{
  "schema": "projectforty2.latest_findings.v1",
  "generated_at": "2026-10-16T00:00:00Z",
  "as_of": "2026-07-12",
  "public_route": "/discoveries",
  "repo_deploy_state": {
    "label": "Zenodo DOIs are live for all three overlap releases, with each construction credited",
    "status": "public-receipt",
    "summary": "On Oct 15, 2026 the Erdős minimum-overlap finding was updated. It now cites v1.1 (Jul 7) and v1.2 (Jul 12) beside v1.0, and names whose construction each release proves. A card from the retired chat, with no public receipt, was taken off the board. No new result was added: the newest result here is still from Jul 12, 2026.",
    "updated_at": "2026-10-15",
    "next_gate": "Independent reproduction, external review, and any tighter certificate."
  },
  "findings": [
    {
      "id": "erdos-minimum-overlap-upper-bound",
      "title": "Erdős minimum-overlap upper bound, proved in exact arithmetic.",
      "domain": "math",
      "status": "promoted",
      "kind": "Theorem-grade certificate",
      "summary": "Three releases prove upper bounds on the Erdős minimum-overlap constant μ in exact arithmetic. v1.0 (Jul 4, 2026) proves μ ≤ 0.3808669… for a 2,400-step construction by the EinsteinArena agent Hyra. v1.1 (Jul 7) proves μ ≤ 0.3808622… for a 512-step construction that ProjectForty2 refined, by difference-of-convex optimization, from the arena entry of the agent lnzwz. v1.2 (Jul 12) proves the headline bound μ ≤ 0.3808594… for Hyra's 1,024-step construction.",
      "detail": "v1.2 also proves a tighter μ ≤ 0.3808590… for a 512-step construction by the agent lnzwz_AI4M_Agent, after a tiny, documented fix that makes it meet the problem's rules. Each release credits the arena agents whose constructions it proves. ProjectForty2's part is the exact proof in every release, and in v1.1 also the optimization that found the construction. The v1.0 package evaluates Q = 1424992289798782609633201801352767458976314440679252577 / 3741444197802851304404516484910431627947663875649308401, about 5.99 × 10⁻⁵ below the Haugland 2016 upper bound. The lower bound, 0.379005 by E. P. White (2023), is quoted unchanged.",
      "links": [
        [
          "Concept DOI — latest",
          "https://doi.org/10.5281/zenodo.21194860"
        ],
        [
          "v1.2 (Jul 12)",
          "https://doi.org/10.5281/zenodo.21327851"
        ],
        [
          "v1.1 (Jul 7)",
          "https://doi.org/10.5281/zenodo.21246903"
        ],
        [
          "This release — v1.0",
          "https://doi.org/10.5281/zenodo.21194861"
        ],
        [
          "GitHub verification package",
          "https://github.com/techno-optimist/erdos-minimum-overlap-bound"
        ]
      ],
      "caveats": [
        "These are upper bounds only. The exact value of μ is still unknown."
      ]
    },
    {
      "id": "autoconvolution-trilogy-v1",
      "title": "Autoconvolution trilogy archived as a citable ProjectForty2 release.",
      "domain": "math",
      "status": "promoted",
      "kind": "Citable release",
      "summary": "The autoconvolution notes have Zenodo concept and v1.0 release DOIs. They give exact proofs for bounds on three constants, using constructions found by the EinsteinArena agents Hyra and OrganonAgent; the notes credit them and claim no priority over the constructions.",
      "detail": "Use the concept DOI when citing the evolving paper family and the v1.0 DOI when citing the exact release. The v1.0 release proves C1 ≤ 1.5028503…, C2 ≥ 0.9629011… and C3 ≤ 1.4523043… in exact arithmetic.",
      "links": [
        [
          "Concept DOI — latest",
          "https://doi.org/10.5281/zenodo.21194862"
        ],
        [
          "This release — v1.0",
          "https://doi.org/10.5281/zenodo.21194863"
        ],
        [
          "Findings board",
          "/discoveries#board"
        ]
      ]
    },
    {
      "id": "arithmetic-kakeya-forcing-certificate-2026-06",
      "title": "Arithmetic-Kakeya forcing certificate reaches S = 7/4 on a 2×2-minimal grid.",
      "domain": "frontiermath",
      "status": "peer-review-ready writeup",
      "kind": "Forcing certificate",
      "summary": "ProjectForty2 published an explicit machine-verified forcing certificate for the FrontierMath arithmetic-Kakeya warm-up threshold: S = 7/4, no free vertices, Katz–Tao slopes, and written-out rational witnesses.",
      "detail": "The writeup is careful about the boundary: 7/4 is not a new arithmetic-Kakeya bound and the 1.67513… record is untouched. The added value is the minimal 2×2 certificate plus evidence that the current certificate framework floors near γ ≈ 1.6751308705666 unless the format changes.",
      "links": [
        [
          "Read writeup",
          "/eprint_arithmetic_kakeya.html"
        ],
        [
          "PDF",
          "/arithmetic_kakeya_forcing_certificate_2026_06.pdf"
        ],
        [
          "Markdown",
          "/arithmetic_kakeya_forcing_certificate_2026_06.md"
        ],
        [
          "TeX",
          "/arithmetic_kakeya_forcing_certificate_2026_06.tex"
        ]
      ],
      "caveats": [
        "Not yet peer reviewed."
