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Counterexample · failed completeness claim

CRD-RW-F1

Disjoint squares do not classify overlapping histories

Exact statement

In the terminating confluent system aa→aaa\to a, swapping consecutive disjoint redex contractions does not connect all complete reductions with the same source and normal-form endpoint: the two reductions of a3a^3 begin at overlapping redexes and lie in different square classes.

StatusRefuted in the stated occurrence-labelled model; repaired by adding contextual overlap moves
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Confluence identifies reachable normal forms; it does not automatically identify the histories used to reach them. This record isolates the first place where overlap, rather than scheduling, matters.

Definitions

  • An occurrence-labelled reduction records which occurrence of the redex is contracted at each step.
  • A square class identifies histories only by swapping adjacent contractions whose residual supports are disjoint.

Hypotheses and scope

  • The one-rule string-rewriting system R={aa→a}R=\{aa\to a\}.
  • Complete reductions from ana^n to the normal form aa, with occurrence identity retained.

Proof or evidence

From a3a^3, contracting the left aaaa first and contracting the right aaaa first produce two complete histories with the same endpoints. Their initial redexes overlap, so neither history contains a disjoint adjacent pair to swap; hence they are distinct square classes. The source further classifies ana^n into Cn−1C_{n-1} such classes and reconnects them with ((AB)C)↔(A(BC))((AB)C)\leftrightarrow(A(BC)).

Self-contained counterexample and repair →

Verification notes

The public statement was reconciled against both the durable claim ledger and CRD-F-5. The witness, model, minimum-within-model qualification, surviving Catalan classification, and repair all agree. No broader impossibility or novelty claim is made.

Limitations

  • The a3a^3 witness is minimal by source length only inside this fixed rewriting system.
  • The counterexample does not refute confluence or term-level reachability.
  • Adding overlap moves proves connectedness of complete histories, not higher coherence.

Open work

Retain the counterexample as the gate against square-only history calculi; audit the all-arity overlap calculus before any coherence claim.