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After Current M1 And M2 Cycles Route-D Stops At Higher-Arity Obstruction Versus Nonprefix Global Construction

Claim/Theorem

Keep the notation of [[current-expressive-power-package-stops-before-higher-arity-family-obstruction-on-connected-multi-parallel.md]], [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]], and [[route-d-now-reduces-to-two-ordinary-hidden-vertex-methodologies-on-the-first-connected-family.md]].

After the current methodology-closure cycle, the remaining compiler-native frontier has been compressed as far as current sources support.

More precisely:

  1. The expressive-power route M1 now stops at one exact missing theorem: a higher-arity weighted-polymorphism or multimorphism obstruction stronger than the current Sep/F_{\mathrm{sep}} package.

  2. The constructive route M2 now stops at one exact missing theorem: an explicit non-prefix global hidden-vertex construction with cross-class coupling or richer growing auxiliary state, beyond the fixed local prefix-state recursion now ruled out.

  3. Therefore the surviving ordinary-hidden-vertex frontier on the first connected post-threshold family is no longer a broad search over constructions or obstructions. It is exactly the dichotomy:

    higher-arity expressive-power obstruction versus non-prefix global hidden-state construction.

  4. Relative to that sharpened stop point, the next strongest Route-D continuations are both secondary:

    • D1 would need a more global basis-robust semantic impossibility theorem than the current source base supports;
    • D3 would need a new positive connected family beyond threshold-lift that beats the already-isolated connected multi-parallel boundary.

Hence the global Route-D ranking is now:

\[ \text{higher-arity obstruction or non-prefix global gadget on connected multi-parallel} \;>\; \text{D1 strengthening} \;>\; \text{D3 new positive subclass}. \]

So the frontier has sharpened one more time:

  • the surviving target is still the ordinary hidden-vertex theorem on the first connected boundary;
  • but the exact next theorem types are now explicit on both the negative and positive sides.

Dependencies

  • [[current-expressive-power-package-stops-before-higher-arity-family-obstruction-on-connected-multi-parallel.md]]
  • [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]]
  • [[route-d-now-reduces-to-two-ordinary-hidden-vertex-methodologies-on-the-first-connected-family.md]]

Conflicts/Gaps

  • This node does not decide the surviving ordinary-hidden-vertex theorem.
  • It records only the strongest supportable stop point from the current source base after completing both methodology cycles.
  • A future stronger expressive-power theorem or a future explicit global construction would change this ranking again.

Sources

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