Canonical Tree Distinguishes All Matroid Tangles

Claim/Theorem

Every finite matroid has a canonical tree decomposition, invariant under automorphisms of the matroid, that efficiently distinguishes all distinguishable tangles.

Equivalently, all matroid tangles can be separated simultaneously by a single automorphism-invariant nested set of separations. So whenever the Conjecture 3 frontier needs a symmetry-respecting decomposition adapted to many tangles at once, such a canonical choice already exists.

Dependencies

• None.

Conflicts/Gaps

• This theorem distinguishes different tangles from each other. It does not classify all low-order separations internal to one fixed tangle.
• The statement is qualitative and canonical; by itself it does not produce a lower bound on balanced cut rank or syndrome-extraction depth.
• For Conjecture 3, its main use is structural: if Quantum Tanner parity-check matroids have strong automorphisms coming from the left-right Cayley geometry, then any tangle-adapted decomposition can be required to respect those symmetries.

Sources

• 10.1007/s00493-017-3595-y