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.
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. For a one-off setup without a global install, run:
npx jacobian setup
For repeated use, install the launcher persistently and use its commands:
npm install -g jacobian
jacobian setup
jacobian upgrade
jacobian doctor
For the Python distribution, install the stable package directly with:
python -m pip install jacobian
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
./scripts/setup-agent --client codex --profile full-python --yes
This performs a locked full-Python sync and configures the selected agent to
start MCP from the absolute source and state paths with --no-sync. It also
records a doctor report containing the Git revision, package version, catalog
digest, and provider availability. See
Configure an agent from a source checkout
for the core, full-python, lean, and external-proof profiles, dry-run,
repeatability, and rollback behavior.
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.”
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.
- Values compose directly. Small, bounded mathematical results stay inline; artifacts carry reusable objects, replayable evidence, and large payloads.
- 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 the two
capability entry points, capability.describe and capability.invoke.
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 |
| Testing strategy | Validation layers, commands, and CI responsibilities |
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.
MCP clients and deployment
jacobian setup registers the local server with one or more supported clients.
jacobian upgrade refreshes the pinned Python kernel in the launcher's managed
environment; use npm install -g jacobian@latest to upgrade the npm launcher
itself.
For a clone, jacobian setup --source <checkout> --state-dir <path> --profile full-python explicitly binds the client to that source environment;
the maintained scripts/setup-agent wrapper performs the required locked sync
and doctor checks first.
The server advertises only the capability entry points;
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
flintextra 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
COREandMATHLIBenvironments 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 0.6.0 is a pre-stable release. Its published package, capability, and artifact contracts describe the current supported surface; ongoing capability research may change experimental contracts between releases.
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. 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-unit
make check
Read CONTRIBUTING.md before changing code. It documents focused test commands, verification rules, documentation placement, and pull-request expectations.