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Higher-Arity Theorem16 Would Close Negative Side On Connected Size-2 Family

Claim/Theorem

Keep the notation of [[current-expressive-power-package-stops-before-higher-arity-family-obstruction-on-connected-multi-parallel.md]], [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]], and [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]].

After the new family-level fan-cone obstruction, the negative Route-D side now has one exact theorem target rather than a generic “stronger higher-arity obstruction” placeholder.

More precisely:

  1. Živný-Cohen-Jeavons prove the arity-4 equivalence

    \[ f\in \langle \Gamma_{\mathrm{sub},2}\rangle \iff F_{\mathrm{sep}}\in \operatorname{Mul}(\{f\}) \iff f\in \mathrm{Cone}(\Gamma_{\mathrm{fans},4}), \]

    in Theorem 16.

  2. The same paper records the higher-arity refinement conjecture

    \[ \Gamma_{\mathrm{sub},k}\cap \langle \Gamma_{\mathrm{sub},2}\rangle = \mathrm{Cone}(\Gamma_{\mathrm{fans},k}) \qquad \text{for all } k, \]

    as Conjecture 20.

  3. By [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]], every size-2 multi-parallel function f_t satisfies the direct higher-arity F_{\mathrm{sep}} necessary condition.

  4. By [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]], every connected member f_t with t\ge 3 lies outside the modular-plus-fan cone.

Therefore any higher-arity extension of Theorem 16 strong enough to characterize \langle \Gamma_{\mathrm{sub},2}\rangle by fan-cone membership on this family would immediately prove nonexpressibility for the entire connected size-2 family.

So the exact expressive-power theorem now missing is not merely “some stronger multimorphism.” It is one of the following equivalent theorem types:

  • an all-arity Theorem-16-style characterization of \langle \Gamma_{\mathrm{sub},2}\rangle on Boolean submodular functions;
  • Zivný's Conjecture 20 itself, or a family-specific specialization of it;
  • a weighted-polymorphism or multimorphism theorem whose force on this family is exactly to upgrade “outside the fan cone” into “outside \langle \Gamma_{\mathrm{sub},2}\rangle.”

Consequences for the current frontier:

  • the negative side is now pinned to one explicit theorem schema already named in the primary source;
  • F_{\mathrm{sep}} is now concretely known to be too weak on an infinite connected family, not just on isolated witnesses;
  • any future expressive-power advance should be evaluated against this exact target rather than against a generic higher-arity search.

Dependencies

  • [[current-expressive-power-package-stops-before-higher-arity-family-obstruction-on-connected-multi-parallel.md]]
  • [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]]
  • [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]]

Conflicts/Gaps

  • This node does not prove Conjecture 20 or ordinary-hidden-vertex nonexpressibility for the connected size-2 family.
  • It identifies the exact higher-arity theorem that would close the negative side if proved.
  • A positive hidden-vertex realization beyond the fan cone would still remain compatible with the current source base until such a theorem appears.

Sources

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