MathKernel-MCP

MathKernel

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.

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Table of contents

• 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 tool surface
• Configuration
• Repository layout
• Skill packages
• Testing
• Safety boundaries
• License

Why

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.

Architecture

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.

Feature matrix

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

Typed functionality surface

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.

Installation

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.

Quickstart — MCP server

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:

math_job_submit("collatz", {"n_max": 14})  ->  math_job_status(job_id)  ->  math_job_result(job_id)

Quickstart — Python library

The MCP server is a thin transport layer; everything is available in-process:

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.

Trust model

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 symbolic mathematics

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)"

Finite dynamics & PRNG analysis

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.

Engineering mathematics

MathKernel provides typed engineering mathematics for signals, control systems and constrained optimization while preserving the same evidence and persistence contracts as the symbolic core.

Signals and spectra

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.

Control systems

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.

Optimization and certificates

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.

Geometry and topology

Differential geometry and tensor calculus

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.

Computational geometry

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.

Algebraic topology

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.

Statistics and stochastic modeling

Samples and descriptive statistics

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.

Generalized linear models

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.

Nonparametric tests, permutation tests and bootstrap

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.

Survival analysis

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.

Time-series models and forecasting

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.

Stochastic processes

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.

Stochastic differential equations

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.

Statistical evidence and persistence

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.

PDEs and adaptive finite elements

PDE representation and classification

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.

Weak forms

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.

Meshes, reference elements and finite-element spaces

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.

Assembly and algebraic solves

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.

Error estimation and adaptivity

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.

Relation and information-geometry inference

Composite relation inference

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.

Robust relation inference

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.

Relation visibility, sensor design and information geometry

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.

Performance: numba · CUDA · parallelism

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.

Correctness-preserving optimization

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.

Visualization & portable artifacts

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.

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:

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.

Shared multimodal projections

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.

Artifact lineage and scientific presentation

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.

Scientific sonification (mathkernel-sonify)

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.

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.

Unified multimodal artifacts (mathkernel-multimodal)

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.

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.

MCP tool surface

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.

Configuration

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

Repository layout

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

Skill packages

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.

Testing

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.

Safety boundaries

• 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.

License

Copyright © 2026 Maarten Boone.

Released under the MIT License.