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

An evidence-aware multi-engine mathematics kernel — usable some arsenic a Python room (mathkernel) and arsenic an MCP server (mathkernel-mcp) — truthful applications and LLMs tin do precocious mathematics while preserving assumptions, provenance, and claim-specific evidence.

The LLM interprets intent; the MathKernel establishes mathematical evidence.

Mathematical results transportation an definitive trust level, an engine tag, and a derivation trail. Exact computation, checked certificates, symbolic results, certified enclosures, empirical evidence, and general proofs are chopped claims. Exact arithmetic unsocial is not a general proof; approximate-input ancestry must not silently disappear.

• Why
• Architecture
• Feature matrix
• Installation
• Quickstart — MCP server
• Quickstart — Python library
• Trust model
• Continuous symbolic mathematics
• Finite dynamics & PRNG analysis
• Engineering mathematics
• Geometry and topology
• Statistics and stochastic modeling
• PDEs and adaptive finite elements
• Relation and information-geometry inference
• Performance: numba · CUDA · parallelism
• Visualization & portable artifacts
• Shared multimodal projections
• Scientific sonification
• Unified multimodal artifacts
• MCP instrumentality surface
• Configuration
• Repository layout
• Skill packages
• Testing
• Safety boundaries
• License

LLMs are bully astatine mathematical intent and bad astatine mathematical arithmetic. MathKernel inverts the section of labor: the exemplary parses, plans, and interprets; the kernel computes and records claim-specific evidence. Some claims usage independent certificates aliases cross-checks; others are nonstop computations successful 1 engine. Engine statement unsocial is not a proof, and a azygous spot explanation does not switch the grounds bundle.

MathKernel is simply a typed orchestration furniture alternatively than a azygous solver. The public facade owns parsing, contexts, entity identity, persistence, grounds composition, resource argumentation and derivation tracking; domain adapters ain the existent mathematics. Presentation layers beryllium downstream and cannot silently alteration the declare being made.

This separation is deliberate: a renderer whitethorn coming evidence, but it does not create stronger mathematical grounds simply by producing a polished crippled aliases audio artifact.

The generic MCP devices math_object_create, math_object_get, and math_apply expose the pursuing compositional operations. This is the afloat typed-operation inventory; math_capability_query is the unrecorded root of parameter schemas, output types, limits, engines and verification methods.

Source objects usage the aforesaid 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 endurance 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 pinch deterministic exact, numerical, aliases seeded-stream replay. TimeSeriesAnalysis, TimeSeriesFit, and TimeSeriesForecast, FiniteDimensionalDistribution, GaussianProcessPosterior, and CTMCTransition travel the aforesaid output-only replay boundary. PDEClassification, PDECompatibilityReport, and WeakForm replay their principal-part, represented-trace, aliases complete weak-identity result from the root problem. FEMMesh links that anemic shape and an optional verified triangulation. ReferenceElement, BasisFunctionSet, QuadratureRule, and FiniteElementSpace are output-only pinch replayable single- aliases multi-source ancestry. AssembledSystem retains section and sparse world contributions plus its space/quadrature sources; output-only FEMSolution retains the exact assembled-system root and replayable solver diagnostics. G.5 output-only FEMErrorEstimate, RefinementMarking, RefinedMesh, MeshTransfer, and FEMConvergenceObservation records clasp 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 beryllium forged through public input.

From a root checkout:

Optional extras:

Lean 4 + Mathlib is installed by default connected first mathkernel-mcp commencement and via mathkernel-lean-setup (elan + a pinned reservoir workspace). Skip with MATHKERNEL_SKIP_LEAN_INSTALL=1 (CI/wheel smoke).

GPU note: CuPy wheels vessel nary CUDA libraries. The cuda other installs the matching nvidia-*-cu12 pip packages — without them, cuBLAS/NVRTC DLL loads fail even though import cupy succeeds. GPU readiness is probed astatine runtime pinch a real matmul, truthful a surgery stack degrades gracefully to CPU. Verify your stack with python scripts/gpu_smoke.py.

The server speaks MCP complete stdio (FastMCP 3) and ships core instructions to the client astatine initialize time: observe → parse → discourse → spot subject → async jobs → provenance. 162 tools, each prefixed math_.

Typical supplier session:

Long-running sweeps are async:

The MCP server is simply a bladed carrier layer; everything is disposable in-process:

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 destruction when you don't request derivation tracking.

Overall spot is constricted by the weakest grounds required to found the claimed result — ne'er the maximum spot emitted by immoderate azygous node. Independent backend disagreement is preserved arsenic an definitive conflict, not averaged away.

Every MathResult besides carries an evidence_bundle pinch abstracted computation, proof, certificate, numerical, exemplary and empirical evidence. claim_evidence retains those bundles per conclusion alternatively of flattening dissimilar claims into one score. The bequest spot section remains a blimpish summary and is automatically capped by the grounds required for the result. A producer-supplied justified_trust is simply a ceiling, ne'er an override; an unverified impervious aliases certificate supports only unknown.

Semantic statuses separate impervious aliases certification spot from mathematical outcomes specified arsenic does_not_exist, undefined, infeasible and unsupported. These distinctions past MCP serialization, asynchronous occupation retrieval, derivation replay, visualization and multimodal artifact assembly.

The capacity registry separates advertised spot levels from verification methods. Query it by domain, input/output type, operation, spot level, verification method aliases engine; capacity records besides place their execution handler and meaningful costs dimensions. Expression plans grounds the resolved capability way earlier the existing responsibility organizer runs it.

Exact and numeric paths are strictly separated: koopman/finite-dynamics devices default to exact=true (proof-grade rational/cyclotomic values); exact=false selects the vectorized numeric way (CuPy GPU erstwhile usable) and downgrades spot to numeric.

Decimal literals are approximate observations. A decimal (RealNode) anyplace in an look caps its spot astatine numeric from parse onward — 0.1 + x parses as numeric, 1/2 + x arsenic symbolic. Formal certificates (Lean) and nonstop SMT counterexamples are refused for approximate inputs, because the backends would encode decimal syntax arsenic nonstop rationals — silently proving a different statement. Use exact rationals aliases interval certification erstwhile proof-grade grounds is needed.

Continuous domains usage typed objects and the compositional object_create → use exemplary alternatively than exposing a level CAS surface. Every cognition records a four-obligation DAG: typed-input validation, candidate computation, domain-invariant verification and blimpish evidence reconciliation.

• Integral transforms — Laplace, Fourier, Mellin and bilateral Z transforms with definitive conventions, assumptions and regions of convergence. Inverse Z uses annulus-aware Laurent/residue extraction erstwhile justified. Verification records round-trip, linearity, convolution, differentiation, value-theorem and ROC obligations separately; unresolved obligations stay unknown.
• Complex analysis — derivatives, analyticity candidates, zeros, singularities, Laurent series, residues, contour integration, winding numbers, argument-principle accounting, blimpish personality continuation and domain-aware conformal maps. Branch conventions, cuts, excluded points, contour predisposition and bound incidents stay explicit.
• Continuous probability — typed univariate, random-variable, associated and conditional distributions; PDF/CDF/survival/quantile, moments, transforms, entropy, truncation, convolution, mixtures, divergence, marginals, conditioning/Bayes, covariance/correlation and bid statistics. Support, parameter constraints, Jacobians and inverse branches are retained.

Symbolic readiness is campaigner evidence, not independent proof. Same-engine identities are capped astatine symbolic; decimal ancestry remains capped at numeric. does_not_exist (for example, a Cauchy mean) is chopped from an unsupported method aliases an unresolved convergence question.

Conventions and assumptions are portion of the object. Fourier motion and normalization, toggle shape source/target variables, analyzable branches/cuts, probability supports and parameter constraints are ne'er selected silently. Contour predisposition and singularity accounting are mandatory wherever the theorem depends connected them.

Verification is operation-specific. Transforms clasp each checked aliases unresolved identity and ROC obligation. Residues are compared pinch defining limit/derivative or Laurent-coefficient formulas; contour claims clasp enclosed singularities, cuts and winding numbers. Probability verifies normalization, support-aware nonnegativity, CDF boundaries/derivative/monotonicity erstwhile decidable, and Jacobian branches. These are symbolic checks unless an nonstop certificate or separate numerical grounds says otherwise.

Failures usage semantic statuses: candidate, unknown, unsupported, does_not_exist, and correction are distinct. Known limitations include non-product associated supports, continuation without an definitive overlapping source domain, branch-sensitive argument-principle inputs, transforms whose ROC SymPy cannot establish, and wide multivariate changes of variables without supplied inverse branches/Jacobians.

Continuous symbolic activity is bounded by the world 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_* worth explicitly erstwhile a larger petition is intentional.

A unique capability: nonstop spectral study of finite dynamical systems (X, μ, T, O) — built for (and validated on) PRNG building analysis.

