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Route D Operation Extension Frontier Now Reduces To Full Rank-Order Or New Generator

Claim/Theorem

After the operation-extension and fallback tests, the Route-D certificate-lift frontier has been compressed to one exact obstruction-side target and one constructive escape.

The obstruction-side target is:

solve the full rank-preserving order-preserving extension instance for the f_3 certificate support, or replace y by an integer certificate whose signed support yields a valid higher-arity conservative Hamming-nonincreasing operation inequality.

Equivalently, produce an explicit all-arity weighted-polymorphism generator or conic decomposition beyond the ordinary meet/join pool.

The current run rules out several weaker continuations:

  • another fan-cone separation is irrelevant, by [[fan-cone-certificates-are-not-hidden-vertex-invariants.md]];
  • another pairwise Hamming check is too weak, by [[f3-certificate-passes-pairwise-rank-order-extension-but-not-full-closure.md]];
  • ordinary meet/join submodularity inequalities are too weak, by [[strict-hamming-replacement-and-meet-join-decomposition-do-not-close-certificate-lift.md]];
  • restricted coordinate-permutation or nested canonical extensions are too narrow, by the operation-extension report.

The constructive counter-route remains exactly the one already isolated in [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]]: an unrestricted nonseparable hidden energy coupling u and all parallel classes before the two threshold events become separately exposed.

No currently formalized smaller nonseparable subclass becomes live again:

  • separable selected-threshold constructions reduce to the selected-OR obstruction;
  • fixed-prefix recursion fails at transition and readout submodularity;
  • modular-plus-fan closure is already separated;
  • low-budget 1/2 auxiliary searches and the first symmetric or visible-symmetric 3/4 hidden ansatze are already ruled out on the graph;
  • the previous unrestricted 3-hidden timeout is not theorem-level evidence.

Thus the next theorem-level task is not a broad construction search. It is one of:

  1. solve the full rank-order lattice-extension instance;
  2. derive a higher-arity weighted-polymorphism generator class strong enough to decompose a replacement certificate;
  3. prove a normal-form theorem reducing every nonseparable hidden realization of f_t to one of the already failed patterns, or construct an explicit unrestricted global hidden energy.

Dependencies

  • [[f3-certificate-passes-pairwise-rank-order-extension-but-not-full-closure.md]]
  • [[strict-hamming-replacement-and-meet-join-decomposition-do-not-close-certificate-lift.md]]
  • [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]]
  • [[certificate-lift-frontier-is-operation-extension-not-cone-duality.md]]

Conflicts/Gaps

  • This node is a frontier-compression status node, not a nonexpressibility theorem.
  • The global hidden-energy constructive route remains open at unrestricted auxiliary budget.
  • The exact full rank-order extension instance may require a stronger SAT/ILP encoding, Hall-type obstruction, or orbit quotient before it becomes computationally decidable.

Sources

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