Twin Prime Proof Claim Unverified; Maynard's Bound Remains Best — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve bounds gaps, but no proof exists. One arXiv claim (1708.07884v1) asserts a constructive proof via sieve of Eratosthenes, but is not peer-validated. | MATH: Twin primes: pairs (p, p+2) with p prime. Maynard's theorem: liminfn→∞ (pn+1 − pn) ≤ 246 (unconditionally; 6 with Elliott–Halberstam). Conjecture: infinitely many n with pn+1 − pn = 2. The arXiv paper claims: for each n, at least three new twin pairs appear between (6n+5)2 and (6(n+1)+5)2 — a density claim, not a proof of infinitude. | CONNECTION: Twin primes modulo 6: all pairs > 3 are (6k−1, 6k+1). This is a residue class symmetry in ℤ/6ℤ — a cyclic group of order 6, related to hexagonal lattice symmetry (root system A2). The gap 2 is the smallest nonzero difference in the even lattice 2ℤ. No direct golden-ratio or base-60 link appears in the given s
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Twin Prime Proof Claim Flawed; Maynard's Bound Remains Best — E8 Intelligence Research
- Twin Prime Proof Unsubstantiated; Maynard's 246 Bound Remains Strongest — E8 Intelligence Research
- Twin Prime Proof Claim Unverified; Bounded Gaps Remain Best Result — E8 Intelligence Research
- Twin Prime Progress: Bounded Gaps, Sieve Claims, and Unverified Construction — E8 Intelligence Research
- Twin Prime Conjecture: Unproven, With Recent Sieve Progress and Unverified Claim — E8 Intelligence Research
BibTeX
@misc{caldin2026twin,
title = {{Twin Prime Proof Claim Unverified; Maynard's Bound Remains Best — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635252},
url = {https://zenodo.org/records/23635252},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twin Prime Proof Claim Unverified; Maynard's Bound Remains Best — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635252
UR - https://zenodo.org/records/23635252
AB - FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve bounds gaps, but no proof exists. One arXiv claim (1708.07884v1) asserts a constructive proof via sieve of Eratosthenes, but is not peer-validated. | MATH: Twin primes: pairs (p, p+2) with p prime. Maynard's theorem: liminf_n→∞ (p_n+1 − p_n) ≤ 246 (unconditionally; 6 with Elliott–Halberstam). Conjecture: infinitely many n with p_n+1 − p_n = 2. The arXiv paper claims: for each n, at least three new twin pairs appear between (6n+5)^2 and (6(n+1)+5)^2 — a density claim, not a proof of infinitude. | CONNECTION: Twin primes modulo 6: all pairs > 3 are (6k−1, 6k+1). This is a residue class symmetry in ℤ/6ℤ — a cyclic group of order 6, related to hexagonal lattice symmetry (root system A_2). The gap 2 is the smallest nonzero difference in the even lattice 2ℤ. No direct golden-ratio or base-60 link appears in the given s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -