{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R663",
  "content_hash": null,
  "slug": "rm28-claim-cubic-witness-distance-88",
  "type": "claim",
  "title": "An eight-term cubic has exact second-order nonlinearity 88",
  "summary": "The cubic f=abc+adg+afh+agh+bdh+beg+ceh+def has minimum distance exactly 88 from every 8-variable Boolean polynomial of degree at most two.",
  "relevance": "For Covering radius of the second-order Reed-Muller code RM(2,8), record rm28-claim-cubic-witness-distance-88 (“An eight-term cubic has exact second-order nonlinearity 88”) records a bound, answer, status fact, or structural consequence. The record states: The cubic f=abc+adg+afh+agh+bdh+beg+ceh+def has minimum distance exactly 88 from every 8-variable Boolean polynomial of degree at most two.",
  "relevance_source": "recorded",
  "body": "Brier and Langevin record the following cubic; this packet checks its distance independently. Use variables \\(a,b,c,d,e,f,g,h\\), and set\n\\[\nF=abc+adg+afh+agh+bdh+beg+ceh+def.\n\\]\nSplit on \\(a\\). With \\(y=(b,c,d,e,f,g,h)\\), write \\(F=G(y)+aP(y)\\), where\n\\[\nG=bdh+beg+ceh+def,\\qquad P=bc+dg+fh+gh.\n\\]\nEvery quadratic correction has the form \\(H(y)+A_0(y)+aA_1(y)\\), where \\(H\\) is homogeneous quadratic and \\(A_0,A_1\\) are affine. The two affine corrections can therefore be optimized independently on the slices. If \\(H\\) ranges over the 21-dimensional space of homogeneous quadratics in seven variables, this gives\n\\[\nd_2(F)=\\min_H\\bigl(d_1(G+H)+d_1(G+P+H)\\bigr).\n\\]\nShifting \\(H\\) by \\(P\\) swaps the two summands. Translating a coordinate of \\(y\\) shifts \\(G\\) by its coordinate derivative and shifts each quadratic by an affine function, which leaves \\(d_1\\) unchanged. Thus the pair score is invariant under \\(P\\) and the seven coordinate derivatives of \\(G\\). These eight quadratics are independent, leaving a quotient of dimension 13. Exhaustive Walsh-transform evaluation of its 8,192 representatives gives the score histogram\n\\[\n\\begin{array}{c|rrrrrrr}\n\\text{score}&88&92&96&100&104&108&112\\\\\n\\text{count}&28&1016&2968&3024&1092&56&8.\n\\end{array}\n\\]\nThe minimum is 88. The quadratic \\(Q=ad+bc+bd+bg\\) gives \\(\\operatorname{wt}(F+Q)=88\\) directly, with slice weights 40 and 48. The replay also fixes the truth-table ordering and checks both table digests.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "one displayed 8-variable cubic against every degree-at-most-two Boolean correction",
    "bounds": {
      "variables": {
        "min": 8,
        "max": 8
      },
      "correction_degree": {
        "min": 0,
        "max": 2
      },
      "witness_count": {
        "min": 1,
        "max": 1
      },
      "quotient_representatives": {
        "min": 8192,
        "max": 8192
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact derivation and exhaustive inline replay in rm28-artifact-exact-cubic-distance"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact derivation and exhaustive inline replay in rm28-artifact-exact-cubic-distance"
  },
  "relations": [
    {
      "slug": "R657",
      "title": "Exact quotient and Walsh replay for the distance-88 cubic",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R658",
      "title": "Full 2^21-quadratic C cross-check of the cubic distance",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R662",
      "title": "The full covering radius satisfies 88 <= rho(2,8) <= 96",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R660",
      "title": "A 2026-07-28 source audit confirms the current 88 to 96 interval",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R661",
      "title": "A monolithic Z3 distance minimization timed out",
      "object_type": "attempt",
      "relation": "attempts",
      "direction": "incoming"
    },
    {
      "slug": "reed-muller-rm2-8-covering-radius",
      "title": "reed muller rm2 8 covering radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}