Large Language Models (LLMs) often memorize and regenerate sensitive or protected content that requires inference-time suppression. While machine unlearning has emerged as a highly active area of research, existing unlearning methods often degrade performance on retain dataset or require costly retraining. In this paper, we introduce Exponential-Tilted Unlearning (ETU), an inference-time method that formulates unlearning as the KL I-projection of the base next-token distribution under a user-specified upper bound 𝜀 on the forget-token mass, with a closed-form exponential tilt whose scalar multiplier 𝜆 is set analytically by 𝜀. Under a single shared protocol, (i) ETU satisfies the 𝜀 constraint across nine instruction-tuned dense and Mixture-of-Experts backbones on TOFU, RWKU, and WPU while preserving retain utility; (ii) the same calibrated family covers the full 𝜀 spectrum with the strict zero-mass case as a limiting operating point; and (iii) representative weight-update methods expose no such 𝜀-calibrated residual-mass control.