The Clock Rate Is the Dark Matter Pressure: a Closed-Form Coefficient History for a Self-Critical MOND Sector, and a Prediction of Positive w_dm

A cuscuton-clock MOND sector is driven by its own gradient instability onto a critical surface where its stress is isotropic and its sound speed vanishes, making it a clustering cold component without tuning (doi:10.5281/zenodo.22727111, doi:10.5281/zenodo.22727860). That construction requires the clock to run faster than proper time at every epoch where the coldness is needed, and no coefficient history satisfying it had been exhibited. We exhibit one in closed form, and find the requirement is not free. Imposing the clock's own field equation together with current and energy conservation collapses the freedom to a single parameter and forces s0 − 1 = w/mrel, where w is the sector's equation of state and mrel the logarithm margin: the clock's excess rate over proper time IS the sector's pressure divided by the margin. The family is U ~ a-3(1+w), d ~ a-3(1-w), q ~ a-3w, with constant margin and conserved clock charge. Imposing Friedmann at the same time closes the system and yields LambdaCDM with one change, the cold sector's exponent -3(1+w) in place of -3, so the free parameter is the dark matter equation of state, a measured quantity. Integrating the acoustic scale from that history (which reproduces the measured 100 theta_s to 0.16% at w = 0) bounds w below about 1e-4, a decade tighter than the algebra alone allowed, and at that bound the zero-gradient sound speed is still negative so the criticality still operates. Since criticality requires w strictly positive -- at w = 0 the clock sits at proper time, the instability vanishes and with it the attractor -- the construction predicts 0 < w_dm <~ 1e-4 where LambdaCDM predicts exactly zero. We also show the depletion trigger cannot be the clock-rate condition itself, which fails the Lyman-alpha bound by a factor of 92 on the candidate's own history, while a rate going as the fourth power of the dark-energy fraction does satisfy it. Not derived: the amount of the sector, since Friedmann is one equation for two dark unknowns. AI-assisted research programme; not peer reviewed.