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Beyond-Fan-Cone Construction Requires Nonseparable Selected-Threshold Energy

Claim/Theorem

Keep the size-2 connected multi-parallel notation from [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]. For each class P_i=\{a_i,b_i\}, write

\[ S_i=\mathbf 1[a_i\ne b_i], \qquad C_i=\mathbf 1[a_i=b_i=0], \qquad D_i=\mathbf 1[a_i=b_i=1], \]

and let

\[ C_t=\mathbf 1[\exists i:\ C_i=1], \qquad D_t=\mathbf 1[\exists i:\ D_i=1]. \]

Then the size-2 family has the form

\[ f_t(u,a,b) = \sum_{i=1}^t S_i + u\,C_t + (1-u)\,D_t. \]

After [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]], the current source-faithful positive route cannot be a modular-plus-fan construction. The next obvious stronger route would be to realize the two global threshold indicators C_t and D_t separately, then select between them using u.

That entire separable selected-threshold architecture is blocked.

Indeed, any construction that factors through separately exposed indicators (u,C_t,D_t) must implement the ternary selector

\[ g(u,C,D)=uC+(1-u)D \]

as a hidden-vertex binary-submodular gadget. But this is exactly the selected-OR obstruction from [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]] up to swapping the names of C and D, and hence is not expressible even with arbitrary auxiliary variables.

Consequently, a positive beyond-fan-cone construction for the connected size-2 family, if it exists, must be genuinely nonseparable:

  • it cannot first build independent hidden summaries for the two global events \exists 00 and \exists 11 and then attach a selector controlled by u;
  • it cannot be a fixed-prefix-state recursion, already ruled out by [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]];
  • it cannot lie in the modular-plus-fan cone, already ruled out by [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]];
  • it must instead use one hidden energy in which u and all parallel classes are coupled before the two global threshold events become separately exposed.

Thus the exact missing constructive theorem is:

Construct a nonseparable selected-threshold hidden-vertex energy for f_t, or prove that every hidden-vertex representation of f_t can be normalized to a separable selected-threshold architecture and is therefore impossible.

This is stronger than the earlier selector-quotient barrier because it identifies the surviving positive construction shape after fan-cone failure. It still does not rule out all possible global hidden-vertex energies.

Dependencies

  • [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]
  • [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]]
  • [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]]
  • [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]]
  • [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]]
  • [[parallel-class-affine-basis-family-is-hidden-vertex-graph-cut-representable.md]]

Conflicts/Gaps

  • This node does not prove ordinary hidden-vertex nonexpressibility for f_t. It rules out the separable selected-threshold construction class and identifies what any positive theorem must do next.
  • The gap is constructive category (i) global coupling architecture: the missing object is a single nonseparable hidden energy coupling u and all parallel classes before the 00 and 11 global events become independent summaries.
  • Auxiliary-budget growth remains allowed. The obstruction is not a bounded-budget statement.
  • A fully asymmetric, nonseparable global construction remains live until either an explicit gadget or a normalization theorem is found.

Sources

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