This repository contains the source associated with:
J. Beau, Breakdown of Bell Factorization from Non-Injective Effective Descriptions, Quantum Reports 8 (2026), 44, DOI 10.3390/quantum8020044.
The article must not be cited as proving that non-injectivity of an observable map is sufficient to obstruct Bell factorisation. That implication is false. The author has notified the journal and requested the appropriate linked post-publication procedure.
The current scientific audit is recorded outside this repository in "../quantum-structure/notes/bell-paper-central-claim-audit.md".
Let
[ \Omega={-1,+1}^4\times{0,1}, \qquad \Pi_{xy}(A_0,A_1,B_0,B_1,r)=(A_x,B_y), ]
with the uniform measure. Every (\Pi_{xy}) is non-injective, with fibres of size eight. Nevertheless, (\lambda=\omega) gives deterministic local responses, a setting-independent measure, and CHSH value zero. More generally, adjoining an independent erased label makes any Bell-local model non-injective without changing its observable statistics.
The formal appendix in the article restricts (\lambda) to a function of (\Pi(\omega)). Bell factorisation permits the complete underlying configuration itself as the conditioning variable, so that restriction does not prove the claimed theorem.
The proposed global assignment requires
[ a_0b_0=a_0b_1=a_1b_0=+1, \qquad a_1b_1=-1. ]
The product of the four constraints gives (+1=-1). The constrained support is empty, so no normalised measure reproduces the claimed PR-box distribution.
The article defines
[ S_\Pi=-\sum_o\nu(o)\log\nu(o), \qquad \nu(o)=\mu(\Pi^{-1}(o)). ]
This is the Shannon entropy of the output distribution, not a measure of non-injectivity. A constant, maximally non-injective map has (S_\Pi=0), while an injective map can have maximal output entropy. The proposed entropy--CHSH interpolation therefore has no derived endpoints.
- Bell violation excludes a common Bell-local factorisation under the standard assumptions.
- Global contextual constraints can obstruct a setting-independent joint assignment.
- Non-injectivity can accompany either Bell-local or Bell-nonlocal statistics.
The missing Cosmochrony input is a derived obstruction to a setting-independent global coupling across measurement contexts. Bare non-injectivity does not provide that obstruction.
The repository source and Zenodo record are not silent replacements for the journal version of record.