The Clock Stability Theorem: the Sub-Horizon Sound Speed of a Cuscuton-Clock MOND Sector Is Set by the Clock Rate

A relativistic MOND candidate built from a cuscuton clock, a dynamical scalar with a projected-gradient (MOND) sector and a cubic coupling has a homogeneous cosmological branch on which the clock sector gravitates like dust. We derive, in the uniform-clock gauge and the sub-horizon limit, the sound speed of the clock-scalar mode: cs2 = [2PX(1−D) − 2s0WY]/[B(1−D)], B = 2PX + 4q2PXX, D = 2q2WY/W. The cuscuton constraint and the lapse response enhance the MOND gradient term by 1/(1−D); the naive k-essence form is wrong by factors of −28 to +4. Under the candidate's coefficient closure the result collapses to cs2 = (1 − s0) mrel/(2 − mrel), with s0 the clock rate relative to proper time and mrel the logarithm margin. Verified against the candidate's exact 6x6 finite-wavelength transfer operator at ten points of its branch to 1.1e-3 (general) and 2.2e-3 (closure). Consequences, each machine-checked in Lean 4: the sector is gradient-stable if and only if the clock does not run faster than proper time (s0 ≤ 1; on the branch s0 crosses 1 once, at a = 0.29); a sound speed small enough for the Lyman-alpha forest (cs2 ≤ 1e-9) forces mrel ≤ 2e-9/(1 − s0) ~ 1e-8, five orders below the branch; and no sound speed can supply the host-mass-dependent dark-fraction depletion that galaxies require, since the forest-compatible cs = 9.5 km/s has a Jeans length below 7 kpc at galactic densities. The theorem turns the candidate's stability into an a-priori condition on its coefficient history and shows that its depletion mechanism, if any, must be dynamical rather than acoustic. AI-assisted research programme; not peer reviewed.