Good Codes Have Weakly 4-Connected Log-Branchwidth Minor¶

Claim/Theorem¶

Let \(C\) be an [n,k,d] binary linear code. Then the associated binary matroid has a minor \(N\) that is weakly \(4\)-connected and has branchwidth

\[ \operatorname{bw}(N) \;\in\; \Omega\!\left(\frac{k(d-1)}{n\log n}\right). \]

In particular, every asymptotically good code family has a weakly \(4\)-connected minor with branchwidth Omega(n/log n).

Proof sketch:

\[ \Omega\!\left(\frac{k(d-1)}{n\log n}\right). \]

By [[tangle-order-equals-branchwidth.md]], this gives a tangle of the same order.
By [[large-tangle-yields-weakly-4-connected-minor.md]], that tangle can be concentrated in a weakly \(4\)-connected minor without losing its order.

So the logarithmic intrinsic width of good codes survives after stripping away all low-order decomposition artifacts.

This theorem is about a minor of the associated binary matroid, not directly the original code family or the original stabilizer presentation.
Weakly \(4\)-connected still does not imply linear balanced-cut rank; it only shows the width survives beyond low-order decomposition.
To use this for Conjecture 3, one still needs either:
a minor-monotone obstruction relating weakly \(4\)-connected high-branchwidth minors to large hardware-balanced interface state, or
a proof that the relevant Quantum Tanner parity-check matroids themselves, not just their minors, inherit stronger connectivity.