Axiomatic AI

For scientists

You do the science. Lemma does the rest.

Lemma is Axiomatic AI's web-based platform for science and engineering challenges. Lemma combines AI with the tools and workflows that scientists and engineers already use, to bring them reliable, reproducible results faster.

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.

Try Lemma free How Lemma works

Animation: six point vortices moving through the velocity field they induce, drawn as streamlines over a map of flow speed, with filled markers rotating counter-clockwise and open markers clockwise. — The 2D Ising model

Spins $s_i = \pm 1$ on an $n \times n$ lattice with periodic boundaries, coupled ferromagnetically:
$E = -J \sum_{\langle i,j \rangle} s_i s_j, \qquad J = 1$
A Metropolis sweep attempts $n^2$ spin flips; a flip that raises the energy by $\Delta E$ is accepted with probability $e^{-\Delta E / k_B T}$ . Below $T_c = 2J / k_B \ln(1+\sqrt{2}) \approx 2.269$ the lattice magnetizes. Lemma compiled the sweep with Cython and checked it bit-for-bit against a pure-Python reference before running the physics.

Point vortices in a 2D flow

$N$ point vortices in an unbounded inviscid flow, each advected by the velocity every other vortex induces:
$u_i = -\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{y_i-y_j}{r_{ij}^2}, \qquad v_i = \frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{x_i-x_j}{r_{ij}^2}$
A Hamiltonian system since Kirchhoff (1876). Lemma integrated it with RK4 and checked the integrator against the conserved $H$ , $Q = \sum_i \Gamma_i x_i$ , $P = \sum_i \Gamma_i y_i$ , and $I = \sum_i \Gamma_i (x_i^2 + y_i^2)$ , and against the two-vortex cases it could solve by hand.

The six trajectories plotted in three dimensions with time on the vertical axis: helices where pairs orbit each other, near-straight lines where pairs run off together. — The 2D Ising model

Spins $s_i = \pm 1$ on an $n \times n$ lattice with periodic boundaries, coupled ferromagnetically:
$E = -J \sum_{\langle i,j \rangle} s_i s_j, \qquad J = 1$
A Metropolis sweep attempts $n^2$ spin flips; a flip that raises the energy by $\Delta E$ is accepted with probability $e^{-\Delta E / k_B T}$ . Below $T_c = 2J / k_B \ln(1+\sqrt{2}) \approx 2.269$ the lattice magnetizes. Lemma compiled the sweep with Cython and checked it bit-for-bit against a pure-Python reference before running the physics.

Point vortices in a 2D flow

$N$ point vortices in an unbounded inviscid flow, each advected by the velocity every other vortex induces:
$u_i = -\frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{y_i-y_j}{r_{ij}^2}, \qquad v_i = \frac{1}{2\pi}\sum_{j\neq i}\Gamma_j \frac{x_i-x_j}{r_{ij}^2}$
A Hamiltonian system since Kirchhoff (1876). Lemma integrated it with RK4 and checked the integrator against the conserved $H$ , $Q = \sum_i \Gamma_i x_i$ , $P = \sum_i \Gamma_i y_i$ , and $I = \sum_i \Gamma_i (x_i^2 + y_i^2)$ , and against the two-vortex cases it could solve by hand.

Our technology

Axiomatic Intelligence

Axiomatic AI’s core technology, Axiomatic Intelligence (AxI), is an AI system that combines formal mathematical reasoning, models of the physical world, engineering constraints, and experimental evidence to turn technical questions into solutions.

It helps scientists and engineers model systems, evaluate designs, expose assumptions, and identify errors earlier, reducing time and experimentation required to reach credible results., drawing on the scientific method.

Axiomatic Intelligence produces trustworthy outputs with formal, checkable guarantees.

