Two-Element Multi-Parallel-Circuit Family Satisfies Direct Fsep

Claim/Theorem

Let M_t be the binary matroid from [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]] in the special case

\[ E=P_1\sqcup\cdots\sqcup P_t\sqcup\{u\}, \qquad |P_i|=2\ \text{for all }i. \]

Write

\[ f_t(x):=\lambda_{M_t}(L_x), \qquad x\in\{0,1\}^{2t+1}. \]

Let F_{\mathrm{sep}}:{0,1}^5\to{0,1}^5 be the Boolean 5-ary multimorphism from Živný-Cohen-Jeavons, Fig. 2 / Theorem 16.

Then for every t\ge 1,

\[ F_{\mathrm{sep}}\in \operatorname{Mul}(\{f_t\}). \]

Equivalently: the whole size-2 multi-parallel-circuit family, including the first connected \chi\ge 3 cases t\ge 3, survives the direct higher-arity F_{\mathrm{sep}} necessary condition for hidden-vertex graph-cut expressibility.

This is a derived exact finite-state theorem.

1. By [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]],

\[ f_t(x) = \sum_{i=1}^t \mathbf 1[p_i(x)=1] + u(x)\,\mathbf 1[\exists i:\ p_i(x)=0] + (1-u(x))\,\mathbf 1[\exists i:\ p_i(x)=2], \]

where u(x)\in\{0,1\} is the bit on the singleton element u, and p_i(x)\in\{0,1,2\} is the number of selected elements in P_i. 2. Fix any 5 input assignments x^{(1)},\dots,x^{(5)}. For one class P_i=\{a_i,b_i\}, the whole input-output behavior under F_{\mathrm{sep}} is determined by the two input columns

\[ \alpha,\beta\in\{0,1\}^5 \]

on a_i,b_i, together with the two output columns

\[ \alpha':=F_{\mathrm{sep}}(\alpha), \qquad \beta':=F_{\mathrm{sep}}(\beta). \]

1. For such a class, record the masks

\[ I_0:=\{j:\alpha_j=\beta_j=0\},\quad I_2:=\{j:\alpha_j=\beta_j=1\}, \]

\[ O_0:=\{j:\alpha'_j=\beta'_j=0\},\quad O_2:=\{j:\alpha'_j=\beta'_j=1\}, \]

and the split-defect

\[ d:=|O_{\mathrm{split}}|-|I_{\mathrm{split}}|, \]

where I_{\mathrm{split}} and O_{\mathrm{split}} are the masks on which the two class bits differ. 4. Enumerating all 32^2=1024 input column pairs (α,β) produces only 402 distinct class summaries

\[ \sigma=(I_0,I_2,O_0,O_2,d). \]

1. For t classes, only the unions

\[ I_0^\cup,\ I_2^\cup,\ O_0^\cup,\ O_2^\cup \]

and the total split-defect

\[ D_t:=\sum_{i=1}^t d_i \]

matter. Define a dynamic-programming table that assigns to each union state

\[ (I_0^\cup,I_2^\cup,O_0^\cup,O_2^\cup) \]

the maximum achievable D_t. 6. Exact enumeration gives the following reachable-state counts:

\[ 402,\ 8349,\ 12192,\ 12411,\ 12419 \]

for t=1,2,3,4,5, and the DP reaches a fixed point at t=5, meaning the t=6 table is already identical to the t=5 table. 7. For the singleton element u, let U_1,U_0 be the input masks on which the u-bit is 1,0, and let V_1,V_0 be the corresponding output masks after applying F_{\mathrm{sep}}.

Then the full multimorphism defect is

\[ \Delta = D_t + |V_1\cap O_0^\cup| + |V_0\cap O_2^\cup| - |U_1\cap I_0^\cup| - |U_0\cap I_2^\cup|. \]

1. Exhaustive evaluation over all 32 input patterns on u and all reachable DP union states gives

\[ \max \Delta = 0 \]

for each t=1,2,3,4,5. 9. Since the DP has already stabilized at t=5, the same maximum remains 0 for all t\ge 6.

Therefore F_{\mathrm{sep}}\in \operatorname{Mul}(\{f_t\}) for every t\ge 1.

As an internal consistency check on the local page-image decode of Fig. 2, the same reconstructed F_{\mathrm{sep}} reproduces Proposition 18 of Živný-Cohen-Jeavons: the quasi-indecomposable function \theta_{(1,1,0,0)} has a direct F_{\mathrm{sep}} violation with defect 1.

Consequences for the current frontier:

• [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]] is not the end of the low-arity story: even the direct higher-arity F_{\mathrm{sep}} test still does not separate this connected family;
• the first connected \chi\ge 3 regime therefore survives the selector quotient obstruction, the one-branch-bit local-gadget obstruction, the full exact 4-ary closure toolkit, and now also the direct higher-arity F_{\mathrm{sep}} test on an infinite subfamily;
• this still does not prove hidden-vertex graph-cut representability for the family, because F_{\mathrm{sep}} remains only a necessary condition.

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]]
• [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]]

Conflicts/Gaps

• This node does not prove a hidden-vertex realization for the multi-parallel-circuit family.
• The theorem is proved only for the subfamily with all |P_i|=2; it does not cover arbitrary parallel-class sizes.
• The argument is derived from exact finite-state computation using the F_{\mathrm{sep}} operation read from the local Živný page image, not from a verbatim theorem statement in the literature.
• Even a future positive hidden-vertex theorem for this connected family would still need a separate bridge to routing-style CD(T_n,G) semantics.

Sources

• 10.1016/j.dam.2009.07.001
• 10.48550/arXiv.2109.14599
• Kashyap 2008 preprint: A Decomposition Theory for Binary Linear Codes