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

Paper and formal verification for "Order-Convex Subsets of Grid Posets." We study order-convex subsets of the product poset [m]×[n] with the componentwise partial order. The Endpoint Classification Theorem gives a complete characterization: a subset is order-convex if and only if each row fiber is an interval and the endpoints satisfy a rectangle-fill condition. Convex subsets decompose uniquely into maximal contiguous row blocks separated by anti-monotone gaps. A Supermultiplicativity Theorem (|CC([m+n]²)| ≥ |CC([m]²)|·|CC([n]²)|) gives the existence of the growth constant ρ via Fekete's lemma. The upper bound ρ ≤ 16 follows from an ideal/filter injection; the matching lower bound establishes ρ = 16 = 2⁴ exactly (upper bound fully verified; lower bound verified except for one auxiliary lemma). Since 16 > 6⁶/5⁵ ≈ 14.93, the sequence defines a new exponential combinatorial class distinct from all generalized Catalan/Raney families. A transfer-matrix algorithm with exactly C(n+2,2) reachable states enables computation of 50 exact terms. The dimension law log|CC([m]^d)| = Θ(m^{d−1}) is proved for all d ≥ 2 via tiling inequalities. The deposit includes the manuscript and formal proofs in Lean 4 / Mathlib. This is Paper II of a four-paper series. Paper I: "Causal-Algebraic Geometry: A Grothendieck-Type Framework for Partial Orders, with Applications to Manifold Detection and Arithmetic." Paper III: "Black Hole Thermodynamics from Counting Convex Subsets of a Lattice." Paper IV: "Jackiw-Teitelboim Gravity and Hagedorn Density of States from Counting Causally Convex Subsets."