Does the radial-acceleration scale depend on a galaxy's distance from us? SPARC tests and a registered prediction to tell distance, cosmic age and expansion rate apart

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 [email protected], since Zenodo has no space for comments. This record. In galaxies the observed acceleration follows the baryonic one through a single scale (the radial acceleration relation, RAR). The author's premise asks what can make two galaxies with the same baryonic acceleration different, and answers: «the distance from the principal point of observation». We translated it into three rules for the RAR scale and tested them on SPARC with learning and verification galaxies kept apart. The parameters were frozen on the learning galaxies before any verification measure was computed. The program read all curves to check admission, split and number of points, but the verification velocities entered no fit and no simulation. Within 130 Mpc every translation is indistinguishable from MOND with a constant scale: the largest gain on the 51 verification galaxies, for the power of the distance, is +1.55 % (interval −1.46 … +3.31 %), and the others are +0.71 % and +0.06 %, all below the criterion of ≥ 5 % with the interval all above zero. This is an operational outcome, not a proof of physical equivalence. A small trend remains: farther galaxies show a slightly larger scale, s = +0.115 ± 0.104 (log scale per log distance), but in SPARC distance and luminosity go together and do not separate. The registered prediction. With time and SHA-256, before any new data, we registered the shape that the distance rule predicts, with the light-travel distance d = c·t_lt and s in the band 0.039–0.115 (two SPARC point estimates, chosen for the prediction): between 30 and 300 Mpc, at equal luminosity and baryonic acceleration, the scale grows by +0.04 … +0.11 dex, where a scale tied to the expansion rate (a₀ ∝ H(z)) grows by +0.013 and one tied to the cosmic age (a₀ ∝ 1/t) by +0.028; between z = 1 and 2.5 it almost stops (+0.006 … +0.017 dex), where those grow by +0.32 and +0.35; the far step is 0.15 times the near step for any s ≠ 0 (25 times for a₀ ∝ H(z), 12.5 times for a₀ ∝ 1/t). MM's material thus holds three registered readings of how the scale changes when looking far: with the cosmic age (MM 2.42), with the expansion rate (MM 2.44) and with the distance from the observer (this record). The prediction is designed to tell them apart; a future test may favour one reading, or be undecidable. The near test separates the distance rule from a constant scale; the far test separates it from a₀ ∝ H(z) and a₀ ∝ 1/t, not from a constant scale. The test procedure is fixed in advance and written as programs: sample, distances, luminosity, fit, bootstrap, mass systematics, and the outcomes in a fixed order for each test. For the near test the order is: falls, only if the interval corrected for distance errors also falls; otherwise undecidable; undecidable; different amplitude; holds; otherwise grows, with the amplitude loosely constrained. 36 tests cover every branch, and 12 more tests of the rules for edge cases were written after the final note. About 700 galaxies with rotation curves at 10–300 Mpc, with distance and luminosity not tied together, would detect s = 0.115 at 3σ; s = 0.039 would need about 6000. BIG-SPARC (announced, about 4000 galaxies) would be large enough for s = 0.115, not for s = 0.039. A published hint, read after the registration. Vărăşteanu et al. (2025, MNRAS; arXiv:2504.