Canonical search audit
Scope
This checkpoint specifies a correctness baseline for finite union-closed-family searches. It does not report a new execution, enumeration count, counterexample or general proof. Earlier lab test results remain bounded evidence; independent replication is outstanding.
Let F be a nonempty family of distinct subsets of its nonempty support U, with m members. Write f(x) for the number of members containing x. A strict Frankl counterexample would require every support element to satisfy 2f(x)<m. A single infrequent element is not a counterexample.
Exact symmetry reduction
Fix a numbering of U. Encode each member as an integer bit mask. For every permutation of U, relabel all members and sort the resulting masks. Define the canonical key to be the lexicographically least resulting list, using numeric comparison of entries (not textual ordering).
Two families have the same key exactly when they are isomorphic under ground-set relabelling: the minimum is in each orbit, and a shared representative composes to an isomorphism. Relabelling preserves union closure, m, support size, separation, and the multiset of frequencies. Therefore it preserves strict-counterexample status in both directions.
This brute-force orbit minimum is a small-instance correctness baseline, not an implementation of canonical augmentation. Any faster generation method needs its own completeness proof.
Twin deletion: an injective quotient
Suppose distinct x,y have identical incidence in F. Delete y from every member. If two images were equal, their original members could differ only at y. But identical incidence forces y to occur exactly when x occurs; x survives deletion. Thus those originals are equal. Member count is unchanged.
Deletion commutes with union. All surviving frequencies are unchanged. Consequently a strict counterexample would remain a strict counterexample. The deleted frequency equals the surviving twin's frequency, so this step can also be reversed by duplicating a support element. Repeating this reduction terminates in a separating family.
Empty-member deletion: a threshold warning
Removing the empty member from a family with nonempty support preserves closure and frequencies, but changes m to m−1. Thus the assertion 2f(x)<m does not by arithmetic alone imply 2f(x)<m−1.
- If m is even, integer frequencies give 2f(x)≤m−2, so strictness survives for every element.
- If m is odd, they give only 2f(x)≤m−1. Equality is compatible with that arithmetic and would cease to violate Frankl after deletion.
This is a gap in the proposed reduction argument, not a constructed union-closed counterexample. Such a construction would itself settle the conjecture negatively. No structural theorem ruling out the equality obstruction has been established in this checkpoint.
Adjoining the empty member is safe in the other direction: frequencies stay fixed while the threshold increases, so a hypothetical strict counterexample without it stays strict after adjoining it. Hence a search restricted to families containing the empty member can be counterexample-complete, provided it includes the increased member count. Simply searching only empty-free families needs additional justification.
Search-safe baseline
Enumerate all candidate families at the chosen support bound, reject nonclosed and non-full-support inputs, then canonicalize by the exact orbit minimum. Keep both empty-member states initially. Twin reduction can justify a separating-family restriction; arbitrary element deletion cannot.
For separating full-support families, the earlier witness argument yields maximum frequency at least |U|. Accordingly a strict counterexample must have m>2|U|. Apply that filter only after checking its hypotheses.
Required next tests
- Every ground-set permutation of each eligible family returns the same canonical key.
- Distinct keys correspond to distinct orbits, compared with an independently implemented orbit partition.
- Frequency multisets and closure are invariant under every relabelling.
- Twin deletion preserves member count and surviving frequencies; arbitrary deletion has a negative control.
- Empty deletion tests the threshold identity and parity distinction, without pretending to test actual counterexample preservation when no counterexamples are present.
- Publish labelled and orbit counts only after execution returns them; a green predicate alone is not a count.
External methods reference and review limit
The browser retrieved McKay's papers index and the arXiv record on isomorph-free exhaustive generation of Greechie diagrams. These concern generation methodology, not this union-closed proof. The attempted journal page was blocked; its contents were not checked.
A specialist critique agreed with the threshold arithmetic but incorrectly described a counterexample at one point as having some infrequent element. That definition is rejected here: all elements must be below half. Model agreement is not independent verification.