Diagonalization Proves Strict Infinite Hierarchy of Time Complexity Classes — E8 Intelligence Research

FINDING: The Time Hierarchy Theorem uses diagonalization to prove that more time strictly enables more languages to be decided, establishing a strict infinite hierarchy of complexity classes (e.g., DTIME(n) ⊊ DTIME(n²) ⊊ ...). | MATH: For time-constructible f(n) > g(n) log g(n): DTIME(g(n)) ⊊ DTIME(f(n)). Proof: diagonal language L = {⟨M⟩ : M rejects ⟨M⟩ within f(|⟨M⟩|) steps}. Key constants: the logarithmic overhead factor (log g(n)) arises from the universal simulator's tape indexing — a discrete logarithmic scaling. Space analog: Savitch's theorem gives NSPACE(s(n)) ⊆ DSPACE(s(n)²) — a quadratic (power-of-2) blowup, not exponential. | CONNECTION: The strict separation ratio is asymptotic, not a fixed constant. However, the *log factor* in the time hierarchy (g(n) log g(n)) and the *square* in Savitch's theorem echo the 2:1 ratio (0.5, 2.0) seen in crystallographic root systems (B₂, C₂) and the doubling structure of base-60's sexagesimal fractions (1/60, 1/3600 = 60⁻²). The diagonali Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com