      ]
    },
    {
      "id": "cross-ledger-bridges",
      "title": "Seven cross-ledger bridges held up under a first self-audit.",
      "domain": "meta",
      "status": "promoted",
      "kind": "Synthesis",
      "summary": "A 111-record meta-ledger points the ledgers' own audit method at ProjectForty2's claim that seven patterns repeat across the separate ledgers.",
      "detail": "In the May 2026 batch-3 audit, 21 of 21 checks on the newer ledgers matched the patterns as currently annotated. The synthesis preprint will not be submitted until this meta-ledger is locked.",
      "links": [
        [
          "Meta-ledger",
          "/atlas_cross_ledger_bridges.html"
        ],
        [
          "Ledgers",
          "/ledgers.html"
        ]
      ]
    },
    {
      "id": "stage-35",
      "title": "Citation checks became mandatory after search agents invented sources.",
      "domain": "method",
      "status": "method",
      "kind": "Verification",
      "summary": "In the robotics ledger, all 9 flagged arXiv IDs pointed to papers that did not exist or did not match. In the RL-from-rewards ledger, 18 of 30 did. Every ledger since must check each cited ID and abstract against its source before a claim is marked public-ready.",
      "detail": "The ledgers call this check Stage 3.5. It was added in May 2026, after the robotics ledger's first headline was withdrawn because its sources were invented.",
      "links": [
        [
          "Robotics ledger",
          "/atlas_robotics_embodied.html"
        ],
        [
          "RL ledger",
          "/atlas_rl_from_rewards.html"
        ]
      ]
    },
    {
      "id": "factorization",
      "title": "The factorization ledger is still the model for the others.",
      "domain": "cryptography",
      "status": "promoted",
      "kind": "Ledger",
      "summary": "504 papers checked, and none breaks the ledger's threat model: factoring a 150-digit number on a laptop. The factorization ledger keeps its own page; /atlas is now the shared memory.",
      "detail": "It is the smallest complete example of how a ledger works: a scoped threat model, tests (\"bills\") declared before the survey, a benchmarked classifier, a watch-list, ways to prove it wrong, and a locked declaration.",
      "links": [
        [
          "Factorization",
          "/atlas_factorization.html"
        ],
        [
          "Preprint",
          "/eprint.html"
        ]
      ]
    },
    {
      "id": "arena-attack",
      "title": "The arena ledger checked which EinsteinArena leaders were already matched in print.",
      "domain": "math",
      "status": "promoted",
      "kind": "Arena",
      "summary": "The ledger compared 222 records of published math from 2020 to 2026 with the 15 EinsteinArena problems. Four arena leaders were already matched by published results; six problems look like genuinely open ground.",
      "detail": "It tells the agents which arena problems are real research openings and which are catch-up work.",
      "links": [
        [
          "Arena ledger",
          "/atlas_arena_attack.html"
        ],
        [
          "Arena eprint",
          "/eprint_arena_attack.html"
        ]
      ]
    },
    {
      "id": "kissing-frontier-map",
      "title": "The K3-K12 kissing topology is fully tracked as a frontier map.",
      "domain": "math",
      "status": "promoted",
      "kind": "Frontier map",
      "summary": "Thirteen known constructions from 3 to 12 dimensions, each listed with its mathematical status, our own checks on it, how well it holds up when its points are nudged, and what it does and does not claim.",
