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Large Binary Tangles Dominate Grid Minors But Grid Domination Stops Before Dense Breadth

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[tangle-concentration-spine-stops-before-order-to-density-theorem.md]], [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]], and [[tangle-order-equals-branchwidth.md]].

For binary matroids there is a currently sourced tangle-side invariant that is genuinely stronger than order alone, but still stops before dense breadth: domination of a large grid minor.

More precisely:

  1. Geelen--Gerards--Whittle prove that for every positive integer k and finite field F there exists an integer \theta(k,F) such that, if an F-representable matroid M has a tangle \mathcal T of order at least \theta(k,F), then \mathcal T dominates a minor N isomorphic to the cycle matroid of the k \times k grid.

    For the intrinsic Conjecture 3 setting this applies to the binary case F = GF(2).

  2. Their domination criterion is concrete. If N=M(G_k) and P_1,\dots,P_k are the vertical grid paths, then \mathcal T dominates N if and only if each path edge-set E(P_i) is independent in the tangle matroid M(\mathcal T).

  3. So, conditional on obtaining large original-matroid tangle order, the strongest currently sourced Route A upgrade is not merely a weakly 4-connected minor. It is the stronger statement that the original tangle already controls a large grid-pattern of independent path-sets in its tangle matroid.

  4. However, the current graph still contains no theorem of the form

    \[ H_{\mathrm{grid}\to\mathrm{dense}}(\beta): \quad \text{if an original-matroid tangle dominates a } k \times k \text{ grid minor with } k=\Omega(|Q_n|), \text{ then } M_n \text{ has dense tangle breadth or a sufficiently dense linear } k_n\text{-connected set.} \]

  5. The obstruction is sharp:

    • grid domination is still minor-side and tangle-matroid-side data, not dense concentration on the original qubit ground set;
    • the domination criterion only gives independence of the path-sets E(P_i) inside M(\mathcal T), not a spanning uniform restriction on more than (1-\beta)|Q_n| elements;
    • the currently loaded lift theorems still move from the original matroid to minors, not from a dominated grid minor back to dense original-matroid connected mass, exactly as in [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]].

Therefore Route A now has one exact missing theorem:

upgrade dominated large-grid structure in a binary tangle to dense original-matroid concentration.

Within the current graph, this is the strongest sourced candidate for an invariant strictly stronger than order alone but weaker than dense breadth. It is also still non-closing.

Dependencies

  • [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
  • [[tangle-concentration-spine-stops-before-order-to-density-theorem.md]]
  • [[weakly-4-connected-minor-route-stops-before-dense-connected-set-lift.md]]
  • [[tangle-order-equals-branchwidth.md]]

Conflicts/Gaps

  • This node does not prove that the target Quantum Tanner / left-right-Cayley family already has linear-order tangles in the original qubit matroid.
  • It does not prove that H_{\mathrm{grid}\to\mathrm{dense}}(\beta) is false; it isolates that theorem as the exact current missing lift.
  • A future source-grounded theorem converting grid domination into dense original-matroid connected mass would immediately reopen Route A.

Sources

  • 10.1016/j.jctb.2007.10.008
  • 10.37236/12467