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Theorem · exact finite-horizon compactness and escape modulus

FPRD-T169

Every fixed causal-selector budget either works globally or fails at a finite horizon

Exact statement

For a tagged language A and letter projection pi, let Sel_n(A,pi) contain the selector-count and selector-state budgets that cover all accepted observations through length n. Then Sel(A,pi)=intersection_n Sel_n(A,pi). Each set is upward closed, the horizon sequence decreases, and every infeasible finite budget has a least finite escape length. Applied to the hard visible lift A_C, every fixed finite selector budget fails on some finite input horizon.

StatusSelf-contained finite-candidate compactness proof with exhaustive bounded audit
External reviewThe compactness argument is self-contained and uses a finite candidate universe. No external or specialist review of the selection-spectrum formulation is recorded. The hard-lift application retains T168's imported Kim-Park premise.

Context

T168 rules out one global finite family of online finite-state selectors. T169 upgrades that global nonexistence to finite obstructions for every fixed selector-count and selector-state budget.

Hypotheses and scope

  • The observed and tagged alphabets are finite, and the projection is letter-to-letter.
  • Selectors are total deterministic synchronous Mealy transducers.
  • Both selector count and selector state size are bounded by fixed finite integers.

Proof or evidence

For fixed resources there are finitely many labeled selector families. Candidate families correct through horizon n form nested subsets of this finite universe, so nonempty feasibility at every horizon yields one family correct at every horizon. Upward closure gives a finite Pareto basis by Dickson's lemma.

Causal selection spectrum theorem →

Verification notes

The checker exhausts 32,768 tagged unary acceptance relations through horizon three, verifying 196,608 nesting comparisons, 65,536 finite-intersection identities, 131,072 resource-upward cases, and 32,832 finite escapes.

Limitations

  • The theorem is qualitative and does not make the escape modulus computable or give a closed-form upper bound.
  • It does not cover causal selectors with unbounded memory or asynchronous output.
  • An empty selection spectrum is presentation-relative and does not alone imply SCA or PEG non-membership.
  • The hard-lift corollary depends on T168 and therefore on the imported Kim-Park separation.

Open work

Bound or estimate the escape modulus on positive allocator tiers and seek an intrinsic growth law for the Kim-Park visible lift.