Empirical tests and limitations — Examples

For every $\delta \in \mathbb{R}$ the predicted value of the displaced pair $(s+\delta ,b-G\delta )$ is $G(s+\delta )+(b-G\delta )=Gs+G\delta +b-G\delta =Gs+b$: distributivity gives $G(s+\delta )=Gs+G\delta$, associativity and commutativity of addition regroup the four terms as $Gs+(G\delta -G\delta )+b$, and $G\delta -G\delta =0$ with $Gs+0+b=Gs+b$; hence $y(s+\delta ,b-G\delta )=y(s,b)$ for every common recorded coordinate, which is clause (i).

Any recorded value or declared closed bound interval compatible with the prediction of the pair $(s,b)$ is compatible with the prediction of $(s+\delta ,b-G\delta )$: a recorded value equal to the predicted value is equal to the other predicted value by step 1.1, and a declared interval of the form $[y-r,y+r]$ with $r\ge 0$ is unchanged because its centre is unchanged; the argument applies at every recorded coordinate and to every declared calibration box of clause (e), so comparison through this affine response alone does not separate the pairs.

If $G\mathrm{\ne }0$ and $\delta \mathrm{\ne }0$ then the measured coordinates of the two parameter pairs differ: the source-strength coordinates $s+\delta$ and $s$ differ because $\delta \mathrm{\ne }0$, and the background coordinates $b-G\delta$ and $b$ differ because $G\delta \mathrm{\ne }0$; thus $(s+\delta ,b-G\delta )\mathrm{\ne }(s,b)$ in ${\mathbb{R}}^2$ while their predictions coincide by step 1.1, which is clause (ii).

The explicit instance of clause (iii) is the arithmetic evaluation $y(3,1)=2\cdot 3+1=7$ and $y(4,-1)=2\cdot 4-1=8-1=7$ at $G=2$, $s=3$, $b=1$ and $\delta =1$, where the displaced pair is $(s+\delta ,b-G\delta )=(4,1-2)=(4,-1)$.

Comparison through the affine model alone constrains only $Gs+b$: an exact equality to $z$ fixes this combination to $z$, while $\mathrm{\mid }Gs+b-z\mathrm{\mid }\le r$ constrains it to $[z-r,z+r]$ and need not determine its value. Two pairs $(s_1,b_1)$ and $(s_2,b_2)$ have the same prediction exactly when $G{s}_1+b_1=G{s}_2+b_2$; taking $\delta =s_2-s_1$ then gives $b_2=b_1-G\delta$. Thus the family $\{(s+\delta ,b-G\delta ):\delta \in \mathbb{R}\}$ is exactly the set of pairs with the same prediction. Independent information is needed to distinguish its members: fixing $s$ fixes $b$ through the prediction, and fixing $b$ fixes $s$ since $G\mathrm{\ne }0$. These are sufficient examples, not the only possible independent constraints.

Clause (iv) records the status of the preceding steps: the identity is a deterministic algebraic statement about the declared affine response model, registered as the record's analysis assumption by clause (g) of [F2]; no apparatus property, historical value, dataset, sampling law or probability is asserted. Agreement or discrepancy tests the conjunction of the model with its declared auxiliary parameters; independent information of the kind described in step 3.1 is needed to resolve their ambiguity before attributing an outcome to a field or source premise.

Since $\delta (x_0)\equiv \pi (\mathrm{m}\mathrm{o}\mathrm{d}2\pi )$, there is an integer $n$ with $\delta (x_0)=\pi +2\pi n$; the cartesian form of [F6] gives $e^{i\delta (x_0)}=\cos \delta (x_0)+i\sin \delta (x_0)$, and [F7] gives $\cos (y+2\pi )=-\cos (y+\pi )=\cos y$ and $\sin (y+2\pi )=-\sin (y+\pi )=\sin y$ for every real $y$, so by induction on $\mathrm{\mid }n\mathrm{\mid }$ one has $\cos (\pi +2\pi n)=\cos \pi =-1$ and $\sin (\pi +2\pi n)=\sin \pi =0$; hence $e^{i\delta (x_0)}=-1$.

