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Current Expressive-Power Package Stops Before Higher-Arity Family Obstruction On Connected Multi-Parallel

Claim/Theorem

Keep the notation of [[five-qubit-stabilizer-cut-rank-satisfies-fsep.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 [[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]].

The strongest expressive-power machinery currently sourced on the graph still stops before any family-level ordinary-hidden-vertex obstruction for the connected multi-parallel boundary.

More precisely:

  1. Živný--Cohen--Jeavons provide two load-bearing expressive-power facts already used on the graph:

    • F_{\mathrm{sep}} is a multimorphism of every binary-submodular language, so it is a universal necessary condition for ordinary hidden-vertex expressibility;
    • in arity 4, condition Sep is exact, equivalently F_{\mathrm{sep}} becomes a complete characterization there.

  2. The connected multi-parallel boundary already escapes both sourced instantiations of that machinery:

    • 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]];
    • the infinite size-2 multi-parallel family satisfies direct F_{\mathrm{sep}}, by [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]].

  3. Iwamasa's broader Boolean network-representability framework does not add a stronger positive or negative route here, because for nontrivial stabilizer/binary cut-rank functions it collapses back to ordinary hidden-vertex expressibility by [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]].

  4. No stronger explicit higher-arity multimorphism, weighted polymorphism, or exact higher-arity characterization theorem is currently sourced on the graph for the ordinary hidden-vertex class on Boolean functions, let alone for the connected multi-parallel subfamily.

Therefore the exact expressive-power theorem missing for family closure is now:

a higher-arity ordinary-hidden-vertex obstruction theorem, such as a weighted-polymorphism or multimorphism criterion stronger than the current Sep/F_{\mathrm{sep}} package, that can separate M_{2,2,2} or an infinite connected multi-parallel subfamily.

Consequences for the current frontier:

  • the negative ordinary-hidden-vertex route is no longer waiting on another 4-ary reduction or another direct F_{\mathrm{sep}} computation;
  • any family-level nonexpressibility theorem from expressive-power methods must use machinery genuinely stronger than what is currently loaded;
  • this isolates the exact M1 stop point more sharply than the earlier generic “need a deeper obstruction” phrasing.

Dependencies

  • [[five-qubit-stabilizer-cut-rank-satisfies-fsep.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]]
  • [[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]]

Conflicts/Gaps

  • This node does not prove ordinary-hidden-vertex nonexpressibility for the connected multi-parallel family.
  • It proves only that the currently sourced expressive-power package stops before such a theorem.
  • A future higher-arity weighted-polymorphism computation or exact characterization theorem could still close the negative route.

Sources

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