  We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci
flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that
arise as curvature tensors of space-time connections satisfying the Ricci flow
with respect to the natural space-time degenerate metric. As a special case, we
employ scaling arguments to derive a linear-type matrix LYH estimate. The new
LYH quadratics obtained in this way are associated to the system of the Ricci
flow coupled to a 1-form and a 2-form evolving by heat-type equations. In the
case of a Kaehler solution, a special case of our linear-type trace LYH
estimate is weaker than but qualitatively equivalent to Hamilton's trace
estimate.
