We study the problem of logic privacy in game anti-cheat systems: hiding verification thresholds and rule structure from clients while still enabling correct verdict computation. We propose a practical framework that achieves logic privacy through server-side fully homomorphic encryption (FHE). Clients encrypt game state under a public key, and the server evaluates its detection function directly over ciphertexts, returning only a binary verdict. Because the detection logic never leaves the server, static reverse engineering is eliminated by architecture. We give a formal security model with four attack classes (logic extraction, cryptographic state forgery, replay, and adaptive oracle queries) and prove logic secrecy and soundness under the LWE assumption. We identify adaptive oracle queries as the primary residual threat, prove that threshold predicates are learnable in O(log N) queries, and propose a randomized response mechanism that forces Omega(1/epsilon^2) queries for epsilon-precision recovery. We implement and evaluate the FHE core using Zama's Concrete/TFHE library across four predicates with three optimization strategies, achieving 2.7x speedup via LUT-based circuit design.