• Koopman suite — carrier matrix Q, observation-transfer C, mode visibility ρ_O, lagged authorities tensors (raw/connected), observed statistics, IPR/entropy diagnostics. Walsh bases for GF(2)^r, characteristic bases for Z_M.
• Stochastic study transportation (library API) — exact FiniteJointLaw contractions for arbitrary finite latent associated laws; ordered Markov way moments/cumulants pinch the required multiplication operators; statewise multiplicativity-defect certificates; and nonstop finite-noise deterministic dilations for logical Markov kernels. The accompanying published primate quartet aviator deliberately records that the earlier K3ST split-zero diagnostic does not past extracurricular its group-based assumptions.
• Branching General Markov tensors (library API) — exact FiniteMarkovTree sum-product laws and cumulants connected heterogeneous rooted trees; nonstop L M R edge-flattening certificates pinch the crisp transition- rank bound; section stochastic study channels arsenic Kronecker transforms; exact left-inverse recovery, collision witnesses, corporate sensor fusion, and channel-conditioned singular-value bounds. The published primate pilot distinguishes algebraic identifiability from finite-sample stability.
• Statistical phylogenetic conclusion (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 root of visibility loss and inverse instability. Two fixed published-data pilots adhd tract and moving-block bootstrap checks without claiming wide competitory accuracy.
• Frozen phylogenetic benchmarking (library API) — FASTA, relaxed PHYLIP, applicable NEXUS and Newick ingestion; portable source SHA-256 manifests; canonical protocol and corpus locks; result-blind quartet sampling from reference-tree splits; complete-case tract 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 root alignments and a 1,920-alignment known-truth accent grid. A separate fastener fixes the first 20 eligible BenchmarkAlignments datasets before acquisition; that outer corpus is explicitly pending alternatively than silently replaced.
• Observable connected-relation discovery (library API) - exact and numerical pure-interaction laws; weighted conditional-expectation singular spectra; mode-specific stochastic visibility; nonstop local-channel transfer of connected amplitude; chi-square and null-Fisher information retention; nonstop invisibility certificates; finite basal and constructive sufficient sample bounds; binary-parity scaling; and complementary sensor fusion. The controlled theorem shows that section visibility losses multiply in amplitude and quadrate successful information, yielding an s^(-2d) detection-cost law successful the homogeneous binary specialization.
• Relation-subspace visibility and sensor creation (library API) - finite multi-parameter section narration laws; latent and observed Fisher Gram matrices; generalized retained-information eigenvalues and principal visibility directions; nonstop observation-blind collision certificates; direction-level accusation and sample multipliers; rank, E-optimal, trace, D-optimal and pseudo-logdet sensor-subset selection; businesslike 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, root hashing and a fail-closed raw-data runner. The bundled SkewDB fixture is explicitly synthetic because the existent afloat payload was not acquired successful this environment.
• Coordinate-invariant narration geometry (library API) - finite-simplex tangent vectors pinch the intrinsic Fisher metric; stochastic tangent pushforward; coordinate-invariant generalized retained-information eigenvalues; nonstop score/tangent equivalence; nonstop section chi-square transfer; worst-direction minimax basal sample bounds; finite Bhattacharyya and retention-based pointwise capable counts; and iid, moving-block and cluster bootstrap intervals for ordered narration spectra. A SHA-256-locked local-resolution SkewDB adapter converts documented cumulative *_fit.csv tracks to model increments and explicitly separates genuine inputs from the bundled source-parameterized generated fixture.
• Finite Fourier — nonstop arithmetic successful ℚ(ζ_L) via cyclotomic polynomials: DFT complete Z_M, output-transfer transforms, two-point quality 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 pinch L1/Hamming weight bounds.
• GF(2^m) from transitions — reconstruct the section (dual-orbit cyclic basis, minimal/reduction polynomial, Rabin-verified) purely from a generator's GF(2)-linear modulation columns.
• State-conditioned dynamics — nonstop per-state orbit entree T^κ(x)(x): least-lag solving, symmetry-to-access conversion, cocycle composition, exhaustive additive closure proofs, symbolic affine entree maps, GF(2) baby-step/giant-step orbit solving, sparse giant-lag predictive closures, and constrained symmetry find wherever numeric probing only ranks candidates — canonical-rewrite aliases exhaustive proofs decide.

The scripts/ character 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 pinch scripts/families/run_all.py and scripts/companion/run_all.py. Reference information ships successful scripts/data/ — no external fixtures required.

MathKernel provides typed engineering mathematics for signals, power systems and constrained optimization while preserving the aforesaid grounds and persistence contracts arsenic the symbolic core.

Continuous and sampled signals transportation definitive domains, sample grids and units. Spectral representations are typed alternatively than treated arsenic anonymous arrays. FIR/IIR filters and select designs clasp coefficients, conventions and root signals, while immutable streaming authorities makes block-by-block processing replayable. Frequency-response and time-response operations grounds whether they utilized nonstop symbolic algebra aliases numerical evaluation.

Typed SISO and MIMO models support state-space and transfer-function representations, continuous/discrete conversion, poles and zeros, stableness checks, discretization, controller building and perceiver construction. LQR, finite-horizon LQR, steady-state Kalman filtering, LQG creation and immutable Kalman prediction/update states clasp plant/model ancestry and abstracted algebraic checks from modeling assumptions.

Constrained finite-horizon MPC keeps feasibility, optimality, terminal invariance, recursive-feasibility and stableness claims separate. Frequency-domain study includes Bode, Nyquist and root-locus representations together pinch checked clip responses.

