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. Standard quantum mechanics has no mass at which a superposition stops: large objects look classical because of decoherence, the environment that keeps recording where they are. Collapse models (Ghirardi–Rimini–Weber, CSL, Diósi–Penrose) add such a limit by hand. The framework says that a threshold exists: below a certain mass level the principal GCD of the point of observation and the GCD of the diffraction that generates the object become equal, and the object spreads over several positions (see the box above). The box gives no number. This deposit tries to give the threshold a number, or to say clearly why it cannot. A second part tests two later remarks of the author: the limit cannot be fixed as a mass, because a calculation of the «fractional distance» is missing; and large distant bodies stay fixed because their size makes up for the distance. It was announced in deposit 2.64 (DOI 10.5281/zenodo.23610071). The main result: for distance, the limit is not a mass alone. In the two readings of the fractional distance that hold, it is a ratio between the size that has to be told apart and the cell of the copies that do the telling, together with the number of copies; the readings of part one with a mass threshold at the Planck scale remain possible, untested, and that the two readings are one rule of the framework is a proposal. For the state of an object the copies are the photons and molecules the environment carries away, and one copy cannot tell apart two positions closer than about half a wavelength: a 16-microgram crystal stays quantum over 2 × 10⁻¹⁸ m because about 10³⁴ thermal copies would be needed to fix it. For the way a distant observer reads an object, the cell is the diffraction cell of the observer's aperture at the object's distance: two stars of almost the same mass fall on opposite sides (the disc of Sirius B, 1.02 solar masses at 2.64 pc, inferred from its radius, is 20–60 times smaller than the finest cell of today's interferometers; τ Ceti, 0.78 at 3.65 pc, and 61 Cyg A, 0.67–0.74 at 3.49 pc, are resolved): with this instrumental classification the sample does not support a mass-only limit growing with distance. Both rules coincide with known physics (decoherence and wave optics): the framework admits them; experiments put them to the test but cannot tell them apart from standard physics. Prior work, first. Decoherence: Joos and Zeh (1985), Zurek (2003), Schlosshauer (2019); measured against separation and number of photons: Chapman et al. (1995), Kokorowski et al. (2001), Hornberger et al. (2003), Hackermüller et al. (2004). Collapse models: Károlyházy (1966), Ghirardi, Rimini and Weber (1986), Diósi (1987, 1989), Pearle (1989), Ghirardi, Pearle and Rimini (1990), Penrose (1996), the review of Bassi et al. (2013), Howl, Penrose and Fuentes (2019). Tests and limits: Nimmrichter and Hornberger (2013), Romero-Isart (2011), Donadi et al. (2021), the Majorana Demonstrator (2022), Carlesso et al. (2022), Gasbarri et al. (2021, MAQRO). Experiments: Kovachy et al. (2015), Bild et al. (2023), Panda et al. (2024), Pedalino et al. (2026). Distance and observers: Bell tests from 398 m to 1 203 km and with cosmic settings, Wigner's-friend tests (Proietti et al. 2019; Bong et al. 2020; Zeng et al. 2025), stellar interferometry (van Cittert–Zernike), the threshold of vision (Hecht, Shlaer and Pirenne 1942; Tinsley et al. 2016). How it was tested. Part one: an inventory of nine experiments with quantum behaviour and of the limits on collapse models, read in 20 papers; ten readings of the thesis, each identifying the two GCDs with physical quantities (choices made by the AI, not the author's words), registered with their outcome rules before any calculation. Part two: an inventory of 22 more sources; eight readings of the «fractional distance», registered before any calculation; a test with existing data registered before its primary sources were read (its main values and expected outcome were known from the abstracts: not a blind test). Do not follow from the data (registered label «excluded»): the internal acceleration compared with the galactic scale a₀ (sodium clusters of 172 000 u bound at 1.3 × 10¹⁶ m/s² interfere, Pedalino et al. 2026); a threshold at the proton or electron mass (a caesium atom kept in superposition for 70 s, Panda et al. 2024); a₀ written as a mass. In part two: «for the state of an object only the distance of the observer counts» (heating fullerenes, with the same apparatus and observer, takes their visibility from 47 % to 0 % through photons nobody collects, Hackermüller et al. 2004); a mass-only limit of the reading growing with distance, in the sample and with the instrumental classification registered; the fragmentation of deposit 2.36.2 as the carrier of images, and three fixed values of its deposited share as the rule of what is sharp; a sharp boundary at the diffraction cell. Alive but not testable today: a threshold at the Planck mass (22 μg) or at m_P/√2 (15.4 μg), at the mass of the observer (70 kg), and of the Diósi–Penrose type; in part two, the distance between a system and whoever records it (no experiment changes only that), a Wigner's friend with a macroscopic mass, and the fractionation tied to the travel
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