Route-D Semantic Separation Now Dominates The Remaining CD Frontier¶

Claim/Theorem¶

Keep the notation of [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]], [[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]], [[pivot-class-best-incidence-load-still-overestimates-connectivity.md]], [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], and [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]].

After the current Route-D stress test, the strongest remaining compiler-native frontier is no longer the search for a first exact hidden-vertex realization or a stronger fixed-basis impossibility theorem. It is the sharper semantic question of whether any nonterminal auxiliary realization can recover routing-style compiler meaning.

More precisely:

Subroute D1 still stops short in an exact way. The current basis-robust impossibility nodes rule out:
- fixed-basis edge-cut or incidence-load readings by [[fundamental-graph-edge-cuts-are-basis-unstable.md]];
- any incidence-local pivot-robust construction by [[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]] and [[local-incidence-lifts-of-fundamental-graphs-are-not-pivot-robust.md]];
- even best-basis optimization over the whole pivot class by [[pivot-class-best-incidence-load-still-overestimates-connectivity.md]];
- exact terminal nonnegative-hypergraph semantics for some binary examples by [[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]].
But D1 does not rule out arbitrary global auxiliary-variable constructions, because hidden-vertex realizations survive on 5- and 6-qubit stabilizer examples.
Subroute D2 now yields the strongest positive-to-negative separation on disk. By [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]], a connected 5-qubit stabilizer cut-rank function can be exactly hidden-vertex representable while still failing every exact terminal routing-style semantics.
Subroute D3 remains secondary. The strongest currently sourced positive family still stops before a nontrivial connected theorem:
- the threshold-lift and direct-sum class never reaches a connected \lambda\ge 3 core, by [[threshold-lift-plus-direct-sum-class-never-reaches-connectivity-three-core.md]];
- the first connected candidate family is [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]];
- but its current positive extension is blocked by the selector reduction, the branch-min local-gadget obstruction, the exact 4-ary minor survival theorem, the one-hidden obstruction on pinned 5-ary shadows, the 1/2-auxiliary box obstruction, and the failure of the first symmetric and visible-symmetric global ansatz classes.
Therefore the Route-D ranking is now:

\[ D2 \;>\; D1 \;>\; D3. \]
The exact best remaining Route-D bottleneck is consequently:

determine whether any auxiliary-variable realization stronger than terminal routing semantics can still be converted into a compiler-meaningful theorem, or whether hidden-vertex representability itself is already too semantically weak to matter beyond witness-level algebraic packaging.

So the Route-D frontier is sharper than before:

D1 is now a boundary package of terminal and basis-robust impossibilities;
D2 is the canonical remaining semantic frontier;
D3 is demoted to a secondary positive-expressibility search unless a genuinely new connected building block appears.

Dependencies¶

[[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]]
[[fundamental-graph-edge-cuts-are-basis-unstable.md]]
[[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]]
[[local-incidence-lifts-of-fundamental-graphs-are-not-pivot-robust.md]]
[[pivot-class-best-incidence-load-still-overestimates-connectivity.md]]
[[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]]
[[threshold-lift-plus-direct-sum-class-never-reaches-connectivity-three-core.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]]
[[branch-min-route-for-connected-family-fails-at-local-submodularity.md]]
[[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.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]]

Conflicts/Gaps¶

This is a Route-D ranking and stop-point theorem for the current graph state, not a universal impossibility theorem for all auxiliary semantics.
It does not prove that D3 can never succeed. It records only that current sources do not yet support a nontrivial connected positive theorem beyond the threshold-lift boundary.
A future whole-family theorem converting hidden-vertex realizability into routing-style meaning, or ruling such a conversion out, would immediately sharpen this ranking again.

Sources¶

10.1016/j.jctb.2005.03.003
10.1016/j.disc.2016.02.010
10.1016/j.dam.2009.07.001
10.48550/arXiv.2109.14599