{ "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/ActuatorConstitutiveReal.lean", "source_sha256": "ca84ad202854c712c05d58d948b16f9bc48e1b7b557613124ebb2920f41c6453", "claims_sha256": "409ce86cb88b8b16ec19bb0ef5c177276b18b62e60356444a877f6e4ee00e66e", "command": "lake --no-cache env lean -DwarningAsError=true proofs/ActuatorConstitutiveReal.lean (from pinned mathlib workspace)", "claims": [ { "id": "actuator-real-activation-interval", "theorem": "Kenoma.ActuatorReal.activation_in_unit_interval", "claim": "The exact real constant-input activation update stays in [0,1].", "assumptions": "Real a,u in [0,1], h>=0 and tau>0.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/elbow.mjs: activation; web/series.mjs: stage activation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-activation-ode", "theorem": "Kenoma.ActuatorReal.activation_hasDerivAt", "claim": "The exact real update has derivative (u-a(t))/tau with respect to elapsed time.", "assumptions": "Real a,u,h; fixed real tau!=0 and constant input u.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/elbow.mjs: exact activation function.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-active-force-bounds", "theorem": "Kenoma.ActuatorReal.active_force_bounds", "claim": "Real Gaussian active force is nonnegative and at most a*F0.", "assumptions": "Real activation a>=0 and maximum force F0>=0; any real fiber/optimal length/width.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/elbow.mjs: activeFiber force shared by the series lesson.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-active-force-derivative", "theorem": "Kenoma.ActuatorReal.active_force_hasDerivAt", "claim": "The real Gaussian derivative equals -2*z*FA/(w*l0).", "assumptions": "Real fixed activation and maximum force; optimal length l0!=0 and width w!=0; z=(f/l0-1)/w.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/elbow.mjs: activeFiber slope; web/series.mjs: implicit derivative denominator.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-gaussian-radius-bound", "theorem": "Kenoma.ActuatorReal.gaussian_weighted_radius_bound", "claim": "The real Gaussian factor obeys 2*abs(z)*exp(-z^2)<=sqrt(2/exp(1)).", "assumptions": "Any real normalized coordinate z.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/elbow.mjs: Gaussian force-length law; exact series slope estimate.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-active-slope-bound", "theorem": "Kenoma.ActuatorReal.active_slope_bound", "claim": "The real Gaussian slope magnitude is at most F0*sqrt(2/exp(1))/(w*l0).", "assumptions": "Real a in [0,1], F0>=0, l0>0 and w>0.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/series.mjs: the exact monotonicity bound used by the runtime guard.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-series-strict-monotone", "theorem": "Kenoma.ActuatorReal.series_residual_strictMono", "claim": "The real compliant-series residual including tension-only passive force is strictly increasing.", "assumptions": "Real a in [0,1], F0>=0, l0>0, w>0, compliance c>=0, passive stiffness k>=0; guard c*F0*sqrt(2/exp(1))/(w*l0)<1.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/series.mjs: H(f)=f+c*(FA+FP)-totalFiber.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-series-unique-root", "theorem": "Kenoma.ActuatorReal.series_equilibrium_unique", "claim": "Two real roots of the compliant-series residual are equal under the sufficient guard.", "assumptions": "Same physical signs and guard as strict monotonicity; both real residuals equal zero. Root existence is not assumed without an explicit root.", "limitations": "Real mathematical law only; no JavaScript refinement, finite-precision residual, solver termination, or biological validation.", "implementation": "web/series.mjs: bracketed series equilibrium.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-series-root-exists", "theorem": "Kenoma.ActuatorReal.series_equilibrium_exists", "claim": "A feasible real residual has a root in the bracket [0,totalFiber].", "assumptions": "Real a>=0, F0>=0, l0>0, w>0, compliance c>=0 and passive stiffness k>=0; totalFiber>=0 and H(0)<=0.", "limitations": "Continuity and endpoint signs prove existence, without asserting uniqueness unless the separate monotonicity guard holds. No minimum-fiber acceptance, finite-precision residual or solver termination guarantee.", "implementation": "web/series.mjs: lower/upper bracket for the quasistatic root.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-series-lower-rejection", "theorem": "Kenoma.ActuatorReal.series_positive_lower_endpoint", "claim": "Positive real H(0) excludes any nonnegative fiber root under the guard.", "assumptions": "Real a in [0,1], F0>=0, l0>0, w>0, compliance c>=0, passive stiffness k>=0 and c*F0*sqrt(2/exp(1))/(w*l0)<1; H(0)>0 and fiber>=0.", "limitations": "Supports the mathematical reason for rejecting the lower-end infeasible case; no binary64 branch equivalence or minimum-fiber acceptance guarantee.", "implementation": "web/series.mjs: H(0)>0 boundary result reported inadmissible.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "actuator-real-series-denominator", "theorem": "Kenoma.ActuatorReal.series_denominator_positive", "claim": "The real reported branch denominator 1+c*(FA_slope+branchK) is positive under the guard.", "assumptions": "Real a in [0,1], F0>=0, l0>0, w>0, compliance c>=0 and passive branch stiffness branchK>=0; c*F0*sqrt(2/exp(1))/(w*l0)<1.", "limitations": "An algebraic denominator bound for either nonnegative passive branch; the passive kink is not claimed differentiable. No floating-point division, implicit-velocity refinement, work/energy or biological validation.", "implementation": "web/series.mjs: denominator uses activeFiberSlope and passiveK or zero.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] } ] }