Gradient-Driven Criticality: a MOND Sector That Cures Its Own Gradient Instability, and a Marginal State That Is Exactly Cold

A relativistic MOND candidate built from a cuscuton clock, a dynamical scalar with a projected-gradient sector and a cubic coupling has a sub-horizon sound speed cs2 = (1 - s0) mrel/(2 - mrel), negative whenever the clock rate exceeds unity (doi:10.5281/zenodo.22717950). That result was obtained on a background with NO field gradient. Since the sector's entire content is its dependence on the projected gradient invariant Y, we recompute with a background gradient carried through. Three consequences follow from the candidate's own closure with nothing added: the destabilising stiffness falls as (1+Y/l)^(-1/2); the logarithm margin GROWS as m0 + 2dY, carrying the background away from its singularity; and cs2 rises monotonically and crosses zero at a unique gradient Y*. Below Y* the sector is unstable so gradients grow; above it expansion dilutes them. Y* is a two-sided attractor and the state it selects has cs2 = 0 exactly. On the candidate's own coefficient history Y* lies at 0.2-2% of the transition scale, and at Y = l the sound speed stays positive even with the clock rate tripled. The surface is not merely dynamical: varying the action with respect to the inverse metric gives det(Tmixedtx - L I) = -s0 b2 W N, with N exactly the numerator whose vanishing defines Y*, so on the critical surface a third eigenvalue joins the transverse pair and the stress takes perfect-fluid form. Isotropic stress together with non-propagating perturbations is precisely a clustering cold component, the ingredient a Lean-certified necessity result requires of any completion, and both properties arrive on one surface fixed by the action rather than by tuning. Seven statements are machine-checked in Lean 4. Not established: the tracking dynamics, since the Lyman-alpha bound requires the gradient to follow Y* to a fractional precision of about 1e-6. AI-assisted research programme; not peer reviewed.