Logarithmic $A$-hypergeometric series ${\textrm I}\! {\textrm I}\! {\textrm I}$
We study the logarithmic coefficients that can occur at a fixed fake exponent in an $A$-hypergeometric series subject to prescribed negative support conditions. Let $L=\operatorname{Ker}_{\mathbb Z}(A)$. For a fixed generic weight $w$, a fixed fake exponent $v_0$, and an ordered negative support family, we first derive a finite system of constant-coefficient differential equations whose solutions encode the admissible logarithmic coefficients. A normalization of the coefficient equations shows that, for $u\in L$, the normalized coefficient associated with $x^{v_0+u}$ depends only on the negative support of $v_0+u$. From this finite system, we identify the annihilator of the coefficient space with an explicitly defined colon ideal. For the negative support family determined by the direction $w$, we further identify this colon ideal with the primary component of the indicial ideal supported at $v_0$, shifted to the origin. We next introduce an ambient perturbation construction in which the fake exponent is perturbed in the full ambient space rather than only within the affine space $v_0+L_{\mathbb C}$. We prove that the ambient perturbation construction produces $A$-hypergeometric series and realizes the full coefficient space. Finally, we compare the ambient perturbation construction with the intrinsic perturbation construction developed in our previous papers. The intrinsic construction always yields a subspace of the full coefficient space, and it realizes the full coefficient space if and only if a natural equality between the corresponding colon ideals holds. We also give several sufficient conditions for this equality.