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Corollary · exact hidden-choice blowup in a deterministic visible lift

FPRD-T168

A one-stack visible language can require infinitely many causal selectors after tag hiding

Exact statement

There is a deterministic visibly pushdown language A_C with a degree-one, distance-two scaffold such that its forgotten-action projection C has empty stationary transport spectrum. Moreover, no fixed finite family of synchronous finite-state tag selectors covers all accepting observed words: csel(A_C,pi)=infinity.

StatusSource-attributed corollary with explicit transport and selector-cover proof
External reviewThe SCA non-membership of C is imported from Kim and Park, arXiv:2608.29592v1. FPRD Lab has not independently reverified their lower bound or Lean artifact.

Context

The hard forgotten-action lift is not difficult because its tagged runs need exotic storage. The tagged language is a deterministic one-stack machine and has a minimal persistent-stack presentation. Difficulty appears only after hidden histories must be merged.

Hypotheses and scope

  • A_C is the deterministic visibly pushdown lift of the Kim-Park language C supplied by FPRD-T91.
  • The persistent-stack compiler applies to the letter-synchronous push, keep, or pop dynamics of a deterministic visibly pushdown automaton.
  • Kim-Park Theorem 4.19 supplies C outside SCA.

Proof or evidence

A_C has finite transport by the one-stack compiler. If finitely many causal selectors covered its accepting lifts, FPRD-T167 would give a finite transport tuple for C, contradicting Kim-Park. Thus its causal selection number is infinite.

Hard visible-lift selector theorem →

Verification notes

The local checker verifies the finite-cover implication orientation and 256 raw-cover growth cases. It does not reverify the visible-lift construction, full stack compiler, or external separation.

Limitations

  • This is a corollary of an imported SCA separation, not an independent lower bound.
  • Infinite causal selection number alone does not imply SCA non-membership for an arbitrary projection.
  • The theorem establishes nonexistence of a fixed finite selector family, not a quantitative finite-length lower bound optimized for C.

Open work

Find an intrinsic growth measure for accepting selector families that can be bounded on the positive allocator tiers and forced to diverge on the C lift.