The Relativization Barrier: Why Oracles Cannot Settle P vs NP — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-10
DOI
https://doi.org/10.5281/zenodo.22689371
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
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preprint

The Relativization Barrier: Why Oracles Cannot Settle P vs NP — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

The Relativization Barrier: Why Oracles Cannot Settle P vs NP — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Baker-Gill-Solovay (1975) relativization barrier shows P vs NP cannot be resolved by diagonalization/oracle-based techniques; random oracles separate P^A ≠ NP^A with probability 1 (Bennett-Gill), yet specific oracles can collapse them. | MATH: For oracle A, P^A ≠ NP^A with probability 1 over random A (measure-theoretic density 1); ∃A: P^A = NP^A (e.g., A = PSPACE-complete oracle); ∃B: P^B ≠ NP^B (e.g., B = random oracle). Time Hierarchy Theorem: P ⊊ EXPTIME (since DTIME(n^k) ⊊ DTIME(2^n) for all k). No constants/ratios emerge — the structure is set-theoretic, not metric. | CONNECTION: None direct to 0.382/0.618/1.618 or base-60. However, the "probability 1" result is a measure-theoretic density — akin to how rationals have measure zero in reals; the random oracle space is a Cantor set with Haar measure, and the separating set has full measure. This mirrors the ubiquity of irrational ratios (golden ratio is irrational, hence "generic" in measure) — but this is an analogy, not a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
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The Relativization Barrier: Why Oracles Cannot Settle P vs NP — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS