MathKernel: An evidence-aware multi-engine mathematics kernel and MCP server

MathKernel, an evidence-aware multi-engine mathematics kernel usable as a Python library and MCP server, has been released to let applications and LLMs perform advanced mathematics while preserving assumptions, provenance, and claim-specific evidence. The kernel assigns each result a trust level, engine tag, and derivation trail, distinguishing exact computation, checked certificates, symbolic results, certified enclosures, empirical evidence, and formal proofs. It is designed so that the LLM handles intent and the kernel handles computation, with a typed orchestration layer separating parsing, contexts, and evidence composition from domain-specific solvers.

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. Python / MCP | v MathKernel facade |-- parser + contexts + typed objects |-- execution/evidence contract |-- persistence + derivation graph | +-- symbolic / exact / certified / formal / numerical engines | +-- MathResult and derived mathematical objects | +-- MultimodalProjection |-- mathkernel-viz |-- mathkernel-sonify +-- unified portable artifacts 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. | Domain | Compute surface | Engines | Verification / evidence ceiling | |---|---|---|---| | Symbolic algebra | parse, substitute, simplify/expand/factor, solve, systems | SymPy | SYMBOLIC; input ancestry may lower it | | Calculus | differentiation, integration, limits, series, sums, products | SymPy | SYMBOLIC + conditions | | Integral transforms | Laplace/Fourier/Mellin/bilateral Z, inverses, ROC and property obligations | typed transform adapter + SymPy | SYMBOLIC; NUMERIC for approximate ancestry | | Complex analysis | branches/domains, zeros/singularities, residues, Laurent series, contours, argument principle, continuation, conformal maps | typed complex adapter + SymPy | SYMBOLIC defining identities; EXACT winding certificates only for exact geometry, ancestry-capped otherwise | | Continuous probability | typed univariate/joint/conditional distributions, transformations, marginals, Bayes, covariance, divergence, order statistics | typed probability adapter + SymPy | SYMBOLIC normalization/identity evidence; mathematical nonexistence retained | | Exact graphs | typed simple/directed/weighted/multi graphs, traversal, components, shortest paths, MST, max-flow/min-cut, bipartite matching, Euler trails, coloring, topological sort, cycles, centrality, isomorphism | deterministic exact graph algorithms over Fraction + njit CSR traversal kernels | EXACT witness certificates; NP-hard optimality is OPTIMUM/CANDIDATE/IMPOSSIBLE/UNKNOWN, never heuristic nonexistence | | Exact combinatorics | combinatorial classes, exact counts, lazy generation, ordinary/exponential generating functions, recurrences | exact integer/ Fraction enumeration + SymPy + checked njit recurrence kernels | EXACT counts and recurrence/coefficient checks | | Finite algebra | finite groups, permutation groups, abelian groups, homomorphisms, Z/nZ, GF p^m , modules, Smith/Hermite normal forms | exact algebra + SymPy combinatorics + njit Cayley/GF p x kernels | EXACT axiom, homomorphism, irreducibility, and normal-form certificates | | Linear algebra | determinant, inverse, multiply, rank, RREF, eigenvalues, exact solves | SymPy | EXACT for exact arithmetic; otherwise ancestry-capped | | Reasoning | obligation-DAG planning, equivalence, counterexamples | SymPy + Z3 + Lean | SYMBOLIC / EXACT / FORMAL by verifier | | Certified numerics | arbitrary-precision evaluation and interval enclosures | mpmath + mpmath.iv | CERTIFIED NUMERIC or NUMERIC | | Integers | arbitrary precision, gcd/lcm, primality, factorization, CRT, modular arithmetic | exact + numba batch | EXACT | | Code generation | TypeScript/Python/Rust emission, typecheck, symbolic round-trip, sandbox | compilers + SymPy | SYMBOLIC verification; never stronger than source | | Binary fields | GF 2^m arithmetic/construction and Rabin irreducibility | njit n-limb kernels | EXACT certificates | | GF 2 linear algebra | rank, nullspace, powers, Berlekamp–Massey, carry-free columns | bit-packed integers | EXACT | | Discrete transforms | exact FWHT with bigint fallback | numba | EXACT | | Finite dynamics | Koopman/observation transfer, visibility, lagged tensors, diagnostics | exact + NumPy/CuPy | EXACT or NUMERIC, selected explicitly | | Branching Markov tensors | arbitrary finite rooted Markov trees, exact leaf laws/cumulants, true-edge flattening certificates, stochastic leaf observations, channel-rank transfer, exact recovery and collective sensor fusion | exact Fraction sum-product/enumeration + NumPy SVD diagnostics | EXACT algebraic identities/ranks/recovery; NUMERIC singular-value and conditioning evidence kept separate | | Connected-relation detectability | pure connected-interaction laws, stochastic mode visibility, conditional-expectation spectra, exact chi-square/Fisher retention, invisibility certificates, finite sample bounds and sensor fusion | exact Fraction laws + weighted NumPy SVD + exact binomial likelihood-ratio validation | EXACT transfer/information identities and lower/upper bounds; EMPIRICAL Monte Carlo checks remain separately labelled | | Relation-subspace visibility | multi-relation Fisher Gram transfer, generalized visibility spectra, blind-combination collision certificates, cost-constrained sensor design, empirical partitions and long-run-covariance correction | finite probability algebra + weighted NumPy generalized eigensystems + exact finite sensor enumeration | EXACT local transfer/data-processing/collision identities; NUMERIC spectra and EMPIRICAL dependence/SkewDB checks retain explicit scope | | Intrinsic observation information geometry | finite-simplex Fisher tangents, coordinate-invariant retained-information spectra, exact local chi-square transfer, worst-direction testing lower bounds, finite Bhattacharyya upper bounds, iid/block/cluster spectrum bootstrap, local-resolution SkewDB adapter | finite probability algebra + weighted generalized eigensystems + SciPy exact-binomial validation + seeded resampling | EXACT finite tangent/data-processing/divergence identities and finite simple-testing bounds; NUMERIC eigensystems and EMPIRICAL uncertainty checks remain separately labelled | | Composite relation inference | one direction-agnostic relation-subspace test, dimension-aware finite bound, nuisance-efficient Fisher geometry, eigenspace regions, studentized/block bootstrap, HAC and misspecification diagnostics | finite Fisher algebra + NumPy eigensystems + optional SciPy chi-square calibration + seeded resampling | EXACT nuisance/data-processing identities and conservative bounded-score guarantee; ASYMPTOTIC composite calibration and EMPIRICAL bootstrap/dependence checks are labelled | | Finite Fourier | cyclotomic DFT/transfer/coefficient/orbit calculations | exact + NumPy FFT | EXACT or NUMERIC cross-check | | Closure search | cyclic/XOR irreducible closure relations | njit meet-in-the-middle | EXACT witness/exhaustive evidence | | Conditioned dynamics | orbit access, cocycles, closures and symmetry synthesis | exact enumeration + canonical rewrite | EXACT witnesses | | Cumulants | moments/cumulants and connected sample statistics | exact + NumPy | EXACT algebra or EMPIRICAL samples | | Sets & logic | set algebra, membership, quantified truth and elimination | SymPy sets + Z3 | EXACT SMT witnesses where established | | Polynomial algebra | Gröbner bases, division, resultants, factorization, ideal membership | exact SymPy polynomial algorithms | EXACT algebraic certificates | | Discrete probability | rational RVs, Bayes, Markov quantities, seeded sampling | Fraction + NumPy | EXACT distributions; EMPIRICAL sampling | | Statistics and stochastic systems | typed samples, GLMs, rank/resampling inference, survival/time-series analysis; Poisson/Wiener/GP/CTMC laws; typed Itô SDEs, Euler–Maruyama/scalar Milstein paths and coupled convergence studies | typed statistical/survival/time-series/stochastic/SDE adapters + SymPy + NumPy/SciPy/mpmath | EXACT identities remain separate from labelled NUMERIC fits/conditioning/exponentials and seeded EMPIRICAL resampling/simulation; no implied process/model validity, convergence theorem, population inference or causality | | Tensors | sparse tensors, contraction and sparse solves | exact + njit + CuPy | EXACT or NUMERIC by arithmetic path | | ODEs / PDE | symbolic ODE classification/dsolve; numerical IVP/named PDE solvers; typed PDE systems, weak forms, oriented simplex meshes, P1 spaces, sparse assembly, checked algebraic solves, residual–jump indicators, marking, conforming refinement, nodal transfer and observed estimator rates | typed PDE/FEM/adaptivity adapters + SymPy + SciPy sparse + mpmath + njit + CUDA/CuPy | estimators and empirical rates retain ancestry and never become rigorous continuum bounds or convergence theorems | | Optimization | critical points, KKT, exact LP, numerical nonlinear/multistart | Fraction + njit + process pool | EXACT LP certificates or NUMERIC candidates | | Units | SI dimensions, rational conversions and semantic-unit propagation | exact Fraction | EXACT | | Assurance | interval obligations, Lean replay, Arb balls, persistence and fuzzing | mpmath.iv + flint + Lean | CERTIFIED NUMERIC / FORMAL / differential evidence | | Theorem proving | SMT portfolio and Lean certificates | Z3 + Lean | EXACT SMT witness or FORMAL kernel-checked proof | | Exhaustive sweeps | Collatz and cuboid searches | numba + CUDA + process pools | EXACT only when coverage is exhaustive | | Async jobs | submit/status/result/list with evidence-preserving retrieval | job pool | Preserves underlying evidence | | Visualization | renderer-neutral interactive/static mathematical artifacts | Python SVG + vendored three.js | No new evidence; preserves source trust | | Sonification | declarative scientific audio mappings and deterministic WAV | Python PCM + WebAudio | Candidate observation only | | Multimodal artifacts | synchronized visual/audio artifact assembly | shared artifact schema | Weakest included claim/evidence | | Differential geometry | manifolds, oriented charts, metrics, coordinate maps, tensor fields, forms, curvature, covariant/Lie/exterior derivatives, wedge/interior/pullback/Hodge