Bounded Masks for Finite-Alphabet Phase Retrieval: Sharp Precision, Photon Cost, and Calibration

This paper studies exact recovery of discrete signals from one passive spatial mask and the original n Fourier intensities. It quantifies three measurement resources: intensity precision, detected photons, and calibration accuracy. Recovery is modulo global sign, and all intensity and photon formulas use the unnormalized Fourier transform. The constructions use a single fixed mask for each stated signal class, without zero padding or additional frequency samples. For every fixed integer alphabet {−A,…,A}, an explicit affine-amplitude mask with a phase ramp recovers every signal at strict absolute intensity-error radius 16/[289A²(n−1)], independently of support size and energy. The ternary radius improves to 2/[9(n−1)]. A singleton packing bound proves that the n⁻¹ length order is optimal. The constructive decoder uses O(n log n + s² + As + A) arithmetic operations for a signal with s occupied sites. Under independent Poisson acquisition with known common exposure, the optimal worst-case detected-photon order at fixed alphabet and energy is n² log(1/δ), where δ is the allowed error probability. For ternary weight at most s, with 1 ≤ s ≤ floor(√n)+1, a quadratic-phase mask attains the sharp order sn² log(1/δ) on arbitrary supports of nonwrapping span L ≤ floor(√n). It allows arbitrary signs and gaps, does not require the actual weight, and tolerates unknown complex mask errors of optimal order n⁻¹. A mask tuned to a supplied factor-two weight band extends complete recovery to every L < n/2. Its stated calibration bound depends on the supplied weight. Both decoders have polynomial expected bit complexity. Further sharp photon laws cover binary intervals and specified reflection classes under their stated weight information. A separate random-phase construction addresses candidate generation. A polynomial-expected-time Las Vegas algorithm fixes one certified mask independently of the unknown signal and supplied sparsity bound. At exposure 2²⁴n, its decoder returns at most 6n⁴+n+1 candidates and contains the true ternary signal with probability at least 7/8 whenever L < n/2. The list-size bound and polynomial computation hold for every observation. This is a proposal theorem: statistical selection from the list and proposal generation for the cyclically aliased range remain open. For unrestricted ternary supports, the proved photon bounds are Ω(sn² log(1/δ)) and O(s²n² log(1/δ)). The paper also treats integer amplitudes with growing or unbounded energy. At every odd length n ≥ 3, explicit curved phases recover unknown supports of size at most two at energy-independent radius 1/[32(n−1)²]. At odd prime lengths this improves to 1/(32n), which has optimal length order. At length three, one fixed algebraic mask recovers the entire class ‖x‖₂² ≤ h at strict radius 1/(2000√h), with bit complexity polynomial in log h and the input length. The proof combines a cubic algebraic kernel, an integer companion determinant, and fixed-dimensional integer programming. A structural classification of the three-site polarization determinant identifies necessary conditions for energy-independent separation; that question remains unresolved. At length four, one fixed complex mask recovers every integer signal at radius 1/16 with polynomial bit complexity. Every real or affine-phase mask instead has signed-energy radius O(1/√h), demonstrating a strict advantage for nonlinear phase control. Additional calibration theorems quantify robustness to bounded variation, unknown gain, and uniform background. The deposit contains the main paper, a companion supplement with specialized results, alternative measurement models, and obstructions to specific proof approaches, LaTeX sources, certificate programs, and reproducible acquisition experiments. Written proofs establish the universal theorems; exact finite checks and floating-point experiments support reproducibility without replacing those proofs. The unrestricted sparse photon law, the full three-site constant-radius problem, and the general joint length–energy precision law remain open.