1 The model and exact structure
The two-species interacting particle system on \(\mathbb Z\) with states \(\{ 0,1,2,3\} \) per site and the \(n=\infty \) jump-rate table of eq. (eq:rates): hops at rates \(1, q^2\), pair moves at \(1, q^4\), swap at \(q^2\), merge at \(1-q^2\), split at \(q^2(1-q^2)\).
The finite-\(n\) table of eq. (eq:nrates) with \(\beta _n = q^{1-2n}+q^{2n-1}\), \(\sigma _n = (q^{n-1}-q^{1-n})^2\), encoded through the real parameter \(r = q^n\) (so every real \(r\), not only integer \(n\), is covered).
At every \(n\) (and every real \(q^n\)), each species’ marginal current is that of an autonomous single-species ASEP: the species-\(1\) transfer rates are independent of the species-\(2\) occupancy, with the \(n\)-dependence a pure time change \(\beta _n\).
Under the blocking measure the cross mobility and cross compressibility vanish identically in the fugacities: \(\mathbb E_\nu [V_x] = 0\) and \(\mathrm{Cov}_\nu (\eta _{1,x},\eta _{2,y}) = 0\).
The \(n=\infty \) type D ASEP is self-dual with respect to the orthogonal \(q\)-Krawtchouk product duality; the \(\alpha \to 0\) coefficient extraction yields the bounded triangular duality \(D^{\mathrm{tri}}\). (Sorry-free assembly from the cited CFG20/REU interlacing inputs, stated in TypeDDecouplingDuality.lean.)