Axiomatic Intelligence's foundations

01

Verification

Uses formal methods when possible and physical law and known results otherwise to determine if the reasoning is mathematically sound.
02

Validation

Compares a result against the world, with checking against measured data, physics-based rules, and the benchmarks of the domain.
03

Uncertainty quantification

Clearly states the assumptions made in the reasoning, and the range of confidence in the result.

For organizations

Bring Axiomatic Intelligence into your organization.

We offer private deployments tailored to your tools, workflows, and organizational knowledge—and integrated directly into the environments your design, engineering, and testing teams already use.

Built first for photonics, our enterprise suite connects the full design-to-test loop. It combines AI-driven design and independent verification with automated, traceable characterization and physics-based analysis.

Explore more possibilities, catch issues earlier, and reduce costly iteration before tape-out—all with direct support from our team and a system that evolves alongside your organization.

Wafer map of measured film thickness, zoomed to a die of ring resonators and then to a single resonance in the transmission spectrum, from which the device parameters are extracted. — Wafer-scale characterization

Measured film thickness across a 300 mm wafer, zoomed to a die of ring resonators and then to a single resonance in the transmission spectrum.

Each resonance is compressed to a few characteristics, group index, coupling, and loss, and the map shows the film thickness inferred from them.

Ring resonator model

From the depth of the resonance dips $T(\lambda_{\mathrm{res}})$ and the baseline loss $L$ , the system fits
$T(\lambda_{\mathrm{res}}) = \left|\frac{t-a}{1-at}\right|^2 - L + \epsilon$ $\kappa(\lambda_{\mathrm{res}}, g) = \kappa_0\, e^{-g/\ell_d}\bigl(1 - e^{-(g-g_i)/\ell_i}\bigr) + \delta_\kappa(\lambda_{\mathrm{res}} - \lambda_0)$
with $\epsilon \sim \mathcal{N}(0,\sigma^2)$ , $t = \sqrt{1-\kappa(\lambda_{\mathrm{res}}, g)^2}$ , $a = 10^{-2\alpha\pi r}$ , and parameters $\theta = (\kappa_0, \delta_\kappa, \ell_d, \ell_i, g_i)$ .

Animation: devices on a wafer coloured by expected information gain, with the chosen device marked each iteration, beside the posterior over the coupling curve narrowing as measurements accumulate. — Wafer-scale characterization

Measured film thickness across a 300 mm wafer, zoomed to a die of ring resonators and then to a single resonance in the transmission spectrum.

Each resonance is compressed to a few characteristics, group index, coupling, and loss, and the map shows the film thickness inferred from them.

Ring resonator model

From the depth of the resonance dips $T(\lambda_{\mathrm{res}})$ and the baseline loss $L$ , the system fits
$T(\lambda_{\mathrm{res}}) = \left|\frac{t-a}{1-at}\right|^2 - L + \epsilon$ $\kappa(\lambda_{\mathrm{res}}, g) = \kappa_0\, e^{-g/\ell_d}\bigl(1 - e^{-(g-g_i)/\ell_i}\bigr) + \delta_\kappa(\lambda_{\mathrm{res}} - \lambda_0)$
with $\epsilon \sim \mathcal{N}(0,\sigma^2)$ , $t = \sqrt{1-\kappa(\lambda_{\mathrm{res}}, g)^2}$ , $a = 10^{-2\alpha\pi r}$ , and parameters $\theta = (\kappa_0, \delta_\kappa, \ell_d, \ell_i, g_i)$ .

Discipline

Photonics engineering

We bring AI-driven design and optimization into the tools your photonics engineers already use—then independently verify every design through a process the AI cannot influence or self-grade.
Discipline

Test engineering

Our agentic test software turns measurement objectives and GDS data into executable test plans, runs them on your lab instruments, fits the results to physics models, and keeps every step traceable to raw data.

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Latest from research

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Verifiable AI for science and engineering.

You do the science. Lemma does the rest.

Axiomatic Intelligence

Verification

Validation

Uncertainty quantification

Bring Axiomatic Intelligence into your organization.

Photonics engineering

Test engineering

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