Fisher zeros of the second temporal Rényi moment in a gravitational model of BTZ microstates at finite central charge

We study the second temporal Rényi entropy S₂ = −log Tr ρ_A² of a heavy primary state in a 2d CFT, for an interval of temporal extent Δt and spatial extent Δφ. For BTZ microstates we define a gravitational model F_grav, built from the Gaussian-ensemble average over heavy OPE data: a sum of exact finite-c Virasoro vacuum blocks, continued to the Lorentzian sheet, in which the integer-spin projection (a sum over Dehn-twist images) is truncated to five images. Within F_grav, Tr ρ_A² has zeros at real (Δt, Δφ) near the transition between the same-sheet and winding saddles. All 18 zeros found for 46 ≤ c ≤ 241 pass a numerical validation by a degree argument. For the zeros at c = 46 and c = 61 we give computer-assisted proofs. Interval arithmetic with analytic bounds on every omitted series term shows that F_grav has Brouwer degree −1 on a disk of radius 0.01 around each. Analytic bounds on all omitted images show that on each disk the complete image sum differs from F_grav by less than 1/160 of a rigorous lower bound on |F_grav| at the boundary, so it also has a zero there. Assuming mainly that only two saddles compete near the transition, that their large-c expansion is uniform there and that their relative phase is monotonic, the zeros follow a Lee-Yang law. We test it numerically: it predicts their number in a fixed window and their positions out of sample, to within 0.0033 in Δφ. The spacing of consecutive zeros, which requires only |Tr ρ_A²|, gives the modulus of the mean slope of the relative action phase between them to within 2%. Persistence of the zeros in individual microstates requires a bound on the deviation of the microstate correlator from F_grav, or at c = 46 and c = 61 from the complete image sum, which we do not establish. Files. Robles_Blanquer_Fisher_zeros_BTZ.pdf is the manuscript. ancillary_files.zip contains its ancillary files (the folder anc/). certificate_c46/ and certificate_c61/ contain the programs, the coefficient data and the stored interval certificates of the computer-assisted proofs of Theorem 1 (c = 46) and Theorem 2 (c = 61), with checkers, pinned requirements and SHA-256 manifests. independent_checks/ contains independent checks, which are not inputs to the proofs. The other files are the numerical pipeline of the paper, the series coefficients, the validation records and the figure scripts. README.md lists the script of each step, table, figure and check. To check the stored certificates, use Python 3.12 with the pinned requirements of each folder (python-flint 0.9.0, mpmath 1.4.1, numpy 2.3.5, scipy 1.17.0). From the folder where the archive is unpacked, run:cd anc/certificate_c46/F6_boundary && python work/check_certificate.pycd anc/certificate_c46/image_sum && python work/check_image_tail.pycd anc/certificate_c61 && python work/check_image_tail.py 61 This preprint has not been peer reviewed. The computations, the code, the computer-assisted proofs and the manuscript were developed with substantial assistance from AI systems, under the author's direction. The author takes full responsibility for the content.