Linear and quadratic programs tin return exact/checkable optimality witnesses wherever the supported part permits it. Infeasible LPs tin expose Farkas certificates and unbounded problems tin expose recession rays. MILP hunt results transportation replayable impervious trees alternatively than only an incumbent value. Conic and quadratic-constraint workflows support bounded SOCP/SDP merchandise cones and Lagrangian-style certificates successful their declared fragments.

External autochthonal campaigner solvers are isolated successful caller processes pinch bounded requests and difficult timeout termination. Candidate procreation and certificate verification are chopped steps: a solver uncovering a constituent does not by itself found a stronger declare than the verifier tin check.

Immutable Manifold, Chart, and Metric objects provender 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 screen inverse identities, torsion freedom, metric compatibility, Riemann symmetries, the first Bianchi personality and the contracted Bianchi identity. Chart domains and metric nondegeneracy conditions stay explicit.

Directional CoordinateMap objects transportation definitive 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 see graded commutativity, d²=0, pullback commutation pinch d, metric compatibility, coordinate-map creation and the Hodge double-star motion erstwhile metric signature is supplied. Orientation and signature are ne'er guessed.

Point, PointSet, Polygon, half-space Polytope, Triangulation, and derived VoronoiDiagram objects supply nonstop orientation, incircle and segment-intersection predicates, monotone-chain convex hulls, winding containment, nonstop squared-distance nearest neighbors, certified receptor clipping, convex polygon clipping, empty-circumcircle Delaunay triangulation and finite Voronoi duals pinch definitive unbounded rays. Decimal predicates usage blimpish floating-point correction filters; erstwhile topology cannot beryllium established, the consequence is explicitly ambiguous alternatively than promoted to an nonstop classification.

Exact finite SimplicialComplex, CubicalComplex, and integral ChainComplex objects grow cells to canonical look closures and deduce oriented bound matrices. Complexes verify boundary[k-1] * boundary[k] = 0 earlier homology is attempted. homology computes free ranks and integer torsion complete Z done certified Smith-kernel/quotient reductions, and nonstop Betti numbers positive typical cycles complete Q aliases GF(p). boundary_matrix, chain_complex, and euler_characteristic expose ordered bases and the Euler–Poincaré cross-check.

Verified nonstop triangulations tin beryllium converted into canonical simplicial complexes and composed straight pinch homology operations; numeric aliases refuted triangulations cannot transverse that exactness boundary. Closure description is bounded earlier combinatorial maturation tin transcend configured topology limits. Persistent homology, cohomology products and infinite/CW-complex conclusion are not claimed.

StatisticalSample stores a rectangular nonempty matrix of finite actual existent observations, unsocial adaptable labels, optional unsocial study IDs and definitive asserted sampling/population/design metadata. picture derives nonstop aliases ancestry-capped numeric moments and type-7 bid statistics; covariance derives centered cross-products pinch sample aliases organization normalization; empirical_distribution preserves nonstop wave counts and logical probabilities; and evidence_profile audits the grounds bound itself.

The required grounds establishes only calculations connected the stored observations. Sampling metadata, empirical support and exemplary assumptions enactment successful abstracted diagnostic grounds records, while organization generalization and exemplary validity stay explicitly unestablished. Missing values, unresolved symbolic observations and silent imputation are refused. Decimal input cannot upgrade, assets limits are checked earlier costly work, and each derived entity retains its root crossed persistence and restart.

Immutable GeneralizedLinearModel objects nexus to stored samples and nutrient derived-only GLMFit objects. Supported canonical pairs are Gaussian/identity, binomial/logit and Poisson/log. verify checks consequence domain, creation rank and residual degrees of freedom; fresh reports ordered coefficients, covariance/standard errors, fitted conditional means, deviance, null deviance, dispersion, convergence, people residual and conditioning. Fits independently support verify, diagnostics, and predict.

Exact-input Gaussian models usage capable cross-products and nonstop normal equations. Numeric Gaussian fits usage checked float64 slightest squares; logistic and Poisson fits usage deterministic float64 IRLS. Rank deficiency, invalid aliases degenerate consequence domains, non-convergence, singular/ill-conditioned accusation and detected complete/quasi separation neglect closed without a fresh object. No ridge term, statement deletion, imputation aliases family/link substitution is silent. Coefficient, covariance, deviance and prediction claims stay conditional connected the stored sample/design; exemplary validity, organization generalization and causal effects are not inferred.

Stored samples support mann_whitney, wilcoxon, kruskal_wallis, ks_2samp, spearman, and kendall, pinch definitive mean ranks and necktie corrections. method="auto" performs complete nonstop sign/label/permutation enumeration only erstwhile some authorities and activity estimates fresh configured bounds; different the consequence names its normal, chi-square, Kolmogorov aliases Student-t approximation. Thus an nonstop p-value is an nonstop conditional null calculation for the stored observations, while an asymptotic p-value remains numerical grounds without a finite-sample correction theorem.

permutation_test supports mean/median differences utilizing nonstop enumeration aliases explicitly seeded PCG64 Monte Carlo pinch an add-one p-value. bootstrap supports mean/median percentile intervals pinch a mandatory uint64 seed, bounded draws and memory-bounded batches. Simulated results grounds random algorithm, seed, tie count and replay configuration. Exchangeability, sampling design, asymptotic validity, organization sum and causal mentation stay abstracted assumptions aliases unestablished claims.

