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Route-D Terminal-Side Semantics Now Stop Before Nonterminal Auxiliary Frontier

Claim/Theorem

Keep the notation of [[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-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]].

After the current Route-D cycle, the remaining compiler-native frontier is no longer about exact terminal routing-style semantics on the natural connected families already on disk. It is about whether any nonterminal auxiliary semantics can still carry compiler meaning.

More precisely:

  1. The witness-level semantic separation already holds on a connected example by [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]].

  2. The same separation already lifts to an infinite connected threshold-lift family by [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]].

  3. On the first connected post-threshold \chi\ge 3 test family, the odd size-2 subfamily already fails every exact terminal nonnegative-hypergraph semantics by [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]].

  4. Therefore terminal graph/packet/path/nonnegative-hypergraph semantics are no longer a serious surviving Route-D frontier on the current connected family stock.

  5. The exact Route-D question that remains is:

    can some genuinely nonterminal auxiliary-variable semantics still support a compiler-meaningful theorem, or is hidden-vertex representability itself already too weak even after terminal semantics are exhausted on the current connected families?

This re-ranks the remaining Route-D subfrontiers as follows:

\[ \text{nonterminal auxiliary semantics on the multi-parallel boundary} \;>\; \text{basis-robust D1 strengthening} \;>\; \text{new positive connected subclass search beyond threshold-lift}. \]

So the global Route-D frontier has moved again:

  • terminal-side semantics are now a largely exhausted negative package;
  • the live question is the semantic value of nonterminal auxiliaries;
  • the first concrete test case for that question remains the connected multi-parallel-circuit family.

Dependencies

  • [[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-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]]
  • [[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]]

Conflicts/Gaps

  • This is a Route-D frontier-ranking theorem for the current source base, not a universal impossibility theorem for all conceivable compiler semantics.
  • It does not prove that nonterminal auxiliary semantics fail. It proves only that terminal-side semantics are no longer the canonical open Route-D target on the current graph.
  • A future whole-family theorem converting a nonterminal auxiliary model into compiler meaning, or ruling that out, would sharpen this ranking again.

Sources

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