Conclusion

A packet operator representation records which block maps exist, how they are stored, and what output they denote. Exact block evaluation and composition turn the support graph into algebra. Arithmetic, storage, and one pass data movement follow from the active block dimensions. Block diagonal plus low rank operators remain closed under multiplication before recompression. Their inverse is reduced by Woodbury, and their correction can be recompressed optimally through a small singular value problem.

Sparse transfer graphs predict path support, triangular schedule depth, and possible fill under elimination. Packet diagonal structure gives local Krylov spaces and a certified block Jacobi condition bound. Packetwise perturbations give a direct mixed precision certificate. Continuous tolerances follow from a convex allocation, while finite data type choices lead to an exact multiple choice budget program.

A finite candidate refinement tree supports a genuine packet compiler. The Lagrangian optimum, the quantized error budget optimum, and local recompilation are exact under additive hypotheses. This is narrower than unrestricted automatic discovery, but it turns one broad compiler conjecture into a proved algorithmic statement.

The measured corpus shows the practical distinction between regular and irregular structure. Equal block and BDLR operators gained substantially on the recorded GPU. Recursive HODLR, Strassen, iterative refinement, a small transform, and an attempted overlap lost to recorded baselines. Discovery, transfer, capture, and materialization changed the break even point.

The computational principle is therefore precise. Remove work through a verified representation. Preserve the declared output. Compile only inside a model whose costs and errors are stated. Assign precision by sensitivity. Amortize one time costs. Keep negative controls. Treat the final timing as a property of the representation, device, kernel, and reuse regime together.

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