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Coordinate-Zero Toggle Is Not Single Interface Transport

Claim/Theorem

The coordinate-0 compression of the f_3 operation-lift problem does not reduce to a one-commodity or one-cut transport problem across the generator A_0.

The exact toggle pairing is:

\[ B=P\oplus e_0. \]

Equivalently, for each positive pattern \alpha\in P, the paired negative label is \beta=\alpha\oplus e_0, and the boundary constraints are:

\[ h_{\alpha\oplus e_0}(A_0)=1-\alpha_0, \]

while

\[ h_{\alpha\oplus e_0}(A_j)=\alpha_j \]

for j=1,\ldots,6.

The structural reason this is not pure transport is that the six preserved generator coordinates all lie in the A_0-incomparable slice of L_P, but each label has its own prescribed 0/1 values on those same-slice generators. A model that only transports unit mass from one side of the A_0 cut loses the six-bit tail constraints.

The fresh toggle audit gives the exact finite data:

  • positive patterns with coordinate 0 equal to 0: 25;
  • positive patterns with coordinate 0 equal to 1: 25;
  • distinct six-bit tails among positive patterns: 50;
  • tails with both coordinate-0 values present in P: 0;
  • missing six-bit tails out of 64: 14;
  • L_P elements assigned to the U\subseteq A_0 slice, with equality assigned below: 9,613;
  • L_P elements strictly above A_0: 9,612;
  • L_P elements incomparable with A_0: 2,830,406;
  • cover transitions below_A0 -> incomparable_A0: 9,612;
  • cover transitions incomparable_A0 -> above_A0: 9,612;
  • one cover transition below_A0 -> above_A0.

Thus the coordinate-0 toggle should be treated as a tail-coupled monotone count-decomposition problem, not as a single interface flow.

Dependencies

  • [[f3-count-decomposition-is-coordinate-zero-toggle-boundary.md]]
  • [[monotone-count-decomposition-is-column-generation-not-single-flow.md]]
  • [[full-rank-order-closure-reduces-to-pattern-upset-lattice.md]]

Conflicts/Gaps

  • This node rules out only a coordinate-0-only transport reduction. It does not rule out full column generation, a sparse dual obstruction, or a higher-arity generator basis.
  • The interface counts are structural; they are not an infeasibility certificate for the monotone count-decomposition.

Sources

  • local computation: docs/project_QEM-QEC/tmp/certificates/toggle_branch_price_report.json
  • local script: docs/project_QEM-QEC/tmp/scripts/route_d_toggle_branch_price.py