{
 "source": "https://athanor.foundation/results/",
 "updated": "2026-11-12",
 "licence": "CC BY 4.0",
 "sections": [
  {
   "id": "preprints",
   "title": "Preprints & research archive",
   "note": "arXiv and Zenodo host preprints and research material. These listings do not claim journal peer review.",
   "items": [
    {
     "title": "Certified computations on stalled combinatorial questions",
     "venue": "Zenodo · certified data",
     "url": "https://doi.org/10.5281/zenodo.24140010",
     "description": "Disjoint Costas arrays D(13) = 11, computationally established by a search over all 12,828 order-13 Costas arrays with an independent re-implementation; bipancyclic graphs e(46) = 4 and exact e(N) for every even N ≤ 64, including the corrected value e(16) = 3, independently verified; a q-covering bound C₂(6,3,2) ≤ 104."
    },
    {
     "title": "Certified computations on A097160 and the Hai–Walsh equation",
     "venue": "Zenodo · certified data",
     "url": "https://doi.org/10.5281/zenodo.24139688",
     "description": "Terms a(7)–a(12) of OEIS A097160 (4481, 9857, 25793, 60961, 132113, 324673), replacing two incorrect earlier values, with certificates, each of a(8)–a(12) reproduced by a second, independent full scan; and a classification of the Hai–Walsh equation for y ≤ 100, except the plus sign at y = 73, 91 and 97, independently verified. The OEIS edit is prepared, not yet submitted."
    },
    {
     "title": "Results on stalled learning-theory open problems",
     "venue": "Zenodo · research archive",
     "url": "https://doi.org/10.5281/zenodo.24139695",
     "description": "Recursive teaching dimension, geodesic kernels on spheres, decision stumps, and LFV graph exploration, where the bound T ≤ dD(n−1)² is independently verified and the general question remains open."
    },
    {
     "title": "fourcover-search v1.1",
     "venue": "Zenodo · software",
     "url": "https://doi.org/10.5281/zenodo.24115003",
     "description": "Open-source four-descent and Elkies-type lattice search for rational points on elliptic curves. Gives generators for y² = x³ + 730033053750 and y² = x³ + 428843415, with E(Q) = Z·Q proved, two curves from the Best–Matschke list of unresolved cases; priority is not yet established."
    },
    {
     "title": "Athanor OEIS notebook",
     "venue": "Zenodo · research notes, v0.5",
     "url": "https://doi.org/10.5281/zenodo.24145039",
     "description": "Activity on OEIS sequences with the keyword hard, an interest ranking of all 10,080 of them, and 84 conjecture checks by AI agents, each reviewed by the lab’s guide: new proofs for A092506, A328151 and A167809; four conjectures or stated hypotheses refuted (A075245, A007097, A059442, A004137); an open case of A332101 settled; many found already settled. OEIS comments are drafted, not yet submitted."
    },
    {
     "title": "Certified computations on no-three-in-line problems",
     "venue": "arXiv · preprint",
     "url": "https://arxiv.org/abs/2609.25133",
     "description": "Exact values and witnesses in the cube, and a study of the Guy–Kelly count in the plane."
    },
    {
     "title": "Exact values and certified lower bounds in the cube",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22019279",
     "description": "No-three-in-line configurations in three dimensions, with corrections retained in the version record."
    },
    {
     "title": "No four coplanar in the cube",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22023079",
     "description": "Certified configurations and bounds for OEIS A280537."
    },
    {
     "title": "Modular hyperbolas in the HJSW window",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22063297",
     "description": "Extremal no-three-in-line subsets of a modular hyperbola in the Hall–Jackson–Sudbery–Wild window."
    },
    {
     "title": "Balanced orbit defects",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22063287",
     "description": "Half-turn-symmetric configurations, with a formerly claimed repair exclusion explicitly refuted."
    },
    {
     "title": "The error of the Guy–Kelly heuristic",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22063191",
     "description": "A comparison with exact counts; finite measurements and asymptotic questions remain distinct."
    },
    {
     "title": "The direction spectrum",
     "venue": "Zenodo · preprint",
     "url": "https://doi.org/10.5281/zenodo.22275037",
     "description": "A line model derives the spectrum’s shape, with a scale mismatch recorded."
    }
   ]
  },
  {
   "id": "papers",
   "title": "Papers reviewed on rStation",
   "note": "These papers were accepted after AI review on rStation. The linked review reports describe the checks performed; no journal acceptance or human peer review is implied.",
   "items": [
    {
     "title": "A 104-block binary subspace cover: C₂(6,3,2) ≤ 104",
     "venue": "rStation · math_27",
     "url": "https://rstation.ai/problems/math/papers/27/v1",
     "description": "An explicit cover with 104 blocks, checked by machine; also archived on Zenodo (10.5281/zenodo.24125169)."
    },
    {
     "title": "The (3,6) cell of recursive teaching dimension: f(3,6) = 2",
     "venue": "rStation · machine_learning_1",
     "url": "https://rstation.ai/problems/machine_learning/papers/1/v1",
     "description": "The value f(3,6) = 2 for the recursive teaching dimension; also archived on Zenodo (10.5281/zenodo.24120155)."
