claim5-analytic/certificate.json

{
  "destructive_controls": {
    "gamma_outside_convex_range_control": {
      "gamma": "6/5",
      "old_hull": "[0,1]",
      "outside_old_hull": true,
      "updated_point": "6/5"
    },
    "matrix_valued_step_control": {
      "coordinate_step": [
        "3/5",
        "7/10"
      ],
      "coordinate_sum": "11/10",
      "old_hull": "conv{(0,0),(1,0),(0,1)}",
      "outside_old_hull": true,
      "updated_point": [
        "2/5",
        "7/10"
      ]
    },
    "singular_B_control": {
      "B": 0,
      "distinct_argmax_not_singleton": true,
      "points": [
        -1,
        2
      ],
      "top_multiplicity": [
        2,
        2
      ]
    }
  },
  "environment": {
    "platform": "Linux-5.15.0-139-generic-x86_64-with-glibc2.35",
    "python": "3.10.12"
  },
  "exact_combinatorial_audit": {
    "arithmetic": "fractions.Fraction; no floating-point comparisons",
    "exact_query_candidate_pair_checks": 487935,
    "functional_leader_maps_times_gamma": 10236,
    "gamma_values": [
      "1/3",
      "1/2",
      "2/3"
    ],
    "n_values": [
      2,
      3,
      4,
      5
    ],
    "row_stochasticity_checks": 50214,
    "skew_part_identity_implies_same_candidate_map": true,
    "symmetric_part_identity_implies_same_candidate_map": true
  },
  "paper": {
    "arxiv": "2508.09628",
    "main_tex_sha256": "84fc972b95015217ba6738618f213f8e2bde186d4b84d3f551bbb1b17ec32a1b",
    "openreview_id": "zrn7rRuvhW",
    "source_lines": {
      "lemma_2_1": "main.tex:285-306",
      "lemma_2_2": "main.tex:346-355"
    },
    "source_tar_sha256": "9d4482cb592eb1a7622508e7134ea5697c727cc61270bf108b09e99282b09f9b",
    "submission_number": 8097
  },
  "proof_certificate": {
    "lemma_2_1": {
      "decision": "verified_by_independent_measure_zero_proof",
      "independent_steps": [
        "Fix a finite leader history through time t. Each current token is A_i X^0, where A_i is a nonnegative coefficient row summing to one, because every update is a scalar convex combination.",
        "A tie for query i between distinct candidate maps j,k is Q(X^0)=<B^t A_i X^0,(A_j-A_k)X^0>=0. Let a=A_i and b=A_j-A_k; then sum(a)=1 and sum(b)=0.",
        "If the symmetric part of B^t is nonzero and Q were the zero polynomial, diagonal coefficients force a_p b_p=0; cross coefficients then force b=0. If B^t is skew-symmetric, zero polynomial coefficients force a_p b_q=a_q b_p, so b=c a; the row sums give c=0. Thus Q is nonzero whenever the candidate maps are distinct.",
        "The zero set of a nonzero real polynomial has Lebesgue measure zero. There are finitely many leader histories, queries and candidate pairs at fixed t, and countably many integer times. Finite and countable unions preserve measure zero. If b=0, the candidate positions coincide identically and represent one element of the set, not a distinct tie."
      ],
      "logical_dependencies": [
        "finite token count",
        "invertible (hence nonzero) B^t",
        "scalar gamma^t in (0,1)",
        "discrete times t in nonnegative integers"
      ],
      "statement": "For invertible B^t, for Lebesgue-almost-every initial configuration the hardmax set is a singleton for every particle and every finite integer time."
    },
    "lemma_2_2": {
      "decision": "verified_by_direct_convexity_proof",
      "independent_steps": [
        "The selected leader y_i^t belongs to K^t. Therefore x_i^{t+1}=(1-gamma^t)x_i^t+gamma^t y_i^t belongs to K^t by convexity.",
        "K^{t+1} is the convex hull of points x_i^{t+1}, all of which lie in K^t. Since K^t is convex, their entire convex hull is contained in K^t."
      ],
      "statement": "K^{t+1} is a subset of K^t for gamma^t in (0,1)."
    }
  },
  "runtime_seconds": 28.2428120393306
}