Local Born geometry and the Tsirelson bound: registered consistency checks, checkerboard interference and a retrocausal singlet model

Abstract

Bell experiments without loopholes give CHSH values above the local bound 2, and high-precision measurements are compatible with Tsirelson's quantum bound 2√2 (Poh et al. 2015: 2.82759 ± 0.00051). This record asks what the author's framework, Mathematical Materiality (MM), must contain to respect both facts, and where it then stands among interpretations of quantum mechanics. No mathematical priority is claimed: the theorems used are published (Tsirelson; Barnum et al. 2010; Sorkin's quantum measures; Craig, Dowker et al. 2007; Aharonov's two-time formalism; Cramer; Argaman 2010; Wharton and Argaman 2020), and the record separates published results, derivations in the working notes and numerical conjectures. Predictions were registered with SHA-256 fingerprints before each test; failed predictions are reported. (1) Local readings and the Tsirelson bound. If each wing is read with the Born geometry (Euclidean state space with all its effects), no no-signalling joint event exceeds 2√2. Replacing the Euclidean ball with ℓ^p balls raises the maximum: 2^(2−1/p) for p ≥ 2 (the value of Ver Steeg and Wehner 2009), above 2√2 also for 1 < p < 2 (exact value conjectural). With marginals of length m the maximum is the quantum 2√(2 − m²). Two registered predictions failed: a hand formula, wrong for p < 2, and a three-dimensional search that stayed below the value already reached by the two-dimensional construction. (2) Two slits on the Feynman checkerboard (MM 2.40). In the 2.40 representation the imaginary turn weight −i sin θ produces destructive interference (dark fringes); with positive weights the two-slit pattern never falls below the sum of the single slits. The fringes fall at −143.2 and +140.4 lattice sites against ±142.1 predicted from the lattice dispersion. For the preparations tested, a real coin gives the same probabilities. A first registered run failed because its "central region" was badly defined; the second run was registered after seeing the first. (3) Can a future boundary give Born frequencies? In a minimal model, a future that selects the outcome reproduces Born frequencies only with some rules (weighting each future by the closure of its loop, or an exponential race), not with others. The measure over futures remains a postulate. (4) A locally mediated retrocausal singlet model. The future reading of one wing fixes a parameter carried back to the source and forward to the other wing, which reads it with the local Born rule. Singlet correlations are reproduced with every link along paths inside light cones. Within this family, no observable signalling holds only for the balanced (Born) weight, as already discussed by Argaman (2010); the angular law is assumed. (5) Self-consistency and the balanced weight. In the pair model the Born weight is put in by hand. With any other weight a bit could be signalled through the loop and, composed with a second link (Tolman's antitelephone, assumed), into the sender's past; a paradox protocol then makes consistent histories as improbable as one likes (0.02 with p = 0.51 and 10 001 pairs), while with the Born weight the channel carries nothing. This does not make non-Born weights impossible: the step from "arbitrarily improbable" to "cannot exist" is stated as the author's postulate. (6) Placement and corrections. MM is placed among retrocausal readings with local mediation (in the Local Friendliness taxonomy: absoluteness of observed events abandoned for readings that leave no copies, No-Superdeterminism violated by future-input dependence, temporal causal arrow abandoned, relativistic causality retained as a programme assumption). Five statements in versions 2.7–2.27 of the status report are corrected, including the claim that MM is non-local "in the sense of Bell's theorem" and a conditional theorem on outcomes, withdrawn. Nothing here distinguishes MM from standard quantum mechanics by experiment, and the Born rule is not derived. python codice/run_all.py --short reproduces all checks except three long programs in about 15 minutes; the full run takes about 2 hours. The author is not a professional physicist; computations were done with AI assistants (Claude and Codex), not reviewed by a human referee.