Exact Fixed and Adaptive Reconstruction of Sparse Complex States from Interference Measurements

Exact Fixed and Adaptive Reconstruction of Sparse Complex States from Interference Measurements Abstract We investigate exact fixed and adaptive reconstruction in a finite, noiseless complex interference model. The admissible states are two-sparse vectors over the Gaussian integers, with two nonzero amplitudes normalized to 1 and one of four relative phases: 1, i, −1, or −i. Each admissible measurement consists of a single-coordinate phase gate followed by a pairwise interference-intensity observation. For this restricted measurement family, we establish the exact minimum number of nonadaptive measurements: F(n) = n(n − 1), for n ≥ 3. We also construct a deterministic adaptive reconstruction algorithm requiring at most n + 1 measurements for n = 3 and n ≥ 5, and at most six measurements for n = 4. These results establish a quadratic-versus-linear separation between optimal nonadaptive reconstruction complexity and an achievable adaptive reconstruction bound. Two independently implemented exact-arithmetic Python programs additionally compute optimal adaptive worst-case measurement depths of 4, 5, 5, and 6 for systems with 3, 4, 5, and 6 sites, respectively. The manuscript provides mathematical proofs, a constructive reconstruction procedure, finite computational verification, and a discussion of related phase-retrieval and adaptive-testing literature. The results concern a precisely specified discrete measurement model. They do not establish general phase-retrieval lower bounds, optimal asymptotic adaptive complexity, noise robustness, quantum computational speedup, or a Fibonacci-specific advantage. Reproducibility The accompanying archive contains the LaTeX manuscript, compiled PDF, bibliography, and two standalone Python 3 verification programs. The programs use exact arithmetic, require no external dependencies, and can be executed offline. Publication status: Preprint, version 1.1. Not peer-reviewed. OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit.