{
  "schema": 1,
  "lean_version": "Lean (version 4.19.0, x86_64-unknown-linux-gnu, commit 6caaee842e94, Release)",
  "mathlib": {
    "repository": "https://github.com/leanprover-community/mathlib4",
    "tag": "v4.19.0",
    "commit": "c44e0c8ee63ca166450922a373c7409c5d26b00b",
    "manifest_sha256": "5c6421b650bc87a2427a892a39e5522dc44543ca4ac1bd6e9f08d536c044a752",
    "toolchain": "leanprover/lean4:v4.19.0",
    "license": "Apache-2.0",
    "qualification": "Fresh compilation required; this lock alone is not proof evidence."
  },
  "source": "proofs/SerialSpecimenReal.lean",
  "source_sha256": "b0b4ccb250f7a10561c66fc32def5aad4f3ec900eeb3575a3e8bd4aacf0f3fbe",
  "claims_sha256": "ae5102fb1f62d0499af6aa4aa7ab7faf0d2ad4830da441d5c4e991fded772ec1",
  "command": "lake --no-cache env lean -DwarningAsError=true proofs/SerialSpecimenReal.lean (from pinned mathlib workspace)",
  "claims": [
    {
      "id": "serial-real-energy-factor",
      "theorem": "Kenoma.SerialSpecimenReal.energy_factor",
      "claim": "The real homogeneous neo-Hookean energy has the exact nonnegative-factor form.",
      "assumptions": "Real mu; lambda>0.",
      "limitations": "W=mu/(2*lambda)*(lambda-1)^2*(lambda+2); signs are not assumed in this algebraic identity. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-energy-sign",
      "theorem": "Kenoma.SerialSpecimenReal.energy_nonnegative",
      "claim": "Real homogeneous stored energy is nonnegative.",
      "assumptions": "Real mu>=0 and lambda>0.",
      "limitations": "W=mu/2*(lambda^2+2/lambda-3); this is local energy, not global specimen stability. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-energy-zero",
      "theorem": "Kenoma.SerialSpecimenReal.energy_zero_iff",
      "claim": "Real homogeneous stored energy vanishes exactly at lambda=1.",
      "assumptions": "Real mu>0 and lambda>0.",
      "limitations": "The zero-energy state of this one-parameter homogeneous law is unique; no global minimization is certified. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-energy-derivative",
      "theorem": "Kenoma.SerialSpecimenReal.energy_hasDerivAt",
      "claim": "The exact real derivative of W with respect to lambda equals nominal axial stress P.",
      "assumptions": "Fixed real mu; lambda>0.",
      "limitations": "HasDerivAt W P lambda, with P=mu*(lambda-1/lambda^2); no numerical differentiation is certified. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-stress-identities",
      "theorem": "Kenoma.SerialSpecimenReal.nominal_factor_and_cauchy",
      "claim": "Nominal stress factors through lambda-1, and axial Cauchy stress equals lambda*P.",
      "assumptions": "Real mu and lambda>0.",
      "limitations": "P=mu*(lambda-1)*(lambda^2+lambda+1)/lambda^2; sigma=mu*(lambda^2-1/lambda)=lambda*P. Lateral pressure and matrix stress equations are not proved here. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-stress-sign",
      "theorem": "Kenoma.SerialSpecimenReal.nominal_sign",
      "claim": "Nominal stress is tensile, zero or compressive exactly when lambda is above, equal to or below one.",
      "assumptions": "Real mu>0 and lambda>0.",
      "limitations": "All three sign equivalences concern this homogeneous local law. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-stress-difference",
      "theorem": "Kenoma.SerialSpecimenReal.nominal_difference",
      "claim": "The real nominal-stress difference has an exact positive-factor identity.",
      "assumptions": "Real mu; positive stretches u>0 and v>0.",
      "limitations": "P(v)-P(u)=mu*(v-u)*(1+(u+v)/(u^2*v^2)); this row states an algebraic identity. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-stress-monotone",
      "theorem": "Kenoma.SerialSpecimenReal.nominal_strictMono",
      "claim": "Real nominal stress is strictly increasing on positive axial stretches.",
      "assumptions": "Fixed real mu>0; stretches in (0,infinity).",
      "limitations": "StrictMonoOn P (Set.Ioi 0); no global multiaxial stability is certified. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-root-unique",
      "theorem": "Kenoma.SerialSpecimenReal.homogeneous_root_unique",
      "claim": "A fixed axial force has at most one positive homogeneous stretch root.",
      "assumptions": "Real mu>0, reference area A>0 and any real N; u>0,v>0 with A*P(u)=N and A*P(v)=N.",
      "limitations": "Uniqueness only; existence, bisection brackets, termination and residual accuracy are not proved. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-volume",
      "theorem": "Kenoma.SerialSpecimenReal.area_length_volume",
      "claim": "The imposed current area and length preserve each block reference volume exactly.",
      "assumptions": "Real reference area A and length L; lambda>0; current area is A/lambda and current length is L*lambda.",
      "limitations": "(A/lambda)*(L*lambda)=A*L. Incompressibility and these geometric formulas are assumptions; mesh boundary integration and det(F) are not certified. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-tension-order",
      "theorem": "Kenoma.SerialSpecimenReal.same_force_tension_order",
      "claim": "Under equal tensile force, the narrower reference block stretches more.",
      "assumptions": "Common real mu>0; 0<A1<A2; N>0; positive stretches lambda1,lambda2 satisfying A1*P(lambda1)=A2*P(lambda2)=N.",
      "limitations": "1<lambda2<lambda1. Equal force is an explicit assumption for separate homogeneous blocks connected through ideal massless lateral-sliding axial fixtures. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "serial-real-compression-order",
      "theorem": "Kenoma.SerialSpecimenReal.same_force_compression_order",
      "claim": "Under equal compressive force, the narrower reference block compresses more.",
      "assumptions": "Common real mu>0; 0<A1<A2; N<0; positive stretches lambda1,lambda2 satisfying A1*P(lambda1)=A2*P(lambda2)=N.",
      "limitations": "lambda1<lambda2<1. Equal force is an explicit assumption for separate homogeneous blocks connected through ideal massless lateral-sliding axial fixtures. Real local homogeneous incompressible material law only; no JavaScript refinement, numerical root or mesh-volume validation, continuous stepped-solid interface, global stability, bulk-penalty calibration or anatomical claim.",
      "implementation": "web/serial-specimen.mjs: nominalStress and energyDensity use the same factored real laws; solveStretch numerically solves the separate homogeneous resultant; measured mesh volume is checked independently.",
      "status": "checked",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    }
  ]
}