Generic Robust Flower Control Still Stops Before Dense Tangle Breadth¶

Claim/Theorem¶

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[robust-tangle-tree-displays-all-nonsequential-separations.md]], [[every-k-flower-is-anemone-or-daisy.md]], [[k-flower-local-connectivity-classification.md]], [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]], and [[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.md]].

Even if one grants the missing tangle-orientation step for Route B, the generic robust/nonsequential/flower package still stops before dense breadth.

More precisely:

Clark--Whittle show that, once a tangle of order k is robust and the relevant k-separations are already known to be nonsequential with respect to that tangle, those separations are displayed by a single tree up to tangle-equivalence.
Aikin--Oxley then show that every k-flower is an anemone or a daisy, and for n\ge 5 petals its local connectivity data is controlled by integers c,d satisfying

\[ k-1 \ge c \ge d \ge \max\{2c-(k-1),0\}. \]

They also compute the local connectivity between arbitrary unions of petals purely from the arc counts and these parameters.
Most importantly for the current frontier, Aikin--Oxley also prove the converse existence theorem: for every admissible parameter pair (c,d) and every petal count n\ge 3, there exists a matroid with a k-flower having exactly that local-connectivity template.
Therefore generic flower-side control after orientation imposes no density concentration on the underlying ground set. The data
- “all balanced low-rank cuts lie in one robust flower regime”,
- “the regime is an anemone or daisy”, and
- “its local connectivities are given by admissible (k,c,d) data”
still does not force dense tangle breadth, a sufficiently dense k-connected set, or linear balanced cut rank.
So the exact post-orientation missing theorem on Route B is stronger than the current node [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]] alone suggests:

\[ H_{\mathrm{flower}}^{\mathrm{dense}}(\beta): \quad \text{in the target family, a } \beta\text{-balanced sublinear-rank cut cannot persist inside an admissible robust flower template} \]

unless that template already yields dense original-matroid concentration.

Equivalently, once balanced low-rank cuts have been oriented into one robust tangle, the remaining gap is not generic flower classification. It is a family-specific petal-mass exclusion or concentration theorem.

Therefore Route B is now sharper in a stronger sense:

even after the missing orientation theorem H_{\mathrm{ns}}^{\beta}, generic robust flower theory is still too permissive to close the dense-breadth frontier.

Dependencies¶

[[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
[[robust-tangle-tree-displays-all-nonsequential-separations.md]]
[[every-k-flower-is-anemone-or-daisy.md]]
[[k-flower-local-connectivity-classification.md]]
[[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]]
[[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.md]]

Conflicts/Gaps¶

This node does not prove that H_{\mathrm{ns}}^{\beta} is false; it sharpens what would still be missing after that theorem.
It does not show that the target Quantum Tanner / left-right-Cayley family realizes any such flower obstruction. The point is that generic flower theory by itself does not rule the obstruction out.
A future family-specific theorem excluding admissible robust flower templates on hardware-balanced cuts would reopen Route B.

Sources¶

10.1016/j.jctb.2013.03.002
10.1016/j.aam.2007.05.004