Threshold-Lift-Plus-Direct-Sum Class Never Reaches Connectivity-Three Core

Claim/Theorem

The current positive hidden-vertex class on the graph, obtained from circuit-plus-parallel-class threshold-lift pieces and closed under direct sums, does not enter the first genuinely post-gluing regime of connectivity at least 3 on a connected core.

More precisely:

1. every single circuit-plus-parallel-class threshold-lift piece M satisfies

   \[ \lambda_M(L)\le 2 \]

   for every cut L;

2. if M=M_1\oplus\cdots\oplus M_t is any direct sum of such pieces, then any cut with

   \[ \lambda_M(L)\ge 3 \]

   must obtain that value by summing contributions from at least two different connected components;

3. consequently, no connected matroid in this positive class can witness the first post-gluing regime where every sufficiently nontrivial cut satisfies \lambda\ge 3.

   In particular, this class cannot contain any weakly 4-connected matroid with more than one component, and its connected members remain confined to the already-classified low-order interface regime.

   Therefore the first genuinely non-low-order regime lies strictly outside the present threshold-lift-plus-direct-sum theorem. Any further positive theorem must add a new connected building block, not just more direct-sum closure of the old ones.

This is a derived obstruction statement.

1. By [[parallel-extension-of-binary-circuit-gives-threshold-lift-cut-rank.md]], every threshold-lift piece has

   \[ \lambda_M(L)= \mathbf 1[p_L\ge 1,\ q_L<s] + \mathbf 1[q_L\ge 1,\ p_L<m]. \]

   This is a sum of two indicator terms, so

   \[ 0\le \lambda_M(L)\le 2 \]

   for every cut.

2. By [[direct-sums-of-threshold-lift-pieces-stay-hidden-vertex-representable.md]], direct sums satisfy

   \[ \lambda_M(L)=\sum_i \lambda_{M_i}(L\cap E_i). \]

   Hence a value \lambda_M(L)\ge 3 can occur only by adding contributions from multiple summands. It never arises from the internal connectivity of a single threshold-lift component.

3. But the post-gluing regime isolated by [[internally-4-connected-forces-cut-rank-at-least-three.md]] and [[large-tangle-yields-weakly-4-connected-minor.md]] is a connected-core regime: once low-order decompositions are stripped away, one studies connected or nearly indecomposable objects whose nonminimal cuts satisfy \lambda\ge 3.

4. Therefore the present positive class does not overlap that regime except vacuously on tiny ground sets where there are no nonminimal cuts to test.

5. Combined with [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]], this means the asymptotically meaningful high-width core of good binary codes already lies outside the threshold-lift-plus-direct-sum class.

Consequences for the current frontier:

• the new direct-sum theorem is mathematically real, but it does not advance into the first connected high-connectivity regime;
• the next positive step, if any, must produce a genuinely new connected hidden-vertex representable building block with some cuts of rank at least 3;
• the next negative step, if any, should search for a counterexample precisely in that first connected post-gluing regime rather than among more disconnected assemblies of threshold-lift pieces.

Dependencies

• [[parallel-extension-of-binary-circuit-gives-threshold-lift-cut-rank.md]]
• [[direct-sums-of-threshold-lift-pieces-stay-hidden-vertex-representable.md]]
• [[internally-4-connected-forces-cut-rank-at-least-three.md]]
• [[large-tangle-yields-weakly-4-connected-minor.md]]
• [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]]

Conflicts/Gaps

• This node does not prove nonrepresentability for any connected \lambda\ge 3 example. It only shows that the current positive class never reaches that regime.
• The argument is structural and uses the exact formula of the threshold-lift pieces; it does not classify all hidden-vertex representable connected binary matroids of cut-rank >=3.
• The routing-style CD(T_n,G) gap remains completely open even if a new connected hidden-vertex class is found.

Sources

• 10.48550/arXiv.2109.14599
• Kashyap 2008 preprint: A Decomposition Theory for Binary Linear Codes
• 10.1016/j.jctb.2007.10.008
• 10.37236/12467
• 10.48550/arXiv.0805.2199