Selection and loss: the precise hypothesis under which the CHSH bound transfers to recorded pairs

Recorded tables are conditional tables. The arrangement's laws ${\nu }_{ij}^{\rho }$ of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models clause (d) describe a declared pair of local sign measurements for every setting pair on the declared preparation. A comparison with recorded data ordinarily uses only the runs in which both stations register an event; the laws of the recorded runs are therefore conditional laws of a larger model, and they are not the arrangement's tables unless a detection and selection model connecting the two is declared. This is part of the preparation and apparatus discipline of P7 clause (i) (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

A sufficient condition for the transfer. If the full tables admit a local-response representation and the recorded runs are exactly those specifications lying in a measurable selection event $D$ of the specification space with $\mathbb{P}(D)>0$ — membership is determined by the specification alone, using the same event for every setting pair — then the conditional tables again admit a local-response representation, with the same response laws over the conditional measure, and hence satisfy the CHSH bound (Conditioning on a setting-independent selection preserves local-response representability and the CHSH bound). This is the mathematical content of a setting-independent detection or fair-sampling hypothesis for the declared model class. It does not require statistical independence of $D$ from the outcomes: reweighting the specifications can change the tables.

The condition is a hypothesis, not a theorem about detectors. The selection-closure lemma is a conditional statement about the declared model. It does not assert that any laboratory detector, coincidence circuit or post-selection rule realizes a selection event of the required form; identifying a concrete apparatus with such a model is a tested identification with uncertainty under P7 and carries its calibration and control-experiment qualifications. Efficiency, dead time, misalignment, background and drift each require their own declared model.

No transfer without a selection model. If the recorded sample is selected by a rule that depends on the settings or on the observed outcomes, the selection-closure lemma does not apply, and no claim is made here that the CHSH bound constrains the recorded tables. Correspondingly, a deviation computed from recorded pairs is a statement about those recorded tables and their own declared model; it becomes a statement about the full arrangement only through an explicitly justified detection and selection model; the condition above is sufficient, not necessary. Finite-sample fluctuations are a further, separate matter. The result Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict assumes $n$ trials per setting pair, all $4n$ outcome pairs mutually independent, with each trial law equal to the arrangement's table ${\nu }_{ij}^{\rho }$. It applies to recorded pairs only when their declared sampling model satisfies those hypotheses. A postselected law different from ${\nu }_{ij}^{\rho }$, random recorded trial counts or correlations induced by selection require a separate analysis; the cited result does not supply one. The singlet contrast itself is the conditional model computation of The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings.

Settings choice. In a specified experiment model, let $Q_{ij}$ be the actual conditional probability law of the specification given that setting pair $(i,j)$ is chosen whenever that conditional law is defined. The common-law condition required by a local-response representation is that these actual conditional laws are all one common law, $Q_{00}=Q_{01}=Q_{10}=Q_{11}$ (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws clause (c)); if a setting pair is never chosen, its law must still be specified by the comparison model before the four-table representation is applied. The possibility that a setting procedure could depend on the specification does not establish that the actual conditional laws differ; an analysis using a local-response representation must establish their equality from its model and the relevant evidence. This page neither assumes nor reports such a setting procedure or its conditional laws.