An archival-style black-and-white photograph of a mathematician working at a chalkboard, with a constant Jacobian determinant and three distinct inputs mapping to one output.

Jacobian

Executable mathematics for agents. Evidence an independent checker can replay.

An MCP server, CLI, and Python library for conjectures, counterexamples, exact computation, and formal proof.

Quickstart · Verification · Capabilities · Documentation · Contributing

Jacobian gives AI agents small, composable mathematical operations rather than
one opaque solver. An agent can construct an object, compute an invariant,
search for a witness, and submit exact evidence to a separate checker. Every
step remains visible as a typed result or artifact.

The trust boundary is deliberate: a search result, solver status, model answer,
timeout, or score is never promoted directly to VERIFIED. Only an
operator-authorized checker may emit a verified record, bound to the exact
claim, candidate, scope, semantics, certificate format, and checker identity.

Quickstart

The npm launcher installs Jacobian and configures supported MCP clients:

npm install -g jacobian
jacobian setup
jacobian doctor

The launcher supports Claude, Codex, Cursor, Gemini, and OpenCode. It requires
Node.js 18 or newer, Python 3.12, and
uv. Run jacobian mcp to start the server
directly.

Install from source

git clone https://github.com/morluto/jacobian.git
cd jacobian
uv sync --dev
uv run jacobian --state-dir .jacobian init

The default init output is a short onboarding summary. Add init --json
when a script needs the complete reference catalog.

Use uv run jacobian --help to inspect the CLI or uv run jacobian-mcp to
start the MCP adapter.

How verification works

Jacobian separates finding evidence from deciding what that evidence proves.
Suppose an agent is testing the claim “F is injective.”

The claim that F is injective leads to a candidate collision, an exact independent check, and a verification record. Missing witnesses, timeouts, cancellation, and errors remain unknown.

Claim → candidate witness → independent check → verification record

Stage	Output	What it establishes
Claim	`F` is injective	The statement to investigate; not yet trusted
Search	A candidate witness `(F, p, q)`	Inspectable evidence, not a conclusion
Independent check	Confirm `p ≠ q` and `F(p) − F(q) = 0` exactly	The candidate is a genuine collision
Record	Bind the checked collision to the original claim and checker identity	The injectivity claim is `FALSE · VERIFIED`

No witness is not proof. A failed search, timeout, cancellation, or error
leaves the claim UNKNOWN.

In the introductory tutorial, the same boundary appears as:

evaluate.batch   →  FALSE  · HEURISTIC
witness.find     →  exact witness artifact
witness.verify   →  FALSE  · VERIFIED

FALSE · HEURISTIC is an evaluation. FALSE · VERIFIED is a conclusion
backed by independently checked evidence. Follow
Find and verify a counterexample
for a runnable example.

Capabilities

Capabilities are discovered at runtime through capability://catalog,
described with capability.describe, and executed with
capability.invoke. The installed catalog is the source of truth because
availability can depend on local backends.

Domain	Agent-visible outcomes
Polynomial maps	Evaluate maps, compute Jacobians, search for collisions, independently verify collisions
Polynomial algebra	Normalize typed expressions, factor univariate polynomials, verify identities, verify exact system solutions
Exact linear algebra	Compute determinants, rank, kernels, and integer row Hermite normal forms; find and independently verify rational solutions or inconsistency certificates for `Ax = b`
Graphs	Construct and inspect graphs, enumerate paths, realize degree sequences, test isomorphism, search colorings
SAT and SMT	Find models or proof artifacts; independently replay assignments, DRAT proofs, and Alethe proofs
Universal algebra	Evaluate finite magma laws and search for countermodels
Polytopes	Compute convex combinations and linear separations
Lean	Discover declarations, retrieve premises, inspect proof states, and check proofs in pinned environments
Research memory	Store revisioned scratch work, findings, attempts, focus, and dependency-linked context

See the tool reference for the public surface and
the atomic capability portfolio
for portfolio design and evaluation gates.

Design

Jacobian keeps four responsibilities separate:

Agents own strategy. The kernel supplies mathematical operations, not a
prescribed research workflow.
Capabilities expose one coherent outcome. Useful intermediate objects,
failures, and proof obligations remain visible.
Artifacts carry context. Results report execution status, provenance,
scope, completeness, exactness, assurance, and available certificates.
Checkers own trust. Plugins and search code cannot authorize a checker or
change verification policy.

The public MCP surface stays small: the capability catalog plus
capability.describe, capability.invoke, and three direct workspace tools.
workspace.open, workspace.write, and workspace.query manage durable
agent-authored state; workspace entries remain UNVERIFIED.

