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Theorem and finite pigeonhole argument

FPRD-T93

Finite-core sewing criterion for parsing-expression languages

Exact statement

Let K0⊆K1⊆⋯\mathcal K_0\subseteq\mathcal K_1\subseteq\cdots be finite families of canonically normalized scaffolding automata whose union contains every finite scaffold source, with the index bounding all finite source data. If (e_L(N)) is the least (k) for which a member of Kk\mathcal K_k agrees with (L^R) through length (N), then L∈PELL\in\mathsf{PEL} exactly when sup⁡NeL(N)<∞\sup_N e_L(N)<\infty.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

A bounded sequence of finite-horizon fits yields one stationary recognizer only because a fixed normalized stratum contains finitely many machines.

Hypotheses and scope

  • Every stratum is finite after canonical renaming.
  • The index bounds states, alphabets, degree, observation distance, tables, initial state, and accepting set.

Proof or evidence

One PEG gives one scaffold for the reversed language and therefore a uniform bound. Conversely, one machine in a fixed finite stratum recurs at unbounded horizons; it agrees with the reversed language on every word, and the scaffold-to-PEG theorem completes the argument.

Finite-core sewing →

Verification notes

The reversal, minimum existence, monotonicity, finite-pigeonhole step, and necessity of finite normalized strata were independently rederived from the written proof and the cited correspondence.

Limitations

  • The theorem is not a compactness principle for arbitrary infinite resource classes.
  • It does not prove that context-free languages have bounded presentation demand, or that any particular language escapes.

Open work

Apply the criterion only to finite normalized strata; test whether projected visible frontiers escape every fixed stratum.

Notes

The forward implication uses LMR Theorem 16. The reverse implication is finite pigeonhole sewing inside one K_k followed by the scaffold-to-PEG direction of that theorem, with reversal explicit. The result requires finite normalized strata; it is not an unrestricted compactness claim and opens no M34.