An Obstruction Map for Relativistic MOND: the Conformal Lensing Barrier and the Cost of Its Repair

Preprint. This deposit timestamps a structural result and its reproducibility scripts; it is not a journal publication and claims no viable MOND completion. Relativistic completions of MOND face a tension between reproducing the modified force law, preserving lensing, and keeping the degree-of-freedom count minimal. We report a systematic closure attempt across seven local architectures and extract two results. (1) A cost table. Exact MOND requires deforming the GR Hamiltonian constraint away from the point where Hojman–Kučař–Teitelboim uniqueness applies, and each relaxation carries a distinct MEASURED cost: deforming the constraint algebra gives γPPN = 0 and α3 ≠ 0; spatial nonlocality damages the tensor sector (cT = 0) or the solar-system quadrupole; altering the momentum dependence propagates a scalar; altering the matter metric misroutes the source. We do not claim a universal no-go — HKT identifies a structural OBSTRUCTION CLASS for local metric Hamiltonians meeting its hypotheses, matched per-architecture. (2) The AeST-collapse theorem (main result). The natural escape — paying for one extra propagating DOF via a screened (k-mouflage/Vainshtein) scalar — does NOT give a cheaper theory. Such a model passes five gates cleanly (3 DOF; cT = 1; γPPN = 1; the exact interpolation μ(y) = 1 − e−y; BTFR/RAR) but fails two, both BLIND TO THE DOF COUNT: (i) a conformally coupled scalar cannot bend light, because null geodesics are conformally invariant — the model predicts galaxies that rotate like MOND and lens like bare baryons, excluded by the observed agreement of the lensing and dynamical radial-acceleration relations; and (ii) the Cassini external-field quadrupole and the wide-binary signal are governed by the same interpolation-transition sharpness at the same external field η ≈ 1.9, so any screening clearing the former flattens the latter (this obstructs the TESTED external-field-kernel mechanism; screening keyed on a different environmental invariant — density, curvature, scalar charge, Vainshtein radius — is not excluded) (committed 175-galaxy scan: μ5/μ10 clear the Cassini ceiling at 0.39/0.08 while predicting γv = 1.0040/1.0001). Repairing (i) requires a disformal matter coupling; in every construction tested, that in turn requires preferred timelike structure of the kind carried by TeVeS and AeST, and a lensing-sized disformal term additionally splits the photon/graviton cones against GW170817. We state this CONDITIONALLY: within the tested class of local metric/disformal constructions, restoring the lensing channel requires additional preferred-structure degrees of freedom of TeVeS/AeST type — i.e. lensing APPEARS to cost a vector, not a scalar. This is NOT claimed as a classification theorem; establishing it needs a systematic classification of admissible couplings and their constraint structures, which we identify as the decisive next calculation. Corollary. In the surviving class a0 is not structural, which relocates the framework's falsifiable content to the measurement side: a0 = κc√(GρΛ) with a0(z) ∝ H(z) is untouched by all seven obstructions (each fails on the COMPLETION, not on the acceleration scale). Gaia DR4 then becomes a sharp discriminator rather than a confirmation channel: Cassini-safe kernels predict γv ≈ 1.000–1.004 against a pre-registered band of 1.16–1.23. Not claimed: a universal no-go for relativistic MOND; that HKT alone implies all seven failures; any derivation of κ or Z (both FITTED throughout); any exclusion of AeST itself — the point is that the 2+1 route offers no cheaper alternative to it. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of large language models (Anthropic Claude, OpenAI and Qwen models). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. Several intermediate claims were found incorrect and withdrawn before this deposit. No AI system is listed as an author.