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  "slug": "sun-hou-primitive-sum-circle",
  "title": "Primitive neighbours around every finite field",
  "shortTitle": "A circle of primitive sums",
  "url": "https://evidencepress.org/releases/sun-hou-primitive-sum-circle/",
  "oneLine": "An exact sieve and 53 explicit cycles close the finite remainder of the Sun–Hou ordering conjecture for every finite field with more than seven elements.",
  "abstract": "For every prime power q>7, the candidate supplies a circular ordering of all elements of Fq in which each adjacent sum is primitive. It inherits She’s large-field and prime-field ranges and treats the remaining 728 field orders: an exact sieved Chvátal–Erdős criterion covers 675, and explicit Hamilton cycles cover 53. The package records field moduli, exact witnesses, frozen cycles and non-destructive checks. A classical primitive-normal-basis Gray construction handles characteristic two; a sufficient deletion corollary follows from the same bound. This is a finite computer-assisted closure, not a new asymptotic theorem or a claim that every individual case was previously unknown.",
  "datePublished": "2026-10-12",
  "dateModified": "2026-10-12",
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  "doi": "10.5281/zenodo.23307587",
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      "note": "Fresh-extraction exact witness, modulus and cycle checks passed with five rejected controls; optimized Python passed. Regeneration matched all 53 cycles. Linux CI passed.",
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  "problem": {
    "name": "Sun–Hou primitive adjacent-sum conjecture, Conjecture 3.5(i)",
    "url": "https://arxiv.org/abs/1309.1679v9"
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  "metadataProvenance": "Supplied candidate and prose review, with source-backed revisions and reconstructed audit supplements on 12 October 2026. The separately referenced reviewer ZIP was unavailable.",
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  "keywords": [
    "finite fields",
    "Hamiltonian cycles",
    "primitive elements",
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    "The primitive-sum graph on all of Fq is Hamiltonian for every prime power q>7, using inherited ranges and a finite closure.",
    "Of 728 remaining orders, 675 satisfy the exact criterion and 53 have frozen explicit cycles.",
    "A sufficient vertex-deletion bound follows from the same proof; over F32 it permits any six deletions."
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    {
      "citation": "Sun, Some new problems in additive combinatorics, arXiv v9, Conjecture 3.5(i) and Remark 3.5(a): source problem and earlier small cases.",
      "url": "https://arxiv.org/abs/1309.1679v9",
      "doi": "10.48550/arXiv.1309.1679"
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    {
      "citation": "She, Hamiltonicity in graphs defined by primes and primitive elements, v1: inherited large-field and prime-field results.",
      "url": "https://arxiv.org/abs/2609.19114v1",
      "doi": "10.48550/arXiv.2609.19114"
    },
    {
      "citation": "Cohen, Oliveira e Silva and Trudgian, On consecutive primitive elements in a finite field: prime-field ingredient and sieve mechanism.",
      "url": "https://doi.org/10.1112/blms/bdv018",
      "doi": "10.1112/blms/bdv018"
    },
    {
      "citation": "Chvátal and Erdős, A note on Hamiltonian circuits: connectivity versus independence criterion.",
      "url": "https://doi.org/10.1016/0012-365X(72)90079-9",
      "doi": "10.1016/0012-365X(72)90079-9"
    },
    {
      "citation": "Cohen and Huczynska, The strong primitive normal basis theorem: classical normal-basis ingredient.",
      "url": "https://doi.org/10.4064/aa143-4-1",
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