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Route-D2 Family Lift Now Stops At First Connected Family Beyond Threshold-Lift

Claim/Theorem

Keep the notation of [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]], [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], and [[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]].

After the current D2 family-lift cycle, the witness-level semantic separation has been lifted as far as current sources support on a connected infinite family. The next unresolved D2 frontier is the first connected family beyond threshold-lift.

More precisely:

  1. Candidate route A, the connected threshold-lift family, now succeeds. By [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]], there is an infinite connected binary/stabilizer subclass with:

    • exact hidden-vertex realizability, and
    • failure of every exact terminal routing-style semantics on the physical terminals.

  2. Candidate route B, the multi-parallel-circuit connected family, remains open in the exact way already isolated on the graph:

    • it is the first natural connected \lambda\ge 3 family beyond threshold-lift, by [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]];
    • the selector quotient route fails by [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]] and [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]];
    • the selector obstruction does not lift to the whole family by standard minors, by [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]];
    • the branch-min local-gadget route fails by [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]];
    • the family still survives the exact 4-ary minor toolkit, the direct F_{\mathrm{sep}} test on the size-2 subfamily, the one-hidden 5-ary shadow boundary, and the first bounded-auxiliary symmetric and visible-symmetric searches.

  3. Candidate route C, fixed-arity stabilizer families such as all 4- or 5-qubit stabilizer cut-rank functions, does not currently give a stronger family lift:

    • the 4- and 5-qubit theorems are strong but arity-bounded;
    • they do not produce an infinite connected family theorem beyond the threshold-lift subfamily already achieved in route A.

  4. Therefore the exact best remaining D2 bottleneck is:

    \[ H_{\mathrm{conn}\,\chi\ge 3}^{\mathrm{D2}}: \quad \text{upgrade the semantic separation from threshold-lift to the first connected post-threshold family,} \]

    for example the multi-parallel-circuit connected family or another genuinely connected \lambda\ge 3 subclass.

Equivalently:

  • the family-lift problem is solved on the last connected family with an exact hidden-vertex theorem currently on disk;
  • the remaining D2 question begins exactly where current positive expressibility first leaves threshold-lift and enters connected higher-connectivity structure.

So the Route-D2 frontier has moved, not just sharpened:

the next target is no longer "find any connected family lift." It is "cross the first connected \lambda\ge 3 boundary."

Dependencies

  • [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]]
  • [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]
  • [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]]
  • [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]]
  • [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]]
  • [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]]
  • [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]]
  • [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]]
  • [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]]
  • [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]]
  • [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]]
  • [[visible-symmetric-hidden-distinguished-ansatze-fail-on-m222.md]]
  • [[five-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]]
  • [[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]]

Conflicts/Gaps

  • This node does not prove that the multi-parallel-circuit connected family is or is not hidden-vertex representable.
  • It does not prove a broad impossibility theorem for all nonterminal auxiliary semantics.
  • It records only the exact stop point reached by the current source base after the new threshold-lift family lift.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.1016/j.disc.2016.02.010
  • 10.48550/arXiv.2109.14599