A discrepancy can test auxiliaries as well as field assumptions

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.