Relativistic MOND Is a Cuscuton Theory: A Candidate de Sitter-MOND Completion, Its Structural Spine, Falsifiable Predictions, and Open Costs

We report the status of a candidate relativistic completion of Milgrom's modified Newtonian dynamics (MOND) in which the acceleration scale is locked to the de Sitter/cosmological constant, a0 = c2/2πLdS, and the modified-inertia sector is carried by a clock — a scalar φ entering only through Q = nμ∂μφ, with n the unit gradient of a primordial clock field. Central structural result (classification theorem): a relativistic MOND scalar carrying a canonical kinetic term cannot close the Hamiltonian (hypersurface-deformation) constraint algebra — matching the spatial-diffeomorphism generator forces the kinetic coefficient to A=1/μ, and the residual cubic obstruction A A'/2 = −μ'/(2μ3) is nonzero for every strictly monotone interpolating kernel. The unique consistent branch is the one on which the MOND scalar is a cuscuton (non-propagating, momentum-constrained); there, constraint closure and ghost-freedom coincide. This subsumes, as one fact, a family of otherwise-separate health results and explains why every viable construction in the programme is of cuscuton type. Phenomenology of the cuscuton completion (an F(Q)Θ/rotated-constitutive clock, developed collaboratively in the public repository), each verified by a committed runnable script and a machine-checked Lean 4 kernel: the galaxy radial-acceleration relation (SPARC, rms 0.16 dex at zero per-galaxy parameters); weak lensing tied to dynamics (Φ=Ψ, no slip); cluster and cosmic-web dark matter as the conserved Noether charge of the clock's shift symmetry (pressureless, clustering like cold dark matter with σ8=0.811, reproducing the Bullet-cluster lensing offset pure MOND fails); tensor gravitational-wave speed cT=c; and Solar-System post-Newtonian bounds — including the preferred-frame parameter that excludes the earlier AeST completion, here structurally suppressed rather than un-tunable. Distinctive falsifiable predictions: a flat a0(z) (z≈2.5 baryonic-Tully-Fisher zero-point 0.00 dex vs +0.33 dex for ΛCDM); an External Field Effect in dwarf-spheroidal dispersions (~1.9× across Galactocentric radius, where dark matter predicts none); a subdominant near-luminal scalar gravitational-wave polarization; a dark matter that cannot decay or annihilate; and a parameter-free 1/r correction above the flat rotation velocity. We do not claim the theory is established. Two costs are proved and stated: the charge-sourced dark matter carries an intrinsic ~24-order Big-Bang-Nucleosynthesis fine-tuning (the same conserved charge sources an a−6 stiff partner, and the ghost-free degeneracy forbids removing it), and the full relativistic constraint algebra of the completion is still being derived. This is the honest state of a candidate: a consistent structural class, a phenomenology-complete instance, sharp predictions data can confirm or kill, and named open costs. Every dimensional number is carried on both a0 footings; no observation is used to assert the theory is true. AI-assisted research programme with a collaborative construction; not peer reviewed.