Apparent mass from a massless, scale-free structure: geometry, thirteen laws and tests from the hydrogen atom to galaxy clusters

We propose that what is called dark matter is the apparent mass of the empty part of a massless, scale-free structure, Asomatia, whose geometry crosses the boundary between aeons of conformal cyclic cosmology and orders the motion of matter within each aeon. Theorems T0–T8 single out the Hopf fibration as the only crossover data of the assumed form; thirteen laws, each labelled as derived, postulated, calibrated or imported, carry the geometry to physical systems with the fundamental constant ħ and two calibrated inputs (the acceleration threshold a* = 1.20×10⁻¹⁰ m s⁻² and the transition function of the radial acceleration relation). The laws agree with the Milky Way rotation curve and 116 SPARC galaxies at the level of the RAR, give γ ≈ 1.34–1.53 for wide binaries near the Sun, and place the main lensing peak of the Bullet Cluster at its galaxies; they differ from MOND through the structure's own centre and through screening of nested structures. This record contains:- the paper in English and Polish (PDF);- the refutation criteria of table 4.1, fixed on 1 November 2026 before any comparison with new data (Markdown, English and Polish, with SHA-256 checksums);- the reproducibility package asomatia-repro (zip): `bash run_all.sh` verifies the input checksums and reproduces every table, the error budget, the order-of-magnitude estimates and figures 1–5 in about 20 minutes (Python 3.13, numpy 2.5, scipy 1.18, sympy, matplotlib); the logs of a full run are included in outputs/ for comparison. Code under the MIT licence; SPARC data under CC BY 4.0 from their authors. Version 1.1 adds animation v3 of the structure (the structure drawn as the trajectories of its geometry, points for matter only) to the reproducibility package; the paper, the criteria and all results are unchanged.