Is what we see a projection? Objects as the common part of two points of view, tested on a quantum-Darwinism model
Ugo Lanciano
Common introduction to all the records of this series. The content of this record follows below. Mathematical Materiality (MM) in brief. Why are the laws of physics the same everywhere and at every instant, even though nothing forces matter to obey them? MM proposes an answer: the laws are not rules imposed on matter from the outside; rather, matter itself is made of mathematical relations. And mathematics cannot change: a relation binds two distinct terms everywhere and always. In logic, nothing exists on its own: to exist means to be in relation to another logical entity, and outside this web of relations nothing can exist, not even other universes. This reading already meets experimental measurements at two points. First, the ratio of dark to ordinary matter is read as 3√3 = 5.196; when left free in the fits, Planck and DESI DR2 data give 5.21–5.22 (as registered in record 2.44, https://doi.org/10.5281/zenodo.23329080). Second, the acceleration scale of galaxies is read as a₀ = cH₀/6, the same scale that appears in Verlinde's emergent gravity: the computed value is 1.09–1.18 × 10⁻¹⁰ m/s² against the measured value of 1.20 (record 2.46.1, https://doi.org/10.5281/zenodo.23506655). Both readings were formulated with the measurements already known; the tests to check them against future data are registered in the same records. In the first logical nexus between space and time — which became homogeneous with Einstein — there is a non-local phase, made only of a pure logical distinction: space is not time. Between the two there is no relation of quantitative calculation, only the logical entity they have in common, which is needed to make them exist with respect to each other. This level is not distant: 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 based on separate properties and free of faster-than-light influences allows, without anything physically 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 have only 1 in common. The local world arises when a relation is counted through this GCD: its multiples and its fractions form a matrix. Seen by an observer inside the system like us, the multiples appear as time and the fractions as space. When that same multiple is directly contained in the matrix of the point of observation, it turns into mass — that is, into concrete matter as we usually understand it. There is therefore a close link between time and mass, in full agreement with Einstein's relativity, which ties mass to spacetime. The observed objects, while being logically autonomous entities in their local relations, enter into relation with the observer through a logical diffraction — that is, a decomposition into 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 instead taken as a starting principle. The speed of light is the fixed exchange rate between the multiples and the fractions of the same GCD, meaning how much space a single step of time is worth: exceeding it would mean, for some observer, covering space without counting time, and thus leaving the local world. Logic always starts from a logical relative, and this exchange rate has nothing absolute in itself either: 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 at all, because outside there is only non-existence: justifying something requires another term to relate it to, and out there there is none. Asking for a reason from outside is like asking what lies north of the North Pole. The same theoretical framework that reads space, time, mass and the observed objects also admits quantum mechanics. The non-local phase and the mirror-like logical 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: for this reason we see it in one precise place and at one precise instant. When instead the object falls below a certain mass threshold, the principal GCD (that of the point of observation) and the GCD of the diffraction that generates the object are forced to become equal, reflecting each other as in a mirror. At that point 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 the number 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 we know that 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 really 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. Tegmark (2008) had already noted that such a simulation need not be run, while Vazza (2025) estimated that running it physically on a machine would require resources that the very laws of the universe do not allow. This record develops one aspect of the theoretical framework. The work is shared openly so that it can be judged by as many readers as possible: anyone who knows someone interested can pass on this single link, valid for the whole framework (with an overview in English and Italian): https://doi.org/10.5281/zenodo.23535916. The entry page, with the complete list of all records, is https://doi.org/10.5281/zenodo.22667516. To cite the framework: Lanciano, U. (2026). The number 1 in front of the mirror: Mathematical Materiality from the nexus to the cosmos. Zenodo. https://doi.org/10.5281/zenodo.23535916. The records have been deposited since 5 August 2026, each with its own date, and are released under CC BY 4.0: anyone may use them by citing the source. Objections and errors found are welcome at riverantares@gmail.com, since Zenodo has no space for comments. 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; t
•Oct 24, 2026•3 citations