What a Clock Can and Cannot Do: a Cold Dark Sector Derived, a Force Law Excluded, and the Rate of Cosmic Time Certified

A cuscuton-clock sector with a projected-gradient scalar produces a cold clustering component by driving itself onto a critical surface, and passes nine successive cosmological and solar-system gates. Every one of those derivations set the cubic coupling to zero. We restore it and report two things. NEGATIVE AND DECISIVE: in the static spherically symmetric limit the action's two gradient operators give first integrals r^2 (chi')^3 and gamma r (chi')^2, hence rotation curves rising as r^(1/6) and r^(1/4); flat curves require r^2 (chi')^2, that is F proportional to Y^(3/2), the AQUAL deep-MOND kinetic term. Both available exponents are strictly positive, so no value of the cubic coupling and no mixture of the two operators reaches zero. This action cannot produce flat rotation curves. A partial consolation is that its two terms cancel exactly at zero gradient, so the deep-MOND boundary condition holds without being imposed; only the exponent is wrong. A health condition the cosmological work could not see also falls out, 4 d l > U. POSITIVE AND CERTIFIED: restoring the cubic leaves the energy density and the clock's charge untouched, since it cancels out of both, and it survives only in the pressure, correcting the clock identity to s0 - 1 = w/m_rel - 2 gamma q^2 qdot/U, a correction three parts in a million of the leading term at the coupling used. That identity, and its consequence that the clock exceeds proper time if and only if the sector's equation of state is positive, are certified in Lean 4. The honest scope: this construction derives a cold dark sector and the rate of cosmic time from a clock, and does not derive the force law. That is a narrower claim than a theory of gravity and a sharper specification for one. AI-assisted research programme; not peer reviewed.