Do wide binaries follow a scale set by the cosmic expansion? A registered protocol for Gaia DR4, with a dry run on DR3

This record registers, before Gaia DR4 (planned for 2 December 2026), a test of wide binaries. It is one of the checks of Mathematical Materiality (MM): the outcome concerns MM's reading of wide binaries, not the framework. Two MM readings are compared. The main one is a chain of correspondences that takes no parameter from the binaries. It links three things that seem unrelated: the acceleration scale a₀ = cH₀/6, from the cosmic expansion (H₀ = 67.4 km/s/Mpc); the law read in galaxies, ν(y) = 1/(1 − e^(−√y)); the external field of the Milky Way, from its rotation. The field equation (QUMOND) is borrowed. The declared alternative is the reading of MM 2.54, which says that wide binaries are Newtonian. MOND with the external field is reported as a reference. The data and the rules are fixed now. The pairs of El-Badry, Rix and Heintz (2021) will be re-measured in DR4, with the cuts frozen on 11 October 2026. Stars with a DR4 non-single-star solution are removed. A forward simulation models hidden companions. The decision rule compares the chain with Newton and has written validity conditions: the test can favour one reading, or come out inconclusive or not valid. A dry run on DR3 (data already seen during development) reproduces the development sample exactly. It gives: the chain 2.4 log-likelihood units below the best MOND fit; Newton 10 units below; a DR3 decision of «chain favoured», with the minimum margin only just met. DR4 reuses DR3 stars and observations, so a favourable outcome would be a confirmation on non-independent data (grade B*), not a blind prediction. Contents: the protocol (Italian, with an English README), the download, conversion, simulation and evaluation code, the selected sample with Gaia identifiers, the dry-run logs, fingerprints and registration times. The orbit libraries (1.3 GB) are not included; they can be regenerated from the code and the seeds. Two verification rounds by Codex/GPT-6: the points raised were corrected, and the last one was tested. The author is not a professional physicist; computations were done with AI assistants (Claude, with checks by Codex), not reviewed by a human referee.