Planck's Constant and the Operational Precision of Pi: Conditional Analysis of the Zhang Cheng Conjecture

We study how finite decimal approximations to pi affect physical predictions under stated error, calibration, and resource constraints. Sharp circumference bounds combine numerical and diameter-calibration errors. At the fixed diameter 9.3 x 10^26 m and a nominal budget equal to the 2022 CODATA central value of the Planck length, exact rational certificates and existing Lean proofs establish minimum precisions of 62 decimal places for truncation and 61 for nearest rounding. Nonlinear predictions require a distinction between first usable and least stable precision. For independent Gaussian calibration and readout samples, we solve the maximum class-conditional minimax problem over all full-data randomized tests, including the interval-dependent resource frontier. An exact constructed example has integer optimum 87, compared with 90 for tests using a specified symmetric contrast. Quantum estimates give sufficient precision under a hard action budget, a positive-generator mean-energy budget, or explicit passive phase-and-loss constraints. The last bound is uniform over finite mode counts and path lengths. These are conditional applications and explicit solutions using established interval, statistical, and quantum methods. The resource assumptions have not been derived universally from Planck-scale physics, so no cutoff common to all physically realizable protocols follows.