Route-D Now Reduces To Two Ordinary Hidden-Vertex Methodologies On The First Connected Family¶

Claim/Theorem¶

Keep the notation of [[route-d-now-centers-on-ordinary-hidden-vertex-theorem-at-first-connected-boundary.md]], [[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]], and [[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]].

After the current Route-D closure cycle, the remaining compiler-native frontier is now centered even more narrowly than before.

More precisely:

Terminal-side semantics are already exhausted on the connected family stock currently on the graph.
Broader sourced Boolean-network semantics are already exhausted as a positive escape hatch.
The first connected post-threshold boundary is now compressed to two surviving ordinary hidden-vertex methodology types by [[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]]:
- a deeper expressive-power obstruction than the current Sep/F_{\mathrm{sep}} package; or
- an explicit global hidden-vertex gadget construction with cross-class coupling and possibly growing auxiliary budget.
Therefore the best remaining Route-D bottleneck is no longer a generic semantic question. It is:

decide ordinary hidden-vertex expressibility or nonexpressibility on the connected multi-parallel boundary by one of those two methodology types.

Hence the Route-D subfrontier ranking is now:

\[ \text{ordinary hidden-vertex theorem on the connected multi-parallel family} \;>\; \text{D1 basis-robust semantic strengthening} \;>\; \text{D3 search for a new positive connected subclass}. \]

So the global frontier has sharpened one more time:

all currently sourced classical semantics beyond CD_{\mathrm{sub}} have now been compressed away;
the surviving frontier is a single ordinary hidden-vertex theorem with two remaining methodology templates.

Dependencies¶

[[route-d-now-centers-on-ordinary-hidden-vertex-theorem-at-first-connected-boundary.md]]
[[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]]
[[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]]

Conflicts/Gaps¶

This is a Route-D ranking and stop-point theorem for the current source base.
It does not decide the surviving ordinary hidden-vertex theorem itself.
A future family-level realization, a future stronger expressive-power obstruction, or a genuinely new semantics class would change this ranking again.

Sources¶

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