A meaningful dimension bound must be effective or uniform

Exact statement

For each fixed polynomial $p$ , the bare existence of a finite bound on witness dimensions is automatic: if any witness exists, the set of its possible state dimensions has a least element. The substantive questions ask for a computable $N(p)$ , a uniform $N(d,D,L)$ , or a dimension-only $N(d)$ .

StatusProved by well-ordering; the intended effective bound remains open

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

Existence, computability, and parameter-uniformity are different logical statements and require separate bounds.

Definitions

N(p) may depend on the complete encoded polynomial.
N(d,D,L) depends only on visible dimension, degree, and an encoding-size parameter.
N(d) is independent of degree and coefficients.

Proof or evidence

If the witness-dimension set is nonempty, its minimum exists by well-ordering; if it is empty, the conditional statement is vacuous.

Which bound is being asked for? →

Verification notes

Existence, computability, and uniformity were kept as three separate quantifier patterns.

Limitations

This logical distinction does not supply any of the effective bounds.

Open work

Determine which effective or uniform parameterization is intended before treating the general question as a decision problem.