Skip to content

Local-Block Route Stops Before Cut-Uniform Dense k-Connected Set

Claim/Theorem

Keep the notation of [[sufficiently-dense-k-connected-set-forces-branchwidth.md]], [[dense-k-connected-set-forces-balanced-cut-rank.md]], [[large-k-connected-set-gives-balanced-cut-rank.md]], [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]], [[local-quotient-image-span-controls-rank-accumulation.md]], [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]].

Fix the subordinate intrinsic route:

chosen-presentation local quotient-image / local-block accumulation should yield a sufficiently dense linear \(k\)-connected set in the original qubit parity-check matroid.

At the current graph state, this route stops before the cut-uniform dense connected-set step.

More precisely:

  1. To trigger [[sufficiently-dense-k-connected-set-forces-branchwidth.md]], one needs a single subset \(Z_n \subseteq Q_n\) and an integer \(k_n = \Omega(|Q_n|)\) such that

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

    together with the density threshold

    \[ |Z_n| > \frac{2|Q_n|}{3} + k_n - 2. \]

    So the target object is not merely a family of large-rank balanced cuts. It is one fixed dense carrier set \(Z_n\) whose connectivity inequality holds simultaneously for every subset of qubits.

  2. The current local-block package is indexed cutwise, not uniformly.

    In [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]] and [[local-quotient-image-span-controls-rank-accumulation.md]], every decisive object depends on the chosen cut \(L\):

    • the local crossing mass \(\mu_H(L)\);
    • the quotient base space \(B(L)=S_L+S_R\);
    • the local quotient images \(W_v(L)\);
    • the surviving mass \(\sigma_H(L)\);
    • the packed surrogate \(\nu_H(L)\).

    These objects are designed to study one cut at a time. They do not define a cut-independent subset \(Z_n \subseteq Q_n\) or any cut-uniform connectivity witness.

  3. Even a strong cutwise accumulation theorem would still be weaker than the connected-set conclusion needed here.

    For example, a theorem of the form

    \[ \forall\,L \text{ balanced},\qquad \lambda_{M_n}(L)\ge c\,\mu_H(L)=\Omega(|Q_n|) \]

    would already imply linear balanced cut rank. But it still would not produce:

    • a single dense set \(Z_n\);
    • the universal inequality for all subsets \(A \subseteq Q_n\) from [[large-k-connected-set-gives-balanced-cut-rank.md]];
    • or the density threshold \(|Z_n| > 2|Q_n|/3 + k_n - 2\) needed in [[sufficiently-dense-k-connected-set-forces-branchwidth.md]].

    So cutwise rank accumulation is not by itself a theorem-sized substitute for a dense \(k\)-connected set.

  4. The obstruction is sharper still: the local-block data may change drastically with the cut, so there is no current mechanism forcing one common dense carrier.

    Because \(B(L)\) and hence every \(W_v(L)\) vary with \(L\), a local block can survive in the quotient for one balanced cut and die for another. Therefore:

    • surviving or independent quotient images for one cut do not persist uniformly across cuts;
    • a large span \(\sum_v W_v(L)\) for one cut does not identify a fixed qubit subset controlling all other cuts;
    • abundant local crossing mass says that many neighborhoods are crossed, not that the same qubits form a dense \(k\)-connected set.

  5. Therefore the exact theorem-level gap for this subordinate route is:

    a cut-uniform densification theorem turning chosen-presentation local-block data into one fixed dense \(k\)-connected set in the original qubit matroid.

    Equivalently, one would need a source-grounded theorem producing \(Z_n \subseteq Q_n\) and \(k_n=\Omega(|Q_n|)\) such that:

    • \(|Z_n| > 2|Q_n|/3 + k_n - 2\);
    • the connectivity inequality for \(Z_n\) holds for every \(A \subseteq Q_n\);
    • and this conclusion is derived from the Quantum Tanner local-block / quotient-image structure rather than inserted as an abstract hypothesis.

So the current local-block package does not yet reach the branchwidth reduction node through this route. It remains valuable as cutwise evidence for dense intrinsic connectivity, but it still stops before the cut-uniform dense connected-set object needed to invoke [[sufficiently-dense-k-connected-set-forces-branchwidth.md]].

Dependencies

  • [[sufficiently-dense-k-connected-set-forces-branchwidth.md]]
  • [[dense-k-connected-set-forces-balanced-cut-rank.md]]
  • [[large-k-connected-set-gives-balanced-cut-rank.md]]
  • [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
  • [[local-quotient-image-span-controls-rank-accumulation.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 chosen-presentation local-block route can never yield such a set \(Z_n\). It isolates only the exact missing theorem not yet on the graph.
  • It does not contradict the possibility that a stronger local-block theorem could imply balanced cut rank directly without passing through a dense \(k\)-connected set. The present claim is only about this one subordinate route.
  • It also does not weaken the canonical dense-breadth target. It sharpens why the current local-block package is still only subordinate evidence for that target.

Sources

  • 10.37236/12467
  • 10.48550/arXiv.2109.14599
  • 10.48550/arXiv.0805.2199
  • 10.48550/arXiv.2206.07571
  • 10.48550/arXiv.2508.05095