Diagonalization Proves More Time Solves More Problems — E8 Intelligence Research

FINDING: The Time Hierarchy Theorem proves that more time strictly enables more computational problems, via diagonalization — a constructive self-reference argument that separates complexity classes. | MATH: For time-constructible \( f(n) > g(n) log g(n) \), \(DTIME(g(n)) DTIME(f(n))\). Proof diagonalizes: simulate all machines of time \(g(n)\) on input \(x\), flip the output of machine \(M_x\) if it halts within \(g(|x|)\), else accept. The separating language is in \(DTIME(f(n))\) but not in \(DTIME(g(n))\). Key constants: the \(log\) factor arises from simulation overhead — the universal machine's tape indexing costs \(O(log t)\) per step. | CONNECTION: The diagonalization step is a binary complement — a reflection symmetry across the diagonal of the time-indexed machine table. This is structurally analogous to the 0.618/0.382 golden ratio split in that it creates a *gap* that cannot be bridged by the lower class — the gap ratio here is not Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com