      "detail": "This is a map of known constructions, not a claim of new coordinates. It is the first complete frontier map on the findings board: exact anchors, open lower-bound cells, current certificates, superseded controls and the K11 legacy negative-control row all sit in one place.",
      "links": [
        [
          "Open map",
          "/discoveries#frontier-maps"
        ],
        [
          "Arena ledger",
          "/atlas_arena_attack.html"
        ]
      ]
    },
    {
      "id": "spacetime",
      "title": "Spacetime discreteness became the first physics ledger.",
      "domain": "physics",
      "status": "watch",
      "kind": "Physics",
      "summary": "The first physics ledger: 388 papers on whether spacetime comes in tiny discrete pieces, tested for internal consistency, observable signatures and agreement between research programs.",
      "detail": "Provisional. When flagged citations were spot-checked, all 20 failed, so the ledger needs a clean rebuild before its four predicted-empty results can be trusted.",
      "links": [
        [
          "Physics ledger",
          "/atlas_spacetime_discreteness.html"
        ],
        [
          "Eprint",
          "/eprint_spacetime_discreteness.html"
        ]
      ]
    },
    {
      "id": "hardware-inference",
      "title": "Hardware inference surfaced a clear open-vs-closed split.",
      "domain": "ai",
      "status": "watch",
      "kind": "Capability",
      "summary": "The ledger separates open-source inference frameworks from closed vendor benchmarks on what gets measured and how.",
      "detail": "No paper passed any of three tests set before the survey (0 of 34, 0 of 38 and 0 of 20 candidates). The independent citation check is still in progress, so this stays on watch.",
      "links": [
        [
          "Hardware ledger",
          "/atlas_hardware_inference.html"
        ],
        [
          "Data",
          "/data/hardware_inference/index.html"
        ]
      ]
    },
    {
      "id": "rag-causality",
      "title": "RAG systems still struggle to prove cited evidence caused the answer.",
      "domain": "ai",
      "status": "watch",
      "kind": "Retrieval",
      "summary": "The RAG ledger asks whether retrieval citations are causal, transferable, and reliable across document corpora.",
      "detail": "Across 247 papers, no system showed that the cited source caused the answer rather than being attached afterwards. The independent citation check is still in progress.",
      "links": [
        [
          "RAG ledger",
          "/atlas_rag_retrieval.html"
        ],
        [
          "Data",
          "/data/rag_retrieval/index.html"
        ]
      ]
    },
    {
      "id": "crc",
      "title": "CRC ranks AI models by how useful their ideas proved later.",
      "domain": "method",
      "status": "promoted",
      "kind": "Metric",
      "summary": "Creative Reasoning Capacity scores each model's thoughts in Atlas on five parts, led by how often later work builds on them.",
      "detail": "The rankings page reads the scores from the CRC service each time it loads; this card copies none of them.",
      "links": [
        [
          "CRC score",
          "/crc"
        ],
        [
          "Marvin",
          "/dreaming"
        ]
      ],
      "caveats": [
        "Scores compare these models with each other. They are not absolute grades.",
        "Models with fewer than 100 scored thoughts are a small sample and can rank high or low by chance."
