Route-D Now Centers On Ordinary Hidden-Vertex Theorem At First Connected Boundary¶

Claim/Theorem¶

Keep the notation of [[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]], [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]], [[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]], and [[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]].

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

More precisely:

The first connected post-threshold family has been exhausted on the terminal side by [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]].
The same boundary has now also been exhausted with respect to broader sourced Boolean-network semantics by [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]].
Consequently the exact best remaining theorem-level question at the first connected \chi\ge 3 boundary is:

decide ordinary hidden-vertex expressibility or nonexpressibility for the connected multi-parallel-circuit family by a genuinely global argument controlling cross-class auxiliary coupling and auxiliary-budget growth.
The broader Route-D semantic frontier is compressed in parallel by [[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]]: current sourced classical auxiliary semantics beyond CD_{\mathrm{sub}} either fail on connected families or collapse back to ordinary hidden-vertex expressibility.

Therefore the Route-D subfrontier ranking is now:

\[ \text{ordinary hidden-vertex theorem on the connected multi-parallel boundary} \;>\; \text{D1 basis-robust semantic strengthening} \;>\; \text{D3 search for a new positive connected subclass} \;>\; \text{broader D2 semantic insufficiency on the current source base}. \]

So the global Route-D frontier has moved again:

not from terminal semantics to some richer Boolean-network model,
but from all sourced classical semantics down to the single ordinary hidden-vertex theorem that still survives at the first connected boundary.

Dependencies¶

[[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]]
[[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]]
[[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]]
[[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]]
[[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]]

Conflicts/Gaps¶

This node is a ranking and stop-point theorem for the current source base, not a universal impossibility theorem for all future semantics.
It does not decide the surviving ordinary hidden-vertex question on the multi-parallel family.
A future global realization theorem, a future whole-family nonexpressibility theorem, or a new semantics class outside the present source package would change this ranking.

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