Representations, Not Revolutions: Czachor's Calculus and Bell's Theorem

M. Sienickia, K. Sienickib aPolish-Japanese Academy of Information Technology, ul. Koszykowa 86, 02-008 Warsaw, Poland bChair of Theoretical Physics of Naturally Intelligent Systems, Lipowa 2/Topolowa 19, 05-807 Podkowa Leśna, Poland

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We examine recent claims that ``non-Newtonian'' arithmetic and calculus topple Bell's theorem. Our basic point regarding such a claim is straightforward: the expectation functional used in those papers is linear only with respect to the deformed sum ⊕, not the ordinary +. Consequently, the familiar Clauser-Horne and Clauser-Horne-Shimony-Holt derivations - which lean on linearity under ordinary addition - do not apply. Within a single arithmetic level, a Bell-type analogue can be formulated if the outcomes and expectation values are defined in that level and satisfy linearity with respect to the level's addition ⊕; however, the standard Clauser-Horne/Clauser-Horne-Shimony-Holt proof for ``+'' is inapplicable. The eye-catching ``beyond-Tsirelson'' effects show up only when levels are mixed - thus, one computes with standard rules on quantities defined in a deformed calculus, producing out-of-range aggregates (e.g., totals exceeding one) rather than single-event probabilities. The touted ``relativity of observed probabilities'' also splices together two different moves, conditioning on a restricted sample space versus pushing everything through a scalar remapping. A simple horizon toy model already shows that there is no single-valued remapping that accomplishes this globally. The analogy with Einstein velocity addition helps a little in one dimension; in three dimensions, it collapses. There, the composition is non-commutative and non-associative, and the right language is a gyrogroup structure, not a pullback of ordinary addition. Bringing in Lambare's measurement-independence critique, we further argue that Czachor's reply (built from a hand-tuned bijection and a non-additive integral) addresses neither that objection nor Bell's own premises. In short, the program amounts to a representational re-encoding, not a counterexample of local hidden-variable.

DOI:10.12693/APhysPolA.148.273 topics: Bell and Clauser-Horne-Shimony-Holt (Bell-CHSH), non-Newtonian arithmetic, deformed expectation, level mixing