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Dense Tangle Breadth Is The Canonical Remaining Intrinsic Target

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-forces-balanced-cut-rank.md]], [[dense-k-connected-set-forces-balanced-cut-rank.md]], [[lean-matroid-bag-gives-rank-connected-set.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]].

At the current graph state, the intrinsic balanced-cut-rank frontier collapses to one canonical missing theorem target:

\[ H_{\mathrm{dense}}(\beta): \quad \text{the original qubit parity-check matroid }M_n \text{ has a tangle of order }k_n=\Omega(n) \text{ and breadth }t_n>(1-\beta)|Q_n|+k_n-2. \]

More precisely:

  1. [[dense-tangle-breadth-forces-balanced-cut-rank.md]] already shows that \(H_{\mathrm{dense}}(\beta)\) implies

    \[ \chi_L(\mathcal S_n)=\Omega(n) \]

    on every \(\beta\)-balanced qubit cut, hence the desired interface-state and SWAP-only cut-congestion lower bounds.

  2. The other currently live intrinsic candidates on disk are not independent closing routes at this stage; they are mechanisms that would have to feed into the same dense original-matroid concentration phenomenon:

    • a dense linear-size \(k\)-connected set is the connected-set shadow of dense breadth, by [[tangle-breadth-gives-k-connected-set.md]] and [[dense-k-connected-set-forces-balanced-cut-rank.md]];
    • a dense large-rank lean bag is only a possible generator of such a connected set, because [[lean-matroid-bag-gives-rank-connected-set.md]] converts a large-rank or large independent bag into a rank-connected set but does not supply the required linear concentration;
    • a linear local-block accumulation theorem is only a construction-level mechanism for the same phenomenon, because [[local-quotient-image-span-controls-rank-accumulation.md]] shows that the real missing step is linearly many independent quotient images in the original quotient space \(S/(S_L+S_R)\), i.e. another dense intrinsic concentration statement rather than a separate closure theorem.

  3. The current Quantum Tanner / left-right-Cayley source package still stops strictly before this target:

    • [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]] shows that the sourced family package reaches tester expansion, local agreement, and adjacent LTC structure, but not dense intrinsic connectivity of the original parity-check matroid;
    • [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]] identifies the exact family-contact gap: no currently sourced theorem upgrades those tester-side statements to \(H_{\mathrm{dense}}(\beta)\) or to any subordinate dense original-matroid concentration theorem.

Therefore the exact theorem-level gap is no longer a menu of incomparable intrinsic invariants. Within the current graph, it is one canonical missing upgrade:

prove dense original-matroid concentration for the target family, with dense tangle breadth the cleanest exact formulation currently on disk.

Equivalently, future intrinsic runs should treat the following as subordinate formulations of the same gap, not as separate frontiers:

  • dense \(k\)-connected-set existence;
  • dense large-rank lean-bag concentration;
  • linear accumulation of independent local quotient images.

If a future run proves one of those directly, it should be recorded as a route to \(H_{\mathrm{dense}}(\beta)\)-level closure, not as a distinct primary branch.

Dependencies

  • [[dense-tangle-breadth-forces-balanced-cut-rank.md]]
  • [[tangle-breadth-gives-k-connected-set.md]]
  • [[dense-k-connected-set-forces-balanced-cut-rank.md]]
  • [[lean-matroid-bag-gives-rank-connected-set.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 \(H_{\mathrm{dense}}(\beta)\) for Quantum Tanner codes.
  • It does not claim that every future intrinsic invariant must literally factor through tangle breadth. The claim is only that, within the currently loaded graph, no other invariant is presently supported as an independent theorem-sized closing route.
  • The node is a synthesis obstruction, not a family counterexample.
  • A future source-grounded theorem that produces dense connected mass directly in the original parity-check matroid could rephrase the frontier without passing through breadth language first.

Sources

  • 10.37236/12467
  • 10.1016/j.jctb.2017.12.001
  • 10.1109/FOCS54457.2022.00117
  • 10.1145/3519935.3520024
  • 10.48550/arXiv.2109.14599
  • 10.48550/arXiv.2508.05095