Exact Compilation and Resource Laws for Hidden Integer Feedback

Hidden randomized rounding couples the privacy loss of successive integer releases. We study a rational-feedback mechanism under predictable signed-unit participation and full observation of every physical output. A preparation's lifetime connects three results. First, preserving complete carries until an observed echo synchronizes the worlds gives exact finite-law point compilation, with support-width exponent $3C-2$ at every fixed budget $C\ge2$. Second, releasing commitments earlier gives a monotone hierarchy whose exact endpoint, strict gaps and numerical brackets quantify calibration loss. At one fixed discrete Gaussian, a loss allowance of 0.09 requires all four commitments. Third, legal regeneration yields an unrestricted KL rate with a square-root testing remainder and a complete asymptotic frontier under joint physical-time and participation constraints. At fixed rational parameters, finite-window occupation programs approximate that frontier to error $2^{-b}$ in polynomial bit time in $b$. Verified countable-transcript semantics and independently checked finite certificates support these analyses. A matched static workload illustrates the resulting design tradeoff: a structured finite integer matrix improves error and fresh randomness, while the scalar feedback instance retains fewer noise-state and source-table bits.