S. J.Thomas, “Gram Decay and Intrinsic Dimensions of Krylov Subspaces,” Numerical Linear Algebra with Applications33, no. 3 (2026): e70094, https://doi.org/10.1002/nla.70094. 1 Introduction This paper establishes two main results for Krylov subspace methods applied to large sparse linear systems A x = b $$ Ax=b $$ . Throughout, SPD denotes symmetric positive definite, and the core decay analysis of Sections 2, 3, 4 is stated for SPD A $$ A $$ with Euclidean inner product.