Internally 4-Connected Forces Cut Rank At Least Three¶

Claim/Theorem¶

Let \(C\) be a \(3\)-connected binary linear code that is internally \(4\)-connected, meaning that every \(3\)-separation of \(C\) is minimal.

Then for every partition

\[ I = J \sqcup J^c \]

with

\[ \min\{|J|,|J^c|\}\ge 4, \]

one has

\[ \lambda_C(J)\ge 3. \]

Equivalently, for any stabilizer space \(\mathcal S\) represented by a full-row-rank matrix with kernel code \(C\),

\[ \chi_J(\mathcal S)\ge 3 \]

for every nonminimal cut.

Proof sketch:

Internal \(4\)-connectedness excludes nonminimal exact \(3\)-separations by definition.
\(3\)-connectedness already excludes \(1\)- and \(2\)-separations.
Hence no cut with both sides of size at least 4 can satisfy \(\lambda_C(J)\le 2\).

So low-order exact decompositions are completely ruled out once internal \(4\)-connectedness is known.

This gives only a constant lower bound.
It does not imply anything close to the linear balanced-cut connectivity required by the present Conjecture-3 frontier.
It is therefore best viewed as a clean threshold: after this node, the unresolved regime begins at cut rank 3 and above.