{
  "subject": "Graffiti 295 proof package",
  "statement_source": {
    "title": "Written on the Wall",
    "page": 80,
    "local_file": "wow-p80-top.png",
    "statement": "If girth is >= 5 then the number of positive eigenvalues of the distance matrix <= n / meangravity."
  },
  "claims": [
    {
      "id": "P1",
      "claim": "Girth >= 5 and n >= 2: sum of gravity entries <= (2m)^2/n, so mean (n^2 entries) <= (n-1)/n and the off-diagonal mean <= 1",
      "type": "proved; formally verified",
      "evidence": "paper section 2, Lemma 1; lean/Graffiti/Basic.lean (sum_gravity_le, meanGravity_le_one, meanGravityOffDiagonal_le_one)"
    },
    {
      "id": "P2",
      "claim": "The distance matrix has at most n positive eigenvalues, counted with multiplicity",
      "type": "proved; formally verified",
      "evidence": "lean/Graffiti/Graffiti295.lean (countP_pos_le_card)"
    },
    {
      "id": "P3",
      "claim": "A graph with no edges has distance matrix 0, hence no positive distance eigenvalue",
      "type": "proved; formally verified",
      "evidence": "lean/Graffiti/Graffiti295.lean (distanceMatrix_eq_zero, countP_pos_eq_zero)"
    },
    {
      "id": "P4",
      "claim": "Graffiti 295 holds for both readings of the mean gravity",
      "type": "formally verified (Lean 4 kernel)",
      "evidence": "lean/Graffiti/Graffiti295.lean (agnt-gg/graffiti-lean commit daf0f85c89c6a7914f7ffd99247a2a1549e26389): graffiti_295, graffiti_295.variants.off_diagonal_mean; Lean v4.33.1, Mathlib v4.33.1; #print axioms: propext, Classical.choice, Quot.sound"
    },
    {
      "id": "P5",
      "claim": "No violation on 346 connected graphs of girth >= 5 (12 named, all 34 with n <= 7, 300 random); named n_+(D) values agree with the July 2026 attack",
      "type": "exact rational gravity, numeric spectra",
      "evidence": "battery_295.py and battery_295_output.txt; ../graffiti-290-artifacts/attack_295_output.txt"
    }
  ],
  "known_risks": [
    "The source does not define the mean of a matrix. Both readings (all n^2 entries; the n(n-1) off-diagonal entries) are proved.",
    "The proof is mathematically complete and formally verified, but has not yet received external peer review."
  ]
}