{
  "abstract": "For the Erdős--Straus equation $4/n=1/x+1/y+1/z$, Vaughan's unconditional bound on the exceptional set $E(N)$ has stood at $|E(N)|\\ll N\\exp(-c(\\log N)^{2/3})$ since 1970. We harness composite divisor witnesses at growing depth to cross this $2/3$ barrier unconditionally. On the full exceptional set $E(N)$, we prove the unconditional bound $|E(N)|\\ll N\\exp(-c'(\\log N)^{5/7}/(\\log\\log N)^{2/7})$, improving the exponent to $5/7$ (with an intermediate stage at $7/10$). On the structured component $E_2^{\\mathrm{small}}(N;J(N))$, an algebraic and spectral container hierarchy combined with a generating-polynomial large sieve reaches the exponent $1-o(1)$: $|E_2^{\\mathrm{small}}(N;J(N))|\\ll_κN\\exp(-(\\frac{1}{40960}+o(1))\\frac{\\log N}{(\\log\\log N)(\\log\\log\\log N)^κ})$, matching the CRT capacity scale. These results resolve the obstructions to composite-witness sieving at growing depth. First, we replace subgroup generation by Kneser's stabilizer $K(M)$, which embeds into a $2$-primary character kernel, reducing container counts from $2^{Θ(J^{4})}$ to $\\exp(\\mathcal{O}(J^{2}\\log\\log J))$. Coupling this with a generating-polynomial large sieve, Markov $1/2$-popular symmetry containers, and Pontryagin--Plancherel character containers yields the exponent $1-o(1)$ on $E_2^{\\mathrm{small}}$. Second, to bypass unlocalized diffuse divisor sheets without subset enumeration, we construct a semi-composite congruence sieve $s=pq\\equiv-1\\pmod{12ua}$ ($p\\in(Z^{9/10},Z]$, $q\\le Q$ rough). Conditioning on $n\\bmod M_{\\mathrm{small}}$ yields CRT mutual independence while decoupling the modulus $12ua\\le 12Z^{1/4}$ from $q$; a Bonferroni majorant and the Bombieri--Vinogradov theorem then yield the unconditional $5/7$ exponent on all of $E(N)$. Finally, we study bilinear dispersion on $4uavw-4u^{2}v-w=n$ and difference sets $D(K_c)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08717",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Dahan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ef67exjzzg5f8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08717",
  "fulltext_url": null,
  "id": "ef67exjzzg5f8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08717",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ef67exjzzg5f8.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Composite witness sieves, Kneser--Plancherel containers, and the exponent $5/7$ for the Erdős--Straus conjecture",
  "title_text": "Composite witness sieves, Kneser--Plancherel containers, and the exponent 5/7 for the Erdős--Straus conjecture",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/ef67exjzzg5f8",
  "venue": "arXiv",
  "versions": []
}