Source Package Stops Before Robust High-Order Original-Matroid Tangles¶

Claim/Theorem¶

Keep the notation of [[tangle-order-equals-branchwidth.md]], [[robust-tangle-tree-displays-all-nonsequential-separations.md]], [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]], [[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.md]], [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].

Fix the intrinsic mechanism:

the relevant high-order tangles in the original qubit parity-check matroid are robust.

At the current graph state, this mechanism stops before robustness itself, because the sourced target-family package does not yet produce the relevant high-order tangles in the original qubit matroid to begin with.

More precisely:

By [[tangle-order-equals-branchwidth.md]], proving a robust tangle of order \Omega(|Q_n|) in the original qubit parity-check matroid M_n first requires proving that M_n has a tangle of order \Omega(|Q_n|), equivalently branchwidth \Omega(|Q_n|).

So any robustness route factors through the antecedent theorem

\[ H_{\mathrm{tan}}^{\Omega}: \quad M_n \text{ has a tangle of order } \Omega(|Q_n|). \]
The current sourced Quantum Tanner / left-right-Cayley package does not supply H_{\mathrm{tan}}^{\Omega} in the original qubit matroid.

As recorded in [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]] and [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]], the available family theorems reach tester expansion, local agreement, local testability, and chosen-presentation expander structure only. They do not produce any exact branchwidth, tangle-order, or dense intrinsic connectivity theorem for the original qubit parity-check matroid itself.
Therefore the current package cannot yet prove the stronger robustness statement

\[ H_{\mathrm{rob}}^{\Omega}: \quad M_n \text{ has a robust tangle } \mathcal T_n \text{ of order } \Omega(|Q_n|). \]

Robustness is not the first missing theorem on this route. The route already fails one level earlier at the existence of a source-grounded linear-order original-matroid tangle.
The external near-bridge literature does not repair this gap.

By [[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.md]], Clark--Whittle and Clark assume robustness rather than deriving it, while the Brettell--Johnson--O'Brien--Semple--Whittle line preserves order and breadth only after passing to weakly 4-connected minors. So the best external structural theorems still do not provide H_{\mathrm{tan}}^{\Omega} or H_{\mathrm{rob}}^{\Omega} for the original qubit matroid of the target family.
Consequently, the exact theorem-level gap before robustness is:

a source-grounded theorem producing linear-order tangles, equivalently linear branchwidth, in the original qubit parity-check matroid of the target family.

Only after that would a genuine robustness theorem become the next live prerequisite for the Clark--Whittle route.

So the present intrinsic frontier is sharper than the previous tangle-orientation obstruction alone suggested. The active gap is not merely “balanced low-rank cuts are not yet known to be nonsequential with respect to a robust tangle.” It is earlier:

no source-grounded theorem yet identifies the required linear-order tangles in the original qubit matroid;
therefore no source-grounded theorem can yet prove those tangles robust;
therefore the Clark--Whittle structural regime remains doubly conditional for the target family.

Dependencies¶

[[tangle-order-equals-branchwidth.md]]
[[robust-tangle-tree-displays-all-nonsequential-separations.md]]
[[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
[[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]]
[[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.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 the target family lacks linear-order tangles or robust high-order tangles. It isolates only the absence of a source-grounded theorem currently on the graph.
It does not supersede [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]]; rather, it sharpens that node by identifying an earlier missing prerequisite.
It also does not claim that linear branchwidth is necessary for every conceivable intrinsic route. The claim is only about the robustness-of-high-order-tangles route fixed in this run.

Sources¶

10.1016/j.jctb.2007.10.008
10.1016/j.jctb.2013.03.002
10.37236/12467
10.1109/FOCS54457.2022.00117
10.1145/3519935.3520024
10.48550/arXiv.1605.06139