State the properties you need — bulk modulus, shear, anisotropy, Poisson ratio — and get the exact stiffness tensor that realizes them, the nearest real material, or a printable metamaterial when none exists yet. Certified for cubic symmetry — every specification provably realized by exactly one tensor — with certified instances in orthorhombic; screened exhaustively over a physicality-audited catalogue of 27,015 known materials.
Stiffness tensors are symmetric positive-definite matrices on a curved manifold — not points in flat space. Treating them as 21-vectors introduces bias worth orders of magnitude.
A material is described equally by its stiffness C or its compliance S = C⁻¹. Under Euclidean distance the same pair of materials can look five orders of magnitude closer or farther apart depending on which one you picked — an arbitrary choice silently reorders your whole database.
The affine-invariant Riemannian metric makes stiffness and compliance exactly isometric — distance ratio 1.000 for every pair. Similarity search, clustering, and inverse design stop depending on representation. The geometry costs nothing where curvature doesn’t matter, and everything where it does.
Real math, running in your browser. Set the targets; the exact realizing tensor, its nearest real material by geodesic distance, and — for auxetic / white-space targets — a printable re-entrant lattice are all computed live. Try the Diamond-stiff auxetic preset.
The target is a tensor; the deliverable is a material. Three bridges cover the gap — and the tool already runs the first two.
If a catalogued compound sits near your target, we return it — ranked by affine-invariant geodesic distance across the full database. Blind tests recover held-out materials from their properties alone, 9 of 9.
When your target is auxetic and no atomic solid applies, we solve the re-entrant lattice geometry that realizes the negative Poisson ratio — with an honest read on the stiffness a real base material and print density can deliver, plus an SVG unit cell.
For targets beyond both, generative crystal models conditioned on the tensor propose new compositions for DFT validation — the research frontier, and the natural home for a Materials-Project–scale collaboration.
Cubic targets solve in closed form; orthorhombic by homotopy continuation with a posteriori certification — provably exactly one realizing tensor, not one lucky local optimum. For cubic symmetry we prove every positive spec (K, G, A) is realized by exactly one tensor.
For cubic symmetry feasibility is never the obstruction, so what remains is occupancy. The atlas maps certified-feasible specs no known material occupies at a stated tolerance: nearer-in gaps become directed synthesis targets (exact tensors a substitute must realize), while extreme corners — like a diamond-stiff auxetic (K=218, G=600 GPa, ν=−0.22), whose c₄₄ would exceed diamond's — are reachable only as architected lattices, not atomic solids.
Affine-invariant geometry + numerical algebraic geometry. Preprint: 10.5281/zenodo.21709402. Pre-registered evaluations (protocols and tolerances fixed before results): 10.5281/zenodo.22265876.
Working engine across six domains, 117 tests. Patent OMNI-2026-007. Cubic at ~10⁵ solves/s; certified homotopy for orthorhombic symmetry.
Screened over Materials Project + JARVIS-DFT (27,015 tensors passing Born-stability validation); the same geometry audits data quality — 2,017 non-physical tensors flagged in MP, 11,480 in JARVIS — and calibrates the DFT-vs-measured gap (median ≈11%).
API access, a database audit, or a certified inverse-design collaboration — including SBIR / DOE partnerships on the metamaterial track.