Oracle Worlds Split: Why Diagonalization Cannot Settle P vs NP — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Baker-Gill-Solovay (BGS) proves P vs NP cannot be settled by any relativizing proof technique — there exist oracles A and B with P^A = NP^A and P^B ≠ NP^B. | MATH: ∃A: P^A = NP^A; ∃B: P^B ≠ NP^B. Diagonalization (the only known lower-bound method pre-1990) relativizes — hence cannot resolve P vs NP. Time Hierarchy: P ≠ EXPTIME (deterministic), but NP vs P remains open. | CONNECTION: The oracle separation is a *binary symmetry* — two distinct worlds (A and B) that are mirror-opposite in their P/NP relation. This is structurally analogous to a *Z₂ symmetry breaking*: the truth of P=NP is not invariant under oracle perturbation. No golden-ratio or base-60 link; the relevant structure is the *lattice of complexity classes* under relativization — a partial order with incomparable elements, reminiscent of root-system Weyl chambers where different choices of simple roots yield different but equally valid geometries. | DEPTH: 7 — Profound for metamathematics: it shows the *independenc
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026oracle,
title = {{Oracle Worlds Split: Why Diagonalization Cannot Settle P vs NP — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553620},
url = {https://zenodo.org/records/23553620},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Oracle Worlds Split: Why Diagonalization Cannot Settle P vs NP — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553620
UR - https://zenodo.org/records/23553620
AB - FINDING: Baker-Gill-Solovay (BGS) proves P vs NP cannot be settled by any relativizing proof technique — there exist oracles A and B with P^A = NP^A and P^B ≠ NP^B. | MATH: ∃A: P^A = NP^A; ∃B: P^B ≠ NP^B. Diagonalization (the only known lower-bound method pre-1990) relativizes — hence cannot resolve P vs NP. Time Hierarchy: P ≠ EXPTIME (deterministic), but NP vs P remains open. | CONNECTION: The oracle separation is a *binary symmetry* — two distinct worlds (A and B) that are mirror-opposite in their P/NP relation. This is structurally analogous to a *Z₂ symmetry breaking*: the truth of P=NP is not invariant under oracle perturbation. No golden-ratio or base-60 link; the relevant structure is the *lattice of complexity classes* under relativization — a partial order with incomparable elements, reminiscent of root-system Weyl chambers where different choices of simple roots yield different but equally valid geometries. | DEPTH: 7 — Profound for metamathematics: it shows the *independenc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -