Cramér's Conjecture: Placeholder, Proof Retracted

Read full text
Authors: Clifford Keeble
Cited by: 6

Abstract

This paper, originally uploaded December 2025, claimed a complete structural proof of Cramér's conjecture g_n = O((log p_n)²). On review in May 2026, the proof is retracted on three grounds: (a) v1.0 Section 2.3's leap from Jacobsthal at log p to prime gaps at p is the conjecture itself, not a proof of it — Jacobsthal bounds gaps among integers coprime to P#, a much larger set than the primes when P# uses only small primes; (b) v1.0 Section 3's forbidden-transitions argument controls residues mod 3, not magnitudes, and therefore cannot prevent unbounded gap growth; (c) the Primary Tension T = √(113/111) ≈ 1.00897 is a finite-range fluctuation from the PNT asymptotic limit of 1, not a derived constant — many small-integer ratios cluster near 1.009 and no derivation in v1.0 specifies these primes from the geometry. The headline proof claim is retracted. The genuine observation in v1.0 Section 3.4 — that consecutive prime gaps cannot both be ≡ 1 (mod 3), nor both ≡ 2 (mod 3) — survives as an elementary structural result about gap residues, and will be re-issued as a short research note in its own slot when written. This v2.1 is a placeholder marking the retraction honestly; the slot is held for the research note when written.

Fields of Study & Subjects

Benford’s Law and Fraud Detection Computability Logic AI Algorithms Probability and Statistical Research

Publication details

Journal / source
Zenodo (CERN European Organization for Nuclear Research)
Publication date
17 May 2026
Document type
Preprint