Proof of `Polynomial semisparse blockades for the house`
What this proof establishes
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
Start with the one-block layout and choose a maximal layout with fewer than blocks. Its conserved weight identifies a block containing a polynomial fraction of the graph. The local blockade theorem either gives at least blocks at once or refines the layout to a larger one, which must cross the -block threshold. The bound on exceptional ordered edges then gives the required semisparsity.