SurvivalDataset stores durations, nonstop binary arena indicators, optional delayed-entry times and optional strata wrong an immutable statistical sample. kaplan_meier constructs nonstop consequence sets and product-limit values together pinch numerical Greenwood modular errors and two-sided log-log intervals. Multi-stratum inputs require an definitive stratum, and survival_at queries the right-continuous measurement curve.

CoxProportionalHazardsModel provides an unstratified Cox aboveground pinch definitive Efron aliases Breslow ties. Its deterministic float64 Newton fresh uses monotone statement hunt and refuses rank-deficient, event-sparse, non-convergent, singular, over-conditioned aliases separation-like cases. CoxPHFit records coefficients/hazard ratios, covariance/standard errors, partial likelihood, people residual, baseline hazard, concordance and Schoenfeld clip correlations, pinch replay verification, diagnostics and bounded partial-hazard prediction. Independent censoring, proportional hazards, organization generalization and causality stay assumptions aliases unestablished.

TimeSeriesDataset preserves statement order, chopped time/value columns, strict timestamps, reject-missing argumentation and detected regular spacing. Exact-source acf uses a communal lag-zero centered denominator and pacf uses Durbin–Levinson recursion. stationarity_test provides a numerical constant-case ADF regression pinch named asymptotic captious values alternatively than inventing an nonstop p-value aliases claiming stationarity is proved.

TimeSeriesModel covers AR, MA, ARMA, ARIMA and GARCH orders, changeless choice, Gaussian innovations and initialization. ARMA-family fits usage bounded conditional-sum-of-squares optimization; GARCH uses constrained Gaussian likelihood pinch affirmative variance and persistence beneath one. Derived fits grounds coefficients, residual/fitted series, conditional variance, roots, likelihood, AIC/BIC and convergence, pinch Ljung–Box/Jarque–Bera diagnostics. Forecasts deduce regular early times, recursive intends and Gaussian intervals utilizing ARIMA impulse responses aliases GARCH variance recursion. Irregular spacing whitethorn beryllium 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 aliases exact. Gaussian-process finite laws support RBF, Matérn-3/2, linear and Brownian kernels pinch numerical PSD checks; conditioning uses bounded float64 Cholesky solves, definitive observation-noise variance and optional stored jitter without silently fitting hyperparameters. CTMC verification checks generator and initial-law axioms exactly; transitions usage a checked matrix exponential, while stationary laws usage an nonstop left-nullspace strategy and sphere nonuniqueness.

Independent/stationary increments, continuity, Gaussianity, kernel suitability and clip homogeneity stay declared exemplary assumptions alternatively than facts established by calculation.

StochasticDifferentialEquation supports vector Itô systems pinch declared awesome scope, drift vector, afloat state-by-noise diffusion matrix, actual first authorities and finite interval. Euler–Maruyama supports vector states and afloat diffusion. Milstein is restricted to scalar state/scalar sound and uses the symbolic diffusion derivative; unsupported multidimensional cases are refused alternatively than silently substituting different scheme.

Simulation records the nonstop measurement grid erstwhile possible, float64 paths, PCG64 algorithm/seed/stream, terminal sample moments and nominal strong/weak orders. Large outputs expose compact metadata positive bounded path/terminal queries. Coupled convergence studies reuse a finest Brownian watercourse crossed aggregate measurement sizes and study observed terminal RMS convergence erstwhile defined. Simulation and convergence stay numerical/empirical; nominal orders, existence, characteristic and regularity are assumptions, not proofs.

Across each statistical/stochastic objects, exact, symbolic, asymptotic, numerical, empirical and exemplary grounds stay distinct. Derived types are output-only, replay operates nether existent limits, decimal ancestry cannot upgrade, persisted JSON is integrity checked earlier decoding, and stored type/class/source fields are reconciled to forestall cross-type root substitution.

Typed PDE problems support scalar and coupled systems, declared independent/dependent variables, derivative multi-indices, coefficients/parameters and definitive initial/boundary conditions. Principal-part study classifies the represented strategy only wrong the declared symbolic fragment, and trace compatibility checks separate represented bound accusation from stronger claims specified arsenic existence, uniqueness, regularity aliases well-posedness.

PDEFunctionSpace, PDEMeasure, WeakIntegralTerm, IntegrationByPartsStep, and output-only WeakForm artifacts correspond anemic formulations explicitly. derive_weak_form requires integration variables, ordered proceedings spaces, trial spaces, boundary-trace indices and selected term/coordinate transfers; it does not conjecture analytic spaces aliases silently merge terms.

