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

Version 4: the kernel swap the theorem selects is carried through the framework's whole constraint ladder, not just the Solar System. The carried kernel enters every gate through one function, J_Y(s) = s/Δ(s), whose saturated slope is e for the exponential carrier and 1.544 for νRAR — one ratio, 1.76, is the entire content of the swap, and it is admissible on every gate. Post-Newtonian α1 is untouched (the screening crushes the scalar drag to under 2% at either kernel's own floor, leaving −4.1×10−5 against a 10−4 bound); the cosmological linear-source screening is untouched, verified by re-deriving it from the action and finding no kernel dependence; and the dark sector's deep-MOND hydrostatic length is IDENTICAL for the two kernels, only its saturated value differing by 0.568, so the admissible |K2| window shifts by that one factor and stays non-empty. The costs are a factor 1.4–1.5 in the coherence length and, through it, an anisotropy sign-flip radius pushed out of the usable separation range. Clusters are NOT rescued, exactly as the theorem requires: the excess is 5.2× νRAR's wider ceiling against 9.1× the carrier's, so the extra-source requirement is a property of the whole kernel class rather than of a choice within it. Version 3: the programme's own wide-binary prediction is recomputed at the corrected floors with the carried kernel, closing the loop v2 opened. The pre-registered estimator returns γv = 1.0375 (canonical, ξ = 0.07 pc) and 1.0275 (alt, 0.10 pc) for the exponential carrier, and 1.0450 (0.10 pc) and 1.0300 (0.15 pc) for νRAR carried, against a stale 1.032/1.040. A control confirms the estimator reproduces the published values to 0.0005 from the original tables, so the change is physics. A decomposition shows the two corrections act in OPPOSITE directions and are of comparable size — the carried kernel raises the boost by +0.0405/+0.0225 because ν is 5.5% larger at the external field ye = 1.9, which lies above the saturation point, while the higher floor lowers it by −0.0350 — so the net change is not monotone in the coherence length and had to be computed rather than assumed. Every value remains about 33σ below the pre-registered band, so the prediction stays sharply distinguishable from standard MOND, while the anisotropy sign-flip radius moves from 15.7/26.0 kAU to 36–77 kAU, out of the survey's usable range. 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.