{ "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/MixedLogVolume.lean", "source_sha256": "d1411e4068ea99b9f7a85fd0a573e9ecd976b260617d0f209b6baebb3ae7e15e", "claims_sha256": "4af9bb7096ea3e76dc3212f95769cddddd414047be9c5b369f165f130a6fad72", "command": "lake --no-cache env lean -DwarningAsError=true proofs/MixedLogVolume.lean (from pinned mathlib workspace)", "claims": [ { "id": "projection-inner-comm", "theorem": "KenomaMixedVolume.inner_comm", "claim": "Weighted inner-product symmetry", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-add-right", "theorem": "KenomaMixedVolume.inner_add_right", "claim": "Right addition linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-add-left", "theorem": "KenomaMixedVolume.inner_add_left", "claim": "Left addition linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-sub-right", "theorem": "KenomaMixedVolume.inner_sub_right", "claim": "Right subtraction linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-sub-left", "theorem": "KenomaMixedVolume.inner_sub_left", "claim": "Left subtraction linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-smul-right", "theorem": "KenomaMixedVolume.inner_smul_right", "claim": "Right scalar linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-smul-left", "theorem": "KenomaMixedVolume.inner_smul_left", "claim": "Left scalar linearity", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-normSq-nonneg", "theorem": "KenomaMixedVolume.normSq_nonneg", "claim": "Nonnegative squared weighted norm", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-normSq-eq-zero-iff", "theorem": "KenomaMixedVolume.normSq_eq_zero_iff", "claim": "Positive-definite squared weighted norm", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-normSq-sub", "theorem": "KenomaMixedVolume.normSq_sub", "claim": "Squared norm of a difference", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-normSq-smul", "theorem": "KenomaMixedVolume.normSq_smul", "claim": "Squared norm under scalar multiplication", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-project-eq-self-iff", "theorem": "KenomaMixedVolume.project_eq_self_iff", "claim": "Projection fixes exactly represented vectors", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-project-idempotent", "theorem": "KenomaMixedVolume.project_idempotent", "claim": "Projection is idempotent", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-inner-project", "theorem": "KenomaMixedVolume.inner_project", "claim": "Projection preserves pairing with admissible vectors", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. Candidate p belongs to Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-project-self-adjoint", "theorem": "KenomaMixedVolume.project_self_adjoint", "claim": "Weighted self-adjoint projection", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-optimizer-mem", "theorem": "KenomaMixedVolume.optimizer_mem", "claim": "Optimizer belongs to the pressure space", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-square-completion", "theorem": "KenomaMixedVolume.square_completion", "claim": "Square completion of the mixed objective", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0. Candidate p belongs to Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-optimizer-value", "theorem": "KenomaMixedVolume.optimizer_value", "claim": "Attained condensed energy", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-objective-le-condensed", "theorem": "KenomaMixedVolume.objective_le_condensed", "claim": "Admissible objective bounded by condensed energy", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0. Candidate p belongs to Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-value-eq-iff-optimizer", "theorem": "KenomaMixedVolume.value_eq_iff_optimizer", "claim": "Maximum value exactly at the optimizer", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0. Candidate p belongs to Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-unique-maximizer", "theorem": "KenomaMixedVolume.unique_maximizer", "claim": "Unique maximizing sampled pressure vector", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0. Candidate p belongs to Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-pythagoras", "theorem": "KenomaMixedVolume.pythagoras", "claim": "Weighted Pythagorean decomposition", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-pointwise-minus-condensed", "theorem": "KenomaMixedVolume.pointwise_minus_condensed", "claim": "Exact pointwise-minus-condensed residual energy", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-condensed-nonneg", "theorem": "KenomaMixedVolume.condensed_nonneg", "claim": "Nonnegative condensed energy", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-gap-nonneg", "theorem": "KenomaMixedVolume.gap_nonneg", "claim": "Nonnegative fixed-vector energy gap", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-gap-zero-iff-represented", "theorem": "KenomaMixedVolume.gap_zero_iff_represented", "claim": "Zero gap exactly for represented vectors", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, { "id": "projection-nested-energy-monotone", "theorem": "KenomaMixedVolume.nested_energy_monotone", "claim": "Nested-space monotonicity at the same field", "assumptions": "Finite vectors Fin n -> Real (including n=0); strictly positive diagonal weights; exact real arithmetic. Supplied real linear projection into Q with residual orthogonal to every vector in Q. One constant K>0. Q is contained in R; both projectors use the same weights, K and fixed g.", "limitations": "Conditional finite-vector algebra only; no numerical-projector refinement, re-equilibrated stability, inf-sup, nonlinear derivative/Hessian, continuum or anatomical guarantee. Vector uniqueness does not imply basis-coefficient uniqueness without injective evaluation.", "implementation": "standalone/pressure-projection-lab.html: fixed four-sample group means; separate numerical checks, not a Lean instantiation.", "status": "checked", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] } ] }