Variable-coefficient integration by parts retains the complete merchandise rule, storing differentiated-test and coefficient-derivative measurement position separately. Every transportation emits oriented bound faces. Boundary position that vanish nether declared zero trial traces stay represented and are marked arsenic such. Dirichlet, Neumann/Robin and periodic indices are recorded arsenic essential, earthy and periodic partitions. WeakForm.verify reconstructs spaces, measures, volume/boundary terms, signs, product-rule derivatives, partitions and derivation steps from the root PDE. The verified declare is the represented integral personality nether declared assumptions—not a theorem of solvability aliases regularity.

FEMMesh supports interval, triangle and tetrahedron simplices. Construction checks bounded connectivity, nondegeneracy, canonical affirmative orientation, boundary/interior facet incidence, induced bound ownership and compartment connected components. A compatible stored Triangulation tin supply triangle connectivity while preserving geometry and weak-form ancestry. Combinatorial replay does not infer geometric non-overlap aliases approximation quality.

reference_element provides canonical portion simplices. ground 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 pinch definitive local-to-global connectivity and basal bound DOFs. Derived objects are replayable and output-only.

AssembledSystem and FEMSolution support scalar linear stationary anemic forms connected affine P1 simplices. Assembly stores dense section matrices/vectors and Jacobian determinants, coalesces the world matrix into ordered sparse entries, integrates supported Neumann/Robin facet position and performs documented symmetric elimination for Dirichlet DOFs while retaining earthy and transformed systems. Concrete substitutions resoluteness remaining PDE parameters done restricted MathIR.

Assembly distinguishes nonstop integration from an nonstop finite quadrature sum. Insufficient-order aliases non-polynomial quadrature whitethorn still specify a replayable algebraic system, but quadrature_exact=false records the limitation. Unsupported beardown 2nd derivatives, clip derivatives, coupled/nonlinear fields, periodic constraints, unresolved parameters and missing bound fluxes neglect closed.

Solves prime nonstop rank/augmented-rank study aliases an definitive SciPy sparse numeric path. FEMSolution records unique, ill_conditioned, singular_inconsistent, singular_underdetermined, aliases singular_least_squares, together pinch residual and conditioning diagnostics. Verification establishes the transformed finite-dimensional strategy and solver result only, ne'er a continuous PDE solution theorem aliases continuum correction bound.

FEMSolution.estimate_error provides residual–jump indicators for complete unsocial aliases ill-conditioned P1 solutions successful its supported scalar stationary diffusion fragment. Each CellErrorIndicator retains diameter-weighted beardown residual, interior conormal-jump contribution, natural-boundary publication and total. FEMErrorEstimate stores local/global estimator values, quadrature-exactness and algebraic residual separately, and ever records rigorous_error_bound=false; reliability and ratio constants are not inferred.

FEMErrorEstimate.mark implements deterministic Dörfler and maximum policies. RefinementMarking.refine applies triangle reddish refinement and propagates conforming closure done shared edges. RefinedMesh records requested/closure cells and child-to-parent mappings; MeshTransfer records refined P1 nodal values arsenic definitive affine combinations of genitor DOFs. Refined meshes tin re-enter the basis, quadrature, space, assembly, lick and estimation chain.

FEMErrorEstimate.compare accepts nonstop parent/child refinement pairs and reports estimator ratios and observed two-mesh rates. FEMConvergenceObservation is explicitly empirical grounds astir an estimator sequence, not a convergence theorem aliases continuum correction bound.

mathkernel.composite_relation_inference provides a quadratic people trial for an full visible narration subspace. Generalized observed scores are whitened nether the nominal rule and the statistic is the squared norm of their sample mean. A finite bounded-score statement supplies a blimpish guarantee pinch definitive dependence connected narration dimension, weakest retained-information eigenvalue, perturbation radius and people bound.

The aforesaid module computes nuisance-adjusted target accusation done latent and observed Fisher Schur complements. It reports nonstop post-observation confounding erstwhile a target guidance tin beryllium reproduced by nuisance variation. For repeated aliases astir repeated accusation eigenvalues, bootstrap uncertainty is attached to invariant eigenspaces done main angles alternatively than arbitrary individual eigenvectors. Studentized ordered-spectrum intervals, dependence-informed circular-block heuristics, nominal/empirical/HAC covariance modes and norm-bounded misspecification guarantees are disposable pinch 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.

These investigation APIs stay numerical/model-scoped unless a stronger finite guarantee is explicitly returned. They do not get formal-proof aliases interval-certification labels simply because they are composed pinch different MathKernel objects.

The relation-analysis stack besides includes nonstop observable-relation visibility, information-retention calculations, sample-cost diagnostics, multi-relation Fisher geometry, sensor-design objectives, coordinate-invariant tangent representations, section testing bounds and uncertainty for accusation spectra. Numerical near-null directions are kept chopped from mathematically nonstop unsighted directions.