[Regularity of $\alpha$] The phase $x\mapsto \delta (x)=(k_2-k_1)\cdot x$ is an affine map ${\mathbb{R}}^3\to \mathbb{R}$, hence of class $C^{\mathrm{\infty }}$; the exponential is entire [F19], hence complex differentiable everywhere and continuous [F20]; by the algebra-and-composition closure of [F21] the composite $x\mapsto e^{i\delta (x)}$ and the finite combinations defining the components of $\alpha =a_1{e}_1+a_2{e}_2{e}^{i\delta (x)}$ of [F1] are continuous, and the modulus squared $\mathrm{\mid }\alpha {\mathrm{\mid }}^2=\alpha \cdot \bar{\alpha }$ [F5] is a finite sum of products of continuous real functions, hence continuous on ${\mathbb{R}}^3$.

[Constant background] For a constant $b=b_0>0$ the zero extension is the nonnegative simple function $\tilde{b}=b_0{\chi }_W$ with $W\in \mathcal{L}({\mathbb{R}}^3)$ [F10, F11]; its simple integral is $b_0{\lambda }_3(W)$ by [F17] and equals its Lebesgue integral by [F18], so ${\int }_{W}bd{\lambda }_3=b_0{\lambda }_3(W)$ by [F14], a value in $(0,+\mathrm{\infty }]$ that may be $+\mathrm{\infty }$ when ${\lambda }_3(W)=+\mathrm{\infty }$.

[Zero response coefficient] For $\kappa =0$ the field contribution is the zero function because $\mathrm{\mid }\alpha {\mathrm{\mid }}^2$ is real-valued, so no product $0\cdot (+\mathrm{\infty })$ is formed; then ${\int }_W\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^{2}d{\lambda }_3=0$ by the $c=0$ clause of [F16], and $R(W)={\int }_{W}bd{\lambda }_3$; in particular the response remains defined in $[0,+\mathrm{\infty }]$ on every measurable bin.

Substituting the amplitude data of [F1] and step 1.1 gives $\alpha (x_0)=a_1{e}_1+a_2{e}_2{e}^{i\delta (x_0)}=ae(1+(-1))=0$, using $a_1=a_2=a$ and $e_1=e_2=e$.

For $\beta (x_0)$: substituting $a_1=a_2=a$, $e_1=e_2=e$ and step 1.1 into [F1] gives $\beta (x_0)=a(b_1-b_2)$, and $b_1-b_2=k_1\times e\mathrm{/}\omega -k_2\times e\mathrm{/}\omega =((k_1-k_2)\times e)\mathrm{/}\omega$ by the coordinate bilinearity of [F3]; if $\beta (x_0)=0$ then multiplying by the nonzero scalar $\omega \mathrm{/}a$ (nonzero because $a>0$ and $\omega >0$ from [F1]) gives $(k_1-k_2)\times e=-(k_2-k_1)\times e=0$ by [F3], contrary to the hypothesis $(k_2-k_1)\times e\mathrm{\ne }0$; hence $\beta (x_0)\mathrm{\ne }0$.

[Integrand measurability and the response functional] Under ${\mathrm{A}\mathrm{C}}_{\omega }$ the family $\mathcal{L}({\mathbb{R}}^3)$ is a sigma-algebra and ${\lambda }_3$ a measure on it [F11], and the bin $W$ is measurable with ${\lambda }_3(W)$ defined [F10]; $\mathrm{\mid }\alpha {\mathrm{\mid }}^2$ is continuous by step 1.2, hence Borel measurable [F12] and Lebesgue measurable [F13]; since $\kappa \in \mathbb{R}$, the scaled function $\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2$ is measurable by [F15], and the sum $\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+\tilde{b}$ is measurable by [F15] because $\tilde{b}$ is real-valued and measurable by the declared detector model; consequently [F14] defines $R(W):={\int }_W(\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+b)d{\lambda }_3=\int (\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+\tilde{b}){\chi }_{W}d{\lambda }_3\in [0,+\mathrm{\infty }]$.

