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Theorem

FPRD-RS-T01

Dimension bound for relative degree

Exact statement

If a dd-dimensional linear representation has a relative degree, then that degree satisfies r<dr<d.

StatusProved by a Krylov-space argument
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The bound turns an unbounded search over prefixes into a finite search controlled by the representation dimension.

Hypotheses and scope

  • An r satisfying the support condition and the nonzero witness exists.
  • The supplied representation has dimension d; minimality is not required.

Proof or evidence

Write v_k=λA^k. The safe-row lemma puts v_0,…,v_{r−1} in W, while the witness keeps v_r outside W. A first Krylov dependence would make the earlier span A-stable and force v_r back into W. Hence v_0,…,v_r are independent and r+1≤d.

Dimension bound →

Verification notes

The argument was checked for nonminimal representations, positive dimension, and the r=0 boundary. The complete one- and two-dimensional exhaustive certificate was rerun without mismatch.

Limitations

  • The strict bound is representation-dependent and need not be sharp for every series.

Open work

Obtain an independent specialist check of the first-dependence argument.