Order-Convex Subsets of Grid Posets: A New Exponential Combinatorial Class with Structural Classication, Transfer-Matrix Enumeration, and a Supermultiplicativity Theorem

We study the number |CC([m]²)| of order-convex subsets of the m×m grid poset. The sequence 2, 13, 114, 1146, 12578, 146581, ... has growth constant ρ = 16 exactly, proved via a Lindström-type domain-splitting injection that bounds crossing pairs of antitone boundary functions, combined with a Fekete squeeze argument. The upper bound ρ ≤ 16 follows from the ideal/filter injection into pairs of antitone functions counted by C(2m,m). The lower bound |CC([m]²)| ≥ C(2m,m)²/(2(m+1)) closes the gap. For d-dimensional grids with d ≥ 3, the 1/m normalization diverges; the correct normalization is c_d = lim log|CC([m]^d)|/m^{d-1}, which exists by Fekete's lemma applied to the superadditive sequence. The dimension law log|CC([m]^d)| = Θ(m^{d-1}) is proved for all d ≥ 2 via an antichain tiling construction. A transfer-matrix dynamic program with O(n²) boundary parameters enables computation of 50 exact terms. The generating function is likely not D-finite (no recurrence of order ≤ 5 with polynomial coefficients of degree ≤ 4 found). Since ρ = 16 > 6⁶/5⁵ ≈ 14.93, this sequence does not belong to the (6,2) generalized Fuss-Catalan family despite matching its first three terms. All structural and asymptotic results are formally verified in Lean 4 (zero sorry). Part II of a four-paper series on causal-algebraic geometry.