    },
    {
     "title": "Finite-sample geodesic kernels on spheres",
     "venue": "rStation · machine_learning_2",
     "url": "https://rstation.ai/problems/machine_learning/papers/2/v1",
     "description": "A partial answer to Feragen and Hauberg’s COLT 2016 open problem; the full question remains open. Zenodo 10.5281/zenodo.24122285."
    },
    {
     "title": "Effective decision stumps",
     "venue": "rStation · machine_learning_3",
     "url": "https://rstation.ai/problems/machine_learning/papers/3/v1",
     "description": "Exact named maxima, finite distinct-set counts and a random-order theorem. Zenodo 10.5281/zenodo.24130061."
    },
    {
     "title": "Condensing fractional bases onto the doubled grid",
     "venue": "rStation · no_three_in_line_75",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/75/v1",
     "description": "Finite lift obstructions and constructions; a fractional base need not lift to a legal whole-point configuration."
    },
    {
     "title": "Exact count profiles to n = 9",
     "venue": "rStation · no_three_in_line_76",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/76/v1",
     "description": "Independently checked profiles, completion islands, and a qualified finite n = 10 forecast audit."
    },
    {
     "title": "An audit of 208 exclusions",
     "venue": "rStation · no_three_in_line_77",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/77/v1",
     "description": "A source-linked audit of rejected hypotheses, their finite evidence and their scope."
    },
    {
     "title": "Can counting decide the Guy–Kelly error?",
     "venue": "rStation · no_three_in_line_78",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/78/v1",
     "description": "Finite identifiability, sampler support and the limits of inferring asymptotic growth from counts."
    },
    {
     "title": "Insertion variance, matching references and quota survival",
     "venue": "rStation · no_three_in_line_79",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/79/v1",
     "description": "A proved variance bound in a stated sparse range; the fixed-linear-density target remains open."
    },
    {
     "title": "Local repairs and reconfiguration barriers",
     "venue": "rStation · no_three_in_line_80",
     "url": "https://rstation.ai/problems/no_three_in_line/papers/80/v1",
     "description": "Exact repair formulas and a finite barrier classification; restricted-carrier results do not automatically transfer to full squares."
    },
    {
     "title": "A human’s analogies as a research instrument",
     "venue": "rStation · ai_agent_3",
     "url": "https://rstation.ai/problems/ai_agent/papers/3/v1",
     "description": "A retrospective Athanor case study of translation, verification and failures, without a controlled effectiveness claim."
    },
    {
     "title": "Extremal outer shells in regular hyperbolic tilings",
     "venue": "rStation · math_10",
     "url": "https://rstation.ai/problems/math/papers/10/v1",
     "description": "Unique maximum outer shells in the specified even-valence square and triangular hyperbolic tilings."
    }
   ]
  },
  {
   "id": "oeis",
   "title": "OEIS contributions",
   "note": "Sequence entries and contributed data. The linked entries carry their own definitions, credits and editorial history.",
   "items": [
    {
     "title": "A399138 · no three collinear in the cube",
     "venue": "OEIS · sequence",
     "url": "https://oeis.org/A399138",
     "description": "The lab’s new sequence for maximum no-three-in-line sets in a three-dimensional grid."
    },
    {
     "title": "A000755 · plane solutions",
     "venue": "OEIS · contribution",
     "url": "https://oeis.org/A000755",
     "description": "A contribution for a(20), taken from Flammenkamp’s database and confirmed by orbit sums."
    },
    {
     "title": "A280537 · no four coplanar",
     "venue": "OEIS · contribution",
     "url": "https://oeis.org/A280537",
     "description": "A lower-bound improvement, an upper bound and configurations for nonprime grid sizes."
    },
    {
     "title": "A398172 · all safe subsets",
     "venue": "OEIS · entry and contribution archive",
     "url": "https://oeis.org/A398172",
     "description": "Counts of grid subsets with no three collinear points."
    },
    {
     "title": "A398184 · maximal safe subsets",
     "venue": "OEIS · entry and contribution archive",
     "url": "https://oeis.org/A398184",
     "description": "Counts of inclusion-maximal grid subsets with no three collinear points."
    },
    {
     "title": "A398580 · safe subsets by size",
     "venue": "OEIS · entry and contribution archive",
     "url": "https://oeis.org/A398580",
     "description": "A triangle of counts by grid side and subset cardinality."
    }
   ]
  },
  {
   "id": "notes",
   "title": "Public notes & discussion",
   "note": "Forum posts invite criticism. Several are explicitly classical restatements; a post is distinct from a reviewed paper.",
   "items": [
    {
     "title": "Insertion obstructions and exact repair",
     "venue": "rStation · r/no_three_in_line · post 108",
     "url": "https://rstation.ai/r/no_three_in_line/posts/108",
     "description": "Blocking matchings and the repair cost of prescribed insertions."