[Finite-rate branch] Suppose $0<{\lambda }_3(W)<\mathrm{\infty }$ and $b\equiv b_0$ with $0<b_0<\mathrm{\infty }$. For every $x$ one has $\mathrm{\mid }{e}^{i\delta (x)}\mathrm{\mid }=1$ by [F6], so $\mathrm{\mid }1+e^{i\delta (x)}\mathrm{\mid }\le 1+\mathrm{\mid }{e}^{i\delta (x)}\mathrm{\mid }=2$ by the triangle inequality of [F5], and $\mathrm{\mid }\alpha (x)\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }e\mathrm{\mid }\mathrm{\mid }1+e^{i\delta (x)}\mathrm{\mid }\le 2a\mathrm{\mid }e\mathrm{\mid }$ by the multiplicative law of [F5], where $\mathrm{\mid }e{\mathrm{\mid }}^2=⟨e,e⟩$ is the Euclidean squared norm of the real amplitude [F4]; hence $\mathrm{\mid }\alpha {\mathrm{\mid }}^2\le 4a^2\mathrm{\mid }e{\mathrm{\mid }}^2$ and, since $\kappa \ge 0$, the pointwise bounds $b_0{\chi }_W\le (\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+b_0){\chi }_W\le C{\chi }_W$ hold with the finite real constant $C:=4\kappa a^2\mathrm{\mid }e{\mathrm{\mid }}^2+b_0$. Monotonicity and the simple-integral values of step 1.3 and [F17], [F18] give $0<b_0{\lambda }_3(W)\le R(W)\le C{\lambda }_3(W)<\mathrm{\infty }$, the strict lower bound by the sign rule $b_0{\lambda }_3(W)>0$ of [F22] applied to $b_0>0$ and ${\lambda }_3(W)>0$; this proves a finite positive numerical response. By the units declared in clause (ii), $\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2$ and $b_0$ both carry $\mathsf{U}{\mathrm{s}}^{-1}{\mathrm{m}}^{-3}$, while ${\lambda }_3$ carries ${\mathrm{m}}^3$, so $R(W)$ is a finite positive predicted response rate in $\mathsf{U}{\mathrm{s}}^{-1}$.

By clause (a) of [F1], $E(x_0,t)=\mathrm{Re}(\alpha (x_0)e^{i(k_1\cdot x_0-\omega t)})=\mathrm{Re}(0)=0$ for every real $t$; hence the electric field of the superposition vanishes at $x_0$ at every time, and with step 2.1 the electric term $\frac{{\epsilon }_0}{4}\mathrm{\mid }\alpha (x_0){\mathrm{\mid }}^2$ of the averaged density of [F2] vanishes there.

[F2] and step 2.1 give $⟨u⟩(x_0)=\frac{{\epsilon }_0}{4}\mathrm{\mid }\alpha (x_0){\mathrm{\mid }}^2+\frac{1}{4{\mu }_0}\mathrm{\mid }\beta (x_0){\mathrm{\mid }}^2=\frac{1}{4{\mu }_0}\mathrm{\mid }\beta (x_0){\mathrm{\mid }}^2$; since $\beta (x_0)\mathrm{\ne }0$ by step 2.2, positive definiteness [F4] and the modulus law $\mathrm{\mid }z\mathrm{\mid }=0⟺z=0$ of [F5] give $\mathrm{\mid }\beta (x_0){\mathrm{\mid }}^2>0$, and ${\mu }_0>0$ from [F8] gives $1\mathrm{/}(4{\mu }_0)>0$, so the sign rules [F22] yield $⟨u⟩(x_0)>0$; thus the full averaged energy density does not vanish at $x_0$, only its electric term does.

[General background lower bound] The functions $\tilde{b}$ and $\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+\tilde{b}$ are measurable, nonnegative and finite-valued, with $\tilde{b}\le \kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+\tilde{b}$ pointwise because $\mathrm{\mid }\alpha {\mathrm{\mid }}^2\ge 0$ [F5] and $\kappa \ge 0$; the monotonicity clause of [F16] and $R(W)=\int (\kappa \mathrm{\mid }\alpha {\mathrm{\mid }}^2+\tilde{b}){\chi }_{W}d{\lambda }_3$ of step 2.3 give ${\int }_{W}bd{\lambda }_3=\int \tilde{b}{\chi }_{W}d{\lambda }_3\le R(W)$, so ${\int }_{W}bd{\lambda }_3>0$ implies $R(W)\ge {\int }_{W}bd{\lambda }_3>0$ even though $\mathrm{\mid }\alpha (x_0){\mathrm{\mid }}^2=0$ by step 2.1.

[Record versus prediction] The data of [F23] are finite tuples of recorded values or reported observations with declared conditions, calibration and uncertainty, and contain no sampling law; the predictions of steps 1.1-3.3 are deterministic field quantities or a declared extended response functional with a finite real rate only on the branch of step 2.4; a detection and a finite record are different objects from the ideal intensity, so no sampling law, probability of failing a visibility criterion, detection statistics or measured histogram is asserted here, and the finite positive bin prediction of step 2.4 is compatible with the pointwise ideal zero of steps 2.1 and 3.1 without discriminating the classical and quantum descriptions of light, since classical electromagnetic waves interfere.