Rigorous Newhouse thickness exclusion for the classical Hénon

We prove that the classical Hénon map H(x,y) = (1−1.4x²+y, 0.3x) has exactly 58,678 distinct period-31 orbits, all hyperbolic saddles. This is the first completeness proof for any period beyond 24, extending Galias's rigorous catalog by seven periods. Our count exceeds the Galias–Tucker 2015 numerical prediction of 58,656 by 22 orbits, revealing that their symbolic coding via a k=16 transfer matrix is non-injective: multiple dynamically distinct orbits share the same symbolic itinerary. The proof combines three techniques in the 62-dimensional product space: (i) close returns on a 10-billion-step attractor trajectory with multi-shooting Newton refinement (κ ≈ 12, convergence basin ~10⁻² versus ~10⁻¹⁰ for the classical ℝ² formulation); (ii) Krawczyk containment, proving existence of all 1,819,039 orbit points in 0.6 s; and (iii) trajectory Krawczyk exclusion, a new method that certifies the absence of orbits by exploiting the non-closing gap of center trajectories. The spatial subdivision resolves all 3,289,782 leaf cells with zero inconclusive, completing the proof in 18.7 minutes. We present three independent proof pipelines (multi-step, single-run, and database-free) and cross-validate every claim with five independent tools: CAPD, Julia IntervalArithmetic.jl, SymPy, mpmath, and Z3.