Relativization Barriers and the Limits of Diagonalization in P vs NP — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Baker-Gill-Solovay (1975) proved relativization cannot resolve P vs NP — oracle separation shows diagonalization is fundamentally insufficient for this problem. | MATH: P^A ≠ NP^A for some oracle A; P^B = NP^B for another oracle B. This implies any proof of P≠NP must be non-relativizing (cannot hold under arbitrary oracle substitution). Time Hierarchy Theorem: P ⊊ EXPTIME (since EXPTIME = ∪_k TIME(2^{n^k}) and P = ∪_k TIME(n^k), strict by diagonalization). | CONNECTION: The oracle construction creates a *lattice of computational universes* — each oracle A defines a distinct "world" with its own P^A vs NP^A truth value. This mirrors how crystallographic root systems (A_n, B_n, C_n, D_n) generate distinct symmetric structures from shared axioms. The relativization barrier is a *symmetry* of the problem space: diagonalization respects oracle substitution, so it cannot break the P-NP symmetry. | DEPTH: 8 — This is a profound structural limitation, but it's a *negative* result abou
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Relativization Barriers and the Non-Relativizing Path to P vs NP — E8 Intelligence Research
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BibTeX
@misc{caldin2026relativization,
title = {{Relativization Barriers and the Limits of Diagonalization in P vs NP — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587307},
url = {https://zenodo.org/records/23587307},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Relativization Barriers and the Limits of Diagonalization in P vs NP — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587307
UR - https://zenodo.org/records/23587307
AB - FINDING: Baker-Gill-Solovay (1975) proved relativization cannot resolve P vs NP — oracle separation shows diagonalization is fundamentally insufficient for this problem. | MATH: P^A ≠ NP^A for some oracle A; P^B = NP^B for another oracle B. This implies any proof of P≠NP must be non-relativizing (cannot hold under arbitrary oracle substitution). Time Hierarchy Theorem: P ⊊ EXPTIME (since EXPTIME = ∪_k TIME(2^{n^k}) and P = ∪_k TIME(n^k), strict by diagonalization). | CONNECTION: The oracle construction creates a *lattice of computational universes* — each oracle A defines a distinct "world" with its own P^A vs NP^A truth value. This mirrors how crystallographic root systems (A_n, B_n, C_n, D_n) generate distinct symmetric structures from shared axioms. The relativization barrier is a *symmetry* of the problem space: diagonalization respects oracle substitution, so it cannot break the P-NP symmetry. | DEPTH: 8 — This is a profound structural limitation, but it's a *negative* result abou Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -