{
  "abstract": "FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods and a claimed (likely flawed) constructive proof using Eratosthenes sieve bounds. | MATH: Twin primes: \\(p, p+2\\) both prime. Maynard's theorem: \\(\\liminf_{n\\to\\infty}(p_{n+k}-p_n) < \\infty\\) for any fixed \\(k\\) (bounded gaps for \\(k\\)-tuples). For \\(k=1\\), best unconditional bound: gaps \\(\\le 246\\) infinitely often (Polymath8). Claimed arXiv proof (1708.07884) asserts at least 3 new twin pairs per increment of \\(n\\) when \\((6n+5)^2\\) is used as a sieve bound — but this is not peer-validated and contradicts known density estimates. | CONNECTION: Twin primes modulo 6: all pairs \\(> (3,5)\\) are of form \\((6n-1, 6n+1)\\) — a lattice of period 6, aligning with hexagonal (crystallographic) symmetry. The ratio of twin primes to primes \\(\\sim \\frac{2C_2}{\\log^2 x}\\) where \\(C_2 \\approx 0.66016\\) (twin prime constant). Note: \\(C_2\\) is not a golden-ratio multiple, but \\(2C_2 \\approx 1.32032\\) — close to\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
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  "author_total": 1,
  "authors": [
    {
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      ],
      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/fh4cspr5pwds3.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23465072",
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  "id": "fh4cspr5pwds3",
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  "language": "eng",
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  "published": "2026-10-19",
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  "ris": "https://research.draconitic.com/paper/fh4cspr5pwds3.ris",
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    {
      "first_seen": "2026-11-02T12:21:49Z",
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  "title": "Twin Prime Progress: Bounded Gaps, Unproven Conjecture, and a Flawed Claim — E8 Intelligence Research",
  "title_text": "Twin Prime Progress: Bounded Gaps, Unproven Conjecture, and a Flawed Claim — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-02T12:21:49Z",
  "url": "https://research.draconitic.com/paper/fh4cspr5pwds3",
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