Skip to content

Certificate-Lift Frontier Is Operation Extension Not Cone Duality

Claim/Theorem

After the certificate-level attack in [[f3-fan-certificate-passes-necessary-multimorphism-tests-but-not-operation-lift.md]] and [[fan-cone-certificates-are-not-hidden-vertex-invariants.md]], the Route-D certificate-lift frontier has one exact next theorem target.

The target is no longer:

\[ \text{find any fan-cone separating functional}. \]

That has already been done.

The target is:

\[ \text{turn a fan-cone separating functional into an operation-level invariant}. \]

More concretely, let y be the reconstructed f_3 certificate. Since y has 50 positive and 50 negative entries and passes all modular and pairwise Hamming necessary checks, the sharp next problem is:

Decide whether the signed support of y extends to a conservative Hamming-distance-nonincreasing operation F:\{0,1\}^{50}\to\{0,1\}^{50} whose multimorphism inequality is violated by f_3, or whether y lies outside the cone generated by valid \Gamma_{\mathrm{sub},2} multimorphism inequalities.

Equivalently, solve the weighted-polymorphism decomposition problem:

\[ y\in \operatorname{Cone}\{ \text{valid multimorphism inequality vectors for } \Gamma_{\mathrm{sub},2} \} ? \]

If the answer is yes, then f_3\notin\langle\Gamma_{\mathrm{sub},2}\rangle; by pinning extra size-2 classes to (1,0), the same nonexpressibility result lifts to all connected f_t, t\ge 3.

If the answer is no, then the current certificate is provably cone-specific. The obstruction side would need a different certificate, and the constructive side would remain focused on the nonseparable selected-threshold energy isolated in [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]].

The computational fallback now supports this ranking:

  • for f_3, the current certificate passes the first necessary operation-level checks but has no operation extension yet;
  • for the 6-qubit positive witness, a fan-cone certificate exists despite known hidden-vertex representability, so cone duality alone is insufficient;
  • an unrestricted 3-hidden-variable quadratic-submodular MILP search for f_3 in the coefficient box [-64,64] did not solve within a 90-second CBC limit, so it gives no theorem-level constructive evidence either way.

Therefore the global frontier moved from a broad "certificate lift" prompt to one finite algebraic decision problem: operation extension or weighted-polymorphism decomposition for the reconstructed signed certificate y.

Dependencies

  • [[f3-fan-certificate-passes-necessary-multimorphism-tests-but-not-operation-lift.md]]
  • [[fan-cone-certificates-are-not-hidden-vertex-invariants.md]]
  • [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]]
  • [[route-d-post-fan-cone-next-target-is-fan-certificate-to-polymorphism-lift.md]]
  • [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]]

Conflicts/Gaps

  • This node is a precise continuation target, not a completed obstruction theorem.
  • The 3-hidden-variable MILP timeout is not a mathematical obstruction and should not be promoted to a theorem node.
  • A successful nonseparable selected-threshold construction would supersede the obstruction-side ranking.
  • A successful operation-extension proof would close the ordinary hidden-vertex question for the connected size-2 family by the existing pinning argument.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.1007/s10878-017-0136-y
  • 10.48550/arXiv.2109.14599
  • local computation: docs/project_QEM-QEC/tmp/certificates/f3_certificate_report.json
  • local computation: docs/project_QEM-QEC/tmp/certificates/six_qubit_certificate_report.json
  • local script: docs/project_QEM-QEC/tmp/scripts/route_d_certificate_lift.py