operations | typed geometry adapter + SymPy | SYMBOLIC identities with explicit domains, Jacobians, signature and ancestry; numeric input stays NUMERIC | | Computational geometry | concrete points/sets, polygons, half-space polytopes, triangulations, hull, containment, intersection, nearest neighbor, Delaunay and Voronoi | exact SymPy determinants + adaptive float filters | EXACT topology for exact coordinates; NUMERIC only when filters decide; otherwise explicit AMBIGUOUS outcome | | Algebraic topology | finite simplicial/cubical/integral chain complexes, exact triangulation conversion, oriented boundaries, Euler characteristic, homology over Z/Q/GF p | exact integer matrices + certified Smith normal form + rational/modular elimination | EXACT face-closure, boundary², rank-nullity, quotient, torsion and Euler–Poincaré certificates | 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. | Domain | Object | Operations | |---|---|---| | Integral transforms | TransformProblem | apply , solve , verify | | Complex analysis | ComplexFunction | analytic continuation , analyticity , argument principle , classify singularity , conformal at , conformal map , contour integral , derivative , laurent series , residue , singularities , zeros | | Complex analysis | Contour | winding number | | Continuous probability | Distribution | cdf , characteristic function , convolve , cross entropy , entropy , expectation , kl divergence , mean , mgf , mixture , moment , order statistic , pdf , quantile , query , survival , truncate , variance , verify | | Continuous probability | JointDistribution | bayes , condition , correlation , covariance , marginal , order statistic , verify | | Continuous probability | ConditionalDistribution , RandomVariable | conditional cdf / mean / pdf / variance / verify ; random-variable transform | | Exact graphs | Graph , MultiGraph | bfs , centrality , coloring , connected components , cycle detection , dfs , euler path , matching , shortest path , verify ; Graph also has isomorphic to | | Exact graphs | DirectedGraph | bfs , centrality , cycle detection , dfs , shortest path , strongly connected components , topological sort , verify | | Exact graphs | WeightedGraph | bfs , centrality , coloring , connected components , cycle detection , dfs , euler path , matching , maximum flow , minimum cut , minimum spanning tree , shortest path , strongly connected components , topological sort , verify | | Combinatorics | CombinatorialClass , GeneratingFunction | class count / generate / verify ; generating-function coefficient / recurrence / verify | | Finite groups | FiniteGroup | center , centralizer , closure , commutator subgroup , conjugacy classes , cosets , generated subgroup , normality , orbits , order , quotient , stabilizers , subgroups , verify | | Finite groups | PermutationGroup | contains , orbits , order , stabilizer chain , stabilizers , verify | | Finite groups | FiniteAbelianGroup , GroupHomomorphism | abelian order / verify ; homomorphism image / kernel / verify | | Finite algebra | FiniteRing , FiniteField | add , inverse , multiply , verify | | Finite algebra | Module | abelian group , hermite normal form , smith normal form , verify | | Signals | ContinuousSignal , DiscreteSignal | continuous sample ; discrete autocorrelation , convolution , correlation , cross spectrum , dft , resample , stft , window | | Signals | Spectrum , Filter , FilterDesign , FilterState | spectrum idft ; filter apply signal / initial state / to transfer function ; design design ; state process | | Control | TransferFunction | bode , feedback , frequency response , impulse response , nyquist , poles , root locus , series , stability , step response , to filter , to state space , to zero pole gain , zeros | | Control | StateSpaceSystem | bode , coefficient units , controllability , discretize , finite lqr , frequency response , kalman , kalman state , lqg , lqr , mpc , nyquist , observability , observer , place poles , poles , stability , state feedback , to discrete control , to transfer function , zeros | | Control | DiscreteControlSystem | bode , controllability , frequency response , nyquist , observability , poles , stability , to state space , to transfer function , zeros | | Control | ZeroPoleGain , TransferMatrix | ZPK bode / nyquist / poles / to transfer function / zeros ; matrix entry | | Sequential control | FiniteHorizonLQR , KalmanState , MPCPlan | LQR control / rollout / verify ; Kalman predict / update ; MPC first control / verify | | Optimization | OptimizationProblem | certify milp , solve , to conic , verify certificate , verify milp certificate | | Optimization | ConicProblem , QuadraticallyConstrainedProblem | solve , verify certificate | | Differential geometry | Metric | inverse metric , christoffel , riemann , ricci , scalar curvature , einstein , geodesic equations | | Differential geometry | CoordinateMap | jacobian , verify | | Differential geometry | TensorField | covariant derivative , lie derivative | | Differential geometry | DifferentialForm | wedge , exterior derivative , interior product , pullback , hodge star | | Computational geometry | Point | distance to | | Computational geometry | PointSet | orientation , incircle , segment intersection , convex hull , nearest neighbor , delaunay , voronoi | | Computational geometry | Polygon | verify , contains , intersection , triangulate | | Computational geometry | Polytope | verify , contains | | Computational geometry | Triangulation | verify , to simplicial complex | | Algebraic topology | SimplicialComplex , CubicalComplex | verify , chain complex , boundary matrix , homology | | Algebraic topology | ChainComplex | verify , boundary matrix , homology , euler characteristic | | Statistical evidence and inference | StatisticalSample | describe , covariance , empirical distribution , evidence profile , mann whitney , wilcoxon , kruskal wallis , ks 2samp , spearman , kendall , permutation test , bootstrap | | Survival analysis | SurvivalDataset | verify , kaplan meier | | Survival analysis | KaplanMeierEstimate | verify , survival at | | Survival analysis | CoxProportionalHazardsModel | verify , fit | | Survival analysis | CoxPHFit | verify , diagnostics , predict partial hazard | | Time series | TimeSeriesDataset | verify , acf , pacf , stationarity test | | Time series | TimeSeriesAnalysis | verify | | Time series | TimeSeriesModel | verify , fit | | Time series | TimeSeriesFit | verify , diagnostics , forecast | | Time series | TimeSeriesForecast | verify | | Stochastic processes | PoissonProcess | verify , pmf , moments , increment distribution | | Stochastic processes | WienerProcess | verify , finite dimensional , increment distribution | | Stochastic processes | GaussianProcess | verify , finite dimensional , condition | | Stochastic processes | ContinuousTimeMarkovChain | verify , transition matrix , distribution , stationary distribution | | Stochastic process results | FiniteDimensionalDistribution , GaussianProcessPosterior , CTMCTransition | verify | | Stochastic differential equations | StochasticDifferentialEquation | verify , simulate , convergence study | | SDE simulations | SDESimulation | verify , path , terminal values | | SDE convergence | SDEConvergenceStudy | verify | | Generalized linear models | GeneralizedLinearModel | verify , fit | | Generalized linear models | GLMFit | verify , diagnostics , predict | | Non-parametric results | NonparametricTestResult , ResamplingResult | verify | | Partial differential equations | PDEProblem | verify , classify , boundary compatibility , derive weak form | | PDE results | PDEClassification , PDECompatibilityReport | verify | | Weak formulations | WeakForm | verify | | Finite-element mesh | FEMMesh | verify , reference element , finite element space | | Reference element | ReferenceElement | verify , basis , quadrature | | Finite-element results | BasisFunctionSet , QuadratureRule , FiniteElementSpace | verify | | FEM algebra | AssembledSystem | verify , solve | | FEM solution | FEMSolution | verify , estimate error | | FEM error estimate | FEMErrorEstimate | verify , mark , compare | | Refinement | RefinementMarking | verify , refine | | Refined mesh | RefinedMesh | verify , reference element , finite element space | | Mesh transfer / convergence | MeshTransfer , FEMConvergenceObservation | verify | 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 cannot be forged through public input. pip install mathkernel Python mathematical core pip install 'mathkernel mcp ' add the optional MCP transport From a source checkout: python -m venv .venv source .venv/bin/activate Windows: .venv\Scripts\activate pip install -e . Python mathematical core pip install -e '. mcp ' add the optional MCP transport Optional extras: pip install -e '. perf ' numba — JIT kernels sieves, GF 2^m , FWHT, closure search pip install -e '. cuda ' CuPy + all nvidia- -cu12 runtime libraries RTX-class GPU pip install -e '. latex ' antlr4 runtime for math parse latex pip install -e '. dev ' pytest Lean 4 + Mathlib is installed by default on first mathkernel-mcp start and via mathkernel-lean-setup elan + a pinned lake workspace . Skip with MATHKERNEL SKIP LEAN INSTALL=1 CI/wheel smoke . GPU note: CuPy wheels ship no CUDA libraries. The cuda extra installs the matching nvidia- -cu12 pip packages — without them, cuBLAS/NVRTC DLL loads fail even though import cupy succeeds. GPU availability is probed at runtime with a real matmul, so a broken stack degrades gracefully to CPU. Verify your stack with python scripts/gpu smoke.py . mathkernel-mcp The server speaks MCP over stdio FastMCP 3 and ships core instructions to the client at initialize time: discover → parse → context → trust discipline → async jobs → provenance. 