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Theorem, construction, or exact resource formula

MMR-023

Claim MMR-023

Exact statement

Two exact realizations of (P,Q,R)(P,Q,R) have the same (N,A,C,D,S)=(3,2,0,3,3)(N,A,C,D,S)=(3,2,0,3,3), labelled arrivals (1,3,1)(1,3,1), and unlabelled fan-out multiset {0,0,1,1,2}\{0,0,1,1,2\}.

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

Explicit gadgets GR:(Q+R)−RG_R:(Q+R)-R and GP:(Q+P)−PG_P:(Q+P)-P in Run 3; schedules and output-count lower bound.

Mathematical overview and evidence boundary →

Verification notes

The definitions, analytic proof chain, seam hypotheses, completion obligation, and likely failure points were reconstructed from the archived proofs on 2026-09-01. Four retained exact checkers reproduced the separator, product-state recurrence, bisimulation quotient, and stutter-trace witness with their recorded hashes. The current-label/external-pending distinction and the dead-gate boundary were checked explicitly.

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

Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.