A Structural Signature Framework for Frontier Mathematical Problems: Evidence for a Universal Hardness Zone in Information-Theoretic Coordinate Space

We introduce a structural analysis framework for mathematical systems grounded in three information-theoretic operators: transition entropy (HT), information drift (DI), and spectral scaling (β). Every mathematical system is embedded into a shared coordinate space via a signature vector Σ = HT, DI, β ∈ ℝ³. Applying this embedding to nine frontier mathematical systems — all seven Clay Millennium Problems plus Fermat's Last Theorem and the Poincaré Conjecture — all signature vectors cluster within a restricted region H ⊂ ℝ³, termed the Hardness Zone. Control systems lie outside H. Four structural signatures are identified across all nine systems without exception: a provable forbidden transition, a contraction/expansion distinction, a power-law spectrum, and finite Hilbert complexity. These findings constitute the empirical foundation for the Molina Structural Conjecture (Molina 2026a), deposited separately. The full computational methodology is documented in a restricted companion deposit (Molina 2026c). SHA-256: b6048c78763a8aee61a0a6022ed79e86bab250c28d48e9453b4e65e61c4aa38a