162 tools, all prefixed math . Typical agent session: math capabilities discover surface, limits, engines math parse "x^2 - 3 x + 2 = 0" - expr id math context create domains={"x": "real"} - context id math reason expr id, context id, formal=true solve + independently verify math derivation trace step id full provenance on demand Long-running sweeps are async: php math job submit "collatz", {"n max": 14} - math job status job id - math job result job id The MCP server is a thin transport layer; everything is available in-process: python from mathkernel import MathKernel kernel = MathKernel symbolic r = kernel.parse "x^2 - 2 = 0" sol = kernel.solve r.data "expr id" , "x" assert sol.ok and sol.trust.value == "symbolic" exact GF 2^m field arithmetic f = kernel.gf2m create 8, "1b" AES polynomial x^8 + x^4 + x^3 + x + 1 hex reduction part kernel.gf2m compute f.data "field id" , "mul", "53", "ca" finite dynamics: an explicit eight-state cyclic permutation transition = 1, 2, 3, 4, 5, 6, 7, 0 fs = kernel.finite system create "uniform", transition km = kernel.koopman matrix fs.data "system id" , {"kind": "walsh", "r": 3} vis = kernel.koopman visibility fs.data "system id" , {"kind": "walsh", "r": 3} Exact zeros certify the requested modes in this declared finite model. closure relations njit meet-in-the-middle kernel.closure search "cyclic", m="97", weight bound=10, multipliers= "1", "5" Standalone modules mathkernel.gf2m , mathkernel.koopman , mathkernel.relations , mathkernel.cumulants , mathkernel.finite fourier , mathkernel.transforms , mathkernel.integral transforms , mathkernel.complex analysis , mathkernel.continuous probability , mathkernel.integers , mathkernel.computational geometry , mathkernel.algebraic topology , mathkernel.collatz , mathkernel.cuboid are usable without the facade when you don't need derivation tracking. formal Lean certificate accepted by the Lean kernel exact exact computation / checked claim-specific certificate symbolic symbolic engine agreement e.g. SymPy residual checks interval certified rigorous enclosure mpmath interval numeric high precision arbitrary-precision numeric numeric float evidence incl. GPU fast paths empirical / heuristic / unknown Overall trust is limited by the weakest evidence required to establish the claimed result — never the maximum trust emitted by any single node. Independent backend disagreement is preserved as an explicit conflict, not averaged away. Every MathResult also carries an evidence bundle with separate computation, proof, certificate, numerical, model and empirical evidence. claim evidence retains those bundles per conclusion instead of flattening unlike claims into one score. The legacy trust field remains a conservative summary and is automatically capped by the evidence required for the result. A producer-supplied justified trust is a ceiling, never an override; an unverified proof or certificate supports only unknown . Semantic statuses distinguish proof or certification strength from mathematical outcomes such as does not exist , undefined , infeasible and unsupported . These distinctions survive MCP serialization, asynchronous job retrieval, derivation replay, visualization and multimodal artifact assembly. The capability registry separates advertised trust levels from verification methods. Query it by domain, input/output type, operation, trust level, verification method or engine; capability records also identify their execution handler and meaningful cost dimensions. Expression plans record the resolved capability route before the existing obligation executor runs it. Exact and numeric paths are strictly separated: koopman/finite-dynamics tools default to exact=true proof-grade rational/cyclotomic values ; exact=false selects the vectorized numeric path CuPy GPU when usable and downgrades trust to numeric . Decimal literals are approximate observations. A decimal RealNode anywhere in an expression caps its trust at numeric from parse onward — 0.1 + x parses as numeric , 1/2 + x as symbolic . Formal certificates Lean and exact SMT counterexamples are refused for approximate inputs, because the backends would encode decimal syntax as exact rationals — silently proving a different statement. Use exact rationals or interval certification when proof-grade evidence is needed. Continuous domains use typed objects and the compositional object create → apply model rather than exposing a flat CAS surface. Every operation records a four-obligation DAG: typed-input validation, candidate computation, domain-invariant verification and conservative evidence reconciliation. - Integral transforms — Laplace, Fourier, Mellin and bilateral Z transforms with explicit conventions, assumptions and regions of convergence. Inverse Z uses annulus-aware Laurent/residue extraction when justified. Verification records round-trip, linearity, convolution, differentiation, value-theorem and ROC obligations separately; unresolved obligations remain unknown . - Complex analysis — derivatives, analyticity candidates, zeros, singularities, Laurent series, residues, contour integration, winding numbers, argument-principle accounting, conservative identity continuation and domain-aware conformal maps. Branch conventions, cuts, excluded points, contour orientation and boundary incidents remain explicit. - Continuous probability — typed univariate, random-variable, joint and conditional distributions; PDF/CDF/survival/quantile, moments, transforms, entropy, truncation, convolution, mixtures, divergence, marginals, conditioning/Bayes, covariance/correlation and order statistics. Support, parameter constraints, Jacobians and inverse branches are retained. Symbolic availability is candidate evidence, not independent proof. Same-engine identities are capped at symbolic ; decimal ancestry remains capped at numeric . does not exist for example, a Cauchy mean is distinct from an unsupported method or an unresolved convergence question. Conventions and assumptions are part of the object. Fourier sign and normalization, transform source/target variables, complex branches/cuts, probability supports and parameter constraints are never selected silently. Contour orientation and singularity accounting are mandatory where the theorem depends on them. Verification is operation-specific. Transforms retain every checked or unresolved identity and ROC obligation. Residues are compared with defining limit/derivative or Laurent-coefficient formulas; contour claims retain enclosed singularities, cuts and winding numbers. Probability verifies normalization, support-aware nonnegativity, CDF boundaries/derivative/monotonicity when decidable, and Jacobian branches. These are symbolic checks unless an exact certificate or separate numerical record says otherwise. Failures use semantic statuses: candidate , unknown , unsupported , does not exist , and error are distinct. Known limitations include non-product joint supports, continuation without an explicit overlapping source domain, branch-sensitive argument-principle inputs, transforms whose ROC SymPy cannot establish, and general multivariate changes of variables without supplied inverse branches/Jacobians. Continuous symbolic work is bounded by the global AST/output/solver-time limits and dedicated contour, joint-dimension, mixture-component, series-order, order-statistic and inverse-branch limits. Raise the corresponding MATHKERNEL MAX value explicitly when a larger request is intentional. PDF → Laplace transform, preserving support and evidence ancestry d = kernel.object create "Distribution", { "family": "exponential", "parameters": "2" , "variable": "x", } r = kernel.apply d.data "object id" , "integral transform", { "transform": "laplace", "transform variable": "s", "convention": "laplace standard", } assert r.data "value" == "2/ s + 2 " A distinctive capability: exact spectral analysis of finite dynamical systems X, μ, T, O — built for and validated on PRNG structure analysis. - Koopman suite — transport matrix Q, observation-transfer C, mode visibility ρ O, lagged state tensors raw/connected , observed statistics, IPR/entropy diagnostics. Walsh bases for GF 2 ^r, character bases for Z M. - Stochastic observation transfer library API — exact FiniteJointLaw contractions for arbitrary finite latent joint laws; ordered Markov path moments/cumulants with the required multiplication operators; statewise multiplicativity-defect certificates; and exact finite-noise deterministic dilations for rational Markov kernels. The accompanying published primate quartet pilot deliberately records that the earlier K3ST split-zero diagnostic does not survive outside its group-based assumptions. - Branching General Markov tensors library API — exact FiniteMarkovTree sum-product laws and cumulants on heterogeneous rooted trees; exact L M R edge-flattening certificates with the sharp transition- rank bound; local stochastic observation channels as Kronecker transforms; exact left-inverse recovery, collision witnesses, collective sensor fusion, and channel-conditioned singular-value bounds. The published primate pilot distinguishes algebraic identifiability from finite-sample stability. - Statistical phylogenetic inference library API — probability-simplex projection; known-channel EM and constrained ridge recovery; held-out regularization selection; multinomial covariance and tangent-space Fisher information; nonnegative-rank multinomial likelihood; covariance-Wald rank diagnostics; and tie-safe quartet scoring. Controlled GM 4 experiments quantify the shared singular-value origin of visibility loss and inverse instability. Two fixed published-data pilots add site and moving-block bootstrap checks without claiming broad competitive accuracy. - Frozen phylogenetic benchmarking library API — FASTA, relaxed PHYLIP, practical NEXUS and Newick ingestion; portable source SHA-256 manifests; canonical protocol and corpus locks; result-blind quartet sampling from reference-tree splits; complete-case site provenance; site, circular-block, partition-stratified and whole-partition resampling; rank-tail, p-distance and normalized log-det baselines; and tie-safe corpus summaries. The bundled execution evaluates 22 predeclared correlated units from two published source alignments and a 1,920-alignment known-truth stress grid. A separate lock fixes the first 20 eligible BenchmarkAlignments datasets before acquisition; that external corpus is explicitly pending rather than silently replaced. - Observable connected-relation detection library API - exact and numerical pure-interaction laws; weighted conditional-expectation singular spectra; mode-specific stochastic visibility; exact local-channel transfer of connected amplitude; chi-square and null-Fisher information retention; exact invisibility certificates; finite necessary and constructive sufficient sample bounds; binary-parity