Decomposition-Concentration Branch Is Demoted Before Subtree-Union Theorem

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[dense-large-rank-lean-bag-yields-dense-tangle-breadth.md]], [[lean-linked-decomposition-route-stops-before-dense-bag-rank-concentration.md]], [[every-matroid-admits-optimal-lean-tree-decomposition.md]], [[linked-branch-decomposition-exists-at-optimal-width.md]], and [[lean-matroid-bag-gives-rank-connected-set.md]].

The final still-plausible decomposition-concentration variant on the current graph was:

replace one dense bag by a bounded connected cluster of adjacent bags in an optimal lean or linked decomposition, and prove that this cluster can be merged source-faithfully into a dense linear k-connected set in the original matroid.

The current sourced package still does not imply any theorem of the following kind for the target original qubit parity-check matroids M_n on qubit sets Q_n:

\[ H_{\mathrm{cluster}}^{\beta}: \quad \text{there exists a bounded connected subtree } S_n \]

of an optimal lean or linked decomposition such that some source-faithful merger of

\[ U_n:=\tau_n^{-1}(V(S_n)) \]

produces a set Z_n\subseteq E(M_n) with

\[ \lambda_{M_n}(A)\ge \min\{|A\cap Z_n|,\ |Z_n-A|,\ k_n-1\} \]

for all A\subseteq E(M_n), where

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

and, if needed,

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

Equivalently, no currently sourced theorem converts a bounded connected cluster of adjacent bags into the kind of dense linear k_n-connected set that would close the canonical dense-tangle-breadth route.

The obstruction is exact.

1. Erde's lean theorem is bag-local or path-local, not subtree-union local.

   On the graph side, Erde recalls Thomas' lean condition:

   • for t=t', the bags are controlled by their external connectivity;
   • for t\ne t', one controls linkage between two chosen sets Z_1\subseteq V_t and Z_2\subseteq V_{t'} unless there is a small separator on the path tTt'.

   This is exactly the source of [[lean-matroid-bag-gives-rank-connected-set.md]] when one specializes to the same bag. But no theorem in the sourced package upgrades these statements to a connectivity inequality for the union of all bags in a connected subtree, or for any basis of that union.

2. Hlineny--Whittle node-width does not localize to unions of adjacent bags.

   The node-width of a matroid tree-decomposition is

   \[ w(t)=\sum_{i=1}^d r_M(E-F_i)-(d-1)r(M), \]

   equivalently rank minus a sum of branch rank defects.

   This quantity is attached to one node and its complementary branches. It does not provide a sourced lower bound on the rank of

   \[ \tau^{-1}(V(S)) \]

   for a connected subtree S, nor any merger law turning several adjacent bags into one internally connected set.

3. Linked branch decompositions remain edge-path objects.

   Geelen--Gerards--Whittle's linked branch decomposition theorem controls minimum edge width along a path between two displayed edges. It does not define bags, and no currently sourced theorem converts linked displayed cuts along a bounded connected tree region into a dense connected set on the underlying elements.

Therefore the exact remaining decomposition-theoretic theorem would have to be genuinely new:

a source-faithful subtree-union theorem that converts bounded connected regions of an optimal lean or linked decomposition into dense original-matroid connected mass.

Since [[lean-linked-decomposition-route-stops-before-dense-bag-rank-concentration.md]] already shows that single-bag concentration is not currently sourced either, the whole decomposition-concentration branch should now be treated as on-demand support only, not as a default-active subordinate frontier.

In other words, the graph has now exhausted both decomposition-side variants currently visible on source:

\[ \text{single dense bag} \qquad\text{and}\qquad \text{bounded adjacent-bag merger}. \]

Any future reopening of this branch should introduce a genuinely new decomposition-side invariant, not merely re-run width, linkedness, or bag-local rank-connectedness.

Dependencies

• [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
• [[dense-large-rank-lean-bag-yields-dense-tangle-breadth.md]]
• [[lean-linked-decomposition-route-stops-before-dense-bag-rank-concentration.md]]
• [[every-matroid-admits-optimal-lean-tree-decomposition.md]]
• [[linked-branch-decomposition-exists-at-optimal-width.md]]
• [[lean-matroid-bag-gives-rank-connected-set.md]]
• [[good-codes-admit-logarithmic-width-lean-decomposition.md]]

Conflicts/Gaps

• This is a demotion node, not a family counterexample. It does not prove that a subtree-union theorem is false.
• The claim is only that no such theorem is currently sourced on the graph, and that the existing decomposition theorems control the wrong objects for deriving it.
• A future run could reopen this branch only by adding a genuinely new cluster-merger invariant or a theorem bounding the connectivity of subtree unions directly.

Sources

• 10.1016/j.ejc.2006.06.005
• 10.1016/j.jctb.2017.12.001
• 10.1006/jctb.2001.2082