Weakly 4-Connected Minor Route Stops Before Dense Connected-Set Lift¶

Claim/Theorem¶

Keep the notation of [[dense-k-connected-set-forces-balanced-cut-rank.md]], [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]], [[high-tangle-order-gives-large-connected-set.md]], [[large-k-connected-set-persists-in-weakly-4-connected-minor.md]], [[robust-tangle-tree-displays-all-nonsequential-separations.md]], [[every-k-flower-is-anemone-or-daisy.md]], [[k-flower-local-connectivity-classification.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].

Fix the dense connected-set mechanism:

obtain a large connected set in a weakly 4-connected minor of the original qubit parity-check matroid, then lift it back to a dense linear-size k-connected set in the original matroid.

At the current graph state, this mechanism stops before the lifting step.

More precisely:

[[dense-k-connected-set-forces-balanced-cut-rank.md]] already shows that a dense k-connected set in the original qubit matroid would close the balanced-cut-rank route.
The currently available minor theorems only move in the wrong direction:
- [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]] is a one-way passage from width in an original matroid to width in a weakly 4-connected minor;
- [[high-tangle-order-gives-large-connected-set.md]] gives a connected set of size \Omega(\operatorname{bw}) only after one is already inside a 3-connected matroid;
- [[large-k-connected-set-persists-in-weakly-4-connected-minor.md]] again goes one way, from an already-existing k-connected set in the original matroid to a minor carrying the same set.
No currently loaded theorem lifts connected mass back from a minor to the original matroid. In particular, the graph contains no statement of the following type:

\[ H_{\mathrm{lift}}^{\mathrm{conn}}(\beta): \]

if a weakly 4-connected minor N\preccurlyeq M contains a k-connected set Z of size t, with t large enough to force balanced cut rank in N, then M itself contains a k'-connected set Z' with

\[ |Z'|>(1-\beta)|E(M)|+k'-2 \]

or at least every \beta-balanced cut of M already has \lambda_M(L)=\Omega(|E(M)|).
The present low-order structure theorems do not supply this inverse lift:
- [[robust-tangle-tree-displays-all-nonsequential-separations.md]] organizes nonsequential low-order separations only after robustness is known in the original matroid, and only up to tangle-equivalence;
- [[every-k-flower-is-anemone-or-daisy.md]] and [[k-flower-local-connectivity-classification.md]] classify fixed-order crossing patterns already present in the original matroid.
These theorems normalize low-order obstruction templates inside M; they do not reconstruct deleted or contracted elements from a minor, and they do not force those elements to join a dense connected set when passing back to M.
Therefore the current source-grounded Quantum Tanner / left-right-Cayley package still stops strictly before this dense connected-set mechanism:
- [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]] already shows that no current family theorem reaches dense intrinsic connectivity in the original qubit matroid at all;
- even if one were handed a favorable weakly 4-connected minor, the graph still lacks the inverse minor-lift theorem needed to convert that minor-level connected mass into a dense k-connected set in the original parity-check matroid.

Hence the exact theorem-level gap for this mechanism is:

an inverse minor-lift theorem for dense connected sets is missing, and none of the current robustness or flower-classification nodes bridges that gap.

So this route is not presently a closing theorem path. It is a boundary obstruction inside the dense connected-set program.

Dependencies¶

[[dense-k-connected-set-forces-balanced-cut-rank.md]]
[[large-k-connected-set-gives-balanced-cut-rank.md]]
[[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]]
[[high-tangle-order-gives-large-connected-set.md]]
[[large-k-connected-set-persists-in-weakly-4-connected-minor.md]]
[[robust-tangle-tree-displays-all-nonsequential-separations.md]]
[[every-k-flower-is-anemone-or-daisy.md]]
[[k-flower-local-connectivity-classification.md]]
[[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]]
[[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]]

Conflicts/Gaps¶

This node does not prove that inverse lifting is impossible; it proves only that no such theorem is currently on the graph.
It also does not prove that the target family presently yields the required minor-level connected set on a source-grounded qubit-matroid route. The obstruction is sharper: even that would not finish the argument with the current graph state.
A future theorem controlling how deletion and contraction affect dense connected sets in binary matroids could reopen this mechanism.

Sources¶

10.37236/12467
10.1016/j.jctb.2014.12.003
10.1016/j.jctb.2007.10.008
10.1016/j.jctb.2013.03.002
10.1016/j.aam.2007.05.004
10.1109/FOCS54457.2022.00117
10.1145/3519935.3520024