Near in space, far in time: galaxy clusters seen through a turned spacetime

This record. The missing mass of galaxy clusters read as part of their spacetime turned for the observer, with tests registered before their computations. Reading path: Galaxies and the cosmic scale a₀ = cH₀/6, 6 of 6 · before: The dynamics of MM · all themes. 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. With the galaxy rule of Mathematical Materiality (MM), the radial acceleration relation with a₀ = cH₀/6, galaxy clusters lack mass. This record follows a premise of the author: the observer reads part of a cluster's spacetime as turned. Read as cos φ = r/R_b, the turn gives an effective scale that grows towards the centre as A = R_b/r. Its exponent is 1 between centre and edge, zero in galaxies, and the same for light and for matter. The main tests were registered before their computations, checks added after an outcome are marked as such, and every step was checked by two AI assistants. What the data say: Exponent close to 1 in the median. In 19 clusters (12 X-COP, 7 from Vikhlinin et al. 2006) the centre-to-edge exponent ranges from 0.62 to 1.62, median 1.02; in disc galaxies it is nearly zero (0.05-0.33). Lensing and gas are compatible with the same reading. In 16 CLASH clusters, lensing gives A ≈ 12.5 and hydrostatic gas A ≈ 8.9. The variant in which only slow matter feels the turn fails. This is a registered re-reading, and dark matter predicts the same. Not exactly 1 in every cluster. With the statistical X-COP errors, «exponent 1 in each cluster» fails: the clusters scatter by about ±0.2. If centre-to-edge hydrostatic systematics differ by 10-15 % from cluster to cluster, this is not decidable. The exponent follows the slope of the total mass (Spearman −0.875), partly by construction. The baryonic slope alone, the stars and the redshift show no robust association. The missing part follows the member galaxies, not the gas, in 6 of 7 X-COP clusters with stellar profiles. Groups, at the same radius, lack slightly less than clusters. What does not hold, or is not decidable: the edge R_b = R200m: amplitude not decidable, per-system prediction not supported; the angle of view of the long axis: not decidable; a rule of the depth from the gas temperature: not decidable; the cluster as a full external context: fails. What is not claimed. The turn is a coherent reading of these data, not a proof. An ordinary explanation (NFW-like mass with a baryon fraction rising outwards) gives the same mean exponent, and what fixes R_b is open. The four registered tests of MM 2.46 for future data are unchanged. Contents: the 22 scripts exactly as registered, with their logs; the author's premises, protocols and outcome notes (Italian, with an English README); the reports of the two AI checkers; a download script with SHA-256 checks and a reproduction script that re-runs everything and compares the logs. The author is not a professional physicist. Computations were done with AI assistants (Claude, with checks by Codex and Gemini), not reviewed by a human referee.