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Every k-Flower Is Anemone Or Daisy

Claim/Theorem

Let \(\Phi=(P_1,\dots,P_n)\) be a k-flower in a polymatroid, hence in particular in a matroid.

Then \(\Phi\) is either:

  1. a k-anemone, meaning every nonempty proper union of petals is exactly k-separating, or
  2. a k-daisy, meaning exactly the unions of cyclically consecutive petals are exactly k-separating.

Equivalently, crossing exact k-separations do not generate an arbitrary zoo of patterns. They collapse to one of two flower templates.

For Conjecture 3, this means that any fixed-order family of mutually crossing low-rank balanced cuts must already live inside an anemone or daisy template.

Dependencies

  • None.

Conflicts/Gaps

  • The theorem classifies the combinatorial shape of crossing exact k-separations, but does not bound how many petals such a flower may have in the relevant Quantum Tanner parity-check matroids.
  • It says nothing about sequential separations, nor about cuts of varying order.
  • Therefore it narrows the fixed-order obstruction to two templates, but does not yet exclude those templates from the Conjecture 3 setting.

Sources

  • 10.1016/j.aam.2007.05.004