A Clock That Must Run Fast: the Solar System as an Inequality on the Rate of Cosmic Time

Abstract

A cuscuton-clock sector coupled to matter disformally along the clock's own unit timelike direction gives a post-Newtonian gamma of exactly one with no added vector field. Boosted, the same coupling gives alpha_1 = 8 f_s, five orders over the bound unless local matter drags the clock into its own rest frame to within 4.6 m/s. That alignment was left uncomputed in the previous paper. We compute it. FOUR STRUCTURAL RESULTS PRECEDE THE NUMBER. (i) Writing the matter metric with independent conformal and disformal strengths and demanding gamma = 1 forces b = 2a exactly: the operator that fixes light bending IS the operator that generates the preferred-frame effect, so the gate cannot be evaded by tuning. (ii) The cuscuton's linearised Lagrangian has a VANISHING coefficient for the squared time derivative, so the clock propagates nothing and its tilt solves an elliptic boundary-value problem whose condition at infinity is the cosmic frame. (iii) The stiffness resisting the drag is the gradient function at the LOCAL invariant, W = f_s^2 G M^2/(8 pi s_0 r^4), free of both the matter coupling and the MOND coefficient: neither can buy alignment, and the clock rate is the only handle. (iv) The gain at a stellar surface is 24 s_0/f_s, independent of the star's mass and radius entirely, and the falling stiffness makes the exterior tilt decay as r^-(sqrt(17)-1)/2 = r^-1.562 rather than the r^-3 of a constant-stiffness dipole. Both help; neither suffices. Carried to Earth's orbit the drag is 0.011 at the minimum clock rate and the residual misses the bound by 7.9e4. Inverting, the gate becomes s_0 >= 1.5e7. THE SOLAR SYSTEM DOES NOT EXCLUDE THIS CONSTRUCTION; IT DEMANDS A CLOCK RUNNING ABOUT TEN MILLION TIMES PROPER TIME. That is the third independent constraint on the clock rate and the third pointing the same way, after the clock identity and positivity of the sector's energy, but the first two want order unity and nothing in the programme explains a number seven orders larger. It is recorded as a requirement, not a derivation. The price, also computed: the margin falls to 6.8e-12 and the scalar's kinetic coefficient rises to 1.5e11, a strong-coupling question this paper states and does not answer. Four new Lean 4 certificates; the file stands at 128 theorems, zero sorry. This is NOT a complete theory of gravity. AI-assisted research programme; not peer reviewed.