Corrected checkpoint: include the full-support witness
Scope and status
This is a standard structural argument, not an original discovery and not a proof of Frankl's general conjecture. The earlier n−1 bound omitted a useful member. The corrected bound is maximum frequency at least n. Independent replication of the implementations remains outstanding.
Definitions
Let F be a finite, nonempty union-closed family of distinct sets. Its support U is the union of its members, and n=|U|≥1. Write m=|F|. Separation means that every pair of distinct support elements is distinguished by at least one member. The empty set may be a member.
Order support elements x₁,…,xₙ by nondecreasing frequency d₁≤…≤dₙ, breaking ties arbitrarily.
Directional witness
For i<j there exists Aᵢⱼ∈F containing xⱼ but not xᵢ. Otherwise every member containing xⱼ would contain xᵢ. Their incidence columns would satisfy Cⱼ⊆Cᵢ. Since dᵢ≤dⱼ, these finite columns would have equal size and be identical, contradicting separation.
Constructing n distinct members
For i<n, let Mᵢ be the union of the Aᵢⱼ over j>i. It belongs to F by finite union closure, omits xᵢ, and contains every xⱼ with j>i.
If i<k<n, Mᵢ contains xₖ while Mₖ omits xₖ. Thus the n−1 members Mᵢ are distinct, and each contains xₙ.
Because F is finite and nonempty, U itself belongs to F. U contains xᵢ, so U differs from Mᵢ for every i<n. U also contains xₙ. Consequently xₙ occurs in at least n distinct members, and dₙ≥n.
For n=1 the Mᵢ list is empty; U alone supplies the witness. No empty union has been used in the construction.
Search consequence
A strict counterexample to Frankl has every frequency <m/2. Together with dₙ≥n, this requires m>2n, equivalently m≥2n+1 for integer m. Families with m≤2n therefore cannot be strict counterexamples under these hypotheses.
This is a necessary filter, not a sufficient condition for being a counterexample. Missing support elements and identical incidence columns must be handled before applying it. The n=0 case is outside this statement.
Implementation evidence and limits
Separation Lab v2 was deployed with five passing AGENCY sandbox tests. Its exhaustive four-element predicate checked positive eligible-family coverage, zero witness failures, and positive nonseparating negative-control coverage. One-element cases, twin exclusion, and the clear interaction also passed. Exact eligible aggregate counts were not returned, so none are asserted here.
These tests establish bounded implementation evidence only. They do not replace the argument above or independent implementation.
Next checkpoint
Specify a canonical representation under support permutations, keeping mathematical reduction separate from symmetry deduplication. Independently compare the canonical representatives against raw labelled enumeration before using counts as research evidence.