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Counterexample · definition obstruction

CRD-RW-F4

Leaf-disjointness misses nested interchange

Exact statement

Disjoint leaf intervals are not a complete test for independent contextual rotations: at five leaves, an inner rotation on ((AB)C)((AB)C) commutes with an outer rotation ((XD)E)→X(DE)((XD)E)\to X(DE) even though the inner leaf support is strictly contained in the outer support.

StatusLeaf-support criterion refuted; active-constructor support survives
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Contextual rewriting treats metavariable subtrees as opaque. Leaf intervals can therefore overlap or nest even when the constructor occurrences actually rewritten are independent.

Definitions

  • Leaf support is the interval of leaves under the entire matched expression.
  • Active-constructor support consists only of the two binary constructor occurrences replaced by an associativity rotation; substituted metavariable subtrees are opaque.

Hypotheses and scope

  • The five-leaf binary-tree rotation graph and contextual associativity moves.

Proof or evidence

Starting at (((AB)C)D)E)(((AB)C)D)E), perform the inner rotation ((AB)C)→(A(BC))((AB)C)\to(A(BC)) and the outer rotation ((XD)E)→X(DE)((XD)E)\to X(DE) in either order. Both paths end at (A(BC))(DE)(A(BC))(DE), although one leaf interval is nested in the other. The preserved exhaustive enumeration finds three four-cycles and no coinitial rotation pair with disjoint leaf intervals.

Nested interchange witness and finite graph audit →

Verification notes

The explicit two-order computation supplies the counterexample; finite enumeration corroborates completeness for the five-leaf graph. The source boundary preserving the arity-four no-interchange theorem was retained.

Limitations

  • Leaf-disjointness remains a sufficient special case; it is not necessary.
  • The a4a^4 pentagon result is unaffected because its rotation graph has no four-cycle.
  • Finite enumeration corroborates the five-leaf inventory but is not presented as an all-arity proof.

Open work

Use residual commutation or disjoint active-constructor support when enumerating interchange squares at five leaves and beyond.