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

Abstract

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.