{
  "schema": 1,
  "lean_version": "Lean (version 4.19.0, x86_64-unknown-linux-gnu, commit 6caaee842e94, Release)",
  "mathlib": "not used; bundled Std only",
  "source": "proofs/MaterialResponse.lean",
  "source_sha256": "8ce7f4b5bf4b68f9eeb0d85b06925f417995f945b64498a4533347ead1864092",
  "claims_sha256": "c997267322002413263b8a7f9d04a41807a5b24e2d95f78285cff3404f08fdc9",
  "command": "lean -DwarningAsError=true proofs/MaterialResponse.lean",
  "claims": [
    {
      "id": "material-confined-volume",
      "theorem": "Kenoma.Material.confined_volume",
      "claim": "With both lateral stretch numerators fixed at their reference scale, a smaller height numerator gives a smaller volume numerator.",
      "assumptions": "Exact integers s and h; s > 0 and h < s. Physical interpretation additionally takes h > 0 and stretches b=s/s=1, height=h/s. Volume numerator is b_numer\u00b2 \u00d7 h_numer over s\u00b3.",
      "limitations": "No real determinant, constitutive equilibrium, finite-element volume or floating-point result is proved.",
      "implementation": "web/material-response.mjs: active walls impose b=1; J=h is separately tested numerically.",
      "status": "checked",
      "axioms": [
        "propext"
      ]
    },
    {
      "id": "material-bulk-sign",
      "theorem": "Kenoma.Material.finite_bulk_penalty_nonnegative",
      "claim": "The exact scaled bulk-penalty numerator k \u00d7 delta\u00b2 is nonnegative for k \u2265 0.",
      "assumptions": "Integer k and delta; delta represents a scaled volume-ratio departure, and unit/denominator scale factors are positive.",
      "limitations": "The full real energy, fractional-power shear term and its derivatives are not formalized. A finite penalty is not a hard volume constraint.",
      "implementation": "web/material-response.mjs: bulkEnergyJ uses the unchanged tissue law; derivatives and finite-bulk volume loss are numerical checks.",
      "status": "checked",
      "axioms": [
        "propext"
      ]
    },
    {
      "id": "material-bulk-zero",
      "theorem": "Kenoma.Material.positive_bulk_zero_penalty",
      "claim": "For a strictly positive bulk coefficient, the exact bulk-penalty numerator is zero exactly when its volume-departure numerator is zero.",
      "assumptions": "Exact integer k > 0 and integer delta; positive external scaling factors.",
      "limitations": "Does not assert that a loaded equilibrium has zero penalty, or prove incompressibility, a real solver or biological validity.",
      "implementation": "web/material-response.mjs: positive bulk energy may coexist with J\u22601; K=0 is an explicit counterexample.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "material-separated-wall",
      "theorem": "Kenoma.Material.separated_wall_zero_reaction",
      "claim": "An assumed exact complementarity product with strictly positive wall gap forces zero wall reaction.",
      "assumptions": "Integer reaction and gap; gap > 0 and reaction \u00d7 gap = 0. Physical reaction nonnegativity and positive SI scales are separate inputs.",
      "limitations": "Complementarity is assumed, not derived or certified for the JavaScript root. No continuous collision or spatial contact guarantee.",
      "implementation": "web/material-response.mjs: unilateral walls allow pull-away; numerical tests verify gap, reaction, feasibility and complementarity work.",
      "status": "checked",
      "axioms": [
        "propext",
        "Quot.sound"
      ]
    },
    {
      "id": "material-lateral-work",
      "theorem": "Kenoma.Material.lateral_virtual_work",
      "claim": "The two lateral stress and wall-reaction contributions factor as 2 \u00d7 V\u2080 \u00d7 (P\u2093 + R) in exact scaled arithmetic.",
      "assumptions": "Integer volume, stress and reaction numerators with compatible positive external SI scales. The association with an energy derivative is assumed separately.",
      "limitations": "Does not prove the real energy derivative, stationarity, root existence, uniqueness, global optimality or force validity.",
      "implementation": "web/material-response.mjs: signed lateral residual; finite-difference tests separately check dU/db=2V\u2080P\u2093.",
      "status": "checked",
      "axioms": [
        "propext"
      ]
    }
  ]
}