gemm

A tensor core multiplies small matrix tiles in one instruction and is 16x faster than the regular lanes, but only for dense matmul at the right precision, with the right shapes and layouts. What the instruction looks like, what it refuses, and how to tell from a profile whether you are on it.

A linear layer at batch 1 runs at a third of a percent of peak and at batch 4,096 near 70%. The curve between has two regions and a sawtooth: the memory-bound slope where intensity equals M, the plateau where the tensor cores are the limit, and the dips where the tile count does not divide the SM count.

The naive kernel reads two bytes per multiply-add; a 128x128 tile with register blocking reads a few hundredths of that. Each step (shared-memory tiles, register blocking, vectorized loads, double buffering, tensor cores) with its intensity arithmetic and speedup, and where a hand kernel stops and cuBLAS begins.