Sufficiently Dense k-Connected Set Yields Dense Tangle Breadth

Claim/Theorem

Keep the notation of [[large-k-connected-set-generates-order-k-tangle.md]], [[dense-tangle-breadth-forces-balanced-cut-rank.md]], [[dense-k-connected-set-forces-balanced-cut-rank.md]], and [[sufficiently-dense-k-connected-set-forces-branchwidth.md]].

Let M be a matroid on a ground set E of size n, fix 0<\beta\le 1/2, and suppose Z\subseteq E is a k-connected set of size

\[ t := |Z| \]

such that

\[ t \ge 3k - 5 \]

and

\[ t > (1-\beta)n + k - 2. \]

Then M has a tangle of order k and breadth at least t. In particular,

\[ \operatorname{breadth}(\mathcal T)\;>\;(1-\beta)n + k - 2, \]

so M satisfies the canonical dense-tangle-breadth hypothesis of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]].

Consequently every \beta-balanced cut L\subseteq E satisfies

\[ \lambda_M(L)\ge k-1. \]

Equivalently, in the stabilizer setting every \beta-balanced qubit cut satisfies

\[ \chi_L(\mathcal S)\ge k-1. \]

Proof.

1. By [[large-k-connected-set-generates-order-k-tangle.md]], the hypotheses k\ge 3 and t\ge 3k-5 give a tangle \mathcal T of order k whose tangle matroid has a spanning uniform restriction on Z. Hence

   \[ \operatorname{breadth}(\mathcal T)\ge t. \]

2. Since t > (1-\beta)n + k - 2, the same tangle satisfies the dense-breadth threshold

   \[ \operatorname{breadth}(\mathcal T) > (1-\beta)n + k - 2. \]

3. Apply [[dense-tangle-breadth-forces-balanced-cut-rank.md]] to conclude that every \beta-balanced cut has

   \[ \lambda_M(L)\ge k-1. \]

This upgrades the dense connected-set route to the canonical breadth language directly: a sufficiently dense k-connected set in the original matroid does not merely imply balanced cut rank or branchwidth, it already yields dense tangle breadth itself.

Special case: the previous branchwidth threshold at \beta=1/3.

If

\[ t > \frac{2n}{3} + k - 2, \]

then the extra condition t\ge 3k-5 is automatic, because n\ge t implies

\[ t > \frac{2t}{3} + k - 2, \]

hence

\[ t > 3k - 6, \]

so t\ge 3k-5.

Therefore the route in [[sufficiently-dense-k-connected-set-forces-branchwidth.md]] actually factors through dense tangle breadth:

\[ \text{sufficiently dense }k\text{-connected set} \Longrightarrow \text{dense tangle breadth} \Longrightarrow \text{balanced cut rank}. \]

Dependencies

• [[large-k-connected-set-generates-order-k-tangle.md]]
• [[dense-tangle-breadth-forces-balanced-cut-rank.md]]
• [[dense-k-connected-set-forces-balanced-cut-rank.md]]
• [[sufficiently-dense-k-connected-set-forces-branchwidth.md]]
• [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]

Conflicts/Gaps

• This is still a conditional intrinsic theorem, not a source theorem for the actual Quantum Tanner / left-right-Cayley family.
• The missing step is now sharper: produce a k-connected set in the original qubit parity-check matroid that is both linearly large and dense enough to cross the hardware balance threshold.
• For general \beta, the theorem keeps the explicit size condition t\ge 3k-5. Only in the \beta=1/3 branchwidth subroute is that condition automatic from the density hypothesis alone.

Sources

• 10.37236/12467
• 10.48550/arXiv.2109.14599