Current Sourced Classical Auxiliary Semantics Still Add No New Compiler Meaning Beyond CDsub¶

Claim/Theorem¶

Keep the notation of [[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]], [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]], [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]], [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]], and [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]].

On the current connected family stock of the graph, the sourced classical auxiliary semantics beyond the canonical submodular lower bound CD_{\mathrm{sub}} still do not produce new compiler meaning.

More precisely:

[[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]] already gives the theorem-level compiler-native lower bound from the canonical object

\[ T_{\mathrm{sub}}(\mathcal S)=(Q,\lambda_{\mathcal S}). \]
Exact terminal classical semantics fail on connected examples and families already on disk:
- a connected 5-qubit stabilizer witness, by [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]];
- an infinite connected threshold-lift family, by [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]];
- an infinite connected odd size-2 multi-parallel subfamily beyond the first connected \chi\ge 3 boundary, by [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]].
The strongest sourced nonterminal Boolean-network generalization also does not enlarge the positive semantics class on this boundary: by [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]], Iwamasa-style generalized Boolean network representability collapses back to ordinary hidden-vertex expressibility.

Therefore, on the current source base, every sourced classical auxiliary semantics beyond CD_{\mathrm{sub}} falls into one of two exhausted buckets:

terminal-side semantics, which already fail on connected families now present on the graph; or
nonterminal Boolean-network semantics, which add no new positive power beyond bare hidden-vertex algebraic representability.

So the remaining compiler-native semantic gap is now exact:

either ordinary hidden-vertex expressibility itself must acquire a new compiler-meaningful interpretation, or one must move to a genuinely different semantics class not yet supported by the current Route-D source package.

This is a broader semantic-compression statement than the earlier terminal-side nodes alone.

Dependencies¶

[[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]]
[[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]]
[[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]]
[[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]]
[[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]]
[[route-d-terminal-side-semantics-now-stop-before-nonterminal-auxiliary-frontier.md]]

Conflicts/Gaps¶

This node does not prove that ordinary hidden-vertex expressibility can never gain compiler meaning.
It proves only that the sourced classical semantics currently on the graph do not yet do so.
A future theorem on canonicality, basis-robustness, approximation, or some non-Boolean auxiliary model could still change this stop point.

Sources¶

10.1007/s10878-017-0136-y
10.1016/j.disc.2016.02.010
10.1016/j.dam.2009.07.001
10.48550/arXiv.2109.14599