6.4 Existence defects and quantum combinatorial realizations
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6.4.1 Metric collapse and existence collapse
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The simulation theorems compare costs of two realizations that both exist. A broader restricted-realization framework also allows the restricted object to be absent. If
then enlarging the realization category changes feasibility rather than merely runtime.
6.4.2 Quantum Latin-square example
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The order-six quantum-Latin-square record supplies an existence-level example. Entangled quantum Latin squares of order six exist, while mutually orthogonal non-entangled quantum Latin squares of order six do not. This is an ambient-versus-restricted realization gap, not a proof of classical simulation and not a gate-count shortcut.
The proof architecture uses support patterns, unitary-pattern constraints, graph orthonormal representations, projective-line closure, dimension contradictions, and a finite reduction to twelve classical Latin-square classes. Its conceptual relevance here is limited but useful: enlarging the allowed quantum resource can change whether a structure exists at all.
6.4.3 Why this does not imply computational advantage
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Existence of an entangled combinatorial design does not by itself provide an algorithmic speedup. One must still specify an input problem, circuit family, output rule, and classical comparison. Likewise, nonexistence of a product realization does not prove classical hardness of the entangled realization.
Scope boundary
The quantum Latin-square result is included as a contextual example of restricted-realization failure. It is not used as a premise in any packet-simulation theorem.
6.4.4 Unified terminology
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For a target \(x\), let \(c_{\rm res}(x)\) and \(c_{\rm amb}(x)\) be minimum costs, with \(+\infty\) assigned when a realization does not exist. Then
includes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical realization because its reachable representation is compressed.