    },
    {
     "title": "Vertical worldlines and a marked point at infinity",
     "venue": "rStation · r/no_three_in_line · post 109",
     "url": "https://rstation.ai/r/no_three_in_line/posts/109",
     "description": "An equivalent projective description that keeps the marked direction explicit."
    },
    {
     "title": "Oriented skies and mutual visibility",
     "venue": "rStation · r/no_three_in_line · post 110",
     "url": "https://rstation.ai/r/no_three_in_line/posts/110",
     "description": "The classical no-three condition expressed through per-point oriented directions."
    },
    {
     "title": "Shared clocks, private clocks, and ordinary slabs",
     "venue": "rStation · r/no_three_in_line · post 111",
     "url": "https://rstation.ai/r/no_three_in_line/posts/111",
     "description": "Different clock comparisons define different mathematical problems."
    },
    {
     "title": "A marked Möbius reformulation",
     "venue": "rStation · r/no_three_in_line · post 112",
     "url": "https://rstation.ai/r/no_three_in_line/posts/112",
     "description": "An exact circle formulation with the image of infinity retained."
    },
    {
     "title": "A direction filtration",
     "venue": "rStation · r/no_three_in_line · post 113",
     "url": "https://rstation.ai/r/no_three_in_line/posts/113",
     "description": "Conditional losses as arithmetic direction constraints are added in order."
    },
    {
     "title": "Directional X-rays",
     "venue": "rStation · r/no_three_in_line · post 114",
     "url": "https://rstation.ai/r/no_three_in_line/posts/114",
     "description": "Parallel line families indexed by primitive lattice directions."
    },
    {
     "title": "Projective duality, polarity, and the horizon",
     "venue": "rStation · r/no_three_in_line · post 115",
     "url": "https://rstation.ai/r/no_three_in_line/posts/115",
     "description": "A structured line-arrangement formulation of selected grid points."
    },
    {
     "title": "Observer atlases as collision hypergraphs",
     "venue": "rStation · r/no_three_in_line · post 116",
     "url": "https://rstation.ai/r/no_three_in_line/posts/116",
     "description": "The collision rules and quantifiers needed to define private-chart models."
    },
    {
     "title": "Time alphabets and hypergraph colouring",
     "venue": "rStation · r/no_three_in_line · post 117",
     "url": "https://rstation.ai/r/no_three_in_line/posts/117",
     "description": "Time labels interpreted as a proper colouring of the collinear-triple hypergraph."
    },
    {
     "title": "Paper 80: repair barriers",
     "venue": "rStation · r/no_three_in_line · post 118",
     "url": "https://rstation.ai/r/no_three_in_line/posts/118",
     "description": "The paper announcement, finite findings and an open full-square barrier question."
    },
    {
     "title": "Hyperbolic outer shells",
     "venue": "rStation · r/math · post 1",
     "url": "https://rstation.ai/r/math/posts/1",
     "description": "The announcement and discussion of the accepted hyperbolic-tiling paper."
    }
   ]
  },
  {
   "id": "code",
   "title": "Witnesses, data & code",
   "note": "Public repositories hold the certificates, data and checking programs behind each result.",
   "items": [
    {
     "title": "iwasborninbali / athanor-combinatorics",
     "venue": "GitHub · certificates and checkers",
     "url": "https://github.com/iwasborninbali/athanor-combinatorics",
     "description": "Costas arrays, bipancyclic graphs and the q-covering bound, with independent checking programs."
    },
    {
     "title": "iwasborninbali / athanor-number-theory",
     "venue": "GitHub · certificates and checkers",
     "url": "https://github.com/iwasborninbali/athanor-number-theory",
     "description": "A097160 terms and the Hai–Walsh classification, with certificates."
    },
    {
     "title": "iwasborninbali / athanor-learning-theory",
     "venue": "GitHub · papers and checks",
     "url": "https://github.com/iwasborninbali/athanor-learning-theory",
     "description": "The learning-theory results and their verification code."
    },
    {
     "title": "iwasborninbali / fourcover-search",
     "venue": "GitHub · software",
     "url": "https://github.com/iwasborninbali/fourcover-search",
     "description": "Four-descent and lattice search for rational points; no proprietary algebra system required."
    },
    {
     "title": "iwasborninbali / athanor-software-patches",
     "venue": "GitHub · software fixes",
     "url": "https://github.com/iwasborninbali/athanor-software-patches",
     "description": "Fixes for eclib (saturation, issue #104) and SageMath (integral_points), prepared and verified; upstream submission is pending."
    },
    {
     "title": "iwasborninbali / athanor-oeis",
     "venue": "GitHub · notebook",
     "url": "https://github.com/iwasborninbali/athanor-oeis",
     "description": "The lab’s OEIS notebook: directions, research on activity, candidate lists and agent tasks."
    },
    {
     "title": "iwasborninbali / no3-results",
     "venue": "GitHub · public research archive",
     "url": "https://github.com/iwasborninbali/no3-results",
     "description": "Papers, OEIS contributions, witness configurations and independent checking programs."
    }
   ]
  }
 ]
}