Diagonalization Proves More Time Solves More Problems — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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)) subsetneq DTIME(f(n)). Proof diagonalizes: simulate all machines of time g(n) on input x, flip the output of machine Mx 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
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BibTeX
@misc{caldin2026diagonalization,
title = {{Diagonalization Proves More Time Solves More Problems — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553837},
url = {https://zenodo.org/records/23553837},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diagonalization Proves More Time Solves More Problems — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553837
UR - https://zenodo.org/records/23553837
AB - 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)) subsetneq 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
ER -