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Theorem, construction, or exact resource formula

MMR-056

Claim MMR-056

Exact statement

In a sealed copy-only graph transformation, a value with dd original consumers needs at least cΔ(d)=max⁡{0,⌈(d−Δ)/(Δ−1)⌉}c_\Delta(d)=\max\{0,\lceil(d-\Delta)/(\Delta-1)\rceil\} identity buffers under fan-out cap Δ≥2\Delta\ge2; for Δ=2\Delta=2, a caterpillar attains d−2d-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

Lemma R8.1; outgoing-capacity count on the source plus buffers, followed by an explicit fan-out-two chain. The bound does not survive arbitrary gate duplication, recomputation, equality sharing, or algebraic resynthesis.

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.