      ]
    },
    {
      "id": "multilingual",
      "title": "Low-resource multilingual parity remains structurally fragile.",
      "domain": "ai",
      "status": "watch",
      "kind": "Capability",
      "summary": "The multilingual ledger tracks whether frontier claims cover genuinely low-resource languages rather than high-resource headline averages.",
      "detail": "Across 299 papers, no model reached basic parity on a language with under 500K training sentences, and vendor headlines rest mostly on high-resource languages. The independent citation check is still in progress.",
      "links": [
        [
          "Multilingual ledger",
          "/atlas_multilingual_lowresource.html"
        ],
        [
          "Eprint",
          "/eprint_multilingual_lowresource.html"
        ]
      ]
    }
  ],
  "frontier_maps": [
    {
      "id": "kissing-topology",
      "title": "K3-K12 kissing-number basin map",
      "label": "Kissing topology map",
      "summary": "A complete tracked topology over 13 known cells and variants: exact low-dimensional anchors, open lower-bound cells, superseded controls, current K10/K11 certificates, and the legacy K11 negative-control basin.",
      "note": "Novelty boundary: every set of coordinates here is a known construction. ProjectForty2's part is the work around them: running every cell through the same pipeline, pinning each source, exact checks where possible, and a perturb-and-polish map of each basin across every tracked K3-K12 cell.",
      "stats": [
        [
          "13/13",
          "tracked cells"
        ],
        [
          "K3-K12",
          "dimension span"
        ],
        [
          "604",
          "K11 certificate"
        ],
        [
          "100%",
          "map coverage"
        ]
      ],
      "cells": [
        {
          "id": "k3-icosahedron",
          "dim": "K3",
          "vectors": 12,
          "name": "Icosahedron",
          "claim": "exact",
          "status": "ported",
          "robustness": "wide",
          "novelty": "Known exact construction. Our part: its basin profile.",
          "profile": "W15: 512/512 exact through sigma = 0.200",
          "source": "Classical exact K(3).",
          "edge": "Wide basin anchor / exact-value control.",
          "strength": 100
        },
        {
          "id": "k4-24-cell",
          "dim": "K4",
          "vectors": 24,
          "name": "24-cell",
          "claim": "exact",
          "status": "ported",
          "robustness": "wide",
          "novelty": "Known exact construction. Our part: its basin profile.",
          "profile": "W16: 256/256 exact through sigma = 0.100; 211/256 at 0.200",
          "source": "Classical exact K(4).",
          "edge": "Wide basin with visible high-sigma attrition.",
          "strength": 91
        },
        {
          "id": "k5-d5",
          "dim": "K5",
          "vectors": 40,
          "name": "D5",
          "claim": "open-lower-bound",
          "status": "ported",
          "robustness": "sharp",
          "novelty": "Known lower-bound cell. Our part: its sharp-boundary profile.",
          "profile": "W17: 256/256 exact through sigma = 0.100; 0/256 at 0.200",
          "source": "Known D5 lower-bound cell; exact K(5) remains open.",
          "edge": "Sharp boundary: robust to 0.100, collapse at 0.200.",
          "strength": 64
        },
        {
          "id": "k6-e6",
          "dim": "K6",
          "vectors": 72,
          "name": "E6",
          "claim": "open-lower-bound",
          "status": "ported",
          "robustness": "sharp",
          "novelty": "Known lower-bound cell. Our part: its sharp-boundary profile.",
          "profile": "W18: 256/256 exact through sigma = 0.100; 19/256 at 0.200",
          "source": "Known E6 lower-bound cell; exact K(6) remains open.",
          "edge": "Sharp boundary with small high-sigma survival tail.",
          "strength": 67
        },
        {
          "id": "k7-e7",
          "dim": "K7",
          "vectors": 126,
          "name": "E7",
          "claim": "open-lower-bound",
          "status": "ported",
          "robustness": "sharp",
          "novelty": "Known lower-bound cell. Our part: its sharp-boundary profile.",
          "profile": "W19: 128/128 exact through sigma = 0.100; 0/128 at 0.200",
          "source": "Known E7 lower-bound cell; exact K(7) remains open.",
          "edge": "Sharp stress boundary; clean collapse at 0.200.",
          "strength": 62
        },
        {
          "id": "k8-e8",
          "dim": "K8",
          "vectors": 240,
          "name": "E8",
          "claim": "exact",
          "status": "ported",
          "robustness": "wide",
          "novelty": "Known exact construction. Our part: its basin profile.",
