Skip to content

Connected Multi-Parallel Ordinary Hidden-Vertex Boundary Now Requires Deeper Multimorphism Or Explicit Global Gadget

Claim/Theorem

Keep the notation of [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]], [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]], [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]], and [[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]].

After the current Route-D closure cycle, the surviving ordinary hidden-vertex frontier on the first connected post-threshold family has only two sourced methodology types left.

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

  1. There is no broader sourced Boolean-network positive route beyond ordinary hidden-vertex expressibility, by [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]].

  2. There is no exact arity-4 obstruction route left:

    • every 4-ary pinning/minimization minor of M_{2,2,2} satisfies Sep, by [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]];
    • hence every such minor already lies in \langle\Gamma_{\mathrm{sub},2}\rangle.

  3. There is no current universal F_{\mathrm{sep}}-multimorphism obstruction route left on the natural infinite size-2 subfamily, by [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]].

  4. The natural positive construction routes currently on disk also fail:

    • the compressed selector route fails by [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]];
    • the selector obstruction does not lift to the full family by closure-preserving minors, by [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]];
    • the one-branch-bit local branch-min implementation fails by [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]];
    • the first symmetric and visible-symmetric global ansatz classes fail on M_{2,2,2}.

  5. The currently known finite negative obstructions remain only bounded:

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

Therefore the exact remaining theorem-level methodologies are now:

either find a deeper ordinary-hidden-vertex expressive-power obstruction, such as a weighted polymorphism or multimorphism stronger than the current F_{\mathrm{sep}} package, or construct an explicit genuinely global hidden-vertex realization with cross-class coupling and possibly growing auxiliary budget.

This is sharper than the previous “global coupling or auxiliary growth” phrasing because the broader Boolean-network escape hatch is now gone, and the arity-4 plus direct-F_{\mathrm{sep}} routes are also explicitly exhausted.

Consequences for the current frontier:

  • the surviving ordinary hidden-vertex question is no longer a diffuse search over many semantics;
  • the negative side must now produce a genuinely higher-arity expressive-power obstruction than the exact 4-ary Sep theorem and the current F_{\mathrm{sep}} multimorphism;
  • the positive side must now produce an explicit global gadget theorem, not another local or symmetry-restricted ansatz.

Dependencies

  • [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]
  • [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]]
  • [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]]
  • [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.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]]
  • [[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]]

Conflicts/Gaps

  • This node does not decide ordinary hidden-vertex representability or nonexpressibility for the connected multi-parallel family.
  • It isolates only the two surviving methodology types supported by current sources.
  • A future stronger multimorphism theorem, weighted-polymorphism computation, or explicit global gadget family could still close the boundary.

Sources

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