A Ceiling Dark Matter Cannot Impose: the Bounded-Boost Theorem for MOND-Class Kernels, and What It Says About Galaxies and Clusters

Version 2: references added, and three results folded in — the carrier theorem is shown to BE the bounded-boost theorem (a matter-sourced scalar can carry a kernel only up to the maximum of Δ, so the saturation is universal and every kernel leaves a constant residual force C a0); a bulgeless control (113 of 144 galaxies, where the disc mass-to-light ratio is the only stellar freedom) sharpens the kernel discrimination to 7.3%/3.5% of points exceeding the exponential carrier's ceiling against 1.2%/0.6% for νRAR; and the Solar-System cost of the kernel swap is priced, with νRAR carried admissible at coherence-length floors of 0.10 and 0.15 pc against the exponential carrier's 0.07 and 0.10. A correction to the programme's own standing numbers is flagged: the previously published floors of 0.03/0.05 pc were computed with the unsaturated inverse partner. Preprint; AI-assisted research draft; not peer reviewed. This deposit timestamps a theorem, its tests, and the committed script that reproduces every number. It claims no completed relativistic theory, no derivation of a0 or of κ = 1/2, and no empirical detection. It does not claim that any data set favours this framework over ΛCDM. Any modified-gravity kernel writing g = ν(y) gN, y = gN/a0, with a Newtonian limit and a deep-MOND limit, bounds the acceleration excess Δ = gobs − gbar above by a pure number times a0. For the exponential carrier gbar = g(1 − e−g/a0) the bound is exact: Δ ≤ a0/e = 0.367879..., attained at gbar = (1 − 1/e) a0, and the completed kernel saturates there so the bound is attained on a plateau. Across five standard kernels the constant lies in [0.250, 1.000], so the ceiling is kernel-independent up to an O(1) factor. A dark-matter halo cannot make this prediction: the same observable is ghalo, set by M200 and concentration, which span decades and are not tied to a0. The ceiling is one-sided and parameter-free, so a single system above it falsifies the kernel. Tests. (i) On 2352 SPARC rotation-curve points beyond 2 kpc from 144 galaxies (the survey's own Q < 3, i > 30° cuts, fixed 3.6 µm mass-to-light ratios), 99.23% obey the widest kernel bound at both a0 footings; the eighteen >3σ exceptions lie in five named galaxies, thirteen in NGC 5985 alone (Q = 1, i = 60°), which is kept and named rather than cut. (ii) The ceiling discriminates between kernels and does not favour the exponential carrier: 9.8% of those points (263, over 32 of 144 galaxies) exceed that kernel's own ceiling at >3σ, and between gbar = 1 and 4.5 a0 the measured excess is 2–3× it against 1.2–1.9× the wider νRAR ceiling; the coherent stellar-normalisation freedom is pinned to +1.8% by the Newtonian limit, and raising a0 would need 12× the canonical value. (iii) In X-COP cluster cores, after correcting a radius-unit error verified directly from the FITS headers (the gas profile's RADIUS is R/R500 with R500 in each file's own header, not Mpc), the ceiling fails by 9.1× the exponential carrier's bound and 3.4× the widest bound of any kernel, so no interpolation function can absorb it. Both escapes close quantitatively: nonthermal support would need σ1D = 673–1155 km/s against the Hitomi/XRISM Perseus measurement of 164 ± 10, and unseen baryons 3.9–5.2× the directly imaged X-ray gas. Clusters therefore require a genuine extra source as a theorem rather than a fit. The required source is baryon-tracing, but so is the published NFW fit (slopes −0.05 versus +0.03), so the profile shape is reported as a null; the ceiling itself is the discriminating statement.