Tangle-Concentration Spine Stops Before Order-To-Density Theorem

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[high-tangle-order-gives-large-connected-set.md]], [[high-tangle-order-gives-large-tangle-independent-set.md]], [[tangle-independent-set-gives-connected-set.md]], [[sufficiently-dense-k-connected-set-yields-dense-tangle-breadth.md]], [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]], [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]], and [[tangle-order-equals-branchwidth.md]].

At the current graph state, the surviving canonical intrinsic route after all demotions is the tangle-concentration spine:

try to upgrade linear-order tangles in the original qubit parity-check matroid into dense original-matroid connected mass strong enough to trigger dense tangle breadth.

Among the currently loaded candidates on that spine, the strongest one is the direct order-to-density mechanism. The other two are already blocked earlier:

1. The weakly 4-connected minor route is weaker, because [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]] already shows that no inverse minor-lift theorem is currently available.

2. The robust-structure route is weaker, because [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]] already shows that no robust tangle orientation / balanced-cut classification theorem is currently available in the original matroid.

So the strongest remaining candidate is:

\[ H_{\mathrm{ord}\to\mathrm{dense}}(\beta): \quad \text{linear tangle order in }M_n \Longrightarrow \text{a }k_n\text{-connected set }Z_n \]

with

\[ |Z_n|>(1-\beta)|Q_n|+k_n-2 \]

and, if needed,

\[ |Z_n|\ge 3k_n-5. \]

However, the currently sourced order-theoretic theorems still stop far before this target.

By [[high-tangle-order-gives-large-connected-set.md]], if M has a tangle of order \theta, then the connected-set size m currently obtainable from source satisfies

\[ \theta \ge 20m-13. \]

Equivalently, the strongest connected-set size guaranteed by the current theorem package is

\[ m \le \left\lfloor \frac{\theta+13}{20}\right\rfloor. \]

The same 20-loss is already present one step earlier in [[high-tangle-order-gives-large-tangle-independent-set.md]], and [[tangle-independent-set-gives-connected-set.md]] does not improve that loss; it only converts the resulting tangle-independent set into a connected set.

Now compare this with the density threshold needed by [[sufficiently-dense-k-connected-set-yields-dense-tangle-breadth.md]]. Any connected set Z capable of closing the canonical breadth route must satisfy, for some k\ge 3,

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

In particular,

\[ |Z|>(1-\beta)|E(M)|+1. \]

Since 0<\beta\le 1/2, this forces

\[ |Z|>\frac{|E(M)|}{2}. \]

So any closing order-to-density theorem must produce connected mass on the scale of a dense linear fraction of the whole ground set.

By contrast, the current order-to-connected-set theorem can yield at most a fixed 1/20 fraction of the tangle order. And tangle order itself cannot exceed the ambient element scale: by [[tangle-order-equals-branchwidth.md]], it equals branchwidth, while every displayed cut value is a connectivity value on the n:=|E(M)| element ground set, so \theta=O(n). Consequently the present theorem family yields at best

\[ m=O(n/20), \]

which is asymptotically incompatible with the dense threshold >(1-\beta)n required for dense breadth when \beta\le 1/2.

Therefore the exact theorem-level gap on the surviving tangle-concentration spine is now sharp:

the graph lacks a genuine order-to-density concentration theorem in the original matroid. Current sourced theorems extract only O(\theta) connected mass with a fixed 20-to-1 loss, and that can never reach the dense linear regime needed by the canonical breadth route.

So the active frontier is no longer “get any connected set from high tangle order.” That part is already done. The missing theorem is specifically a density amplification theorem from order to dense connected mass, or an equally strong direct theorem from order to dense tangle breadth.

Dependencies

• [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
• [[high-tangle-order-gives-large-connected-set.md]]
• [[high-tangle-order-gives-large-tangle-independent-set.md]]
• [[tangle-independent-set-gives-connected-set.md]]
• [[sufficiently-dense-k-connected-set-yields-dense-tangle-breadth.md]]
• [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]]
• [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]]
• [[tangle-order-equals-branchwidth.md]]

Conflicts/Gaps

• This node does not prove that H_{\mathrm{ord}\to\mathrm{dense}}(\beta) is false. It proves only that no such theorem is currently sourced on the graph.
• The argument isolates the strongest surviving candidate on the present spine. A future source-grounded theorem could still bypass this obstruction by proving dense tangle breadth directly from an invariant stronger than order alone.
• The upper bound \theta=O(n) is an elementary inference from branchwidth as a maximum cut value on an n-element ground set, not a new external source theorem.

Sources

• 10.1016/j.jctb.2014.12.003
• 10.37236/12467
• 10.1016/j.jctb.2007.10.008