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MathKernel: An evidence-aware multi-engine mathematics kernel and MCP server
An evidence-aware multi-engine mathematics kernel — usable both as a Python library (mathkernel) and as an MCP server (mathkernel-mcp) — so applications and LLMs can do advanced mathematics while preserving assumptions, provenance, and claim-specific evidence.
The LLM interprets intent; the MathKernel establishes mathematical evidence.
Mathematical results carry an explicit trust level, an engine tag, and a derivation trail. Exact computation, checked certificates, symbolic results, certified enclosures, empirical evidence, and formal proofs are distinct claims. Exact arithmetic alone is not a formal proof; approximate-input ancestry must not silently disappear.
LLMs are good at mathematical intent and bad at mathematical arithmetic. MathKernel inverts the division of labor: the model parses, plans, and interprets; the kernel computes and records claim-specific evidence. Some claims use independent certificates or cross-checks; others are exact computations in one engine. Engine agreement alone is not a proof, and a single trust label does not replace the evidence bundle.
MathKernel is a typed orchestration layer rather than a single solver. The public
facade owns parsing, contexts, object identity, persistence, evidence composition,
resource policy and derivation tracking; domain adapters own the actual mathematics.
Presentation layers sit downstream and cannot silently change the claim being made.
This separation is deliberate: a renderer may present evidence, but it does not create
stronger mathematical evidence merely by producing a polished plot or audio artifact.
The generic MCP tools math_object_create, math_object_get, and math_apply
expose the following compositional operations. This is the full typed-operation
inventory; math_capability_query is the live source of parameter schemas,
output types, limits, engines and verification methods.
Source objects use the same boundary: transform/complex/probability objects,
graphs and combinatorial structures, finite groups/rings/fields/modules,
signals/filters/control systems, optimization problems, and Manifold →
Chart → Metric/CoordinateMap/TensorField/DifferentialForm, plus
Point/PointSet/Polygon/Polytope/Triangulation, and finite
SimplicialComplex/CubicalComplex/integral ChainComplex, and typed
StatisticalSample observations, GeneralizedLinearModel specifications, and
SurvivalDataset/CoxProportionalHazardsModel survival sources, plus
TimeSeriesDataset/TimeSeriesModel ordered-time sources, and
PoissonProcess/WienerProcess/GaussianProcess/ContinuousTimeMarkovChain
process-law sources, StochasticDifferentialEquation Itô models, and structured
PDEProblem equations/domains/conditions.
NonparametricTestResult, ResamplingResult, KaplanMeierEstimate, GLMFit,
and CoxPHFit are derived-only, source-linked records with deterministic exact,
numerical, or seeded-stream replay. TimeSeriesAnalysis, TimeSeriesFit, and
TimeSeriesForecast, FiniteDimensionalDistribution,
GaussianProcessPosterior, and CTMCTransition follow the same output-only
replay boundary. PDEClassification, PDECompatibilityReport, and WeakForm
replay their principal-part, represented-trace, or complete weak-identity result
from the source problem. FEMMesh links that weak form and an optional verified
triangulation. ReferenceElement, BasisFunctionSet, QuadratureRule, and
FiniteElementSpace are output-only with replayable single- or multi-source
ancestry. AssembledSystem retains local and sparse global contributions plus
its space/quadrature sources; output-only FEMSolution retains the exact
assembled-system source and replayable solver diagnostics. G.5 output-only
FEMErrorEstimate, RefinementMarking, RefinedMesh, MeshTransfer, and
FEMConvergenceObservation records retain the complete solution-to-child-mesh
chain, marking policy, parent/child cells, interpolation weights and empirical
rate inputs.
SDESimulation and SDEConvergenceStudy additionally replay
their PCG64 streams and discretizations. Derived-only types can