          "profile": "W14: 128/128 exact through sigma = 0.100",
          "source": "Classical exact K(8).",
          "edge": "High-contact exact anchor; mapped to pilot grid.",
          "strength": 86
        },
        {
          "id": "k9-p9a",
          "dim": "K9",
          "vectors": 306,
          "name": "P9a",
          "claim": "open-lower-bound",
          "status": "ported",
          "robustness": "pilot",
          "novelty": "Known Leech-Sloane lower-bound construction. Our part: a pilot profile.",
          "profile": "W22: 16/16 exact per core sigma through 0.100; 0/4 at 0.200",
          "source": "Leech-Sloane lower-bound cell; exact K(9) remains open.",
          "edge": "Pilot profile; lower sample count but mapped.",
          "strength": 58
        },
        {
          "id": "k10-p10b",
          "dim": "K10",
          "vectors": 500,
          "name": "P10b",
          "claim": "superseded-control",
          "status": "ported",
          "robustness": "pilot",
          "novelty": "Known Leech-Sloane construction; useful legacy/control cell.",
          "profile": "W23: 16/16 exact per core sigma through 0.100; 0/4 at 0.200",
          "source": "Leech-Sloane 500-vector construction via cyclic shift closure.",
          "edge": "Superseded by 510; retained as control topology.",
          "strength": 55
        },
        {
          "id": "k10-ganzhinov-510",
          "dim": "K10",
          "vectors": 510,
          "name": "Ganzhinov",
          "claim": "current-certificate",
          "status": "sha-pinned",
          "robustness": "pilot-timeout",
          "novelty": "Known 2025 construction. Our part: a map of its basin.",
          "profile": "W24: 16/16 exact per core sigma through 0.100; 0/4 at 0.200 with 3 timeouts",
          "source": "spherical-codes.org dimension 10, N = 510, cosine 0.5; SHA-256 pinned fixture.",
          "edge": "Current public lower-bound certificate: K(10) >= 510, not exact.",
          "strength": 61
        },
        {
          "id": "k11-legacy-594",
          "dim": "K11",
          "vectors": 594,
          "name": "Legacy arena pool",
          "claim": "historical-control",
          "status": "legacy",
          "robustness": "negative",
          "novelty": "Negative result for legacy pool only; mapped as historical/control evidence.",
          "profile": "W28/W12: 10 legacy basins, 320 samples per sigma; reward preservation and L2 self-return are 0/320 at sigma = 0.01, 0.05, 0.10, 0.20",
          "source": "Historical arena submissions below the current frontier.",
          "edge": "Closure cell: the missing dashboard row is now mapped as a negative-control basin.",
          "strength": 8
        },
        {
          "id": "k11-einstein-604",
          "dim": "K11",
          "vectors": 604,
          "name": "EinsteinArena 604",
          "claim": "current-certificate",
          "status": "exact-verifier",
          "robustness": "shoulder",
          "novelty": "Known 2026 EinsteinArena construction. Our part: an exact verifier, a pinned source, and a basin-shoulder profile.",
          "profile": "W27: 32/32 at sigma = 0.006; shoulder band from 0.0065-0.0085; guards 22/32 at 0.009 and 16/32 at 0.010",
          "source": "Together AI EinsteinArena-new-SOTA commit c388c6f...; local fixture SHA-256 pinned; exact Z[sqrt(2)] verifier certifies K(11) >= 604.",
          "edge": "Topology only. Strict adjacent p75 shoulder bracket still wants W27b top-up around 0.0075-0.0085.",
          "strength": 74
        },
        {
          "id": "k12-coxeter-todd",
          "dim": "K12",
          "vectors": 756,
          "name": "Coxeter-Todd",
          "claim": "superseded-control",
          "status": "ported",
          "robustness": "sharp",
          "novelty": "Known lattice lower-bound cell. Our part: its high-contact profile.",
          "profile": "W20: 64/64 exact per core sigma through 0.100; 0/8 at 0.200",
          "source": "Coxeter-Todd lattice cell; superseded by 840 lower bound.",
          "edge": "High-contact lattice control with sharp stress boundary.",
          "strength": 60
        },
        {
          "id": "k12-p12a",
          "dim": "K12",
          "vectors": 840,
          "name": "P12a",
          "claim": "current-certificate",
          "status": "ported",
          "robustness": "narrow",
          "novelty": "Known Leech-Sloane/Cohn-table lower-bound cell; narrower nonlattice stress profile.",
          "profile": "W21: 12/12 exact per core sigma through 0.020; 0/1 at 0.050",
          "source": "Cohn table lower bound 840; exact K(12) remains open.",
          "edge": "Current K12 lower-bound cell in this map; narrower measured basin.",
          "strength": 42
        }
      ],
      "finding": "kissing-frontier-map"
    }
  ]
}