k-Flower Local Connectivity Classification

Claim/Theorem

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

Then:

1. If

\[ n\ge 5, \]

there exist integers \(c,d\) satisfying

\[ k-1\ge c\ge d\ge \max\{2c-(k-1),0\} \]

such that the local connectivity between distinct petals is

\[ u(P_i,P_j)= \begin{cases} c & \text{if }P_i,P_j\text{ are consecutive},\\ d & \text{otherwise}. \end{cases} \]

Moreover, the flower is an anemone exactly when \(c=d\).

1. If

\[ n=4, \]

there exist integers \(c,d_1,d_2\) with

\[ k-1\ge c\ge d_1\ge d_2\ge \max\{2c-(k-1),0\} \]

such that consecutive petals have local connectivity \(c\) and the opposite-petal local connectivities are \(\{d_1,d_2\}\).

So once one is inside the fixed-order crossing-separation regime, the entire local geometry is controlled by only a few integer parameters.

For Conjecture 3, this is the sharpest current fixed-order reduction on the flower side: excluding many low-rank balanced cuts can be reduced to excluding a small finite-parameter family of petal-local-connectivity templates.

Dependencies

• [[every-k-flower-is-anemone-or-daisy.md]]

Conflicts/Gaps

• This theorem is still purely structural. It does not itself convert expansion, local testability, or Cayley symmetry into a contradiction with those templates.
• It treats fixed k. The conjecture still allows the possibility of slowly growing cut rank unless one proves a stronger exclusion theorem.
• The translation from hardware-balanced cuts to petals of a single flower remains an additional theorem step.

Sources

• 10.1016/j.aam.2007.05.004