The Must-Close Constant κ: Closure from Prime Structure

We introduce the must-close constant κ, defined by κ⁻¹ = ln(P₃₇) = 141.7211, where P₃₇ is the primorial (product of first 37 primes). We derive the index 37 from number theory: the equation 3n + 1 = pₙ has exactly two small solutions, n = 12 and n = 14, where 12 = V (icosahedron vertices). Among Platonic solids, only the icosahedron satisfies this equation. P₃₇ has exactly 62 digits, matching V + E + F = 62. The local thresholds (41 = transit, 62 = locked) manifest experimentally: motor starting torque ≈ 1.5× running torque matches ln(P₂₀)/ln(P₁₅) = 1.504; glass forms when systems cannot transition from 41 to 62; Classical Nucleation Theory's 10²² discrepancy in slow-freeze regimes arises from treating discrete icosahedral thresholds as continuous. Unlike α which governs interaction strength, κ defines where interactions must terminate.