Complete direct sums are neutral under tensor powering

Exact statement

Tensor-powering a complete direct sum is neutral under the asymptotic sum inequality: the powered inequality is exactly the $N$ th power of the one-level inequality.

StatusProved in the FPRD research record

External reviewNo documented external or specialist review of the FPRD synthesis is recorded.

Context

This result belongs to the reconstruction from algebraic matrix-multiplication certificates to explicit arithmetic dependency DAGs and their resource receipts.

Hypotheses and scope

All resource conclusions are restricted to the arithmetic, storage, fan-out, coefficient, and precision models named in the source record.

Proof or evidence

Corollary R13.2 from MMR-114 and $\underline R(D^{\otimes N})\le\underline R(D)^N$ . Laser gains require a selected/merged direct sum from an initially overlapping partition.

Mathematical overview and evidence boundary →

Verification notes

The analytic argument is preserved in the research record. Independent external review is not recorded.

Limitations

The record does not improve the known matrix-multiplication exponent and does not imply that omega equals two.
Construction bounds are not optimality, communication, I/O, or hardware claims unless the exact statement says so.

Open work

Independently re-audit the complete derivation before any further promotion.