scaling; and complementary sensor fusion. The controlled theorem shows that local visibility losses multiply in amplitude and square in information, yielding an s^ -2d detection-cost law in the homogeneous binary specialization. - Relation-subspace visibility and sensor design library API - finite multi-parameter local relation laws; latent and observed Fisher Gram matrices; generalized retained-information eigenvalues and principal visibility directions; exact observation-blind collision certificates; direction-level information and sample multipliers; rank, E-optimal, trace, D-optimal and pseudo-logdet sensor-subset selection; efficient empirical partition transfer; and score-mean long-run-covariance correction. A frozen SkewDB adapter adds source/schema auditing, discovery/validation/challenge splits by held-out taxonomy, discovery-only preprocessing, source hashing and a fail-closed raw-data runner. The bundled SkewDB fixture is explicitly synthetic because the current full payload was not acquired in this environment. - Coordinate-invariant relation geometry library API - finite-simplex tangent vectors with the intrinsic Fisher metric; stochastic tangent pushforward; coordinate-invariant generalized retained-information eigenvalues; exact score/tangent equivalence; exact local chi-square transfer; worst-direction minimax necessary sample bounds; finite Bhattacharyya and retention-based pointwise sufficient counts; and iid, moving-block and cluster bootstrap intervals for ordered relation spectra. A SHA-256-locked local-resolution SkewDB adapter converts documented cumulative fit.csv tracks to window increments and explicitly separates genuine inputs from the bundled source-parameterized generated fixture. - Finite Fourier — exact arithmetic in ℚ ζ L via cyclotomic polynomials: DFT over Z M, output-transfer transforms, two-point difference coefficients, measure Fourier transforms, orbit corrections. - Closure search — short irreducible relations selected by the dynamics: cyclic Σ k j·a^j ≡ 0 mod m and binary ⊕ L^{jK} ᵀ w j = 0 , meet-in-the-middle with L1/Hamming weight bounds. - GF 2^m from transitions — reconstruct the field dual-orbit cyclic basis, minimal/reduction polynomial, Rabin-verified purely from a generator's GF 2 -linear transition columns. - State-conditioned dynamics — exact per-state orbit access T^κ x x : least-lag solving, symmetry-to-access conversion, cocycle composition, exhaustive additive closure proofs, symbolic affine access maps, GF 2 baby-step/giant-step orbit solving, sparse giant-lag predictive closures, and constrained symmetry discovery where numeric probing only ranks candidates — canonical-rewrite or exhaustive proofs decide. The scripts/ tree contains uniform, end-to-end reproductions for 25+ generators xorshift/xoroshiro/xorwow families, MT19937, Melg19937, WELL19937a, MRG32k3a, PCG32/64 +fast , LXM, SplitMix64, SFC64, JSF64, Romu, Philox, Threefry, RXS-M-XS , each runnable from scratch with scripts/families/run all.py and scripts/companion/run all.py . Reference data ships in scripts/data/ — no external fixtures required. MathKernel provides typed engineering mathematics for signals, control systems and constrained optimization while preserving the same evidence and persistence contracts as the symbolic core. Continuous and sampled signals carry explicit domains, sample grids and units. Spectral representations are typed rather than treated as anonymous arrays. FIR/IIR filters and filter designs retain coefficients, conventions and source signals, while immutable streaming state makes block-by-block processing replayable. Frequency-response and time-response operations record whether they used exact symbolic algebra or numerical evaluation. Typed SISO and MIMO models support state-space and transfer-function representations, continuous/discrete conversion, poles and zeros, stability checks, discretization, controller construction and observer construction. LQR, finite-horizon LQR, steady-state Kalman filtering, LQG composition and immutable Kalman prediction/update states retain plant/model ancestry and separate algebraic checks from modeling assumptions. Constrained finite-horizon MPC keeps feasibility, optimality, terminal invariance, recursive-feasibility and stability claims separate. Frequency-domain analysis includes Bode, Nyquist and root-locus representations together with checked time responses. Linear and quadratic programs can return exact/checkable optimality witnesses where the supported fragment permits it. Infeasible LPs can expose Farkas certificates and unbounded problems can expose recession rays. MILP search results carry replayable proof trees rather than only an incumbent value. Conic and quadratic-constraint workflows support bounded SOCP/SDP product cones and Lagrangian-style certificates in their declared fragments. External native candidate solvers are isolated in fresh processes with bounded requests and hard timeout termination. Candidate generation and certificate verification are distinct steps: a solver finding a point does not by itself establish a stronger claim than the verifier can check. Immutable Manifold , Chart , and Metric objects feed typed GeometryTensor , Connection , and GeodesicSystem outputs. Metric operations compute inverse metrics, Christoffel symbols, Riemann/Ricci/scalar/Einstein curvature and affine geodesic equations. Exact symbolic checks cover inverse identities, torsion freedom, metric compatibility, Riemann symmetries, the first Bianchi identity and the contracted Bianchi identity. Chart domains and metric nondegeneracy conditions remain explicit. Directional CoordinateMap objects carry explicit Jacobians and inverse-composition checks. Dense variance-aware TensorField objects and canonical sparse DifferentialForm objects support covariant and Lie derivatives, wedge products, exterior derivatives, interior products, pullbacks and Hodge stars. Checks include graded commutativity, d²=0 , pullback commutation with d , metric compatibility, coordinate-map composition and the Hodge double-star sign when metric signature is supplied. Orientation and signature are never guessed. Point , PointSet , Polygon , half-space Polytope , Triangulation , and derived VoronoiDiagram objects provide exact orientation, incircle and segment-intersection predicates, monotone-chain convex hulls, winding containment, exact squared-distance nearest neighbors, certified ear clipping, convex polygon clipping, empty-circumcircle Delaunay triangulation and finite Voronoi duals with explicit unbounded rays. Decimal predicates use conservative floating-point error filters; when topology cannot be established, the result is explicitly ambiguous rather than promoted to an exact classification. Exact finite SimplicialComplex , CubicalComplex , and integral ChainComplex objects expand cells to canonical face closures and derive oriented boundary matrices. Complexes verify boundary k-1 boundary k = 0 before homology is attempted. homology computes free ranks and integer torsion over Z through certified Smith-kernel/quotient reductions, and exact Betti numbers plus representative cycles over Q or GF p . boundary matrix , chain complex , and euler characteristic expose ordered bases and the Euler–Poincaré cross-check. Verified exact triangulations can be converted into canonical simplicial complexes and composed directly with homology operations; numeric or refuted triangulations cannot cross that exactness boundary. Closure expansion is bounded before combinatorial growth can exceed configured topology limits. Persistent homology, cohomology products and infinite/CW-complex inference are not claimed. StatisticalSample stores a rectangular nonempty matrix of finite concrete real observations, unique variable labels, optional unique observation IDs and explicit asserted sampling/population/design metadata. describe derives exact or ancestry-capped numeric moments and type-7 order statistics; covariance derives centered cross-products with sample or population normalization; empirical distribution preserves exact frequency counts and rational probabilities; and evidence profile audits the evidence boundary itself. The required evidence establishes only calculations on the stored observations. Sampling metadata, empirical support and model assumptions stay in separate diagnostic evidence records, while population generalization and model validity remain explicitly unestablished. Missing values, unresolved symbolic observations and silent imputation are refused. Decimal input cannot upgrade, resource limits are checked before expensive work, and every derived object retains its source across persistence and restart. Immutable GeneralizedLinearModel objects link to stored samples and produce derived-only GLMFit objects. Supported canonical pairs are Gaussian/identity, binomial/logit and Poisson/log. verify checks response domain, design rank and residual degrees of freedom; fit reports ordered coefficients, covariance/standard errors, fitted conditional means, deviance, null deviance, dispersion, convergence, score residual and conditioning. Fits independently support verify , diagnostics , and predict . Exact-input Gaussian models use sufficient cross-products and exact normal equations. Numeric Gaussian fits use checked float64 least squares; logistic and Poisson fits use deterministic float64 IRLS. Rank deficiency, invalid or degenerate response domains, non-convergence, singular/ill-conditioned information and detected complete/quasi separation fail closed without a fit object. No ridge term, row deletion, imputation or family/link substitution is silent. Coefficient, covariance, deviance and prediction claims remain conditional on the stored sample/design; model validity, population generalization and causal effects are not inferred. Stored samples support mann whitney , wilcoxon , kruskal wallis , ks 2samp , spearman , and kendall , with explicit average ranks and tie corrections. method="auto" performs complete exact sign/label/permutation enumeration only when both state and work estimates fit configured bounds; otherwise the result names its normal, chi-square, Kolmogorov or Student-t approximation. Thus an exact p-value is an exact conditional null calculation for the stored observations, while