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Visible-Symmetric Hidden-Distinguished Ansatze Fail On M222

Claim/Theorem

Let M_{2,2,2} be the smallest connected member of the multi-parallel-circuit family, and write

\[ f_{2,2,2}(x):=\lambda_{M_{2,2,2}}(L_x), \qquad x\in\{0,1\}^7, \]

with visible coordinates ordered as

\[ (u,a_1,b_1,a_2,b_2,a_3,b_3). \]

Consider quadratic pseudo-Boolean energies with all pairwise coefficients nonpositive and all coefficients in [-64,64], but impose only the visible symmetry of M_{2,2,2}:

  1. permutation of the three visible pairs P_1,P_2,P_3, and
  2. swapping a_i and b_i inside each pair.

Do not impose any symmetry among the hidden bits themselves.

Then the following two less-symmetric global ansatz classes still fail exactly.

Ansatz C: three distinguished hidden bits

Use hidden bits (h_1,h_2,h_3), all distinguished. Under the visible symmetry above, every quadratic monomial collapses to one of the following orbit types:

\[ 1,\ u,\ x,\ h_r,\ ux,\ uh_r,\ xx_{\mathrm{same}},\ xx_{\mathrm{diff}},\ xh_r,\ h_r h_s, \]

with r,s\in\{1,2,3\} and r<s.

This gives 18 continuous coefficient variables. There is no energy of this form whose hidden minimization equals f_{2,2,2}.

Ansatz D: four distinguished hidden bits

Use hidden bits (h_1,h_2,h_3,h_4), all distinguished. With the same visible symmetry, every quadratic monomial again reduces to the analogous orbit family

\[ 1,\ u,\ x,\ h_r,\ ux,\ uh_r,\ xx_{\mathrm{same}},\ xx_{\mathrm{diff}},\ xh_r,\ h_r h_s, \]

now with r,s\in\{1,2,3,4\} and r<s.

This gives 24 continuous coefficient variables. There is no energy of this form whose hidden minimization equals f_{2,2,2}.

Equivalently: even after breaking the hidden-variable symmetry completely while still respecting only the visible automorphisms of the family, the first natural global constructive route remains infeasible.

This is a derived exact bounded-ansatz obstruction.

  1. By [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]], the target function depends only on the singleton bit u and the three visible pair states, so the above visible symmetry is intrinsic to M_{2,2,2}.
  2. Unlike [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]], the present ansatz classes do not tie hidden bits to the visible classes and do not require any permutation symmetry among the hidden bits.
  3. For each ansatz, exact realizability again reduces to a mixed-integer linear feasibility problem:
  4. one selector binary chooses the minimizing hidden state for each visible assignment x;
  5. for every visible-hidden pair (x,h), impose
\[ E(x,h)\ge f_{2,2,2}(x) \]

and the selected-state upper bound

\[ E(x,h)\le f_{2,2,2}(x)+M(1-s_{x,h}). \]
  1. Because all coefficients are restricted to [-64,64], the same finite-box exactness logic as in [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]] applies.
  2. For Ansatz C:
  3. there are 18 continuous coefficient variables and 128\cdot 8=1024 selector binaries;
  4. HiGHS reports the exact MILP infeasible.
  5. For Ansatz D:
  6. there are 24 continuous coefficient variables and 128\cdot 16=2048 selector binaries;
  7. HiGHS again reports the exact MILP infeasible.
  8. Therefore neither of these visible-symmetric but hidden-distinguished ansatz classes realizes f_{2,2,2}.

Consequences for the current frontier:

  • this closes the first genuinely hidden-symmetry-breaking global constructive route still compatible with the visible automorphisms of M_{2,2,2};
  • any positive theorem for M_{2,2,2} must now either break even the visible pair symmetries or use a still richer hidden coupling pattern than these distinguished-hidden quadratic ansatz classes;
  • this still does not prove nonexpressibility of M_{2,2,2}, because a successful realization could be fully less symmetric or use a more elaborate architecture.

Dependencies

  • [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]]
  • [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]]
  • [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]]

Conflicts/Gaps

  • This node does not prove that f_{2,2,2} is not hidden-vertex graph-cut representable.
  • It proves only that two larger visible-symmetric but hidden-distinguished bounded ansatz classes fail exactly.
  • A fully less-symmetric construction, or a more elaborate hidden architecture, remains open.
  • Even a future explicit realization or nonexpressibility 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