Symmetric cyclic ptychography: exact ambiguities and one-intensity repair

Unrefereed research candidate. Ptychography images an object by scanning a known illumination (the probe) across it and recording only diffraction intensities; the lost phases must be recovered. Symmetric probes, including every real mirror-symmetric probe such as a Gaussian beam, force some coefficients of the data to vanish, so some different objects give identical data. Phasebook (Allain et al. 2025, Section 8.1) asks for recovery guarantees when such zeros occur. This manuscript answers precise finite cyclic subclasses of that question (all scan positions and frequencies recorded, probe known exactly). Results. (1) For real-even probes whose zero pattern is exactly the forced cross, including circular-distance Gaussian probes at all but finitely many widths, every ambiguity is a twin pair: a conjugate pair concentrated on two antipodal coordinates, in the object or in its Fourier transform, with no mixed cases. (2) One extra coherent intensity |g*x|^2 (one extra exposure under a structured illumination, read at the zero-frequency pixel) removes every twin, even among arbitrary positive-semidefinite (mixed-state) competitors. (3) The exceptional Gaussian widths are located by certified interval arithmetic for n = 6,...,16; the exact pattern holds for 0 < t < t*(n), with t*(6) = 2^(-1/2) and t*(n) about 0.74-0.84 for 8 ≤ n ≤ 16. (4) For conjugate-symmetric probes with one forced zero line: exact paired and PSD stability gaps after one added intensity, an added vector optimal for both at fixed energy in a stated deconvolved, orthogonalised metric, and a one-unknown direct reconstruction. (5) Short three-site probes: a two-pixel intensity repairs every nonvanishing signal, with parity-system conditioning of optimal dimensional order. Prior work and scope. The public AI-assisted manuscript VESPER-QUORUM (3 October 2026) already contains the single-line block and fibre classification, the obstruction mechanism and the PSD algorithm; these are credited, not claimed. The subspace-cone angle identity and PSD iteration are also credited to it; it states that the identity was already used in an earlier manuscript of the same credited author. The extra measurement changes the data model. A quadratic chirp (defocus) repair fails when n is divisible by 4, while a twisted geometric added vector works for every even n. Gaussian genericity is an exact-arithmetic statement (multipliers are numerically tiny at small widths). The cross-mask stability constant is qualitative; the n = 6 explicit values are uncertified numerical evidence only. All numerics are synthetic; this is not blind ptychography. Archive: manuscript (LaTeX source and PDF), five verification and certificate programs with their JSON outputs and figures, novelty report, target specification, bibliography with registry records, CI and SHA-256 manifest. Assurance: internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text CC0-1.0; code MIT. Version 0.2.0-candidate corrects the prior-work attribution of 0.1.0-candidate (DOI 10.5281/zenodo.23938688). No theorem, proof, program or result changed.