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Coordinate Toggle Finite Branch Demoted To Computational Frontier

Claim/Theorem

The coordinate-0 toggle finite branch is now demoted from the default active Route-D frontier to appendix/computational frontier evidence.

This is not a disproof of the coordinate-toggle operation-lift possibility. The branch remains unresolved. The stopping reason is purely computational and finite-instance specific:

  • the corrected dual-direction sparse-pricing implementation did not close the 750-row support;
  • the restricted master still had positive slack, with final objective about 64.8269;
  • final pricing still found unseen positive-dual-activity columns, with maximum final dual activity about 143.3414;
  • therefore no sparse Farkas/Hall certificate is valid from the current run;
  • no feasible 50-label monotone toggle decomposition was found;
  • no complete tail-conditioned toggle-interface 3-sorter theorem was proved or verified.

Consequently, future runs should not treat the coordinate-toggle finite branch as the active proof frontier unless they bring one of the following new ingredients:

  1. resumable large-scale column generation with saved columns, reproducible pricing logs, and a closed positive sparse dual;
  2. a precise complete tail-conditioned toggle-interface sorter operation class, including a validity proof over \Gamma_{\mathrm{sub},2} and a completeness theorem for the coordinate-toggle boundary;
  3. an external valued-CSP or multimorphism theorem that supplies a complete generator basis or dual separation theorem for the relevant conservative Hamming-nonincreasing operations.

Until then, the coordinate-toggle branch should be cited only as appendix/computational frontier evidence supporting the statement that the hidden-variable realization question has been stress-tested beyond fan-cone duality and low-order multimorphism checks, but not settled.

Dependencies

  • [[route-d-toggle-frontier-sharpens-to-sparse-pricing-or-tail-sorter-completeness.md]]
  • [[toggle-branch-price-sparse-dual-formulation.md]]
  • [[tail-conditioned-toggle-interface-3-sorter-is-next-generator-basis-target.md]]
  • [[f3-count-decomposition-is-coordinate-zero-toggle-boundary.md]]

Conflicts/Gaps

  • This node is a graph-maintenance classification, not a new mathematical impossibility theorem.
  • The finite branch remains open in principle.
  • The demotion is conditional on the current implementation state and current source base; a future exact column-generation certificate or complete sorter-basis theorem would reopen it.

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