Measurement-model boundaries of the singlet-local comparison: what the no-go excludes and what it does not

What the comparison establishes. For the declared spin correlation arrangement of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models, the four sign tables are the laws ${\nu }_{ij}^{\rho }(\epsilon ,\delta )$ of clause (d) of that definition. If the arrangement's CHSH functional satisfies $\mathrm{\mid }S({\nu }^{\rho })\mathrm{\mid }>2$, then the four tables admit no local-response model: every local-response representation would give $\mathrm{\mid }S\mathrm{\mid }\le 2$ by The CHSH bound for every binary correlation model admitting a local-response representation. The singlet at the directions of The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings is the displayed instance, with $\mathrm{\mid }S\mathrm{\mid }=2\sqrt{2}$. The statement is a conditional no-go for one precisely defined class of models of one precisely defined set of four tables.

What is not excluded. The theorem concerns models of the four tables that consist of one specification space whose probability law is the same for all setting pairs, together with per-station response laws that do not depend on the distant setting (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws clauses (a), (c)). Models outside that class are not addressed: a model whose specification or response is allowed to depend on the distant setting, a model that does not reproduce all four tables simultaneously, or a model of a different arrangement or of a different measurement structure is not excluded. In particular the comparison does not rule out "hidden variables" in general, and it does not claim that no completion of quantum mechanics exists; it excludes the declared local-response class for these tables.

Assumptions used. The deduction rests on the framework readings P1, P2, P3, P5, P6 and P7 fixed by Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models, on the declared composite, preparation and spin-half assignments, on the ideal sharp character of the declared measurements (Declared ideal measurements and instruments of the framework), on 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 the arrangement and singlet-attainment hypotheses, and on the cited mathematics (the finite-dimensional tensor calculus, the CHSH bound for representable models and the operator bound of Tsirelson bound for every declared spin correlation arrangement of the framework). Changing one of these readings voids the conditional conclusion for the changed model; it does not weaken it, and a replacement conclusion needs its own proof.

Measurement-model boundary. The declared sign measurements have exactly two outcomes with sign labels $\pm 1$; detection efficiency, finite resolution, dark counts, background, drift and loss are outside the arrangement and require their own apparatus models. In particular an observed violation on a recorded sub-ensemble does not by itself contradict the declared model class: the recorded tables are conditional tables, and the CHSH bound transfers to them when the selection is determined by a positive-probability event of the specification space that is the same for every setting pair, as in Conditioning on a setting-independent selection preserves local-response representability and the CHSH bound; that lemma is a sufficient condition, not a necessary one, and is not a claim about any real detector. A different selection model may also yield representable conditional tables; its consequences require their own justification. Finite samples add a sampling model and its fluctuations, treated on the examples page. Whether a finite record deductively excludes one of two specified models depends on their sampling supports: a record with positive probability under both is compatible with both, while an observed zero-probability record can exclude one specified ideal model conditional on its assumptions. No universal claim is made about every model pair or every finite sample.

No-signalling is not the separator. Every local-response model is no-signalling (Local-response representability implies setting-independent marginals), and every declared arrangement's marginals are also setting-independent: summing its joint law over $\delta$ gives ${\nu }_{ij}^{\rho ,A}(\epsilon )=\mathrm{tr}(\rho (P_{a_i,\epsilon }\otimes I))$ because ${\sum }_{\delta }{P}_{b_j,\delta }=I$, and the symmetric sum is ${\nu }_{ij}^{\rho ,B}(\delta )=\mathrm{tr}(\rho (I\otimes P_{b_j,\delta }))$ (Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models clauses (a), (d)). For the singlet both marginals are $1\mathrm{/}2$ (The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings clause (d)); a discarded declared local instrument on one factor leaves the other factor unchanged (Local instruments and no-signalling: operations on one factor do not change the other factor's local statistics clause (b)). The contrast between the two classes therefore lies in the joint tables, not in any dependence of one station's marginal law on the distant setting.

Attainment and sharpness. The value $2\sqrt{2}$ is attained for the declared arrangement (Tsirelson bound for every declared spin correlation arrangement of the framework clause (b)), so within the declared class of binary sharp readouts of self-adjoint involutions no smaller uniform bound for the CHSH functional is available. This says nothing about other functionals, other numbers of settings, unsharp measurements or other systems.

Interpretation. The result is a statement about four probability tables. It identifies no mechanism, no influence between the stations and no preferred frame; the arrangement carries no spatial or temporal structure. Interpretations that attach additional physical content to the violation are not part of the deduction and are not asserted here. Whether a laboratory arrangement realizes the declared model and what a recorded sample shows are separate matters under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).