FPRD LabShare an approach

← Claims

Negative result · coherence obstruction

CRD-RW-F3

Connected history graphs need not be coherent

Exact statement

For complete occurrence-labelled reductions of a4a^4 under aa→aaa\to a, the graph generated by disjoint-square and contextual-overlap moves is connected but has π1≅H1≅Z\pi_1\cong H_1\cong\mathbb Z; connectedness therefore does not establish coherence.

StatusConnectedness-to-coherence inference refuted; one higher cell repairs the fixed component
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Two-dimensional moves can connect every pair of histories while still leaving nontrivial loops between transformations. Coherence begins only after those loops are controlled.

Definitions

  • The raw transformation graph has occurrence-labelled complete histories as vertices and disjoint or overlap moves as edges.
  • The schedule quotient contracts the unique disjoint-scheduling edge in the arity-four raw graph.

Hypotheses and scope

  • The fixed source a4a^4 in the occurrence-labelled system aa→aaa\to a.

Proof or evidence

The six raw histories form a cycle C6C_6; contracting the single disjoint edge gives the five-vertex rotation cycle C5C_5. Both are connected and both have fundamental group and first homology Z\mathbb Z. Attaching one arity-four 3-cell along the raw hexagon, equivalently the quotient pentagon, fills the unique loop.

Hexagon/pentagon loop and higher-cell repair →

Verification notes

The durable failure and claim ledgers agree on the two presentations and on the one-cell repair. The site preserves the distinction between the raw hexagon and the quotient pentagon instead of using them interchangeably.

Limitations

  • The result concerns the fixed arity-four component, not every rewriting presentation.
  • One-cell necessity is relative to the fixed one-skeleton and counts newly attached higher cells.
  • No all-arity coherence theorem is claimed.

Open work

Require an explicit higher-dimensional coherence criterion and audit it beyond the fixed arity-four component.