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Route D Toggle Frontier Sharpens To Sparse Pricing Or Tail-Sorter Completeness

Claim/Theorem

The coordinate-0 toggle cycle does not solve or refute the f_3 operation lift, but it sharpens the Route-D frontier.

The strongest supported statement is:

The current f_3 certificate survives the rerun full-rank and monotone-count diagnostics; the coordinate-0 toggle is not a single transport problem; sparse row-supported dual searches reduce exactly to min-weight closure pricing; and the next concrete generator-basis target is the finite class of tail-conditioned toggle-interface 3-sorters.

The rerun data audit confirms:

  • |L_P|=2,849,631;
  • cover relations in L_P: 24,177,188;
  • join-irreducibles: 50;
  • meet-irreducibles: 50;
  • visible coordinate symmetry group order: 3;
  • negative support equals positive support with coordinate 0 toggled;
  • no f_3 interval/Hall violation;
  • no f_3 forced-propagation contradiction;
  • f_3 critical ILP and LP: Optimal;
  • the same pipeline rejects the six-qubit fan-certificate control.

The replacement-certificate route is demoted until a new generator class is available. Bounds 1 through 4 still have feasible continuous strict-Hamming relaxations, the configured integer searches remain unsolved, and the ordinary meet/join decomposition pool remains infeasible.

The constructive hidden-energy route is not promoted. The obstruction route has not failed theorem-level, and no feasible toggle decomposition was found.

Dependencies

  • [[coordinate-zero-toggle-is-not-single-interface-transport.md]]
  • [[toggle-branch-price-sparse-dual-formulation.md]]
  • [[tail-conditioned-toggle-interface-3-sorter-is-next-generator-basis-target.md]]
  • [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]]

Conflicts/Gaps

  • The global frontier moved by sharpening, not by closure.
  • No feasible 50-label toggle decomposition was found.
  • No sparse dual obstruction was certified.
  • The remaining exact bottleneck is to close a positive sparse-pricing dual or prove completeness of the tail-conditioned toggle-interface 3-sorter class.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.1007/s10878-017-0136-y
  • local computation: docs/project_QEM-QEC/tmp/certificates/full_rank_order_extension_report.json
  • local computation: docs/project_QEM-QEC/tmp/certificates/monotone_count_decomposition_report.json
  • 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