Replacement Dynamics of Binary Quadratic Forms

The paper proves that for f(x,y)=Cx^2+Dy^2 (C,D in Q×), the non‑univariate types t4=(L,R,L,R) and t5=(L,L,R,L,R) are governed by explicit eliminants R(y) (quartic) and S(y) (degree 10), with parametrizations x=G(y) and x=H(y), respectively. It shows existence and a bijection between roots and periodic vectors when C≠D (Theorems 3.8 and 3.11), identifies the exceptional degree‑drop loci for S(y) (C=−D or C^2−3CD+4D^2=0), and states the number‑field degree bounds; see Eq. (8) and Theorem statements in the text. The candidate solution reproduces exactly this elimination/Gröbner basis shape and draws the same consequences (bounds, bijections, rationality criteria, and minimality of type), explicitly referencing the same equations. Hence, both the paper and the model follow substantially the same proof strategy and reach consistent conclusions. Key supporting passages include the explicit quartic eliminant and bijection for t4 (Eq. (8) and Theorem 3.8) and the degree‑10 eliminant and bijection for t5, with exceptional loci (Theorem 1.6(2), Theorem 3.11) . The paper’s modular-polynomial construction and zero-dimensionality underpin the Gröbner-based elimination the model describes .