Numerical Conformance

Lattice separates semantic correctness from floating-point reproducibility. Coordinates, relation edges, kernel positions, and artifact tensor layouts are exact contracts. Floating-point reductions are evaluated against a common binary64 reference because valid CUDA and Metal executions can accumulate terms in different orders.

Reference evaluator

lattice_contract.submanifold_conv3d_f32_to_f64 is a scalar, dependency-free reference evaluator for conformance tests. It rounds feature, weight, and bias leaves to binary32, multiplies them in binary64, and uses math.fsum for the final sum. It also owns the canonical (batch, x, y, z) coordinates and z-fastest (x, y, z) kernel-row interpretation.

The evaluator is deliberately unsuitable for model execution. Production backends use their optimized relation and reduction kernels, then tests compare their result to the evaluator with dtype-appropriate tolerances.

Floating-point policy

For unquantized FP32 pointwise projections, the Metal backend uses a native FP32 projection rather than MLX’s accelerated matrix path. The latter can choose reduced-precision hardware, which makes the result depend on matrix size. The native path uses TensorOps for supported 32-channel tiles and a scalar FP32 Metal kernel otherwise. They are both checked against the same binary64 oracle; their reduction order need not be bitwise identical. Lattice does not expose a threshold or compatibility option: identical FP32 inputs receive the same accuracy policy at every size.

Sparse relation kernels and CUDA dataflows may still differ by normal FP32 reduction order. They must preserve coordinate support and stay within the documented oracle tolerance; bitwise equality across devices is not a contract.

Quantized weights are a separate numerical surface. Their affine packed values are exact at storage boundaries, while their runtime output is compared to the corresponding dequantized execution with the quantization tolerance declared by the artifact test.

Migration and validation

Historical TorchSparse checkpoints are not a numerical reference. Convert them once with explicit kernel metadata, inspect the recorded permutation manifest, and validate the converted block against its known input/output fixtures. The runtime never infers legacy row order or silently adapts a checkpoint.

The permanent checks cover the contract evaluator, canonical non-cubic CUDA map builders, Metal convolution, and CUDA-to-MLX artifact replay. This keeps a backend optimization from becoming an undocumented semantic change.