Is what we see a projection? Objects as the common part of two points of view, tested on a quantum-Darwinism model

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. Quantum mechanics offers a further confirmation, as MM reads it. Here the non-local phase and the logical mirror reciprocity between the two ends of the nexus account 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. 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. Four-page summary: DOI 10.5281/zenodo.23535916. Entry page, with the list of all records: DOI 10.5281/zenodo.22667516. This record. When two observers correctly read the same quantity and compare their records, they find them in agreement, within the measurement errors. This deposit proposes to read the object both of them see as the common part of their records: what each reads from its own record, without looking at the other's, with the same value as the other (the Gács–Körner common part, with a very small tolerance when the copies are imperfect). The key word is projection: each record is very rich and mostly private; the reading projects it onto a simpler question, like a shadow, and two records that differ as wholes coincide in their projections. In the author's framework the observed object is a transposition and non-local projection of the observer's "observation matrix" (its record), and what two projections share is the trace of one non-local relation, non-local in the sense of not located in space and time, not of an influence at a distance. How it was tested. Eight formulations, nine test cases and seven criteria were registered before any calculation and tested on a small model of the kind used in quantum Darwinism (a two-valued source, N scattered carriers, two observers; classical and quantum versions). Excluded: "the common part is shared matter" (two people see the same chair with different photons: agreement 1 − 8.6 × 10⁻⁷ with 63 carriers each and no carrier in common); "the common part is having the same statistics" (two independent coins have it too); the exact common part, which is zero as soon as the copies are imperfect. Surviving: the common part with a tolerance δ (provisionally, within the class of readings tested), and a graded measure of how correlated the readings are (their mutual information). In the model it behaves as a yes-or-no: with copies that are wrong 10 % of the time it is zero up to 22 copies per observer and essentially the whole object (≥ 0.9999998 bit) from 23 copies on (δ = 10⁻⁶; 11 and 37 copies for δ = 10⁻³ and 10⁻⁹). At 23 copies the record carries 99.9998 % of the information about the object, and the common part (about 1 bit) is only 8.5 % of the whole record (about 11.8 bits). The number of copies needed depends on their quality, not on how many there are, as in the estimate for many copies of Zwolak, Riedel and Zurek (2014). In the cases examined, no statistical prediction differs from that of quantum mechanics. Prior work, first. Quantum Darwinism and "relatively objective existence" (Zurek 2003–2022; Ollivier, Poulin and Zurek 2004; Riedel and Zurek 2010); spectrum broadcast structures (Korbicz, Horodecki and Horodecki 2014; Horodecki, Korbicz and Horodecki 2015; Le and Olaya-Castro 2019); relational quantum mechanics and cross-perspective links (Rovelli 1996; Di Biagio and Rovelli 2021; Adlam and Rovelli 2022); Wigner's-friend theorems and experiments (Frauchiger and Renner 2018; Brukner 2018; Proietti et al. 2019; Bong et al. 2020; Rivlin, Engineer and Baumann 2025); agreement theorems (Aumann 1976; Contreras-Tejada et al. 2021; Leifer and Duarte 2026, who suggest as a speculation that a shared objective reality emerges from common knowledge; Cepollaro and Di Biagio 2026). As far as we found, no study defines the observed object itself as the common part of two observers' records; the search does not prove that none exists. What is not claimed. Objectivity is not derived and the measurement problem is not solved. The non-local phase is the author's interpretive framework and is not tested by the calculations. The link with spectrum broadcast structures holds only for perfectly distinguishable classical records. Wigner's-friend experiments are discussed, not re-analysed: the tests are on a small model. Contents. README.md (English) and LEGGIMI_IT.md (Italian), with the full text and a glossary; AUTHOR_BACKGROUND.md with the author's sentences; the registered protocol, three corrections and the outcome notes; four Python programs with their results; two verification reports by a second AI (gpt-6-luna via Codex); registers with SHA-256 fingerprints and UTC times; run_all.py, which re-runs everything in under a minute and compares the results with the registered ones. The author is not a physicist; formulations, programs and calculations were made with Claude (Anthropic). The work has not been reviewed by a human referee.