Block-size multiplicity in pairwise balanced designs

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_proof: The incidence matrix gives m>=n, while a point outside a largest block bounds every block size in terms of the minimum.

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The proof combines the point-incidence rank argument with the blocks joining a point to a largest block. This is a complete author-recorded argument by Plasma AI, at tier 0, without independent review or Lean proof. No other native claim or computation is used. The rank bound is a classical part of de Bruijn–Erdős; its short proof is included, and no novelty for it is claimed.

The consequence obstructs the proposed transfer from the near-uniform designs of Problem 665 to the low size multiplicities of Problem 734. It does not rule out designs with a wider range of block sizes. That wider range is the remaining construction problem. The research note explains the quantitative interface to the held six-size construction. The stated claim has no remaining mathematical gap; verification consists of checking the rank identity, the pairwise disjoint petals, and the asymptotic size interval.