an asymptotic p-value remains numerical evidence without a finite-sample error theorem. permutation test supports mean/median differences using exact enumeration or explicitly seeded PCG64 Monte Carlo with an add-one p-value. bootstrap supports mean/median percentile intervals with a mandatory uint64 seed, bounded draws and memory-bounded batches. Simulated results record random algorithm, seed, draw count and replay configuration. Exchangeability, sampling design, asymptotic validity, population coverage and causal interpretation remain separate assumptions or unestablished claims. SurvivalDataset stores durations, exact binary event indicators, optional delayed-entry times and optional strata inside an immutable statistical sample. kaplan meier constructs exact risk sets and product-limit values together with numerical Greenwood standard errors and two-sided log-log intervals. Multi-stratum inputs require an explicit stratum, and survival at queries the right-continuous step curve. CoxProportionalHazardsModel provides an unstratified Cox surface with explicit Efron or Breslow ties. Its deterministic float64 Newton fit uses monotone line search and refuses rank-deficient, event-sparse, non-convergent, singular, over-conditioned or separation-like cases. CoxPHFit records coefficients/hazard ratios, covariance/standard errors, partial likelihood, score residual, baseline hazard, concordance and Schoenfeld time correlations, with replay verification, diagnostics and bounded partial-hazard prediction. Independent censoring, proportional hazards, population generalization and causality remain assumptions or unestablished. TimeSeriesDataset preserves row order, distinct time/value columns, strict timestamps, reject-missing policy and detected regular spacing. Exact-source acf uses a common lag-zero centered denominator and pacf uses Durbin–Levinson recursion. stationarity test provides a numerical constant-case ADF regression with named asymptotic critical values rather than inventing an exact p-value or claiming stationarity is proved. TimeSeriesModel covers AR, MA, ARMA, ARIMA and GARCH orders, constant choice, Gaussian innovations and initialization. ARMA-family fits use bounded conditional-sum-of-squares optimization; GARCH uses constrained Gaussian likelihood with positive variance and persistence below one. Derived fits record coefficients, residual/fitted series, conditional variance, roots, likelihood, AIC/BIC and convergence, with Ljung–Box/Jarque–Bera diagnostics. Forecasts derive regular future times, recursive means and Gaussian intervals using ARIMA impulse responses or GARCH variance recursion. Irregular spacing may be analyzed but not fitted. Immutable PoissonProcess , WienerProcess , GaussianProcess , and ContinuousTimeMarkovChain objects expose finite-dimensional laws and checked derived artifacts. Poisson count masses/moments and Wiener means/covariances are symbolic or exact. Gaussian-process finite laws support RBF, Matérn-3/2, linear and Brownian kernels with numerical PSD checks; conditioning uses bounded float64 Cholesky solves, explicit observation-noise variance and optional stored jitter without silently fitting hyperparameters. CTMC verification checks generator and initial-law axioms exactly; transitions use a checked matrix exponential, while stationary laws use an exact left-nullspace system and preserve nonuniqueness. Independent/stationary increments, continuity, Gaussianity, kernel suitability and time homogeneity remain declared model assumptions rather than facts established by calculation. StochasticDifferentialEquation supports vector Itô systems with declared symbol scope, drift vector, full state-by-noise diffusion matrix, concrete initial state and finite interval. Euler–Maruyama supports vector states and full diffusion. Milstein is restricted to scalar state/scalar noise and uses the symbolic diffusion derivative; unsupported multidimensional cases are refused rather than silently substituting another scheme. Simulation records the exact step grid when possible, float64 paths, PCG64 algorithm/seed/stream, terminal sample moments and nominal strong/weak orders. Large outputs expose compact metadata plus bounded path/terminal queries. Coupled convergence studies reuse a finest Brownian stream across multiple step sizes and report observed terminal RMS convergence when defined. Simulation and convergence remain numerical/empirical; nominal orders, existence, uniqueness and regularity are assumptions, not proofs. Across all statistical/stochastic objects, exact, symbolic, asymptotic, numerical, empirical and model evidence remain distinct. Derived types are output-only, replay operates under current limits, decimal ancestry cannot upgrade, persisted JSON is integrity checked before decoding, and stored type/class/source fields are reconciled to prevent cross-type source substitution. Typed PDE problems support scalar and coupled systems, declared independent/dependent variables, derivative multi-indices, coefficients/parameters and explicit initial/boundary conditions. Principal-part analysis classifies the represented system only within the declared symbolic fragment, and trace compatibility checks distinguish represented boundary information from stronger claims such as existence, uniqueness, regularity or well-posedness. PDEFunctionSpace , PDEMeasure , WeakIntegralTerm , IntegrationByPartsStep , and output-only WeakForm artifacts represent weak formulations explicitly. derive weak form requires integration variables, ordered trial spaces, test spaces, boundary-trace indices and selected term/coordinate transfers; it does not guess analytic spaces or silently integrate terms. Variable-coefficient integration by parts retains the complete product rule, storing differentiated-test and coefficient-derivative volume terms separately. Every transfer emits oriented boundary faces. Boundary terms that vanish under declared zero test traces remain represented and are marked as such. Dirichlet, Neumann/Robin and periodic indices are recorded as essential, natural and periodic partitions. WeakForm.verify reconstructs spaces, measures, volume/boundary terms, signs, product-rule derivatives, partitions and derivation steps from the source PDE. The verified claim is the represented integral identity under declared assumptions—not a theorem of solvability or regularity. FEMMesh supports interval, triangle and tetrahedron simplices. Construction checks bounded connectivity, nondegeneracy, canonical positive orientation, boundary/interior facet incidence, induced boundary ownership and cell connected components. A compatible stored Triangulation can provide triangle connectivity while preserving geometry and weak-form ancestry. Combinatorial replay does not infer geometric non-overlap or approximation quality. reference element provides canonical unit simplices. basis derives symbolic nodal P1 Lagrange functions and gradients and checks the Kronecker property, partition of unity and gradient sum. quadrature supplies bounded exact-moment rules for the supported simplex degrees. finite element space builds P1 vertex-DOF C0 spaces with explicit local-to-global connectivity and essential boundary DOFs. Derived objects are replayable and output-only. AssembledSystem and FEMSolution support scalar linear stationary weak forms on affine P1 simplices. Assembly stores dense local matrices/vectors and Jacobian determinants, coalesces the global matrix into ordered sparse entries, integrates supported Neumann/Robin facet terms and performs documented symmetric elimination for Dirichlet DOFs while retaining raw and transformed systems. Concrete substitutions resolve remaining PDE parameters through restricted MathIR. Assembly distinguishes exact integration from an exact finite quadrature sum. Insufficient-order or non-polynomial quadrature may still define a replayable algebraic system, but quadrature exact=false records the limitation. Unsupported strong second derivatives, time derivatives, coupled/nonlinear fields, periodic constraints, unresolved parameters and missing boundary fluxes fail closed. Solves select exact rank/augmented-rank analysis or an explicit SciPy sparse numeric path. FEMSolution records unique , ill conditioned , singular inconsistent , singular underdetermined , or singular least squares , together with residual and conditioning diagnostics. Verification establishes the transformed finite-dimensional system and solver outcome only, never a continuous PDE solution theorem or continuum error bound. FEMSolution.estimate error provides residual–jump indicators for complete unique or ill-conditioned P1 solutions in its supported scalar stationary diffusion fragment. Each CellErrorIndicator retains diameter-weighted strong residual, interior conormal-jump contribution, natural-boundary contribution and total. FEMErrorEstimate stores local/global estimator values, quadrature-exactness and algebraic residual separately, and always records rigorous error bound=false ; reliability and efficiency constants are not inferred. FEMErrorEstimate.mark implements deterministic Dörfler and maximum policies. RefinementMarking.refine applies triangle red refinement and propagates conforming closure through shared edges. RefinedMesh records requested/closure cells and child-to-parent mappings; MeshTransfer records refined P1 nodal values as explicit affine combinations of parent DOFs. Refined meshes can re-enter the basis, quadrature, space, assembly, solve and estimation chain. FEMErrorEstimate.compare accepts direct parent/child refinement pairs and reports estimator ratios and observed two-mesh rates. FEMConvergenceObservation is explicitly empirical evidence about an estimator sequence, not a convergence theorem or continuum error bound. mathkernel.composite relation inference provides a quadratic score test for an entire visible relation subspace. Generalized observed scores are whitened under the nominal law and the statistic is the squared norm of their sample mean. A finite bounded-score argument supplies a conservative guarantee with explicit dependence on relation dimension, weakest retained-information eigenvalue, perturbation radius and score bound. The same module computes nuisance-adjusted target information through latent and