Low-Order Template Exclusion Route Stops Before Balanced-Cut Classification¶

Claim/Theorem¶

Keep the notation of [[dense-k-connected-set-forces-balanced-cut-rank.md]], [[robust-tangle-tree-displays-all-nonsequential-separations.md]], [[every-k-flower-is-anemone-or-daisy.md]], [[k-flower-local-connectivity-classification.md]], [[sequential-matroid-has-canonical-left-right-ends.md]], [[normalized-sequential-orderings-have-bounded-end-variation.md]], [[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]], [[well-positioned-nonsequential-3-separation-has-safe-element.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].

Fix the dense connected-set mechanism:

rule out all low-order tree / flower / sequential obstruction templates in the original qubit parity-check matroid, so that balanced low-rank cuts cannot persist and connected mass must densify.

At the current graph state, this mechanism stops before the balanced-cut classification step.

More precisely:

[[dense-k-connected-set-forces-balanced-cut-rank.md]] already shows that if low-rank balanced cuts were excluded in the original qubit matroid, then the dense connected-set route would close.
The present separator-structure nodes are conditional normal-form theorems, not family-contact classification theorems:
- [[robust-tangle-tree-displays-all-nonsequential-separations.md]] applies only after one already has a robust tangle and knows the relevant low-order cuts are nonsequential, and even then only up to tangle-equivalence;
- [[every-k-flower-is-anemone-or-daisy.md]] and [[k-flower-local-connectivity-classification.md]] classify fixed-order crossing separations only after one already knows the cuts lie in one flower regime;
- [[sequential-matroid-has-canonical-left-right-ends.md]] and [[normalized-sequential-orderings-have-bounded-end-variation.md]] apply only after one already knows the matroid or the relevant separator regime is sequential;
- [[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]] and [[well-positioned-nonsequential-3-separation-has-safe-element.md]] apply only in the exact nonsequential 3-separation branch and still require extra gadget-exclusion or well-positionedness hypotheses.
Therefore the graph contains no theorem of the following kind:

\[ H_{\mathrm{class}}^{\beta}: \]

for the target Quantum Tanner / left-right-Cayley family, every \(\beta\)-balanced cut L with sublinear intrinsic rank

\[ \lambda_{M_n}(L)=o(|Q_n|) \]

is forced, in the original qubit parity-check matroid M_n, into one of the known low-order template regimes:
- a nonsequential separation displayed by a robust tangle tree,
- a fixed-order flower with controlled petal parameters,
- or a sequential ordering with canonical ends.
The current source-grounded family package still stops strictly before H_{\mathrm{class}}^{\beta}:
- [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]] shows that the available family theorems reach only tester-side expansion, local agreement, and chosen-presentation structure;
- none of those theorems imply robustness of original-matroid tangles, nonsequentiality of hardware-balanced low-rank cuts, fixed-order control of those cuts, or exclusion of the special gadgets appearing in the exact 3-separation theory.
Consequently, the exact theorem-level gap for this mechanism is not yet exclusion of a particular template. It is earlier:

a balanced-cut classification theorem in the original qubit matroid is missing.

Without such a theorem, the current low-order structure nodes remain a normalization toolbox for already-identified obstruction regimes, not a route that forces dense k-connected mass in the target family.

So the low-order template-exclusion mechanism is not presently a closing theorem path. Its current status is a sharp obstruction: the source package does not yet put balanced low-rank cuts into the template regimes one would need to exclude.

Dependencies¶

[[dense-k-connected-set-forces-balanced-cut-rank.md]]
[[large-k-connected-set-gives-balanced-cut-rank.md]]
[[robust-tangle-tree-displays-all-nonsequential-separations.md]]
[[every-k-flower-is-anemone-or-daisy.md]]
[[k-flower-local-connectivity-classification.md]]
[[sequential-matroid-has-canonical-left-right-ends.md]]
[[normalized-sequential-orderings-have-bounded-end-variation.md]]
[[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]]
[[well-positioned-nonsequential-3-separation-has-safe-element.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 balanced-cut classification theorem H_{\mathrm{class}}^{\beta} is false. It proves only that no such theorem is currently on the graph.
It also does not show that one cannot exclude a specific template family directly in the target codes. The sharper point is that the current source package does not yet place balanced low-rank cuts into any such family.
A future theorem proving robustness plus nonsequentiality, or otherwise classifying all sublinear balanced cuts into one canonical low-order regime, would reopen this mechanism.

Sources¶

10.1016/j.jctb.2013.03.002
10.1016/j.aam.2007.05.004
10.1016/j.ejc.2005.10.005
10.1016/j.aam.2005.01.003
10.1016/j.aam.2010.10.009
10.1109/FOCS54457.2022.00117
10.1145/3519935.3520024