Documentation

Start here	When you need detail
Documentation home	Tutorials, how-to guides, reference, and explanation
Architecture	System shape and the independent verification boundary
Product model	Capability contracts, ownership, artifacts, and assurance
Product goals	Active priorities and research direction
Tool surface	MCP resources, tools, and invocation contracts
Domain operation library	Built-in producer, bounded-search, artifact, and exact-replay contracts
Provider runtime	Backend availability, compatibility, and identity
v0.2 specification	Last frozen release snapshot and conformance baseline
Testing strategy	Validation layers, commands, and CI responsibilities
Capability development handoffs	Evidence-preserving agent handoffs between discovery, implementation, checking, and evaluation

Specialized contracts cover
SAT artifacts,
SMT/Alethe artifacts,
exact rational linear-system evidence,
exact rational matrix determinants,
integer matrix HNF, and
Lean declaration discovery.
The domain-capability how-to
demonstrates discovery, computed invocation, bounded-result interpretation,
and exact replay. The
Lean formal-intermediates reference
covers proof states, premise retrieval, dependency graphs, and checked edits.
Architecture decisions are recorded in the
ADR index.

MCP clients and deployment

jacobian setup registers the local server with one or more supported clients.
The server advertises the capability entry points and direct workspace tools;
capability.describe(query=...) searches compact installed outcomes before an
agent inspects an exact contract and invokes it. This is a toolbox interface:
agents own mathematical decomposition, exploration, and composition.

Clients with MCP resource support can read jacobian://instructions for the
operating guide and capability://catalog for the complete machine inventory.
Clients with prompt support can optionally request jacobian-discover or
jacobian-check-evidence for protocol scaffolding.

Remote clients can connect through Streamable HTTP or SSE with bearer-token
authentication and subject-bound tenant state. See
Deploy the remote MCP server. Static tokens
are intended for controlled deployments, not as a hosted identity system.

From a clean clone on a systemd host, the maintained installer can deploy a
localhost endpoint, a Caddy-managed public domain, or Tailscale Funnel:

sudo ./deploy/install.sh
sudo ./deploy/install.sh --mode domain --domain math.example.org
sudo ./deploy/install.sh --mode tailscale

Run ./deploy/install.sh --help or add --dry-run to inspect the plan first.
The public modes require a reviewed Caddy installation; Funnel additionally
requires a connected Tailscale installation. Authentication is enabled by
default, and a newly generated bearer token is printed once.

Optional backends

Some capabilities use backends that are not installed by default:

CaDiCaL finds SAT models and UNSAT proof artifacts.
cvc5 produces SMT UNSAT proofs; Carcara independently checks Alethe.
The flint extra provides Python-FLINT/Arb operations for exact rational
systems, integer matrices and lattices, polynomials, and validated numerical
computation. Individual capabilities and independent replay support depend
on the installed catalog.
Pinned Lean CORE and MATHLIB environments check formal certificates.

Backend availability is not verification authority. Provider output remains
unverified until the appropriate independent checker accepts its bound witness
or certificate.

Lean certificates

The lean.check capability binds an exact proposition and proof body to its
result. The bundled environments pin Lean, imports, and their allowed trust
bases; model-supplied imports and packages are rejected.

Prepare the pinned runtime with:

elan toolchain install leanprover/lean4:v4.31.0
cd lean
lake update
lake build

Proof-state interaction and premise retrieval are exploration aids. Their
output cannot become VERIFIED without a successful lean.check. See the
guided declaration-discovery tutorial.

macOS and Z3

The locked environment uses z3-solver 5.0.0.0. Its upstream macOS wheels
target macOS 13 or newer on Apple silicon and Intel. On an older release, uv
falls back to a source build that requires CMake, make, and a C++20 compiler.

Install the Xcode Command Line Tools and CMake before retrying uv sync --dev.
These commands report the relevant environment without changing it:

sw_vers -productVersion
uname -m
xcode-select -p
clang++ --version
cmake --version
make --version

See the
z3-solver 5.0.0.0 files on PyPI
for the upstream wheel tags.

Status

Jacobian is pre-stable. Experimental contracts may change between releases;
release specifications describe supported snapshots, not the order of ongoing
capability research.

The Python distribution contains the mathematical kernel, CLI, and MCP server.
The npm package is a thin launcher and MCP client installer for that same
implementation; it is not a separate JavaScript API.

About the hero image

The visual motif comes from the three-dimensional counterexample to the
Jacobian conjecture: an exact constant Jacobian determinant alongside three
distinct rational inputs with the same output. The equations are unusually
good shorthand for Jacobian's purpose—surprising candidates are valuable, but
exact computation and independent checking establish what can be trusted.

Terence Tao gives an
accessible mathematical account.
The determinant identity and collision have also been
independently formalized in Isabelle/HOL.
The two-dimensional conjecture remains open.

Project boundaries

Jacobian does not aim to put a universal mathematical ontology, a
natural-language-to-formal-mathematics translator, distributed search
infrastructure, or an opaque generic solver into the kernel. It does not
reimplement theorem provers or SAT/MIP solvers, accept arbitrary
model-supplied executable bundles, or treat floating-point scores, timeouts,
and solver labels as proofs.

Contributing

Jacobian uses Python 3.12, uv, and a small Makefile:

make setup
make test-fast
make check

Read CONTRIBUTING.md before changing code. It documents
focused test commands, verification rules, documentation placement, and
pull-request expectations.

License

MIT