observed Fisher Schur complements. It reports exact post-observation confounding when a target direction can be reproduced by nuisance variation. For repeated or nearly repeated information eigenvalues, bootstrap uncertainty is attached to invariant eigenspaces through principal angles rather than arbitrary individual eigenvectors. Studentized ordered-spectrum intervals, dependence-informed circular-block heuristics, nominal/empirical/HAC covariance modes and norm-bounded misspecification guarantees are available with their assumptions recorded. mathkernel.robust relation inference provides model-scoped quadratic inference, learned nuisance projections, orthogonal residual relations and VAR-prewhitened long-run covariance estimation. | Python API | Function and evidence boundary | |---|---| | quadratic minimax bounds | Gaussian-sequence lower/upper rates using the inverse information spectrum; separate finite iid U-statistic bound under a justified covariance envelope | | gaussian quadratic test | Weighted-square test with finite Gaussian Chernoff threshold | | quadratic u test | O Nr unbiased pair statistic; finite Cantelli calibration for iid known-null scores | | prewhitened long run covariance | VAR 1 , automatic Bartlett bandwidth, recoloring and persistence diagnostics; consistency assumptions remain necessary | | quadratic moment test | Full-rank asymptotic Wald test with empirical or supplied covariance; singular covariance is rejected | | relation folds | Reproducible iid, group-preserving or contiguous folds | | crossfit nuisance projection | Out-of-fold nuisance-projection estimation in a declared candidate span | | crossfit residual relations | Orthogonal residual cross-moments with learned conditional means, custom learners and exclusion gaps | These research APIs remain numerical/model-scoped unless a stronger finite guarantee is explicitly returned. They do not acquire formal-proof or interval-certification labels merely because they are composed with other MathKernel objects. The relation-analysis stack also includes exact observable-relation visibility, information-retention calculations, sample-cost diagnostics, multi-relation Fisher geometry, sensor-design objectives, coordinate-invariant tangent representations, local testing bounds and uncertainty for information spectra. Numerical near-null directions are kept distinct from mathematically exact blind directions. | Workload | CPU fast path | GPU path | Parallel | |---|---|---|---| | Collatz sieve | njit n ≤ 31 | CUDA RawKernel | persistent process pool | | Cuboid sweep | njit leg-pair scan + QR prefilter | CUDA RawKernel | process pool | | GF 2^m ≤ 1024 | njit n-limb uint64×N kernels | — | — | | Integer batch | njit array kernels | — | persistent process pool, adaptive chunksize | | Graph BFS/components | njit CSR traversal, certificate re-verified | — | — | | GF p^m , p < 2^24, m ≤ 64 | njit uint64 polynomial mul/mod | — | — | | Cayley-table validation | njit axiom scan | — | — | | Recurrence extension | checked int64 njit, bigint fallback | — | — | | FWHT | int64 njit butterfly | — | — | | Closure search | njit MITM int64/uint64 | — | — | | Koopman / finite dynamics | numpy complex128 | CuPy matmul | — | | Obligation DAG | — | — | thread waves | | Long sweeps | — | — | async job pool | Exact symbolic types Fraction , CyclotomicNumber are deliberately pure Python — a visibility zero or closure cancellation must remain a proof . Numeric twins exist where scale demands it and always carry trust: numeric . Expansion contract. New domains must design verification and performance tiers together from the start: exact typed semantics and limits, an independently checkable certificate for every VERIFIED claim, and — where the workload is regular enough — a Numba/process/GPU fast path behind a narrow exactness fragment with automatic Python fallback. Fast paths must be re-verified or differential-tested against the reference implementation and must record the selected backend in evidence metadata; they may never raise trust beyond the underlying proof. GPU offload is mandatory only for regular device-exact workloads; irregular arbitrary-precision algorithms document the considered tiers instead. MathKernel optimizes only where the mathematical contract survives the optimization. Regular bounded integer/array workloads use Numba, process or GPU paths with differential checks and guarded fallbacks. Exact symbolic workloads stay on exact representations when converting them to floating point would weaken the claim. Profiling is used to remove repeated symbolic work, hoist invariant computations, cache replayable certificates and replace avoidable superlinear verification passes without changing stored mathematical evidence. Backend selection is recorded in evidence metadata and never raises trust above the underlying computation or certificate. mathkernel viz turns MathKernel objects and results into evidence-carrying interactive artifacts. Visualization is downstream of mathematics: it consumes typed source data or a MultimodalProjection , records presentation transformations, and never upgrades the source evidence merely because a particular graphical form is used. python import mathkernel projection as mkp import mathkernel viz as viz projection = mkp.create projection "matrix", {"matrix": 1, 2 , 3, 4 }, trust="exact", doc = viz.from projection projection viz.export html doc, "matrix.html", mode="portable" The lower-level dashboard API remains available for direct composition: python import mathkernel viz as viz doc = viz.dashboard "My result", cols=2 viz.add point cloud doc, points, trust="numeric" viz.add histogram doc, values, bins=128 viz.add select doc, "lag", {"label": "k=4", "value": {"embed": {"lags": 0, 4, 8 }}} viz.export html doc, "out.html", mode="portable" - Building blocks, not monoliths — artifacts compose reusable panels such as point cloud 3d , trajectory 3d , surface 3d , vector field 3d , plot2d , histogram , heatmap , dag , metric grid , data table , text and select . - Renderer-neutral IR — the versioned VisualizationDocument is consumed by pure-Python SVG, optional matplotlib PNG/PDF, and the HTML+Three.js renderer. - Interactive 3D — orbit/pan/zoom and hover inspection of identity and trust. - Portable HTML — one self-contained .html with embedded datasets, provenance, reproducibility metadata and viewer runtime; no server or CDN is required. - Evidence-preserving — block/series/dataset trust is inherited conservatively; interval-certified display is only used when the source itself carries that support. - Integrity & determinism — payload and per-dataset SHA-256 are exposed, and identical inputs produce deterministic artifacts. - Secure presentation boundary — CSP, escaped labels, no eval , dataset limits, and MathIR treated as data rather than executable code. The shared mathkernel projection layer defines canonical mathematical projection families that can feed visualization, sonification, or a combined research artifact. This prevents each renderer from inventing its own interpretation of a matrix, mesh, graph, field, distribution or high-dimensional object. A MultimodalProjection records: - source lineage SourceRef ; - projection family and structured payload; - coordinates, units and labels; - assumptions and evidence references; - deterministic transformation provenance; - explicit basis, slice, traversal or ordering parameters; - output dimensionality and declared information loss. The canonical families cover scalar/vector fields; point sets/clouds; curves, surfaces and trajectories; sequences and distributions; matrices and tensors; graphs, evidence graphs, expression trees and certificate trees; spectra and complex-valued fields; regions and implicit sets; meshes and geometric complexes; ODE/PDE solutions and dynamical systems; optimization and statistical-inference objects; finite-field/GF 2 structures; relation/information geometry; sets, partitions and piecewise objects; quantities with units; ensembles; and explicit higher-dimensional projections. For source dimension greater than three, a projection method and output dimensionality must be explicit. Coordinate selection, a declared basis, PCA-like reduction or a domain-specific spectral projection are transformations that must be recorded; a renderer cannot silently decide which view is canonical. A registry of result adapters mathkernel projection.result adapters maps stored typed objects and flat result payloads onto these families automatically. Adapters are pure extraction functions: they never recompute mathematics, never upgrade trust, and declare any presentation choice sampling grids, magnitude-only spectra, channel selection, covariance-to-band reduction in parameters and information loss . math visualize object id=... and math projection create source object id=... use the registry to choose the canonical projection for signals, spectra, filters, pole-zero maps, frequency responses, root loci, time responses, distributions symbolic densities are sampled on a declared window , empirical/discrete distributions, statistical samples, GLM fits, Kaplan-Meier estimates, Cox baseline hazards, ACF/PACF diagnostics, time-series fits, graphs and traversal trees, optimization results, ODE/SDE ensembles, FEM meshes/solutions/error indicators/convergence observations, assembled-system sparsity patterns, PDE grids, point sets, polygons, triangulations, Voronoi diagrams, generating functions, Cayley tables, contours, singularity maps, subgroup/coset/orbit partitions, combinatorial counts, and unit quantities. Unregistered object types fail with a typed error rather than an invented view. Evidence graphs are first-class: claim - evidence - assumption/source relationships can be visualized directly, making MathKernel's verification structure inspectable rather than hiding it in metadata. Complex-valued projections retain magnitude/phase structure, and mesh/field projections preserve the geometric entity to which each value belongs. mathkernel viz , mathkernel sonify and mathkernel multimodal share the mathkernel artifacts semantic layer. MathKernelArtifact carries typed source lineage, evidence/certificates, presentation transformations, scientific/perceptual annotations, reproducibility metadata and visual/audio