Exactly one unstable mode: a computer-assisted spectral count for Choptuik's critical spacetime under spherically symmetric perturbations

Choptuik's spacetime, the discretely self-similar solution at the threshold of black hole formation in the collapse of a massless scalar field, is expected to have exactly one unstable mode, whose growth rate λ determines the critical exponent γ = 1/λ ≈ 0.374. The existence of this spacetime was proved by Reiterer and Trubowitz with a computer-assisted argument. We prove, also with computer assistance, that in their realization the solution has exactly one unstable mode, modulo gauge, under spherically symmetric linear perturbations that are smooth in time and analytic in the radial variable up to and across the light cone. The mode is real and simple, with λ ∈ [2.674040, 2.674115], so that 1/λ ∈ [0.373955, 0.373966]. More precisely, per Floquet class the linearized operator has exactly four roots with positive real part: a gauge mode, which translates the time of the singularity; two roots whose eigenvectors violate the constraints; and the physical mode. The count uses an operator version of Rouché's theorem with a finite-dimensional reference, a rank-two deflation of the gauge roots, Newton–Kantorovich enclosures, and bounds in rational and ball arithmetic. The passage from the geometric problem to the operator rests on a global completeness result for the gauge and an explicit reconstruction of the perturbations. The certificates, a verification program and the data are publicly available. Preprint, version 1 (October 2026). Code, certificates and verification program: github.com/edivaldo-amaral/choptuik-espectro. Large data files: doi:10.5281/zenodo.23222363. The use of AI assistants in this work is described in Section 7.6 of the paper.