OneBoot GRU: Validation-Locked Polynomial Calibration for Level-Invariant Encrypted Inference

Encrypted recurrent inference must preserve both ciphertext depth and the numerical behavior of a feedback model. We develop OneBoot GRU, which couples the validation-locked selection of degree-three Sigmoid and degree-seven Tanh domains with a CKKS GRU cell that refreshes one hidden-state ciphertext after each step. On TwoPatterns, the locked polynomial model completed finite official-test trajectories with 92.431% macro accuracy, versus 98.114% for exact activations, and 0.830% mean domain violations. On SyntheticControl, polynomial and exact macro accuracies were 89.067% and 90.133%. Direct state-error comparisons show that improved domain coverage does not always reduce recurrent error; in a separate exploratory domain extension, screening reduced qualification tail error by 74.1% and 94.6% on ECG200 and ItalyPowerDemand. A parameter profile audited against pinned 128-bit sparse-secret references completed four encrypted recurrent steps, prefix pooling, and the trained head with maximum logit error 2.91 × 10^-6 relative to the plaintext polynomial circuit. In three matched four-step runs, GRU median circuit latency was 226.343 s versus 336.981 s for an implemented polynomial LSTM. Twelve additional distinct 128-step inputs were completed under an intentionally insecure structural profile, with 12/12 prediction agreement against the plaintext polynomial reference. These experiments provide measured calibration, arithmetic-fidelity, and system-cost operating points within their stated parameter and sampling scopes.