{
  "schema": 1,
  "lean_version": "Lean (version 4.19.0, x86_64-unknown-linux-gnu, commit 6caaee842e94, Release)",
  "mathlib": "not used; bundled Std only",
  "source_sha256": "6b45c1d9a32fbd0cf6e034d3e4d15d26dc720215afe6845bbbb9d7de07bff2fd",
  "claims_sha256": "e5aee4e575fd3325ea676b0234801b31c9003b46943b3f13dc84ee987298e7b6",
  "source": "proofs/Mechanics.lean",
  "command": "lean -DwarningAsError=true proofs/Mechanics.lean",
  "claims": [
    {
      "id": "force-pair",
      "theorem": "Kenoma.force_pair_cancels",
      "claim": "An equal and opposite scalar force pair cancels.",
      "assumptions": "Exact integer forces in a common positive unit scale; the force pair is assumed.",
      "limitations": "Does not prove Newton's third law from biology or establish a system boundary.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "torque-linearity",
      "theorem": "Kenoma.torque_linear",
      "claim": "Planar torque is additive in applied force at a fixed point.",
      "assumptions": "Exact integer planar coordinates and forces; shared positive length and force scales.",
      "limitations": "No arbitrary real coordinates, JavaScript refinement or rounding bound is established.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "torque-origin",
      "theorem": "Kenoma.torque_origin_shift",
      "claim": "Changing the torque origin subtracts origin cross force.",
      "assumptions": "Exact integer planar coordinates and forces in common positive scales.",
      "limitations": "This is an algebraic identity, not a claim that torque is independent of origin.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "central-pair",
      "theorem": "Kenoma.central_pair_torque",
      "claim": "Equal opposite central forces produce zero summed planar torque.",
      "assumptions": "Force at A is k(B-A); force at B is its negative; integer k and positions.",
      "limitations": "Noncentral pairs can form a couple. Tissue/contact models are not verified by this theorem.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "kinetic-sign",
      "theorem": "Kenoma.kinetic_numerator_nonnegative",
      "claim": "The exact numerator m(vx\u00b2+vy\u00b2) of twice kinetic energy is nonnegative.",
      "assumptions": "Integer mass m \u2265 0; integer velocity coordinates; positive SI scale factors.",
      "limitations": "The factor 1/2 and unit scales are interpreted in prose; no integrator conservation or biological validation.",
      "axioms": [
        "propext"
      ],
      "status": "checked"
    },
    {
      "id": "torque-example",
      "theorem": "Kenoma.torque_worked_example",
      "claim": "300 mm crossed with -49050 mN gives -14715000 scaled torque units.",
      "assumptions": "r=(300,0) mm; F=(0,-49050) mN; product scale 10^-6 N m.",
      "limitations": "Only this exact arithmetic example is checked, not trigonometric slider states.",
      "axioms": [],
      "status": "checked"
    },
    {
      "id": "activation-bound",
      "theorem": "Kenoma.activation_weighted_bound",
      "claim": "A nonnegative integer-weighted mixture of bounded inputs has a bounded numerator.",
      "assumptions": "Exact integer a and u in [0, scale]; weight w in [0, d]. For a normalized mixture, additionally d > 0.",
      "limitations": "Does not prove exp() bounds, floating-point activation, or physiological validity; the numerator is bounded by scale*d.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "virtual-power",
      "theorem": "Kenoma.virtual_power_identity",
      "claim": "The signed generalized-force power equals negative tension times path-rate under the chain-product assumption.",
      "assumptions": "Exact integer force, path derivative dl and angular velocity omega in compatible positive SI scales; derivative and path-rate relations are assumed.",
      "limitations": "Only multiplication/sign algebra is checked; calculus, the geometric derivative and browser arithmetic require separate numerical verification.",
      "axioms": [
        "propext"
      ],
      "status": "checked"
    },
    {
      "id": "element-resultant",
      "theorem": "Kenoma.element_gradient_resultant",
      "claim": "A tetrahedral volume-force contribution has zero resultant when its four coordinate gradients sum to zero.",
      "assumptions": "Exact integer coordinate gradients g1,g2,g3, g0 = \u2212(g1+g2+g3), and a common scalar force factor.",
      "limitations": "The analytic volume derivative, JavaScript implementation, torque, attachments and contact require separate checks. This is the partition algebra used in spatial.mjs.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "centroid-transfer",
      "theorem": "Kenoma.centroid_contact_power",
      "claim": "Equal centroid barycentric weights transfer nodal power with the transpose of the point-motion map.",
      "assumptions": "Exact integer scalar force and four velocity coordinates. Both sides are numerators with a shared division by four in physical interpretation.",
      "limitations": "No division or floating-point equality is proved. Applies coordinatewise to the fixed centroid sample; no nonlinear moving embedding or biological claim.",
      "axioms": [
        "propext"
      ],
      "status": "checked"
    },
    {
      "id": "compliance-denominator",
      "theorem": "Kenoma.compliant_denominator_positive",
      "claim": "A nonnegative inverse-mass quadratic plus strictly positive regularization has a positive scalar denominator.",
      "assumptions": "Exact integer w \u2265 0, gradient g, alpha > 0 in common positive scales.",
      "limitations": "A one-coordinate special case supporting the compliant solver denominator; finite precision, material assembly, time scaling and iterative accuracy are not proved.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    },
    {
      "id": "armijo-decrease",
      "theorem": "Kenoma.accepted_energy_nonincrease",
      "claim": "An accepted sufficient-decrease step cannot increase the exact scaled objective.",
      "assumptions": "Exact integer old/new energy, nonnegative scaled step coefficient, nonpositive directional slope and the stated Armijo acceptance inequality.",
      "limitations": "Does not prove gradients, direction generation, line-search termination, global convergence, float bounds or biological validity. The spatial solver tests its actual accepted energies separately.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "status": "checked"
    }
  ]
}