Exact symbolic types (Fraction, CyclotomicNumber) are deliberately axenic Python — a visibility zero aliases closure cancellation must stay a proof. Numeric twins exist where standard demands it and ever transportation trust: numeric.

Expansion contract. New domains must creation verification and performance tiers together from the start: nonstop typed semantics and limits, an independently checkable certificate for each VERIFIED claim, and — wherever the workload is regular capable — a Numba/process/GPU accelerated way down a constrictive exactness fragment pinch automatic Python fallback. Fast paths must beryllium re-verified or differential-tested against the reference implementation and must grounds the selected backend successful grounds metadata; they whitethorn ne'er raise spot beyond the underlying proof. GPU offload is mandatory only for regular device-exact workloads; irregular arbitrary-precision algorithms archive the considered tiers instead.

MathKernel optimizes only wherever the mathematical statement survives the optimization. Regular bounded integer/array workloads usage Numba, process aliases GPU paths pinch differential checks and guarded fallbacks. Exact symbolic workloads enactment connected nonstop representations erstwhile converting them to floating constituent would weaken the claim. Profiling is utilized to region repeated symbolic work, hoist invariant computations, cache replayable certificates and switch avoidable superlinear verification passes without changing stored mathematical evidence. Backend action is recorded successful grounds metadata and ne'er raises spot supra the underlying computation aliases certificate.

mathkernel_viz turns MathKernel objects and results into evidence-carrying interactive artifacts. Visualization is downstream of mathematics: it consumes typed root information or a MultimodalProjection, records position transformations, and ne'er upgrades the source grounds simply because a peculiar graphical shape is used.

The lower-level dashboard API remains disposable for nonstop composition:

• Building blocks, not monoliths — artifacts constitute reusable panels specified as point_cloud_3d, trajectory_3d, surface_3d, vector_field_3d, plot2d, histogram, heatmap, dag, metric_grid, data_table, matter 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 personality and trust.
• Portable HTML — 1 self-contained .html pinch embedded datasets, provenance, reproducibility metadata and spectator runtime; nary server aliases CDN is required.
• Evidence-preserving — block/series/dataset spot is inherited conservatively; interval-certified show is only utilized erstwhile the root itself carries that support.
• Integrity & determinism — payload and per-dataset SHA-256 are exposed, and identical inputs nutrient deterministic artifacts.
• Secure position boundary — CSP, escaped labels, nary eval, dataset limits, and MathIR treated arsenic information alternatively than executable code.

The shared mathkernel_projection furniture defines canonical mathematical projection families that tin provender visualization, sonification, aliases a mixed investigation artifact. This prevents each renderer from inventing its ain mentation of a matrix, mesh, graph, field, distribution aliases high-dimensional object.

A MultimodalProjection records:

• source lineage (SourceRef);
• projection family and system payload;
• coordinates, units and labels;
• assumptions and grounds references;
• deterministic translator provenance;
• explicit basis, slice, traversal aliases ordering parameters;
• output dimensionality and declared accusation loss.

The canonical families screen scalar/vector fields; constituent sets/clouds; curves, surfaces and trajectories; sequences and distributions; matrices and tensors; graphs, evidence graphs, look 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 pinch units; ensembles; and definitive higher-dimensional projections.

For root magnitude greater than three, a projection method and output dimensionality must beryllium explicit. Coordinate selection, a declared basis, PCA-like simplification aliases a domain-specific spectral projection are transformations that must beryllium recorded; a renderer cannot silently determine which position is canonical.

A registry of consequence adapters (mathkernel_projection.result_adapters) maps stored typed objects and level consequence payloads onto these families automatically. Adapters are axenic extraction functions: they ne'er recompute mathematics, ne'er upgrade trust, and state immoderate position prime (sampling grids, magnitude-only spectra, channel selection, covariance-to-band reduction) successful parameters and information_loss. math_visualize(object_id=...) and math_projection_create(source_object_id=...) usage the registry to take the canonical projection for signals, spectra, filters, pole-zero maps, frequency responses, guidelines loci, clip 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, constituent sets, polygons, triangulations, Voronoi diagrams, generating functions, Cayley tables, contours, singularity maps, subgroup/coset/orbit partitions, combinatorial counts, and portion quantities. Unregistered entity types neglect pinch a typed correction alternatively than an invented view.

Evidence graphs are first-class: declare -> grounds -> assumption/source relationships can beryllium visualized directly, making MathKernel's verification building inspectable rather than hiding it successful metadata. Complex-valued projections clasp magnitude/phase structure, and mesh/field projections sphere the geometric entity to which each worth belongs.

mathkernel_viz, mathkernel_sonify and mathkernel_multimodal stock the mathkernel_artifacts semantic layer. MathKernelArtifact carries typed root lineage, evidence/certificates, position transformations, scientific/perceptual annotations, reproducibility metadata and visual/audio synchronization. mathkernel_viz.visualize(result) attaches deterministic system lineage to ocular datasets and series, while mathkernel_viz.to_artifact(doc, result=...) promotes a ocular archive into the aforesaid evidence-carrying artifact exemplary utilized by multimodal exports. Presentation remains downstream of mathematics and cannot upgrade root trust.

mathkernel_sonify is the auditory related of mathkernel_viz. It consumes the same source lineage and MultimodalProjection contract, while SonificationDocument owns the auditory mapping itself. The mathematical consequence remains untouched.

