Geometry from Order and Count

Two companion papers derive the integers of spacetime from the double-entry act of recognition: the refinement records of a recognizer are dense in a solid cube whose dimension is forced to three, and a costly process carries one discrete clock, unique up to unit and origin, with a causal cone from per-act bounds. Both papers state the same boundary: the reading of the derived order, cone, and count as a Lorentzian metric is adopted from classical theorems, not derived. This paper derives it, up to one carried classical statement. First, the completed order is the cone order: acts at refinement depth n advance the clock by exactly tau_0/2^n and move a record within a cost budget, and the closure of these reach relations over all depths is exactly {(t,x) <= (s,y) <=> ||y-x|| <= c(s-t)} with c = l_0/tau_0; the move set is a level set of the framework's cost, and its roundness is proved from the cost function, not assumed. Second, the count is the volume: the number of depth-(m+k) records in a dyadic box, divided by the record density, converges to the box's Lebesgue measure, and ticks are counted exactly. Against these two theorems we set the classical recovery statement, that a metric field with the causal structure of Minkowski space is conformally flat (Zeeman; Hawking, King, and McCarthy; Malament), carried as an explicit named hypothesis and never re-proved. The composition is then a theorem: the volume normalization kills the conformal factor, so a metric field with the derived order and the derived count is the Minkowski metric eta = diag(c^2, -1, -1, -1) everywhere. Two decoys show the pair is exactly the needed data: a dilation preserves every cone and changes the volume element, and an explicit unit-determinant stretch preserves the volume element and changes the cones. Order fixes the shape, count fixes the scale, and neither alone fixes the metric. As a corollary, the symmetries that survive are the Poincare maps: Lorentz invariance is an output. The metric obtained is kinematic and flat; curvature belongs to dynamics, which this paper does not treat. The order-completion, counting, algebraic, and composition steps are machine-verified in Lean 4 over the Mathlib library; an appendix maps each statement to its formal declaration.