Real quantum theory in the entanglement-swapping network: an exact strategy at 4√3 and machine-checked upper bounds

Renou et al. (2021) showed that real and complex quantum theory can be told apart in a network with two independent sources: a Bell-type quantity T reaches 6√2 in complex quantum theory, while real quantum theory with independent sources obeys T ≲ 7.66. We narrow the real-theory value from both sides.

(i) An explicit real strategy with independent sources reaches T = 4√3 ≈ 6.9282 exactly; the value is verified in Lean. (ii) T ≤ 7.6568542496 for real quantum theory, below the published 7.6605: an exact integer certificate, accepted by a checker whose soundness is proved in Lean, bounds a symmetry-reduced semidefinite relaxation, and a written proof shows every real strategy with independent sources is feasible for it. (iii) With two measurements for Alice, T ≤ 7.1927385035 for all strategies with independent sources, real or complex, by a scalar-extended two-copy inflation; a single shared source reaches 2+4√2 there. (iv) Explicit hidden cross-source correlations that let a real description exceed the bound, with thresholds of 0.56, 0.50 and 0.47 bit for three constructions.

An explicit point shows that this relaxation cannot give less than 2+4√2, so its value is fixed to within 1.3e-11. The optimal real value remains open in [4√3, 7.6568542496]. The deposit contains the note, the certificates (text and binary), a dependency-free Python checker, the Lean sources of the proved checker and of the strategy's value, and the scripts that build, export and attack the certificates.

Version 3: the real-theory bound is tightened from 7.6568542701 to 7.6568542496 (a certificate of the same kind on a more accurate solve; the earlier certificate is kept), and an explicit point of the relaxation at exactly 2+4√2 is added. Nothing stated in versions 1 and 2 is withdrawn. Version 2 had made the archive's Lean folders complete Lake projects.