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. In logic, nothing exists on its own: to exist means to exist in relation to another logical entity, and outside this web of relations nothing can exist, not even other universes. In the first logical nexus between space and time, which with Einstein became homogeneous, there is a non-local phase, made only of a logical distinction: space is not time, with no relation of quantitative calculation between the two, except the common logical entity they share, which is needed to make them exist for each other. 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 distinct terms, space and time, share only one logical element, which we shall call their greatest common divisor (GCD), just as the numbers 3 and 5 share only 1. The local world arises when a relation is counted through this GCD: its multiples and its fractions form a matrix. Read by an observer inside it like us, the multiples appear as time and the fractions as space. That same multiple, when it is directly contained in the matrix of the point of observation, turns into mass, that is, into concrete matter as it is ordinarily understood. Time and mass are therefore closely related, in agreement with Einstein's relativity, which ties mass to spacetime. The observed objects, while being logically autonomous entities in their local relations, are related to the observer through a logical diffraction, in fractions, of the observer's matrix. This reading also gives a reason for the limit of the speed of light, which in Einstein's relativity is a starting principle. The speed of light is the fixed rate of exchange between the multiples and the fractions of the same GCD, that is, how much space one step of time is worth: exceeding it would mean, for some observer, covering space without counting any time, that is, leaving the local world. Logic always starts from a logical relative, and this rate too has nothing absolute about it: it is a necessary relative, which inside the system becomes an absolute hypothesis, the same for everyone and impossible to exceed. Outside the system, by contrast, it has no justification, because outside there is only non-existence: justifying something requires another term to relate it to, and there is none there. Asking for a reason from outside is like asking what lies north of the North Pole. The same framework that reads space, time, mass and the observed objects also admits quantum mechanics. The non-local phase and the logical mirror reciprocity between the two ends of the nexus give a reason for the probabilistic behaviour of matter. When the observed object is large enough, its mass fixes it in the matrix of the point of observation: we see it in one place and at one instant. When it falls below a certain mass level, the principal GCD, that of the point of observation, and the GCD of the diffraction that generates the object are forced to become equal, as in a mirror: the object no longer occupies a single point of the spacetime generated by the observer but spreads over several possible positions, and we observe it as probability. It is like having to describe a 10 using only the number 1: the only way is to distribute the 1 over ten positions, one for each unit of the 10. Today even objects made of thousands of atoms behave in this way, and no experiment has yet found the boundary. The idea that the passage from quantum to classical behaviour depends on mass is close to that of Penrose and Diósi; in MM the threshold is the point where the two GCDs, the principal one and that of the local diffraction, become equal. For all these reasons 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 the framework. Overview of the framework: DOI 10.5281/zenodo.23535916. Entry page, with the list of all records: DOI 10.5281/zenodo.22667516. This record gives the dynamics of Mathematical Materiality (MM). In the weak field it is linearised Einstein gravity, and the homogeneous model is Friedmann's. The junction to the a₀ = cH₀/6 layer is read as a boundary set by the Galaxy (the rule L*), and a premise of the author gives that boundary with two declared choices. The record also registers, before Gaia DR4 (planned for 2 December 2026), a test of this reading on wide binaries, with rules fixed before the data. It is one of MM's checks: the outcome concerns one reading of MM, not the framework. The dynamics: The linearised Bianchi identity follows from MM's lattice action, with a binding stress. The open node is the junction between MM's closure and the a₀ = cH₀/6 layer: local versions (QUMOND, AQUAL, combinations, an extra field) fail on Cassini or on granularity, or are not decidable; the non-local rule L* passes, but it is posited, not derived. With the full vector force of the external field, L* and QUMOND differ in wide binaries mainly through the anisotropy described below. The prediction. In MM's reading L*, a pair of stars reads only its own field inside a boundary, where its own Newtonian field exceeds the Galactic one (7 773 AU for 1.5 solar masses near the Sun). The MOND-like boost is then the same in every direction. QUMOND with the external field predicts instead, inside that boundary, that pairs aligned with the Galactic field move more slowly. Newton predicts no difference. The statistic is the ratio of the mean perpendicular velocities of aligned and crossing pairs, in the band 0.3-0.86 r_M. Simulated before the data, it gives: QUMOND: −3.1 % to −0.4 %; L*: −1.2 % to +1.2 %; Newton: around zero. A first look at DR3 (El-Badry et al. 2021 pairs, cuts frozen on 11 October 2026): +1.8 % ± 2.7 %. This is not decidable, as written down before the data (expected power 0.8σ). A control band shows +2.7 % ± 1.3 % (2.1σ): a possible systematic to understand before DR4. For DR4 the record fixes an exact replica: same pairs, band, statistic, predictions and rule. The download chain is the one of MM 2.56, and it is dry-run on DR3. With this observable DR4 will most likely give «not decidable» as well; the record fixes the test in time. A premise that gives the rule. The author's premise is that the galaxy a system sits in acts on the system's observation scale, which in turn determines its own field. Translated with two declared choices (a scalar reading; the boost with which a system is read cannot exceed the one with which the Galaxy reads its position), it gives the boundary and the isotropy that L* posited. Outside the boundary it predicts an isotropic boost of 1.455 instead of 1.26-1.33. Put through the registered machinery of MM 2.56 on DR3 development data, this law fits best of eight hypotheses. Against MM's QUMOND chain it is not decidable (+3.4, threshold 5); against Newton it is favoured (+11.1). This is a posteriori coherence: the idea arose partly from these data. A rule for DR4 is registered, with a registered estimate of its power: with the same pairs DR4 will probably not separate the two readings (power 0.12 or 0.03). A compatible reason is that they differ mainly in amplitude, which a free mass scale can absorb. Contents: protocols, simulations, data sample, code and logs, with registration fingerprints and UTC times; the final notes of each level of the dynamics (in Italian, with an English README); the reports of two AI verifiers. The author is not a professional physicist. Computations were done with AI assistants (Claude, with checks by Codex and Gemini), not reviewed by a human referee.
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Ugo Lanciano (2026) studied this question.
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