synchronization. mathkernel viz.visualize result attaches deterministic structured lineage to visual datasets and series, while mathkernel viz.to artifact doc, result=... promotes a visual document into the same evidence-carrying artifact model used by multimodal exports. Presentation remains downstream of mathematics and cannot upgrade source trust. mathkernel sonify is the auditory sibling of mathkernel viz . It consumes the same source lineage and MultimodalProjection contract, while SonificationDocument owns the auditory mapping itself. The mathematical result remains untouched. python import mathkernel projection as mkp import mathkernel sonify as son projection = mkp.create projection "spectrum", {"amplitudes": 1.0, 0.42, 0.17 , "phases": 0.0, 0.3, -0.2 }, trust="numeric", audio = son.projection sonification projection son.write wav audio, "spectrum.wav" son.export html audio, "spectrum.html" The IR records every value-to-audio mapping as declarative provenance. Structured objects are never silently flattened: matrix scans record row/column ordering; tensor sonification records the selected slice/order; graphs record traversal or degree reduction; meshes record the geometric reduction; complex objects preserve magnitude and phase mapping; optimization traces, bootstrap/null distributions, relation spectra and ensemble orderings are likewise explicit. Built-in adapters cover harmonic/Fourier additive synthesis, sequential scans, prediction-vs-observation stereo comparison, residual sonification and projection-aware structured mappings. Offline PCM/WAV rendering is deterministic, rejects silent Nyquist aliasing, and applies explicit normalization/peak limits. The WebAudio exporter is a single offline HTML file with no network dependency. Scientific rule: an audible pattern is a perceptual candidate, not mathematical evidence. Any pattern discovered by listening must be validated quantitatively, exactly, formally or empirically through MathKernel. mathkernel multimodal combines visualization and sonification derived from the same source/projection into one portable MathKernelArtifact . Shared SourceRef ancestry allows automatic cross-modal synchronization without weakening the mathematical trust model. python import mathkernel multimodal as mkm artifact = mkm.build artifact title="Result", visualizations= viz doc , sonifications= son doc , mathkernel version="current", mkm.export html artifact, "result.html" - visual blocks can highlight during linked audio playback and linked audio can seek from a visual block; - one inspector surface exposes Result, Evidence, Provenance, Data, Reproduction, Visual Mapping, Audio Mapping, Sync and Annotations; - payload verification and document integrity hashes remain available in the exported artifact; - portable output works from file:// , with no running MathKernel server required; - artifact trust remains the weakest justified source/member trust. Via MCP, research artifacts can be assembled from stored visualization and sonification objects and exported as a single self-contained file. All 167 tools click to expand | Group | Tools | |---|---| | Discovery | math capabilities , math capability query , math result resource get | | Typed mathematics | math object create , math object get , math apply — complete compositional surface tabulated above, including geometry, signals/control, certified optimization, statistics/stochastic systems and general PDE representation | | Parsing | math parse , math parse latex , math get , math substitute , math infer structure | | Algebra | math simplify , math solve , math solve system | | Calculus | math differentiate , math integrate , math limit , math series , math summation , math product | | Numeric | math numeric evaluate , math interval evaluate | | Matrices | math matrix create , math matrix get , math matrix det , math matrix inverse , math matrix transpose , math matrix multiply , math matrix rank , math matrix rref , math matrix eigenvalues , math matrix solve | | Context | math context create , math context infer , math context check | | Reasoning | math analyze , math plan , math plan get , math execute plan , math reason , math execution get , math prove equivalence , math counterexample | | Codegen | math codegen , math verify code , math execute code | | Integers | math integer analyze , math integer compute , math integer batch | | Sweeps | math collatz sieve , math cuboid sweep | | Jobs | math job submit , math job status , math job result , math job list | | GF 2^m | math gf2m create , math gf2m from transition , math gf2m compute , math gf2m coords , math gf2m root jump rows , math gf2m closure roots , math gf2m jump rows | | GF 2 | math gf2 rank , math gf2 nullspace , math gf2 carryfree cols , math gf2 minpoly | | Transforms | math fwht | | Finite dynamics | math finite system create , math koopman matrix , math koopman transfer , math koopman visibility , math koopman lagged , math koopman observed , math koopman diagnostics , math finite fourier compute , math closure search , math cumulant compute | | Conditioned dynamics | math conditioned access solve , math conditioned symmetry access , math conditioned access compose , math conditioned closure , math symbolic conditioned access , math affine conditioned access , math gf2 conditioned access , math gf2 predictive closure , math synthesize conditioned closures , math synthesize gf2 vector conditioned access , math discover structural conditioned closure , math discover factor swap conditioned closure | | Multimodal projections | math projection catalog , math projection create , math projection describe | | Visualization | math visualize , math visualize dag , math render koopman , math visualize projection , math export artifact | | Sonification | math sonify , math sonify compare , math sonification describe , math sonify projection , math export audio | | Multimodal artifacts | math research artifact create , math export research artifact | | Sets & logic | math set create , math set op , math set membership , math quantifier check , math quantifier eliminate , math quantifier eliminate batch | | Polynomials | math poly groebner , math poly divide , math poly resultant , math poly discriminant , math poly factor , math ideal membership , math poly groebner batch | | Probability | math prob rv create , math prob expectation , math prob variance , math prob covariance , math prob bayes , math prob markov stationary , math prob markov hitting time , math prob sample , math prob distribution | | Statistics | math stats moments , math stats order , math stats regression , math stats correlation , math stats ttest , math stats chi2 , math stats confidence interval , math stats batch moments | | Tensors | math tensor create , math tensor get , math tensor contract , math tensor solve | | Numerics | math root find , math root scan , math quadrature | | ODE/PDE | math ode solve , math ode solve numeric , math ode ensemble , math pde heat 1d , math pde heat 2d , math pde wave 1d , math pde advect 1d , math pde ensemble , math pde mol heat | | Optimization | math optimize critical points , math optimize kkt , math lp solve , math optimize minimize , math optimize multistart | | Units | math unit check , math unit convert , math unit simplify | | Assurance | math store status , math replay , math fuzz differential , math certified enclose | | Proving | math prove , math prove batch , math prove replay | | Provenance | math derivation get , math derivation trace | Every tool docstring is written LLM-facing: parameter formats, exact-vs-numeric semantics, limits, and follow-up hints are documented in-place. All settings are environment-driven with the MATHKERNEL prefix Settings.from env , introspectable via math capabilities : | Variable | Default | Purpose | |---|---|---| | MATHKERNEL MAX INPUT LENGTH | 100000 | parser input cap | | MATHKERNEL MAX OUTPUT SIZE BYTES | 256000000 | whole-response byte budget; oversized payloads are preserved as integrity-checked resources and returned by receipt | | MATHKERNEL SOLVER TIMEOUT SECONDS | 30 | symbolic operation budget using bounded cancellable subprocess workers | | MATHKERNEL ENABLE EXECUTION | false | sandboxed codegen execution opt-in | | MATHKERNEL YOLO MODE | false | unlocks math yolo settings to mutate live MATHKERNEL settings typed coerce; default off | | MATHKERNEL Z3 TIMEOUT MS | 10000 | SMT budget set on every Z3 solver instance | | MATHKERNEL LEAN BINARY / MATHKERNEL LEAN TIMEOUT SECONDS | lean / 90 | Lean adapter timeout passed to every lake env lean check | | MATHKERNEL SKIP LEAN INSTALL | unset | skip the default Lean 4 + Mathlib download | | MATHKERNEL LEAN CACHE | platform cache | elan + lake workspace root | | MATHKERNEL ENABLE PARALLEL / MATHKERNEL MAX WORKERS | true / cpu count | process & thread pools | | MATHKERNEL MAX ITERATIONS | 10000 | iteration cap for simplex / Nelder-Mead | | MATHKERNEL TOLERANCE | 1e-12 | numeric convergence tolerance | | MATHKERNEL MAX ODE STEPS | 100000 | RK45 integration step cap | | MATHKERNEL STORE PATH | unset | opt-in SQLite persistence for expressions/derivations + math replay | | MATHKERNEL PROVE PORTFOLIO SIZE | 3 | SMT encodings raced per math prove call | | MATHKERNEL MAX PDE GRID | 1000000 | PDE solver grid-cell cap | | MATHKERNEL MAX PDE FIELDS / MATHKERNEL MAX PDE DIMENSIONS | 16 / 8 | typed PDE field and independent-variable caps | | MATHKERNEL MAX PDE EQUATIONS / MATHKERNEL MAX PDE TERMS | 32 / 1024 | typed PDE system and total-term caps | | MATHKERNEL MAX PDE CONDITIONS | 1024 | total typed boundary/initial-condition cap | | MATHKERNEL MAX PDE DERIVATIVE ORDER / MATHKERNEL MAX PDE NONLINEAR POWER | 4 / 8 | derivative and represented-power caps | | MATHKERNEL MAX PDE WORK | 2000000 | typed PDE construction/replay work cap | | MATHKERNEL MAX PDE SPACES / MATHKERNEL MAX PDE SPACE ORDER | 64 / 8 | weak-form space-count and regularity-order caps | | MATHKERNEL MAX PDE WEAK TERMS / MATHKERNEL MAX PDE IBP STEPS | 4096 / 256 | derived integral-term and integration-by-parts caps | | MATHKERNEL MAX PDE WEAK WORK | 5000000 | weak-form derivation/replay work cap | | MATHKERNEL MAX FEM POINTS / MATHKERNEL MAX FEM CELLS | 100000 / 200000 | simplex mesh vertex/cell caps | | MATHKERNEL MAX FEM DOFS | 200000 | finite-element-space DOF