The candidate remains unresolved. Vulakh and Finkel describe order 32 as unsolved in their 2022 peer-reviewed paper. Their search produced near solutions rather than a Costas permutation. The 2022 paper `The Density of Costas Arrays Decays Exponentially` likewise states that existence at order 32 remains unknown.

A fresh 2026 paper by Gulec and Abolghasemi reports that exhaustive enumeration reaches order 29 and bases its higher-order experiments on Beard's public collection of known arrays. Beard's database page says that its generated collection reaches order 1030, while exhaustive-search additions stop at orders 28 and 29. These records provide a current cross-check against a published witness. Database absence cannot establish nonexistence.

The acceptance condition therefore remains unchanged. A positive resolution needs a 32-entry permutation whose 496 displacement vectors pass an exact duplicate check. A negative resolution needs an exhaustive search certificate that covers every symmetry class.

Vulakh and Finkel applied their parallel ant-colony solver to order 32 for more than 10,000 CPU-days. The best permutation reported in the paper has 17 Costas-property violations. The Appendix also lists near solutions with 18 and 19 violations.

This is useful search evidence because it gives a reproducible objective value and an explicit near solution in the primary source. The run was heuristic and did not cover the full permutation space. Seventeen repeated-vector violations therefore supplies neither a witness nor a lower bound on the minimum possible number of violations.

The principal Welch \(W_1\) construction has order \(p-1\) for a prime \(p\). At order 32 it would require \(p=33\), which is composite. The principal Golomb \(G_2\) construction has order \(q-2\) for a prime power \(q\). At order 32 it would require \(q=34\), which is not a prime power. These arithmetic checks explain why the two main infinite families do not immediately settle this order.

Published construction tables list no successful construction at 32. Beard's archive includes generated arrays at many larger orders and searched arrays through order 29, yet gives no order-32 permutation. Gulec and Abolghasemi's 2026 study uses the same known-array record and still describes complete enumeration only through order 29.

This audit covers known records and the two principal algebraic families. It does not exclude a sporadic array, an unrecorded computation, or a new construction.

For a candidate \(\pi\), the verifier first checks that its entries are exactly \(0,\ldots,n-1\). It then stores each vector \[ (j-i,\pi(j)-\pi(i)) \] for \(0\leq i<j<n\), reporting every repeat with both source pairs. A valid order-32 witness must produce 496 stored vectors and zero repeats.

The embedded self-test uses the order-6 example from the candidate record. It checks all 15 pairs and returns `costas=True`. Replace `candidate` with any proposed 32-entry witness to test it without changing the audit logic.