to T_{p_2}M$ of (a) satisfying $\mathrm{\Phi }(n_1)=n_2$. Put $q_i:=h_{n_i}(W(s_i))$, the spatial projection of clause (c) of Observers, rest spaces and the spatial projection at an event, and suppose $q_1,q_2\mathrm{\ne }0$. Then $\mathrm{\mid }{q}_i{\mathrm{\mid }}_{n_i}:=\sqrt{g_{p_i}(q_i,q_i)}>0$, because $g_{p_i}$ is positive definite on the rest space (The observer rest-space decomposition and the spatial projection), and $\mathrm{\Phi }(q_1)$ lies in $n_2^{\perp }$ with $g_{p_2}(\mathrm{\Phi }{q}_1,\mathrm{\Phi }{q}_1)=g_{p_1}(q_1,q_1)$. The deflection cosine and deflection readout are

   $\kappa :=\frac{g_{p_2}(\mathrm{\Phi }(q_1),q_2)}{\mathrm{\mid }{q}_1{\mathrm{\mid }}_{n_1}\mathrm{\mid }{q}_2{\mathrm{\mid }}_{n_2}}\in [-1,1],\mathrm{\Theta }:=\arccos \kappa ,$

   where the bound is the Cauchy-Schwarz inequality in the real inner product space $(n_2^{\perp },g_{p_2})$ (Real and complex inner product spaces, with the inner product linear in the first argument, Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors) and $\arccos$ is the principal inverse cosine of Principal inverse sine and inverse cosine.

4. Azimuthal increment and perihelion advance. Suppose $D$ contains a worldline entry $h:I\to M$, a chart entry $(U,x)$ with $h([t_1,t_2])\subseteq U$ for $t_1<t_2$ in $I$, and an azimuth entry $\phi :[t_1,t_2]\to \mathbb{R}$. Put $r(t):=x^1(h(t))$ on $h^{-1}(U)\subseteq I$. Require $t_1,t_2$ to be strict local minima of $r$ (relative to its domain in $I$), with no strict local minimum between them; these are the designated successive perihelion passages. For every $t\in [t_1,t_2]$ require a smooth chart $(V,y)$ containing $h(t)$, whose dimensionless coordinate $y^2$ is designated as a local azimuth, and a relative parameter neighborhood $J_t$ of $t$ on which $h(J_t)\subseteq V$ and $\phi (u)-y^2(h(u))\in 2\pi \mathbb{Z}$ for every $u\in J_t$. Here $2\pi \mathbb{Z}:=\{2\pi k:k\in \mathbb{Z}\}$, and the designations specify a common angular orientation and period $2\pi$. The supplied continuous $\phi$ is the unwrapped azimuth along this traversal; local azimuth representatives alone are not sufficient input. The azimuthal-increment readout and the perihelion-advance readout are

   $\mathrm{\Delta }\phi :=\phi (t_2)-\phi (t_1),A:=\mathrm{\Delta }\phi -2\pi \mathrm{.}$

   This convention subtracts one positive circuit; for a negatively oriented traversal it still uses the displayed signed formula. The radial and angular designations are part of the declared configuration, not a dynamical prediction or a chart-invariance assertion. Adding a constant $2\pi k$ to the supplied lift leaves the increment unchanged. A traversal with $\phi (t_2)=\phi (t_1)+2\pi$ has $A=0$ even when its endpoint local azimuth representatives coincide. No lift-existence theorem is assumed: the readout is defined only when the required lift is supplied.

5. Arrival times. Suppose $D$ contains worldline entries $h_E,h_R$, a finite family of ray entries ${\eta }_1,\dots ,{\eta }_N$ and parameter values such that ${\eta }_k$ joins an event of $h_E$ to $h_R(t_k)$, with $t_1<\cdots <t_N$ in $I_R$. The arrival-time readout is the tuple $(t_1,\dots ,t_N)$ of receiver proper-time values.

6. Strain. Suppose $D$ contains a detector entry of (a) and a parameter value at which the quantity under the square root of clause (c) of The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path is positive. The strain readout is the value $S_{\delta }(s)$ of the strain function of that clause.

(c) Operational models. 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 whose coordinates carry declared units; $\theta :S\to \mathrm{\Theta }$ is a surjection, the parameter map; $D$ assigns to each $s\in S$ a model configuration $D(s)$ of (a); and for some integer $d\ge 1$ a 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 value that the framework's definitions, postulates and proved consequences assign to a readout of (b) of the configuration $D(s)$, with every intermediate quantity, event, frame, sign and unit declared. In particular, a component asserted to be a frequency ratio, redshift, radar, deflection, azimuthal, advance, arrival-time or strain value must be the corresponding readout of (b) for entries that the configuration contains, and a component read from a cited prediction theorem must be a value that theorem assigns to entries of $D(s)$ satisfying its hypotheses. 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 any component of $F$ is measured.

(d) 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 as in (a). The record declares: for each $k$, the value of the readout 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.

(e) 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 as in (a); it declares that for every $z\in Z$ obtained under the condition annotated by $h$, the value of the declared readout lies in the closed box ${\prod }_{j=1}^d[c_j(z)-{\delta }_j(z),c_j(z)+{\delta }_j(z)]$ in the declared units. A calibration record is a data record of (d) whose declared readout 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 $\delta$.

(f) Uncertainty vocabulary. For a data record of (d), 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 (d) and (e); no probability law, coverage probability, confidence level, p-value or posterior is defined by this vocabulary.

(g) 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)-(f): it requires its own statement, an explicitly declared comparison rule, and adequate primary-report and analysis prerequisites. A declared rule of the form "$t\in \mathrm{\Theta }$ is consistent with the one-point hard-bound record $R$ exactly when $F(t)$ lies in the record's closed box", together with the resulting consistency set $C(R,M):=\{t\in \mathrm{\Theta }:F(t)\in {\prod }_j[z_{1,j}-{\epsilon }_{1,j},z_{1,j}+{\epsilon }_{1,j}]\}$, is a comparison rule in this sense; it is conditional on the record's declared bounds and on the interpretation $(\ast )$, and it establishes neither the truth of a postulate nor the correctness of any apparatus. The source design's research prerequisite labelled M20, the primary-report and analysis layer, is not supplied by the items of this page: no reported measurement and no experiment item occurs here. The framework's model-side prediction content for the readouts of (b) currently consists of the restricted redshift theorems pthm-gr-schwarzschild-stationary-redshift (static observers in the Schwarzschild exterior model) and pthm-gr-flrw-cosmological-redshift (comoving observers in a homogeneous cosmological model) for readout 1, and of the ideal-detector definition def-gr-linearized-wave-regime-and-detector-coupling with the response theorem pthm-gr-plane-wave-tidal-detector-response for readout 6; the propagation-delay, deflection, perihelion-advance and arrival-time readouts of 2-5 have no in-framework prediction theorem yet and are recorded as report destinations in rem-gr-primary-report-research-queue. These item names are nonlogical status pointers; they add no premise to this definition.

(h) Well-definedness and scope. All components of (a)-(g) are finite tuples of sets, maps, curves and real numbers with declared domains and units. The constructions use no choice principle beyond those already assumed by the cited suppliers: Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model uses countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth structures of tangent bundles, and nothing here uses any further choice. No measurability, integrability, completeness, differentiability or convergence beyond the cited items is used; the continuous unwrapped azimuth in (b)4 is supplied explicitly. Clause (c) evaluates a prediction only at parameter values for which the cited items supply its ingredients, and each readout of (b) is defined only for configurations carrying the stated entries and incidence conditions, so this definition asserts no existence of worldlines, rays, charts, detectors or configurations. 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.