Multi-Parallel Boundary Has No Broader Boolean-Network Positive Route Than Hidden-Vertex¶

Claim/Theorem¶

Keep the notation of [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], [[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]], and [[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]].

For the connected multi-parallel-circuit family, Iwamasa's broader Boolean network-representability framework does not enlarge the current positive frontier beyond ordinary hidden-vertex graph-cut expressibility.

More precisely, let M be any nontrivial member of the family from [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], and let

\[ f_M(x):=\lambda_M(L_x). \]

Then the following are equivalent:

f_M is expressible by binary submodular functions with auxiliary variables, equivalently by an ordinary hidden-vertex graph-cut reduction.
f_M is classically Boolean network representable.
f_M is generalized (k,\rho,\sigma)-network representable in Iwamasa's sense for some k,\rho,\sigma.

Therefore the first connected post-threshold boundary has no stronger sourced nonterminal Boolean-network positive route than the ordinary hidden-vertex route itself.

Proof sketch:

By [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], every such M is connected, so f_M is not identically zero.
By [[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]] and the binary-matroid interpretation already used throughout Route D, f_M is a nontrivial symmetric submodular Boolean function with

\[ f_M(\varnothing)=f_M(E)=0. \]
[[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]] proves that for any nonzero stabilizer cut-rank function, generalized Boolean network representability collapses back to ordinary hidden-vertex graph-cut representability because the only extra Boolean cases in Iwamasa's framework are monotone, while symmetric cut-rank functions with f(\varnothing)=f(E)=0 cannot be nonzero and monotone.
The multi-parallel family lies inside that nontrivial stabilizer/binary-matroid cut-rank subclass, so the same equivalence applies here.

Consequences for the current frontier:

the multi-parallel boundary is not waiting on some broader Boolean-network theorem beyond hidden vertices;
any positive theorem on this boundary must already be an ordinary hidden-vertex realization theorem with genuinely global auxiliary structure;
any negative theorem on this boundary must therefore obstruct ordinary hidden-vertex expressibility itself, not merely some narrower Boolean-network ansatz.

Dependencies¶

[[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]
[[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]]
[[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]]
[[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]]

Conflicts/Gaps¶

This node does not decide whether the connected multi-parallel family is hidden-vertex representable.
It sharpens the positive frontier only by ruling out a larger Boolean-network escape hatch.
The possibility of non-Boolean, non-network, signed, or otherwise more algebraic auxiliary semantics remains outside the present source base.

Sources¶

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