Scope and empirical status of the Bell-models page: conditional model results and the deferred experiment destination

What the page contains. Mathematical vocabulary for binary correlation models and their local-response representations (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional, Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws), the CHSH bound for every representable model (The CHSH bound for every binary correlation model admitting a local-response representation), the preservation of representability and of the bound under a setting-independent selection (Conditioning on a setting-independent selection preserves local-response representability and the CHSH bound), the exact joint laws of the two-factor singlet (Singlet pairings: the joint laws of the two spin-half factors along two directions), the declared spin correlation arrangement of the framework (Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models), the singlet computation with the failure of local-response representability at violating settings (The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings) and the Tsirelson bound with its attainment (Tsirelson bound for every declared spin correlation arrangement of the framework). All are conditional model statements; the spin-arrangement and spin-comparison claims carry the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and the Axiom of Choice (The Axiom of Choice) stated in their supplier hypotheses.

What the page does not contain. No experiment item, no reported count, no calibration, no detector efficiency, no loss model, no background, no trial-level uncertainty analysis, no p-value, no confidence level and no measured correlation. None of the computations establishes that the framework or any of its postulates describes nature, and none of them asserts that any laboratory procedure realizes the declared arrangement. The marginals, tables and correlations are model probabilities for the declared preparation; the identification of a concrete apparatus with the declared measurements is the tested identification with uncertainty of clause (iii) of the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Treating repetitions as independent and identically distributed draws is the additional statistical model of clause (ii), to be declared and checked for the apparatus and procedure; it is not implied by the state postulate.

Deferred experiment destination. The design of this pair reserves an experiment item for a Bell test. Such an item is not built here: it would require reading a primary or authoritative full report and recording its source-specific preparation and apparatus, calibration, measured outputs, trial count where reported, detection efficiency, loss and post-selection, background, the reported statistical treatment and systematics, and the source's omissions; none of the sources read for this page supplies that level of detail. The sources' experimental remarks motivate the model comparison only: Binney's section 6.1.4 refers to the 1972 Freedman-Clauser test and cites a 1998 photonic test, and the MIT chapter closes with the statement that experiments have confirmed violations; no numerical value from those secondary accounts is reproduced here as an observation, and no interval, count or precision is asserted. The statistics and primary-report gap named in the design is recorded as an open destination.

Source qualifications not imported. The same sources contain general conclusions that the comparison rules out hidden variables as such or that quantum mechanics cannot be superseded by a theory with a definite spin direction. Those conclusions are stronger than the conditional statement proved here — which excludes only the declared local-response class of the declared tables — and are not imported. Likewise the operator bound used on this page is proved by the operator-norm route of the cited mathematics (the identity for the square of the CHSH operator and the norm bound, invoked in Tsirelson bound for every declared spin correlation arrangement of the framework fact F6 and proof step 1.2). That route uses $\mathrm{\parallel }[A_0,A_1]\mathrm{\parallel }\le 2$ and $\mathrm{\parallel }[B_0,B_1]\mathrm{\parallel }\le 2$ for the supplier's self-adjoint involutions $A_i,B_j$, where $[A,B]:=AB-BA$ is the commutator; no reversed variance inequality of the form $⟨C^2⟩\le ⟨C{⟩}^2$ is used as a premise.

What a comparison with data could show. Under declared preparation, detection, settings, selection and sampling models, agreement of recorded frequencies with the arrangement's tables supports the combined model and apparatus assumptions. The bound $\mathrm{\mid }S(\mu )\mathrm{\mid }\le 2$ constrains the correlations of a local-response model's probability tables, rather than every finite sample estimate $\widehat{S}$. A finite recorded estimate with $\mathrm{\mid }\widehat{S}\mathrm{\mid }>2$ can have positive probability under that model; finite-sample fluctuations are treated in Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict ↗. Evidence against a specified model class requires an explicit statistical test with its sampling assumptions, uncertainty and error control, together with the apparatus and selection qualifications. A threshold crossing alone does not deductively exclude the class, and a p-value is not the probability that the class is false. Agreement does not prove the framework, and a null or reduced result does not by itself falsify it.

Interpretation and scope of the model. The arrangement assigns no spatial positions, separation or temporal order to the two factors, so no relativistic locality principle is modelled here; the exclusion is a statement about the four tables. The boundaries of the comparison are recorded in Measurement-model boundaries of the singlet-local comparison: what the no-go excludes and what it does not, and no interpretation of the violation beyond those boundaries is asserted.