Comment on 'The axiom of choice and the no-signalling principle'
Pith reviewed 2026-05-10 00:34 UTC · model grok-4.3
The pith
Deterministic no-signalling resources are not stronger than probabilistic ones once measurability is required in infinite-party games.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The main claim of the work being commented upon is that functional (deterministic) no-signalling resources can be stronger than probabilistic ones in a certain nonlocal game on a Bell scenario with countably many parties. We disagree and argue that (i) under standard definitions, deterministic no-signalling resources are always probabilistic no-signalling resources; (ii) the deterministic strategy considered can be promoted to a genuinely probabilistic strategy with similar properties and (iii) a key step in the derivation, claimed to hold for all no-signalling strategies, implicitly assumes measurability, leaving a gap in the argument. We propose measurability assumptions which we conjectur
What carries the argument
The implicit measurability condition on no-signalling strategies in countably infinite Bell scenarios, which separates resources that satisfy the disputed derivation from those that do not.
If this is right
- Deterministic no-signalling resources qualify as probabilistic ones under standard definitions.
- The deterministic strategy examined extends to a probabilistic strategy preserving comparable properties.
- The original derivation applies only after an unstated measurability condition is imposed.
- The reported phenomenon amounts to a difference between measurable and non-measurable no-signalling resources.
Where Pith is reading between the lines
- Similar hidden measurability requirements may exist in other derivations involving infinite collections of parties.
- Imposing measurability could align infinite-party results more closely with finite-party treatments of no-signalling.
- The distinction suggests that physical realizability of strategies may hinge on avoiding non-measurable constructions.
Load-bearing premise
The key step in the original derivation holds for every no-signalling strategy without any restriction to measurable ones.
What would settle it
An explicit construction of a non-measurable no-signalling strategy for which the disputed derivation step fails to hold.
read the original abstract
The main claim of arXiv:2206.08467 is that "functional (deterministic) no-signalling resources can be stronger than probabilistic ones" a certain nonlocal game on a Bell scenario with countably many parties. We disagree and argue that (i) under standard definitions, deterministic no-signalling resources are always probabilistic no-signalling resources; (ii) the deterministic strategy considered in arXiv:2206.08467 can be promoted to a genuinely probabilistic strategy with similar properties and (iii) a key step in the derivation in arXiv:2206.08467, claimed to hold for all no-signalling strategies, implicitly assumes measurability, leaving a gap in the argument. We propose measurability assumptions which we conjecture would make this derivation rigorous. Taken together, the phenomenon highlighted in arXiv:2206.08467 is best understood as a difference between measurable and non-measurable no-signalling resources.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript is a comment on arXiv:2206.08467 claiming that functional (deterministic) no-signalling resources can be stronger than probabilistic ones in a nonlocal game on a Bell scenario with countably many parties. The authors disagree and present three arguments: (i) under standard definitions, deterministic no-signalling resources are always probabilistic no-signalling resources; (ii) the deterministic strategy considered in the original work can be promoted to a genuinely probabilistic strategy with similar properties; and (iii) a key step in the original derivation, claimed to hold for all no-signalling strategies, implicitly assumes measurability without stating it. They propose measurability assumptions as a conjecture that would make the derivation rigorous and conclude that the phenomenon is best understood as a difference between measurable and non-measurable no-signalling resources.
Significance. If the arguments hold, the comment is significant for clarifying a foundational subtlety in resource theories of no-signalling in infinite-party scenarios. It correctly notes that deterministic strategies form a subclass of probabilistic ones and identifies an unstated measurability gap in universal claims over all no-signalling strategies. The constructive proposal of conjectured measurability assumptions provides a clear path to strengthen related derivations, which is valuable for future work on Bell scenarios with countably infinite parties.
minor comments (2)
- The three-part argument in the abstract is logically consistent with standard definitions, but the manuscript would benefit from explicitly quoting or citing the precise step from arXiv:2206.08467 that is alleged to assume measurability (e.g., the relevant equation or paragraph in the original derivation).
- The conjecture on measurability assumptions is presented clearly but could be stated more formally, perhaps as a numbered conjecture, to facilitate future verification or adoption by the community.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of our manuscript, their accurate summary of our arguments, and their recommendation to accept. We are pleased that the significance of clarifying the role of measurability in infinite-party no-signalling scenarios has been recognized.
Circularity Check
No significant circularity
full rationale
The comment paper advances three definitional and conceptual points: (i) deterministic no-signalling resources are formally a subclass of probabilistic ones under standard literature definitions, (ii) the specific deterministic strategy lifts to a probabilistic one preserving the relevant properties, and (iii) the original derivation's universal claim over all no-signalling strategies leaves an unstated measurability gap. These steps rest on external standard definitions and an explicit conjecture rather than any self-referential construction, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or claim reduces to its own inputs by construction; the argument is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Under standard definitions, deterministic no-signalling resources are always probabilistic no-signalling resources.
- ad hoc to paper A key step in the original derivation implicitly assumes measurability.
Reference graph
Works this paper leans on
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discussion (0)
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