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. Cassini ranging data bound the external-field quadrupole of the Solar System at Q₂ = (1.6 ± 1.8)·10⁻²⁷ s⁻² (Park et al. 2026). The wide-binary chain registered in MM 2.56 has these ingredients: a₀ = cH₀/6; the law ν(y) = 1/(1 − e^(−√y)); the Milky-Way field from its rotation; QUMOND. Taken down to the Sun with the same field equation, it predicts Q₂ = 3.22·10⁻²⁶ s⁻², 17σ above the measurement (17.0–19.4σ over eight input choices; the MM 2.56 solver gives the same number within 3·10⁻⁵). A local reading of the external field. Here the anomaly lives in the relation between two bodies. Who reads the external field. The body whose local MCD prevails reads the external field for its whole system: the Sun for planets and comets, a galaxy for its stars and gas. The local MCD is R_s/r, compared at the two ends of the relation, so the comparison is the mass ratio. The two stars of a wide binary read the field each for itself. A dominant compact body reads the relation itself, at its surface, where the relation is Newtonian. A relation without a dominant body is read as the time average along its orbit. How the fields combine. Interior and exterior fields add as magnitudes inside ν. Direction. The Galactic MCD difference across a pair enters through a tensorial alignment (cos²θ); its amount w is an observational input, compatible with zero on DR3 once the close-pair level is subtracted. Light reads the curvature with the same law, and not the external field. What it gives. Saturn: Q₂ ≈ 10⁻³⁰ s⁻². Planets and comets on any orbit: Newtonian. The star S2 around Sgr A*: no anomalous boost (general relativity remains). The 919 low-acceleration wide binaries of MM 2.56: a boost of 1.27 in radial gravity. Disc galaxies: the radial acceleration relation is unchanged. In 78 SPARC galaxies with the environment fields of Chae et al. 2021, the field entering in full (Σχ² = 12 490.8) is preferred to no field (13 447.5) and to a diluted field (13 341.2). Weak lensing (KiDS-1000, 60 points): χ² = 219.9, against 1 687.8 for Newton. An environment field of 0.003 a₀ in the lensing reading is excluded (Δχ² ≥ 552). Around galaxies in denser environments light reads two to three times more at large radii than around isolated galaxies of the same mass. Among isolated galaxies, red and blue differ by a nearly constant ratio, so this excess is compatible with the satellites of the full sample rather than with a reading of light that grows with density. Wide-binary anisotropy in the data of August 2026: once the close-pair level is subtracted, compatible with zero (w < 0.024); the registered DR4 program reads the same statistic. A sky dipole that the sample systematics do not reach. Close pairs: +0.0001 ± 0.0018 dex, while their quadrupole shows +0.0041 ± 0.0018. The anisotropy form is symmetric: the boost is present also when the companion lies behind the star, so the predicted dipole is zero. On DR3 it is +0.0027 ± 0.0046 dex. Systems without rotation. Classical dwarf spheroidals agree within ±0.15 dex. Ultra-faint dwarfs move 2.5–20 times faster than the reading predicts. Distant globular clusters (Palomar 14, NGC 2419) are Newtonian. At the same gravitational state (19 dwarfs and 11 clusters in the common range) galaxies carry the boost and clusters do not. In the Local Volume Database all 38 systems with a metallicity spread (stars of different generations) carry the extra mass. Its amount follows a common profile set by the radius, M_dyn ∝ r_½^1.54±0.11 (81 dwarfs; 1.40 ± 0.11 without gas-rich systems), the same for satellites and isolated dwarfs, with an intrinsic scatter of 0.48 dex. Further checks: matter reads the baryonic external field (the total field worsens SPARC by Δχ² = 2 757); no significant dependence of the boost on the mass ratio; the DR3 gain comes from the fast pairs; the Gaia value of the Galactic field lowers the excess by 6–7 %. Registered for Gaia DR4. The reading has its own orbit library, the unchanged forward model of MM 2.56, and a decision rule against Newton (threshold ±5 in log-likelihood, subsets ±2). The copied evaluation reproduces the MM 2.56 dry run exactly. A draft protocol with three addenda registers the anisotropy test: w in three separation bands; a consistency requirement; the dipole as the statistic between a symmetric reading and one read by the dominant star. On DR3 (data already seen), in the registered simulation series, the local reading has the highest likelihood of the seven hypotheses: +16.1 over Newton, +6.1 over MOND simple and +8.5 over the QUMOND chain. A registered noise test with new seeds and new libraries gives: the local reading over Newton: +11.0 on average over four series (from +8.2 to +16.1), above the threshold in every series; the chain over Newton: from −0.9 to +12.1; the local reading over the chain: from −2.0 to +9.0. A rule over several simulation series is therefore registered for DR4, with its programs, validity checks and the two new orbit libraries already generated. Predictions. Wide-binary boost and anisotropy in DR4, with a zero dipole. Newtonian distant globular clusters and boosted dwarf galaxies. Newtonian light companions at wide separation. Newtonian comets and Oort-cloud bodies. No external-field effect in weak lensing. The MM 2.56 protocol is unchanged. Contents: inventories, protocols and outcome notes (Italian, with an English README), the structure of the local reading and the numbers of its checks, code with exact diffs against the MM 2.56 programs, the MM 2.56 sample, logs, JSON results, the DR3 simulations, fingerprints and registration times. The cited papers, the larger data files and the orbit libraries (about 210 MB each, regenerable) are not included. Five verification rounds by Codex/GPT-6; the last two covered the later additions, and their corrections changed no outcome. The author is not a professional physicist; computations were done with AI assistants (Claude, with checks by Codex), not reviewed by a human referee.
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Ugo Lanciano (2026) studied this question.
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