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Theorem · finite obstruction duality with explicit certificate bound

FPRD-T171

Every infeasible finite selector budget has a bounded-cardinality obstruction suite

Exact statement

For every tagged language A, letter projection pi, and infeasible selector budget (m,s), there is a finite set W of accepted observed words that defeats every family of at most m selectors with at most s states. Its cardinality is at most the number N_(m,s)(pi) of candidate labeled selector families. The minimum obstruction cardinality is an exact finite set-cover number, and the T169 escape length equals the minimum possible maximum word length of an obstruction suite.

StatusSelf-contained finite-universe proof with exhaustive bounded duality audit
External reviewThe proof is a finite candidate-universe argument. No external or specialist review of the obstruction formulation is recorded. The optional hard-lift application retains T168's imported premise.

Context

T169 says every infeasible budget fails at a finite horizon. T171 extracts a potentially much smaller finite test suite and separates certificate radius from certificate cardinality.

Hypotheses and scope

  • The observed and tagged alphabets are finite and pi is letter-to-letter.
  • Selectors are deterministic synchronous finite-state transducers.
  • The count m and state size s are fixed finite resource ceilings.

Proof or evidence

Each candidate selector family misses some accepted word. Choosing one witness per family gives the cardinality bound. Failure sets D_w turn the minimum certificate into an exact set-cover problem. Minimizing the largest word length recovers the escape length from T169.

Finite obstruction duality →

Verification notes

The checker exhausts 32,768 tagged unary relations and 65,536 budgeted duality cases, including 32,832 obstruction instances, cardinality-bound checks, and exact escape-radius identities.

Limitations

  • The certificate is existential and need not be computable from an arbitrary language description.
  • The labeled-machine bound can substantially exceed the number of distinct selector behaviors.
  • The theorem bounds the number of witness words, not their lengths.
  • The hard-lift application depends on T168 and its imported Kim-Park separation premise.

Open work

Seek structural bounds on obstruction cardinality and radius for positive allocator classes, then compare them with the hard visible lift.