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F3 Count Decomposition Is Coordinate-Zero Toggle Boundary

Claim/Theorem

For the reconstructed f_3 certificate, the monotone count-decomposition instance from [[route-d-full-rank-order-frontier-is-monotone-count-decomposition.md]] is not an arbitrary 50-label boundary problem. The negative support patterns are exactly the positive support patterns with coordinate 0 toggled.

Let A be the 50 realized positive support patterns and let B be the 50 realized negative support patterns. The computation in monotone_count_decomposition_report.json verifies

\[ B=\{\alpha\oplus e_0:\alpha\in A\}. \]

Therefore, after identifying a negative label \beta with the positive pattern \alpha=\beta\oplus e_0, the required boundary values become

\[ h_\alpha(A_0)=1-\alpha_0, \]

and, for every j=1,\ldots,6,

\[ h_\alpha(A_j)=\alpha_j. \]

Equivalently, the live operation-extension question is:

Can the closure-rank function on the 2,849,631-element upset lattice be decomposed into monotone Boolean label functions that toggle the generator-0 trace and preserve the other six generator traces?

This is a strict compression of the previous formulation. It isolates the obstruction as a single-coordinate generator-boundary toggle, not as an unstructured search over the 50 negative labels.

The same report gives the structural baseline for this compressed instance:

  • L_P has 2,849,631 elements.
  • The Birkhoff join-irreducible poset has 50 elements.
  • The meet-irreducible count is also 50.
  • The closure rank is the unit-weight sum over the join-irreducibles below a lattice element.
  • The largest rank layer occurs at rank 13 with 154,490 elements, while exact lattice width is not certified.
  • The visible coordinate-permutation group preserving the positive pattern set has order 3, with generator-coordinate orbits {0}, {1,4,5}, and {2,3,6}.

Dependencies

  • [[route-d-full-rank-order-frontier-is-monotone-count-decomposition.md]]
  • [[full-rank-order-closure-reduces-to-pattern-upset-lattice.md]]
  • [[conservative-hamming-operation-extension-is-rank-order-lattice-extension.md]]

Conflicts/Gaps

  • This is a structural compression, not a feasible decomposition.
  • The coordinate-0 toggle identity does not by itself produce a conservative Hamming-nonincreasing operation.
  • The remaining problem is still the full monotone count-decomposition over L_P, now with a sharper boundary target.

Sources

  • 10.1016/j.dam.2009.07.001
  • local computation: docs/project_QEM-QEC/tmp/certificates/monotone_count_decomposition_report.json
  • local script: docs/project_QEM-QEC/tmp/scripts/route_d_monotone_count_decomposition.py