A complete horseshoe for the second iterate of the Hénon map: full shift, uniform hyperbolicity, and optimality

We prove that H2 for the classical Hénon map H(x,y) = (1−1.4x2 + y, 0.3x) has a topological horseshoe with a full shift on 2 symbols, giving htop(H) ≥1 2 log 2 ≈0.347. This completes Galias’s 1998 partial result, which achieved only a golden mean subshift (3 of 4 transitions). The invariant set Λ is uniformly hyperbolic: DH4 preserves the quadratic cone field Q(u,s) = u2 −4s2 on a forward-invariant domain, verified on 36,680 interval boxes with zero failures. We prove that 2 strips is the algebraic maximum for H2 with any h-set alignment, making 12 log 2 the optimal bound achievable by the topological horseshoe method. The proof uses CAPD interval arithmetic and runs in 1.24 seconds total. We identify the geometric root cause for the 28-year gap between Galias’s partial result and ours: his quadrangle framework makes the missing transition N2 →N2 geometrically impossible (the image maps entirely to one side of the target), while our strip-within-h-set framework naturally captures the parabolic fold. All results are cross-validated by five independent tools: CAPD (C++), SymPy, Julia BigInt rationals, Julia IntervalArithmetic.jl, and Z3 SMT solver.