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Lemma

FPRD-RS-L01

Safe-row criterion for relative degree

Exact statement

Put A=MaA=M_a, B=MbB=M_b, V=span⁡{Mxγ:x∈Σ∗}V=\operatorname{span}\{M_x\gamma:x\in\Sigma^*\}, and W={p:pγ=0 and pBV=0}W=\{p:p\gamma=0\text{ and }pBV=0\}. Then supp⁡(f)⊆arΣ∗\operatorname{supp}(f)\subseteq a^r\Sigma^* if and only if λAk∈W\lambda A^k\in W for every 0≤k<r0\le k<r.

StatusProved with finite-dimensional linear algebra
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Over the binary alphabet, a word outside the cone beginning with r copies of a either stops inside the initial a-block or first deviates through b. The safe-row criterion covers exactly these two cases.

Definitions

  • The reachable column space VV is the least subspace containing γ\gamma and invariant under all McM_c.
  • The condition pBV=0pBV=0 means pBv=0pBv=0 for every v∈Vv\in V.

Hypotheses and scope

  • The relative-degree convention and representation are those of FPRD-RS-D01.
  • The alphabet is {a,b}\{a,b\}, so bb is the only possible deviation from the initial aa-corridor.

Proof or evidence

For each k below r, pγ=0 excludes the word a^k and pBV=0 excludes every continuation beginning a^k b. Conversely, every prohibited word has one of those forms under the stated convention, so the row conditions force its value to zero.

Safe-row criterion and proof →

Verification notes

Both directions were reconstructed from the source proof. An explicit three-letter hostile example shows that checking only B fails unless the alphabet is exactly {a,b}.

Limitations

  • For a larger alphabet, W would need the analogous annihilation condition for every non-a letter.

Open work

Obtain an independent specialist check of the two directions.