Can primordial deuterium fix the dark-matter density? DESI DR2 distances read with Ω_c/Ω_b = 3√3 and no CMB, the Hubble constant, a₀ = cH₀/6, and a registered test for DESI beyond DR2

Mathematical Materiality (MM) in brief. MM starts from one thesis: the laws of physics are not rules that something makes matter obey; matter is the relations of mathematics itself. Nothing exists on its own: to exist means to exist for another logical entity, and outside this web of relations there is nothing, not even other universes. In the first logical nexus between elements that are homogeneous in Einstein's sense, namely space and time, there is a non-local phase, made only of relations that sit in no place. It is not a distant level: it is present in every phase of reality, in every place and at every instant. It shows in entanglement, where two distant systems remain correlated more strongly than any explanation with separate properties and no influence faster than light allows, without anything travelling between them. In this non-local phase the two logical distincts exchange only a common logical greatest common divisor (GCD), the single logical element that makes the two entities logically exist for each other. The local world arises when a relation is counted through this GCD: its fractions or multiples form a matrix, and, read by an observer inside it like us, the multiples appear as time and the fractions as space. That same multiple, immediately contained in the matrix of the point of observation, generates mass, that is, matter as it is ordinarily understood. The observed objects, while being logically autonomous entities in their local relations, are related to the observer through a local fractional diffraction of the observational matrix. This is why the universe behaves like a simulation without being one. It follows exact rules, made of counts, everywhere and at every instant, but no machine runs them and no programmer wrote them: a structure of mathematical relations does not need to be computed in order to exist. That such a simulation need not be run had already been noted by Tegmark (2008); Vazza (2025) estimated that running it on a machine would require resources that the laws of the universe do not allow. This record develops one aspect of this framework. Four-page summary: DOI 10.5281/zenodo.23535917. Entry page, with the list of all records: DOI 10.5281/zenodo.22667516. Standard big-bang nucleosynthesis, used here as a dictionary of laboratory data, constrains the baryon density ω_b with the deuterium measured in quasar absorbers (and helium, in the primary dictionary). If the dark-matter density is ω_c = 3√3·ω_b, the 13 DESI DR2 BAO distances are read with one cosmological parameter fewer than in flat ΛCDM. No CMB data enter the fits. Results. The constraint costs Δχ² = 0.09 (0.02-0.42 with five BBN dictionaries, one of them deuterium only). H₀ = 68.52 ± 0.59 km s⁻¹ Mpc⁻¹, the known BAO + BBN value: a compatibility, not a new number. a₀ = cH₀/6 = 1.110·10⁻¹⁰ m s⁻², 0.4σ from the SPARC value; with SPARC's systematic error this does not separate H₀ = 68.5 from 73. Read the other way, the BAO alone with 3√3 fix ω_b and predict D/H = (2.40 ± 0.31)·10⁻⁵, against 2.51 measured. D/H does not enter that fit, but the check shares the BAO and the constraint with the ratio test, so it is not independent of it. A premise of the framework on time treats the standard account of the first minutes as a reading in the present. Six translations into measured quantities were registered before computing, and run with PRyMordial (NACRE II rates with LUNA): a common clock leaves the dictionary unchanged; D/H shows no significant variation among 12 absorbers, at the sensitivity of the tests; the expansion rate is compatible with the local one (ΔN_eff = −0.10 ± 0.18); a hidden mirror sector at the same temperature, with gravitating radiation, does not hold in the two equivalent cases examined (helium +20σ and more; the full thermal history is not simulated); helium gives a weak hint for the bottle neutron lifetime; the baryon-to-photon ratio read in deuterium is compatible with the CMB value (the only comparison with CMB data, made after the fits). The predicted lithium stays about 3.5 times the observed value in every reading that holds. Registered test, before the data: the first DESI release beyond DR2, read with the same chain without the CMB. Δχ² ≤ 4 «holds», 4-9 «tension», > 9 «the reading 3√3 does not read BAO + BBN». A secondary block compares a₀ = cH₀/6 with the first g† measured with a total error ≤ 0.06·10⁻¹⁰. An addendum registered before the data reports the same computation with evolving dark energy next to the main block, with a qualifier fixed in advance. Decision programs with tests are included. With evolving dark energy (w₀w_a, DESI DR2 + BBN, without and with four supernova samples) 3√3 stays compatible in all five combinations (Δχ² 0.02-1.85), with a wider error on the ratio; in ΛCDM the supernovae push the ratio upwards (cost of 3√3 up to 3.1 with DES-SN5YR). 3√3 is a reading chosen knowing the data, not a derivation; the outcome of the test concerns that reading. BAO + BBN → H₀ is a known method (Addison et al. 2018; Cuceu et al. 2019; Schöneberg et al. 2019, 2022; DESI 2025). Specific to this analysis is the imposed ratio, q = Ω_c/Ω_b = 3√3, in place of a measured baryon fraction. Contents: protocols and registers (SHA-256 and UTC time before each run), code, logs, the prediction and its decision program, verification reports by Codex (GPT-6) with the corrections, and run_all.py, which reruns everything and compares it with the reference logs. Third-party data and code (DESI likelihood files, PRyMordial) are not redistributed; scarica_dati.py downloads them and checks their SHA-256. The author is not a professional physicist; computations were done with AI assistants (Claude, with checks by Codex), not reviewed by a human referee.