Grammar of Time GT-04: Distance as Non-Restitution Cost

Cosmology begins too late if it begins with a metric. GT-04 is a technical sector module of the Grammar of Time programme. It does not claim to derive all of cosmology, general relativity, dark matter, dark energy, or spacetime dynamics. Its narrower claim is that primitive metric distance can be declassified in the geometrization regime. Given a positive response generator L and its kernel K_s = exp(-sL), GT-04 defines normalized restitutability R_s(x,y) and non-restitution cost F_s(x,y) = -log R_s(x,y). In standard heat-kernel regimes, the short-scale limit of this cost recovers geodesic distance: d(x,y)^2 = lim_{s -> 0} 4s F_s(x,y). Thus metric geometry is not introduced as a primitive arena; it is reconstructed as the stabilized short-scale cost of failed restitution. The eliminative claim holds only when L is supplied as an operational response generator rather than as the Laplacian of an already primitive metric. GT-04 treats the scalar positive kernelic branch of geometrization; phase-sensitive, bundle-valued, gauge, Lorentzian, or oscillatory kernels require separate sector modules. The mathematical novelty claimed here is not Varadhan's formula. It is the primitive audit: distance enters as cost, not as initial arena. GT-04 does not solve cosmology; it declassifies the primitive ruler on which cosmological models are built. This is a technical preprint, version v1.0, and has not been peer-reviewed.