Who enforces the laws of physics? No one, if matter is the relation itself

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. Version 2.68.1. Section 6 now reports a single calculation that puts the three sentences of the author on time into one model: the loop of 2.48, in which the future chooses the whole history of readings; the record of 2.64.1, which keeps only the past; and the direction of time as the growth of the record. With the future choosing by the closure of the loop, the sentences do not contradict each other: the first readings have the same statistics whether the history stops there or continues (cells already written do not change, by construction), the record grows at every reading and gives the Born frequencies, and in a heat testbed it grows while heat flows backwards, on average and along the most probable history, not in every history. Two outcomes are weaker than expected: on the checkerboard the growth stays below the threshold registered beforehand, and «only the past» does not by itself select the Born rule: at the registered threshold only a deterministic future violates it, and only in the translation of the weight of the future used in 2.48 (the «vettore» convention of the calculation); in the other («blocco») no rule does. The author's answer on what the future chooses closes the first open question of section 10. Protocol, programs, results and two rounds of checks by Astra and Gemini are in the folder CONTO_UNICO. Everything else is the text of 2.68. This record. Philosophers ask who or what makes the laws of physics hold everywhere and always. This deposit, written for philosophers of physics, sets out the author's answer: matter does not obey relations, it is the relations, so the laws need no guardian, and their necessity is a necessity of identity, as in arithmetic. It shows the thesis at work on a clock, where two declared premises give Einstein's proper time as the geometric mean of two coprime light sides, and it shows how the same framework reads the dimensions of space, objects, mass, quantum probability and the direction of time. It also says which sense of «non-local», the foundation of the framework, is meant: not located in space and time, with non-separability as a consequence; Bell correlations are read with a retrocausal loop with local mediation, and action at a distance is excluded. For each thesis it says what rests on registered calculations, what is a premise and what is open. Where it stands. The framework is a form of ontic structural realism close to its moderate variant (Esfeld and Lam 2008). It is set beside its oldest neighbour, the Pythagoreanism of Croton; Tegmark's mathematical universe; the debate on laws of nature (Armstrong, Beebee, Maudlin, Bird), where it is closest to Bird's dispositional essentialism, with arithmetic relations in place of dispositions; emergent spacetime (Huggett and Wüthrich 2025); Bondi's k-calculus and Knuth and Bahreyni (2014); quantum Darwinism and relational quantum mechanics; retrocausal readings; mass-dependent collapse; and, on non-locality, Bohm's implicate order, Howard and Teller on non-separability and Van Raamsdonk on space from entanglement. Objections. Four objections a philosopher will raise are answered. What distinguishes a physical structure from a mathematical one: the language, space-time against mental on/off. Why only our structure exists: every existence needs a starting relative, as the speed of light does; Lewis's modal realism is the opposite view, Carnap and Brouwer the nearest precedents. Whether the framework is circular: the material logic of the nexus is the speed of light, mathematics the concept of speed. Relations without relata: the relata have no properties of their own and are weakly discernible. The author's answers are quoted; their development is marked as the AI's reading. Explanatory premises. After an external review, the AI wrote thirteen short bridges, in seven groups, between the author's sentences and their translations into calculations; the author accepted them into the text, where each is marked «explanatory premise (AI)». Three more explain what «non-local» means and how the non-local phase relates to the loop. They are not theses of the author; fourteen open questions are stated in all, and a fifteenth is closed in version 2.68.1. What is not claimed. No new physics is derived here: the arithmetic identities are known, and every calculation cited is in an earlier deposit or, for the single calculation of version 2.68.1, in this record, with its programs and registration logs; the one exception, a test on the dimensions not yet deposited, is declared. The three dimensions of space are admitted, not derived. Single masses are not derived, and the Born rule is obtained only with declared assumptions. Nothing here distinguishes the framework from standard quantum mechanics by experiment. Contents. README.md (English) and LEGGIMI_IT.md (Italian), with the full text, a figure and 70 references, each checked on an opened page; AUTHOR_BACKGROUND.md with the author's sentences, dates and context; REFERENCE_CHECK.md with the corrections found and the review remarks not applied; REVIEW_GEMINI, five review rounds by a second AI system (Gemini), on the explanatory premises, on the package, on the part on non-locality and a final check of the whole text, with what wa