Proof of Critch–Morton Conjecture 8 (SIGMA 10 (2014), 095, Section 2, p. 6): reconstruction from any consecutive three-site improper marginal

We prove Critch–Morton Conjecture 8 (SIGMA 10 (2014), 095, Section 2, p. 6): every consecutive three-site improper marginalization is birational for open-boundary, translation-invariant, binary bond-two matrix product states of length N at least four. An explicit rational inverse over the complex numbers is given on a nonempty principal open domain, with both inverse identities on the affine Zariski closures. A four-by-four linear system reconstructs a three-site representation, and inverse boundary contractions restore omitted sites. Consequences include rationality in dimension seven, reconstruction from every longer consecutive block with arbitrary fixed nonzero site-dependent coherent contractions, minimality of three retained amplitude indices, and exact equations characterizing the reconstruction chart.