omnibias.sos
Sum-of-squares constructions, polynomial positivity checks, and rigorous validation of proposed Gram matrices.
import omnibias.sosUse sos for supported polynomial positivity or moment-based certificate constructions. The monomial basis, coefficient ordering, degree, and domain constraints determine the represented problem. A floating-point optimization routine proposes a candidate; that candidate is not itself a rigorous certificate.
The validation path checks the required matrix and polynomial identities under its stated arithmetic assumptions. Positive definiteness is stronger than semidefiniteness, and near-singular candidates can fail a strict enclosure check even when a floating-point solver reports success. Keep solver output and verified status separate.
A certificate applies to the polynomial and domain encoded in its obligations. It does not certify arbitrary sampled functions or establish an unrestricted analytic theorem. Preserve the complete evidence artifact and inspect optional formal-check availability. This alpha package uses AGPL-or-commercial licensing.
Module contract & mathematical conventions
omnibias-sos: certified universal positivity by optimization.
An SOS decomposition p = z(x)^T Q z(x) with positive-semidefinite Q
certifies p(x) >= 0 for every x. Some nonnegative polynomials are not
sums of squares. A floating-point
semidefinite program *proposes* the Gram matrix Q; the *proof* is a rigorous,
outward-rounded interval LDL^T positive-definiteness certificate from
omnibias.core.verified -- the same finite obligation the Mathlib-free Lean
kernel re-checks, so a sealed certificate can earn theorem_prover_verified.
The optimizer only proposes; the interval algebra and the Lean kernel prove. Everything is *sound by construction*: a failed rounding or positive-definite margin yields an **inconclusive** verdict, never a false positivity claim.
Shared formulas do not imply universal floating-point identity. Verify dtype, shape, derivative-order support, and the stated numerical assumptions for the path you use.
certifyModule ↗combinatorialModule ↗conditionsModule ↗familiesModule ↗formalModule ↗honestyModule ↗inequalityModule ↗monomialsModule ↗positivstellensatzModule ↗problemModule ↗roundingModule ↗solveModule ↗Imports & aliases
These names share a definition with another public module. Follow a name for its full contract, parameters, and examples.
| Import from this module | Canonical reference |
|---|---|
DEFAULT_DENOMINATORS | omnibias.sos.DEFAULT_DENOMINATORS |
Exponent | omnibias.sos.Exponent |
FINITE_DIM_SYSTEM | omnibias.sos.FINITE_DIM_SYSTEM |
GALERKIN_TRUNCATION | omnibias.sos.GALERKIN_TRUNCATION |
GLOBAL_POLYNOMIAL | omnibias.sos.GLOBAL_POLYNOMIAL |
MonomialBasis | omnibias.sos.MonomialBasis |
Polynomial | omnibias.sos.Polynomial |
PolynomialInequalityBackend | omnibias.sos.PolynomialInequalityBackend |
PositivstellensatzCertificate | omnibias.sos.PositivstellensatzCertificate |
RationalPolynomial | omnibias.sos.RationalPolynomial |
SOSCertificate | omnibias.sos.SOSCertificate |
SOSMultiplier | omnibias.sos.SOSMultiplier |
SOSProblem | omnibias.sos.SOSProblem |
SOSScope | omnibias.sos.SOSScope |
arrangement_adapted_basis | omnibias.sos.arrangement_adapted_basis |
certify_nonneg_on_set | omnibias.sos.certify_nonneg_on_set |
certify_sos | omnibias.sos.certify_sos |
certify_sos_rational | omnibias.sos.certify_sos_rational |
degree_reduction_report | omnibias.sos.degree_reduction_report |
gram_products | omnibias.sos.gram_products |
gram_to_poly | omnibias.sos.gram_to_poly |
honesty_labels | omnibias.sos.honesty_labels |
is_nonneg_on_set | omnibias.sos.is_nonneg_on_set |
is_sos | omnibias.sos.is_sos |
is_theorem_prover_verified | omnibias.sos.is_theorem_prover_verified |
lean_available | omnibias.sos.lean_available |
lean_check_sos | omnibias.sos.lean_check_sos |
monomial_basis | omnibias.sos.monomial_basis |
named_adapted_problems | omnibias.sos.named_adapted_problems |
rational_gram | omnibias.sos.rational_gram |
seal_positivstellensatz_certificate | omnibias.sos.seal_positivstellensatz_certificate |
seal_sos_certificate | omnibias.sos.seal_sos_certificate |