Information geometry of chemical reaction networks: Cramer-Rao bound and absolute sensitivity revisited

The paper derives A = g_Y U* [U g_Y U*]^{-1} U, proves A^2 = A and that A is the g_X-orthogonal projection onto T_xV, and establishes the Hessian geometric Cramér–Rao bound g_X ≥ U* g_H U with g_H = [U g_X^{-1} U*]^{-1} (Theorem 4.2 and Theorem 5.2). The model independently reaches the same conclusions via a direct linearization/Lagrange-multiplier calculation. Assumptions (strict convexity so g_X ≻ 0, full rank of U) are consistent in both. The proofs are mathematically equivalent in content but differ in presentation (diagram-chasing vs. explicit block elimination).