Prove the unconditional asymptotic theorem for average pairwise distance in matchgate commutator-graph components and translate it to graph and Krylov complexity. The central component theorem must resolve Conjecture 14 of West et al., A graph-theoretic approach to chaos and complexity in quantum systems (arXiv:2502.16404).
For independent uniform (\kappa_n)-subsets (S=(s_1<\cdots<s_{\kappa_n})) and (T=(t_1<\cdots<t_{\kappa_n})) of ([2n]), set [ D_n(S,T)=\sum_{i=1}^{\kappa_n}|s_i-t_i|. ] The main paper theorem is, whenever (\kappa_n/(2n)\to\rho\in(0,1)), [ \frac{D_n}{n^{3/2}} \Rightarrow 4\sqrt{\rho(1-\rho)}\int_0^1|B(t)|,dt, \qquad \mathbb E D_n\sim \frac{\sqrt{2\pi}}2\sqrt{\rho(1-\rho)},n^{3/2}. ] In particular, for the central component (\kappa_n=n), [ \frac{D_n}{n^{3/2}}\Rightarrow2\int_0^1|B(t)|,dt, \qquad \mathbb E D_n\sim\frac{\sqrt{2\pi}}4n^{3/2}. ]
- Exact matchgate metric and discrete transport identity.
- Brownian-bridge area limit with a justified expectation passage.
- Fixed-density theorem and a variance or concentration estimate.
- Exact translation to graph complexity and the cited Krylov bound.
- Reproducible exact/numerical checks and finite-size analysis.
- Novelty audit and final paper integration.
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