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Six-Qubit Witness Survives All Four-Ary Exact Minors

Claim/Theorem

Let f_{\mathcal S} be the 6-ary Boolean cut-rank function from [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]], coming from the stabilizer matrix

\[ H= \begin{pmatrix} 1&0&0&1&1&1\\ 0&1&0&0&0&1\\ 0&0&1&0&0&1 \end{pmatrix}. \]

Consider any 4-ary minor of f_{\mathcal S} obtained as follows:

  1. choose any 4 visible coordinates out of the 6,
  2. for each of the remaining 2 coordinates, either
  3. fix it to 0,
  4. fix it to 1, or
  5. minimize over that coordinate.

Then every such 4-ary minor satisfies Živný-Cohen-Jeavons condition Sep from Definition 21 / Theorem 22 (p. 3356), and hence every such minor lies in \langle \Gamma_{\mathrm{sub},2}\rangle.

Equivalently: no exact 4-ary obstruction obtainable from the standard expressive-power closure operations of pinning and minimization separates the 6-qubit witness from hidden-vertex graph-cut representability.

This is a derived exact finite theorem.

  1. There are \binom{6}{4}=15 choices of visible coordinate set.
  2. For each such choice, the two hidden coordinates have 3^2=9 status patterns in {0,1,\min}^2.
  3. Hence there are 135 candidate 4-ary minors to test.
  4. For each minor, compute its Boolean polynomial coefficients by Möbius inversion.
  5. For each of the six ordered Sep inequalities from Theorem 22, check exact coefficient nonpositivity.
  6. All 135 minors satisfy all six Sep inequalities.

By Theorem 22, every one of these 4-ary minors is therefore expressible by binary submodular functions with auxiliary variables.

Consequences for the current frontier:

  • the 6-qubit fan-cone obstruction from [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]] is genuinely higher-arity than any exact 4-ary reduction currently available on disk;
  • any proof that this witness is not ordinarily hidden-vertex graph-cut representable must use a genuinely higher-arity obstruction than the exact arity-4 Sep/F_{\mathrm{sep}} theorem;
  • conversely, the present evidence still falls short of a positive theorem, because surviving all 4-ary exact minors does not imply expressibility of the original 6-ary function.

Dependencies

  • [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]]
  • [[four-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]]
  • [[cross-cut-stabilizer-rank-rank-formula.md]]

Conflicts/Gaps

  • This node does not prove ordinary hidden-vertex representability for the 6-qubit witness. It proves only that the known exact 4-ary criterion cannot refute it, even after pinning and minimization.
  • The result is derived computation rather than a literature theorem stated verbatim.
  • The argument still lives inside auxiliary-variable expressibility and does not address routing-style CD(T_n,G) semantics.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.48550/arXiv.2109.14599