Four vertices. Three edges. Zero paths.
Insert the simplices in this order, with each face preceding its coface:
Over the field with two elements, reduce each boundary using earlier columns with the same latest nonzero row:
| Edge | Reduced boundary | Persistence pair | Incident? |
|---|---|---|---|
| cd | c + d | (d, cd) | Yes |
| bd | (b + d) + (c + d) = b + c | (c, bd) | No |
| ac | (a + c) + (b + c) = a + b | (b, ac) | No |
Only the incident pair remains in the matching: VP = {(d, cd)}. A path from the critical edge ac must first descend to a or c. Both vertices are unmatched, so either path stops immediately.
This disproves Conjecture 2 in Knudson’s 2008 Oberwolfach report, printed p. 1629. The signed boundary calculation in the paper extends the example to any field.
The field must stay fixed
The earlier pair (c, bd) has the path bd → d → cd → c. Cancelling it changes the matching and creates a path from ac to b. That is a path in a different field. The conjecture is about the original VP.
Ordered cancellations in related work, including Bauer, Lange, and Wardetzky’s surface simplification results, are consistent with this example.
Check it in seconds
Download and extract the proof package, then run these commands from its folder. Python 3.9 or later is sufficient; no packages or network access are needed.
python3 verification/verify.py
python3 verification/independent_check.pyThe first checker combines boundary reduction, union-find and exhaustive path enumeration. The second reconstructs the barcode from ranks of component maps and checks signed reduction and path reversal. Both passed, including with Python optimization enabled.
Attribution and scope
A bounded priority audit completed on 23 September 2026 found no earlier explicit refutation in the searched literature. It does not guarantee novelty: related work is listed as unpublished, and some full texts were unavailable. The exact original conjecture was checked directly; the current status of the user-supplied catalogue entry OWR-2040-002 could not be verified.
The candidate arose from prior AI-assisted work supplied by the author. OpenAI Codex prepared this publication and its checks, with separate AI audit agents. This is an unrefereed preprint. No minimum-size claim, proof-assistant certification, or resolution of other conjectures is asserted. The Zenodo files are ready for manual upload; no DOI has been assigned by this project.