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Theorem, construction, or exact resource formula

MMR-114

Claim MMR-114

Exact statement

Direct-sum type classes have multinomial multiplicity and satisfy ∑iVol⁡(Di)ρ/3=(∑iviρ/3)N\sum_{\mathbf i}\operatorname{Vol}(D_{\mathbf i})^{\rho/3}=(\sum_i v_i^{\rho/3})^N.

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

Multinomial theorem in Run 13. Finite checker verifies the first ten powers of Schönhage's (16,9)(16,9) volume example.

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.