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Pairwise-Skew Block Statistics Do Not Control Parity Value

Claim/Theorem

Keep the notation of [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]]. Let L \sqcup R = Q be a qubit cut, and consider the represented-matroid block-packing instance on the local quotient blocks W_v(L).

Then pairwise block statistics are too weak to force a large block-packing or linear-matroid-parity value.

More precisely:

  1. Pairwise skewness of 2-dimensional blocks does not imply that they form a direct-sum family. There exist 2-dimensional subspaces

    \[ U,V,W \le \mathbf F_2^4 \]

    such that

    \[ U\cap V = U\cap W = V\cap W = \{0\}, \]

    but

    \[ W \subseteq U+V, \]

    so

    \[ \dim(U+V+W)=4<6=\dim U+\dim V+\dim W. \]

    Thus the maximum direct-sum packing weight on {U,V,W} is only 4, not 6.

  2. Consequently, no criterion that depends only on pairwise block intersections can force linear \nu_H(L). In particular, the following are all insufficient by themselves:

    • the pairwise-intersection graph of the blocks;
    • counts of pairwise-skew blocks;
    • bounded multiplicity of 1-dimensional quotient directions.

    The counterexample above has the strongest possible pairwise behavior:

    • the pairwise-intersection graph is empty;
    • every quotient line is contained in at most one block;
    • yet the full block family is not direct-sum.

  3. Therefore the exact missing invariant on the represented block instance is not pairwise but higher-order:

    \[ H_{\mathrm{union}}(\beta): \]

    for every \beta-balanced cut L, one has a construction-controlled lower bound on the ranks of unions of quotient blocks strong enough to force a direct-sum block family of total weight \Omega(|Q|).

    Equivalently, what is still missing is a Hall/Rado-type union-rank expansion statement for the block family, not another pairwise packing proxy.

  4. The current explicit Quantum Tanner low-cut data fit this boundary exactly:

    • D_6 does admit a saturating family of six 2-dimensional blocks on its current low-lambda witness, by [[nu-saturation-yields-adapted-triangular-basis.md]];
    • D_4 and D_8 already require mixed 1- and 2-dimensional saturating families on their current low-lambda witnesses, again by that node.

    So a pure criterion phrased only in terms of pairwise-skew 2-blocks is already too narrow for the frontier, even before one asks for a family theorem.

Concrete counterexample:

\[ U=\langle e_1,e_2\rangle,\qquad V=\langle e_3,e_4\rangle,\qquad W=\langle e_1+e_3,\ e_2+e_4\rangle \le \mathbf F_2^4. \]

Then:

  • U\cap V=\{0\} is immediate.

  • If x\in U\cap W, then

    \[ x=a e_1 + b e_2 = c(e_1+e_3)+d(e_2+e_4). \]

    Comparing e_3 and e_4 coordinates gives c=d=0, hence x=0. So U\cap W=\{0\}. - The same argument gives V\cap W=\{0\}. - But every vector of W is already a sum of one vector from U and one from V, so

    \[ W\subseteq U+V. \]

Hence the three blocks are pairwise skew but not globally direct-sum.

This isolates the parity frontier more sharply than the previous node:

  • the represented-matroid block-packing formulation is correct;
  • pairwise-skew or bounded-line-multiplicity heuristics are not enough;
  • the genuinely missing object is a higher-order union-rank expansion rule on block families.

Dependencies

  • [[exact-optimal-quotient-family-problem-reduces-to-linear-matroid-block-packing.md]]
  • [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]
  • [[nu-saturation-yields-adapted-triangular-basis.md]]
  • [[local-quotient-image-span-controls-rank-accumulation.md]]

Conflicts/Gaps

  • This node is an obstruction to pairwise criteria, not a lower-bound theorem for \nu_H(L).
  • It does not prove that the explicit Quantum Tanner quotient-block instances actually realize the same higher-order concentration mechanism; it proves only that current pairwise statistics cannot rule it out.
  • The node therefore sharpens the missing invariant but does not yet produce a family-level quantitative bound.
  • What remains missing is a source-grounded or construction-grounded union-rank criterion on the actual Quantum Tanner quotient blocks.

Sources

  • 10.48550/arXiv.2206.07571
  • 10.48550/arXiv.2508.05095
  • 10.1016/0095-8956(80)90066-0