Not at a distance, but nowhere: non-locality beyond entanglement, in twelve phenomena

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. In physics «non-locality» mostly names one fact: the correlations of entanglement. In Mathematical Materiality (MM) it is the foundation: a non-local phase, not located in space and time and present everywhere, of which the local world is the count made from a point. Here «non-local» means not located in space and time; it does not mean action at a distance. This record collects, from MM's published records and the author's premises, twelve phenomena in which MM's reading uses the same non-local phase as the reason, and shows that entanglement is the case in which it rests most directly on a measured datum, not the only one. There are no new calculations: the result is the unity of the reading. The twelve phenomena: the correlations of entanglement beyond Bell's limit, and their Tsirelson ceiling, set by the local reading (MM 2.48, 2.48.1); the same laws everywhere and always, as copies of the same nexus (MM 2.68); identical particles, as the same unit counted twice (a proposal of the AI); the objects two observers see alike, as two counts of one non-local relation (MM 2.62.1); the boundary of the local as a number of copies, not a distance (August status report, reread); probability and the Born rule (MM 2.48.1); the mass below which an object stops being at one point (MM 2.66); the present, and the loop in which the future chooses what is written (MM 2.48, 2.68.1); space and time themselves: a clock counting events without pre-assigned places runs at Einstein's proper time (MM 2.40.2); the limit of light, the same for everyone; dark matter as the ordinary matter of a second mirror, with no reciprocal non-local link to us (MM 2.44); the boundary within which a gravitating system reads only its own field, set by the Galaxy (MM 2.70). How each is set out. For every phenomenon: what is observed, with sources; how known physics reads it; what the non-local phase does; where the link comes from (already written in a deposit, a premise of the author, or a proposal of the AI); what rests on calculations already deposited; what remains open. In ten phenomena the link is already in the deposits or in the author's premises; in one it comes partly from the AI, and in one from the AI alone. Three ways organise them: copies without distance, existence through a link, and a reading that is always local and partial. What is not claimed. The non-local phase transmits no signals and no influences outside light cones. Today it has no distinct experimental signature: where MM gives numbers, they coincide with known physics (the Tsirelson bound, decoherence, wave optics) or come from readings with declared choices. The record also lists the other senses in which «non-local» appears in MM's records, so that they are not confused with this one. Where the reading is tested. Gaia DR4 in December 2026 (rules registered in MM 2.70) and the mass threshold, with a prediction dated 31 December 2035 (MM 2.66); five open questions from MM 2.68. Contents: README.md (English), LEGGIMI_IT.md (the Italian original), the author's sentences with exact words and dates, the reference check and the review reports. There is no code: the calculations are in the records cited. The author is not a professional physicist. The text was written with an AI assistant (Claude) and checked by a second instance of Claude, by Codex (Astra) and by Gemini (Argon); it has not been reviewed by a human referee.