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Multi-Parallel Hidden-Vertex Boundary Now Reduces To Global Coupling Or Auxiliary Growth

Claim/Theorem

Keep the notation of [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]], [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]], [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]], [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]], and [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]].

After the current Route-D compression cycle, the exact unresolved hidden-vertex gap on the first connected post-threshold family is no longer a generic family-lift problem. It is a much sharper global-coupling or auxiliary-growth problem.

More precisely, for the connected multi-parallel-circuit family:

  1. terminal routing-style semantics are already dead on the infinite odd size-2 subfamily by [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]];

  2. the natural compressed selector route fails by [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]];

  3. the selector obstruction does not lift by closure-preserving minors, by [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]];

  4. the natural one-branch-bit local branch-min route fails by [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]];

  5. low-arity positive evidence still survives:

    • every pinned 4-ary shadow of M_{2,2,2} is one-hidden representable, and
    • the whole size-2 subfamily survives the direct F_{\mathrm{sep}} necessary condition;

  6. but genuinely global low-budget hidden realizations already fail on the first connected witness:

    • pinned 5-ary shadows obstruct every one-hidden realization by [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]];
    • the full M_{2,2,2} witness has no 1- or 2-auxiliary realization inside the large coefficient box [-64,64], by [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]].

Therefore the exact missing theorem at the first connected \chi\ge 3 boundary is now:

either construct a genuinely global hidden-vertex realization that uses cross-class coupling and at least nontrivial auxiliary resources on M_{2,2,2} and then lifts to the family, or prove a whole-family nonexpressibility theorem stronger than the present selector, shadow, and symmetric-ansatz obstructions.

This classifies the remaining gap precisely:

  • it is not mainly a failure to lift from positive low-arity shadows;
  • it is a missing theorem about global auxiliary coupling structure;
  • it may also require controlling auxiliary-budget growth beyond the current low-budget searches;
  • on the negative side, it is a missing whole-family obstruction stronger than current selector/pinned-shadow/symmetry-breaking failures.

Consequences for the current frontier:

  • the multi-parallel boundary is now exhausted on the terminal side;
  • the hidden-vertex side is reduced to one exact theorem schema rather than a diffuse menu of local ansatz tests;
  • any further progress on this family must attack global auxiliary structure directly.

Dependencies

  • [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]
  • [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]]
  • [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]]
  • [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]]
  • [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]]
  • [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]]
  • [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]]
  • [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]]
  • [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]]
  • [[visible-symmetric-hidden-distinguished-ansatze-fail-on-m222.md]]
  • [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]]

Conflicts/Gaps

  • This node does not prove family-level hidden-vertex realizability or family-level nonexpressibility for the connected multi-parallel-circuit family.
  • The exact 2-auxiliary case for M_{2,2,2} remains open outside the searched coefficient box, and larger auxiliary budgets remain open.
  • The statement is a sharpened stop-point theorem: it isolates the remaining theorem shape, but it does not resolve it.

Sources

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