You do the science. Lemma does the rest.

All of the science, less of the pain.

Bring Lemma the work that takes a week when you only have a morning. You direct the approach and approve the plan. Lemma does the work: managed, verified, and reported back.

What you can do with Lemma

Understand

Guided explorationA topic built up in a live notebook, at your level, one runnable step at a time.

Try it“Walk me through the quantum Hall effect in graphene, building the Landau-level spectrum step by step.”

DerivationResults derived from first principles, alternative routes compared.

Try it“Derive the Berry curvature of a gapped Dirac cone, and show the Kubo-formula route as a cross-check.”

Problem solvingA concrete answer to a concrete question, worked end to end.

Try it“How many helium balloons would it take to lift a laptop?”

Interrogate

Parameter explorationSweeps, scaling laws, regime boundaries.

Try it“Simulate the Beddington–DeAngelis predator-prey model and draw a phase diagram over its parameters.”

Scenario comparisonDesigns, models, or platforms side by side, with the trade-offs shown.

Try it“Compare a Si and a SiN grating coupler at 1550 nm on efficiency and bandwidth.”

Experiment planningMeasurement plans with error budgets, before spending lab time.

Try it“Plan a magnetotransport run to resolve the fractional states, with an error budget for the field sweep.”

Computational Methods

Model buildingDescribe a device or a phenomenon and get a working model of it, built and run.

Try it“Build a model of a Gunn diode and show its negative differential resistance.”

Model comparisonCompeting models ranked by information criteria.

Try it“Simulate two-carrier magnetoresistance data, then rank single- vs two-carrier Drude fits by AIC.”

Paper reproductionRebuild a paper as a live notebook reproducing its results.

Try it“Rebuild Castilla et al.'s graphene THz detector paper (arXiv:1905.01881) and confirm the key figure.”

Mathematics

Equation checkingDimensional analysis and symbolic verification of derivations.

Try it“Derive the damped-pendulum frequency, checking units at each step, and flag any sign errors.”

Formal proofMachine-checked proofs in Lean 4 with mathlib, for the ultimate layer of rigour.

Try it“Prove in Lean that the energy functional E[ψ] = ∫|∇ψ|² + V(x)|ψ|² dx is convex.”

Optimization

Numerical optimizationConstrained optimization, ODE/DAE simulation, optimal control.

Try it“Find the layer thicknesses that minimize reflection at 1550 nm under a ±20 nm fab tolerance.”

Inverse designGradient-based design through differentiable simulation.

Try it“Inverse-design a compact waveguide taper reaching >99% transmission with differentiable FDTD.”

Solvers

Tidy3DFull-wave FDTD electromagnetics in the cloud. Your Flexcompute account.MEEPOpen-source FDTD electromagnetics.fxfdtdxDifferentiable FDTD in JAX, simulate and take gradients.gdsfactory + SAXPhotonic circuit layout, design rules, S-matrix models.3DOptix3D optical setups and ray tracing.OptilandOptical design: sequential ray tracing, aberrations, optimization.DEVSIMTCAD: drift-diffusion, Poisson, device IV and CV.aestimo1D Schrödinger–Poisson for quantum wells and HEMTs.QuTiPOpen quantum systems, quantum optics, Lindblad dynamics.QiskitQuantum circuits. Runs on IBM hardware when configured.PySpice / ngspiceAnalog and mixed-signal circuit simulation.OCOptiCommPyFiber-optic links, DSP chains, bit-error rates.pbpybindingTight-binding band structures and DOS for 2D materials.

CMCondensed-matter libraryCurated moiré, graphene, and TMD models, ready to run.

+Your solverAnything on PyPI installs on request while you work.

Foundations

NumPyArrays and linear algebra, the base layer.SciPyIntegration, ODEs, statistics, special functions.SymPySymbolic algebra and calculus.JAXAutomatic differentiation and GPU acceleration.pandasTabular data, cleaning and joining.matplotlibEvery plot, publication style.Lean 4 + mathlibMachine-checked mathematics.Z3Satisfiability and constraint solving.

Read

Document analysisAnalyze equations, figures, tables, and claims extracted from any given document.

Try it“Pull every equation, figure, and claim out of Kim et al.'s photonics review (arXiv:1911.12756).”

Validation reportA structured report on a paper's references, equations and claims, scoped to whichever you want checked.

Try it“Generate a validation report on the equations and references of arXiv:1610.02365.”

Write

ManuscriptsLaTeX drafted from your own results, bibliography included, compiled to PDF.

Try it“Simulate a Lindblad master equation, then draft a LaTeX methods section from the results.”

Grant proposalsDrafted against specific calls, grounded in your results.

Try it“Draft an EIC Pathfinder concept for a graphene THz detector, grounded in a feasibility estimate.”

SlidesTalk-ready decks, built from a paper or from your own results.

Try it“Build a .pptx deck reviewing arXiv:1610.02365 (photonic deep learning).”

Tools

LaTeXTypesetting for manuscripts and reports, compiled in the workspace.arXivPreprints, searched and fetched by identifier or topic.OpenAlexThe open catalog of papers, authors, and citations.