The IR records each value-to-audio mapping arsenic declarative provenance. Structured objects are ne'er silently flattened: matrix scans grounds row/column ordering; tensor sonification records the selected slice/order; graphs grounds traversal aliases degree reduction; meshes grounds the geometric reduction; analyzable objects sphere magnitude and shape mapping; optimization traces, bootstrap/null distributions, narration spectra and ensemble orderings are likewise explicit.

Built-in adapters screen 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 definitive normalization/peak limits. The WebAudio exporter is a single offline HTML record pinch nary web dependency.

Scientific rule: an audible shape is simply a perceptual candidate, not mathematical evidence. Any shape discovered by listening must beryllium validated quantitatively, exactly, formally aliases empirically done MathKernel.

mathkernel_multimodal combines visualization and sonification derived from the same source/projection into 1 portable MathKernelArtifact. Shared SourceRef ancestry allows automatic cross-modal synchronization without weakening the mathematical trust model.

• visual blocks tin item during linked audio playback and linked audio tin seek from a ocular block;
• one inspector aboveground exposes Result, Evidence, Provenance, Data, Reproduction, Visual Mapping, Audio Mapping, Sync and Annotations;
• payload verification and archive integrity hashes stay disposable successful the exported artifact;
• portable output useful from file://, pinch nary moving MathKernel server required;
• artifact spot remains the weakest justified source/member trust.

Via MCP, investigation artifacts tin beryllium assembled from stored visualization and sonification objects and exported arsenic a azygous self-contained file.

Every instrumentality 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 pinch the MATHKERNEL_ prefix (Settings.from_env()), introspectable via math_capabilities:

MathKernel ships 2 synchronized agent-skill packages: 1 for nonstop Python usage and one for MCP clients. They archive the aforesaid grounds contract, entity lifecycle and mathematical semantics, while adapting examples to their respective interfaces.

The skills screen 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 system objects and explicit high-dimensional simplification aliases acoustic extraction alternatively than hidden flattening.

Run the complete source-tree suite pinch the optional limitations required by the domains you want to validate:

The repository degrades unavailable optional engines to chartless aliases unavailable alternatively 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 trial modules and research runners tin beryllium executed independently erstwhile validating a peculiar mathematical surface.

Coverage includes parser and ambiguity handling, symbolic algebra and calculus, nonstop integer and finite-field arithmetic, chart algorithms, linear algebra, Numba/CUDA differential paths, asynchronous jobs, codification procreation and checking, GF(2) and finite Fourier methods, Koopman/finite dynamics, PRNG analysis, typed engineering mathematics, geometry/topology, statistic and stochastic systems, PDE/FEM/adaptivity, grounds propagation, persistence integrity, visualization, sonification, multimodal artifacts and the MCP instrumentality surface.

CI targets supported Python versions pinch autochthonal thread fan-out bounded per worker. Distribution checks build the sdist and wheel, verify metadata, instal the instrumentality successful a cleanable environment, corroborate the runtime type and cheque that vendored offline visualization/multimodal assets are present. Portable exports truthful do not require a CDN aft installation.

• No earthy personification look ever reaches sympify()/parse_expr(); restricted grammar, unknown functions rejected, ambiguous notation refused pinch candidates.
• Chunked arbitrary-length integer conversion; big-result output guards; bounded automatic number-theory work; responsibility measurement ceilings; dependency/cycle validation.
• Sandboxed codification execution is opt-in (MATHKERNEL_ENABLE_EXECUTION=1), runs successful an isolated subprocess pinch a timeout, and is ever branded numeric evidence.
• Lean subprocess invocation uses shell=False; optional engines report unknown/unavailable alternatively than fabricating success.
• External autochthonal LP/QP/MILP, conic/QCQP, Riccati/LQG and numerical pole-placement candidate searches tally successful caller interpreters whose process groups are killed on timeout. Requests/results are bounded and BLAS/OpenMP fan-out is capped.
• SQLite persistence checks each JSON payload pinch SHA-256 earlier decoding. canonical typed records additionally reconcile their declared entity type, decoded model class, and source-link section earlier retrieval aliases execution. Corrupt or substituted records neglect closed without producing derived objects.

This termination bound is not a hostile-code sandbox and does not enforce an OS representation quota. Multi-tenant isolation still belongs successful an outer worker or sandbox layer.

Copyright © 2026 Maarten Boone.

Released nether the MIT License.