Twin Prime Progress: Bounded Gaps, Sieve Claims, and Unverified Construction — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Recent progress on twin primes centers on bounded gaps (Maynard–Tao) and a claimed sieve-based construction, but no proof of infinitude exists; one arXiv paper claims a constructive lower bound via (6n+5)². | MATH: Twin primes: pairs (p, p+2) with p prime. Maynard–Tao theorem: lim infₙ (pₙ₊ₖ − pₙ) < ∞ for any k, specifically gaps ≤ 246 unconditionally (Polymath). The arXiv claim (1708.07884) asserts ≥3 new twin pairs per increment n using sieve bounds near (6n+5)² — unverified, likely flawed. No new constants or ratios emerge. | CONNECTION: Twin primes (excluding 2,3) are of form (6k−1, 6k+1) — a residue class modulo 6, which is a lattice in ℤ/6ℤ. This ties to base-6 structure, not base-60. No golden ratio, 0.618, or crystallographic symmetry appears. The modulus 6 is the smallest non-trivial cyclic group with two coprime units (±1), echoing root system A₂ in 2D hexagonal lattices — but that is structural analogy, not proven harmonic link. | DEPTH: 3/10 — The Maynard–Tao bound
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026twin,
title = {{Twin Prime Progress: Bounded Gaps, Sieve Claims, and Unverified Construction — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658277},
url = {https://zenodo.org/records/23658277},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twin Prime Progress: Bounded Gaps, Sieve Claims, and Unverified Construction — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658277
UR - https://zenodo.org/records/23658277
AB - FINDING: Recent progress on twin primes centers on bounded gaps (Maynard–Tao) and a claimed sieve-based construction, but no proof of infinitude exists; one arXiv paper claims a constructive lower bound via (6n+5)². | MATH: Twin primes: pairs (p, p+2) with p prime. Maynard–Tao theorem: lim infₙ (pₙ₊ₖ − pₙ) < ∞ for any k, specifically gaps ≤ 246 unconditionally (Polymath). The arXiv claim (1708.07884) asserts ≥3 new twin pairs per increment n using sieve bounds near (6n+5)² — unverified, likely flawed. No new constants or ratios emerge. | CONNECTION: Twin primes (excluding 2,3) are of form (6k−1, 6k+1) — a residue class modulo 6, which is a lattice in ℤ/6ℤ. This ties to base-6 structure, not base-60. No golden ratio, 0.618, or crystallographic symmetry appears. The modulus 6 is the smallest non-trivial cyclic group with two coprime units (±1), echoing root system A₂ in 2D hexagonal lattices — but that is structural analogy, not proven harmonic link. | DEPTH: 3/10 — The Maynard–Tao bound Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -