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Counterexample / failed approach

RA-08

Literal PEG conversions fail

Exact statement

Both direct orderings of the CFG path rules fail, as do the corresponding outer end-guard repairs; explicit nested counterexamples are given. U's PEG status remains open.

StatusSpecified conversions refuted; general PEG question open
External reviewNone documented for this work.

Context

Research note extending Kim–Park’s explicit separation: the late-query reduction follows their strategy, and the recursive parity rules are adapted from their displayed grammar. The tree-addressed selection and residual-profile comparison are developed here; their novelty is not established.

Proof or evidence

Self-contained proofs and direct reduction in the manuscript. Finite semantic and rank checks corroborate the constructions; they do not prove an asymptotic lower bound.

Manuscript proof and evidence →

Verification notes

Internal combined-proof audit on 5 September 2026; no independent expert review or formal verification claimed.

Limitations

  • Equal profiles do not mean equal residuals or equal transition structures.
  • The access exclusion uses Ko; the PEG consequence also uses Loff–Moreira–Reis.
  • U's PEG membership and an intrinsic scaffold lower bound remain open.

Open work

Find a different construction for U; the failed conversions do not prove non-membership.