cap | | MATHKERNEL MAX FEM WORK | 20000000 | finite-element construction/replay work cap | | MATHKERNEL MAX FEM ASSEMBLY NNZ / MATHKERNEL MAX FEM ASSEMBLY WORK | 2000000 / 50000000 | sparse-entry and assembly-work caps | | MATHKERNEL MAX FEM EXACT SOLVE DOFS / MATHKERNEL MAX FEM NUMERIC SOLVE DOFS | 256 / 100000 | exact dense-diagnostic and numeric sparse-solve caps | | MATHKERNEL MAX FEM ESTIMATOR WORK / MATHKERNEL MAX FEM REFINED CELLS | 50000000 / 500000 | residual-indicator replay work and refined-output cell caps | | MATHKERNEL MAX QE VARIABLES | 16 | quantifier-elimination variable cap | | MATHKERNEL MAX BATCH JOBS | 10000 | integer batch cap | | MATHKERNEL MAX MATRIX DIM | 128 | matrix engine cap | | MATHKERNEL MAX JOBS RETAINED | 100 | async job retention | | MATHKERNEL MAX MATH OBJECTS | 10000 | retained typed-object cap | | MATHKERNEL MAX CONTOUR VERTICES | 4096 | contour complexity cap | | MATHKERNEL MAX JOINT DIMENSIONS | 8 | joint-distribution dimension cap | | MATHKERNEL MAX DISTRIBUTION COMPONENTS | 256 | mixture component cap | | MATHKERNEL MAX SYMBOLIC SERIES ORDER | 128 | Laurent/classification order cap | | MATHKERNEL MAX ORDER STATISTIC SAMPLE SIZE | 1024 | symbolic order-statistic sample cap | | MATHKERNEL MAX GRAPH VERTICES / MATHKERNEL MAX GRAPH EDGES | 4096 / 65536 | typed graph size caps | | MATHKERNEL MAX COMBINATORIAL ITEMS | 10000 | lazy combinatorial generation cap | | MATHKERNEL MAX GROUP ELEMENTS | 4096 | finite-group enumeration cap | | MATHKERNEL MAX FIELD DEGREE | 64 | GF p^m extension-degree cap | | MATHKERNEL MAX NORMAL FORM DIM | 128 | Smith/Hermite matrix dimension cap | | MATHKERNEL MAX INVERSE BRANCHES | 256 | change-of-variable branch/Jacobian cap | | MATHKERNEL MAX OBLIGATION STEPS | 128 | maximum executable plan obligations | | MATHKERNEL MAX FWHT SIZE | 2²⁰ | FWHT length cap | | MATHKERNEL MAX FINITE STATES | 4096 | finite-system enumeration cap | | MATHKERNEL MAX CUMULANT ORDER | 8 | cumulant/connected-tensor order cap | | MATHKERNEL MAX CLOSURE RESULTS | 10000 | closure-search result cap | | MATHKERNEL MAX GEOMETRY DIMENSION | 8 | manifold/chart dimension cap | | MATHKERNEL MAX GEOMETRY RANK | 6 | dense tensor-field rank cap | | MATHKERNEL MAX GEOMETRY POINTS | 10000 | point/vertex count cap | | MATHKERNEL MAX GEOMETRY SIMPLICES | 100000 | halfspace/triangle count cap | | MATHKERNEL MAX GEOMETRY WORK | 1000000 | preflight symbolic geometry work cap | | MATHKERNEL MAX TOPOLOGY DIMENSION | 16 | maximum finite-complex degree/ambient dimension | | MATHKERNEL MAX TOPOLOGY CELLS | 10000 | total simplicial/cubical/chain-basis cell cap | | MATHKERNEL MAX TOPOLOGY MATRIX ENTRIES | 1000000 | stored boundary-matrix entry cap | | MATHKERNEL MAX TOPOLOGY ENTRY BITS | 4096 | integer boundary-entry bit-length cap | | MATHKERNEL MAX TOPOLOGY WORK | 2000000 | exact topology preflight work cap | | MATHKERNEL MAX STATISTICAL VARIABLES | 256 | typed sample column cap | | MATHKERNEL MAX STATISTICAL OBSERVATIONS | 100000 | typed sample row cap | | MATHKERNEL MAX STATISTICAL CELLS | 1000000 | typed sample rectangular cell cap | | MATHKERNEL MAX STATISTICAL WORK | 2000000 | descriptive/covariance preflight work cap | | MATHKERNEL MAX GLM PARAMETERS | 64 | fitted coefficient cap, including the intercept | | MATHKERNEL MAX GLM ITERATIONS | 200 | requested IRLS iteration cap | | MATHKERNEL MAX GLM PREDICTION ROWS | 100000 | conditional-mean rows per prediction request | | MATHKERNEL MAX GLM WORK | 20000000 | GLM rank/matrix/iteration preflight work cap | | MATHKERNEL MAX NONPARAMETRIC GROUPS | 64 | selected Kruskal–Wallis group cap | | MATHKERNEL MAX EXACT RESAMPLING STATES | 100000 | complete sign/label/permutation state cap | | MATHKERNEL MAX RESAMPLES | 1000000 | Monte Carlo permutation/bootstrap draw cap | | MATHKERNEL MAX RESAMPLING BATCH CELLS | 1000000 | generated cells per bootstrap batch | | MATHKERNEL MAX RESAMPLING WORK | 20000000 | rank/enumeration/resampling preflight work cap | | MATHKERNEL MAX SURVIVAL STRATA | 64 | distinct survival-stratum cap | | MATHKERNEL MAX SURVIVAL TIMELINE POINTS | 100000 | selected Kaplan–Meier timeline cap | | MATHKERNEL MAX COX PARAMETERS | 64 | Cox predictor cap | | MATHKERNEL MAX COX ITERATIONS | 200 | requested Cox Newton-iteration cap | | MATHKERNEL MAX COX PREDICTION ROWS | 100000 | partial-hazard prediction-row cap | | MATHKERNEL MAX COX INFORMATION CONDITION | 1000000000000 | observed-information condition ceiling | | MATHKERNEL MAX SURVIVAL WORK | 20000000 | survival risk-set/matrix/iteration work cap | | MATHKERNEL MAX TIME SERIES LAG | 1000 | ACF/PACF/diagnostic lag cap | | MATHKERNEL MAX TIME SERIES DIFFERENCE | 2 | ARIMA differencing-order cap | | MATHKERNEL MAX TIME SERIES PARAMETERS | 32 | AR/MA/GARCH dynamic-parameter cap | | MATHKERNEL MAX TIME SERIES ITERATIONS | 500 | fit-optimizer iteration cap | | MATHKERNEL MAX TIME SERIES FORECAST STEPS | 10000 | forecast-horizon cap | | MATHKERNEL MAX TIME SERIES WORK | 50000000 | analysis/fit/forecast work cap | | MATHKERNEL MAX STOCHASTIC STATES | 256 | CTMC state cap | | MATHKERNEL MAX STOCHASTIC TIME POINTS | 10000 | finite-dimensional/prediction time cap | | MATHKERNEL MAX GP CONDITIONING POINTS | 2000 | GP observation cap | | MATHKERNEL MAX STOCHASTIC MATRIX ENTRIES | 1000000 | covariance/generator workspace cap | | MATHKERNEL MAX GP CONDITION NUMBER | 1000000000000 | GP conditioning ceiling | | MATHKERNEL MAX STOCHASTIC WORK | 50000000 | factorization/exponential work cap | | MATHKERNEL MAX SDE STATE DIMENSION | 32 | SDE state dimension cap | | MATHKERNEL MAX SDE NOISE DIMENSION | 32 | Brownian driver dimension cap | | MATHKERNEL MAX SDE STEPS | 1000000 | simulation/convergence step cap | | MATHKERNEL MAX SDE PATHS | 100000 | simulation path cap | | MATHKERNEL MAX SDE SIMULATION CELLS | 5000000 | stored-path/random-increment cell cap | | MATHKERNEL MAX SDE WORK | 50000000 | SDE update-work cap | | MATHKERNEL MAX SDE QUERY VALUES | 20000 | path/terminal values returned per query | src/mathkernel/ core library, typed mathematics and kernel facade src/mathkernel mcp/ FastMCP server layer and public math tools src/mathkernel projection/ shared typed multimodal projection layer src/mathkernel viz/ visualization IR, viewers and portable renderers src/mathkernel sonify/ scientific sonification IR, PCM/WAV and WebAudio src/mathkernel artifacts/ shared evidence, lineage and synchronization schema src/mathkernel multimodal/ unified visual/audio research-artifact exporter scripts/ reproducibility, GPU checks and demonstrations experiments/ research validation programs and datasets skills/ synchronized Python and MCP agent skills tests/ core, regression, multimodal and domain test suites benchmarks/ correctness-gated performance measurements MathKernel ships two synchronized agent-skill packages: one for direct Python use and one for MCP clients. They document the same evidence contract, object lifecycle and mathematical semantics, while adapting examples to their respective interfaces. The skills cover symbolic/exact work, reasoning and proving, persistence, finite dynamics, probability/statistics, numerics, tensors/units, performance, visualization, scientific sonification and the shared multimodal projection workflow. The viz/audio skills now require projection-first provenance for structured objects and explicit high-dimensional reduction or acoustic extraction rather than hidden flattening. Run the complete source-tree suite with the optional dependencies required by the domains you want to validate: PYTHONPATH=src:. python -m pytest -q python scripts/gpu smoke.py The repository degrades unavailable optional engines to unknown or unavailable rather than fabricating success. FastMCP is required for MCP registration tests, z3-solver for SMT/proving/quantifier-elimination tests, and the compatible ANTLR runtime for SymPy LaTeX parsing. Domain-specific test modules and experiment runners can be executed independently when validating a particular mathematical surface. Coverage includes parser and ambiguity handling, symbolic algebra and calculus, exact integer and finite-field arithmetic, graph algorithms, linear algebra, Numba/CUDA differential paths, asynchronous jobs, code generation and checking, GF 2 and finite Fourier methods, Koopman/finite dynamics, PRNG analysis, typed engineering mathematics, geometry/topology, statistics and stochastic systems, PDE/FEM/adaptivity, evidence propagation, persistence integrity, visualization, sonification, multimodal artifacts and the MCP tool surface. CI targets supported Python versions with native thread fan-out bounded per worker. Distribution checks build the sdist and wheel, verify metadata, install the wheel in a clean environment, confirm the runtime version and check that vendored offline visualization/multimodal assets are present. Portable exports therefore do not require a CDN after installation. - No raw user expression ever reaches sympify / parse expr ; restricted grammar, unknown functions rejected, ambiguous notation refused with candidates. - Chunked arbitrary-length integer conversion; big-result output guards; bounded automatic number-theory work; obligation step ceilings; dependency/cycle validation. - Sandboxed code execution is opt-in MATHKERNEL ENABLE EXECUTION=1 , runs in an isolated subprocess with a timeout, and is always labeled numeric evidence. - Lean subprocess invocation uses shell=False ; optional engines report unknown / unavailable rather than fabricating success. - External native LP/QP/MILP, conic/QCQP, Riccati/LQG and numerical pole-placement candidate searches run in fresh interpreters whose process groups are killed on timeout. Requests/results are bounded and BLAS/OpenMP fan-out is capped. - SQLite persistence checks every JSON payload with SHA-256 before decoding. canonical typed records additionally reconcile their declared object type, decoded model class, and source-link field before retrieval or execution. Corrupt or substituted records fail closed without producing derived objects. This termination boundary is not a hostile-code sandbox and does not impose an OS memory quota. Multi-tenant isolation still belongs in an external worker or sandbox layer. Copyright © 2026 Maarten Boone. Released under the MIT License /Staatsgeheim/MathKernel/blob/master/LICENSE .