Λ-dependent gravitational corrections to QED running on the de Sitter throat: the on-shell coefficient vanishes

In 2008 Toms found that the one-loop gravitational correction to the running of the electric charge acquires, in the Vilkovisky–DeWitt effective action, a term proportional to the cosmological constant: the dimension-four pole (3/8) κ²ΛF² computed on flat space with Λ ≠ 0. Flat space with Λ ≠ 0 is not a solution of the field equations, and on the solution — the Euclidean de Sitter space S⁴ with R_μν = Λg_μν, the "throat" of the no-boundary geometry — the invariant κ²ΛF² is not separable from the three curvature invariants κ²RF², κ²R^μν F_μα F_ν^α, κ²R^μνρσ F_μν F_ρσ of the same dimension. This note computes all four coefficients, with an exact-arithmetic local-momentum-space engine built for the purpose and gated at every step against published values, and finds that on S⁴ their on-shell combination is zero: −1 from the metric–photon fluctuations (the same number at every one of eight gauge points, with Toms' 3/8 reproduced as the flat, off-shell coefficient at each) and +1 from the Faddeev–Popov ghosts. The gauge independence and the independence of the Vilkovisky–DeWitt connection are then proved by two short lemmas, which rest on the tracelessness of the Maxwell stress tensor on a maximally symmetric Einstein space; the value is not explained by the lemmas and was tested by a scalar control: a massless scalar probe with coupling ξRφ² gives an on-shell κ²Λ²φ² coefficient −4ξ²(1 − 2ξ)(1 − 6ξ), gauge- and connection-independent, which vanishes at the conformal coupling and not otherwise — so the mechanism indicated is the Weyl invariance of the probe, not a property of de Sitter space. The same engine run on the Nariai geometry S² × S², the Euclidean throat of the black-hole universe, where the probe can be an exact Maxwell solution and the Weyl tensor does not vanish, gives zero again, for every orientation of the probe; the lemmas extend to it. Every number is an exact rational from an over-determined fit; eleven method errors found on the way — nine in the instrument, two in the design of the scalar control — are retained in an appendix, with the gate that caught each. This record contains the paper (PDF and LaTeX source) and the complete working record cited from it as [43]: the gate scripts for G0–G2 and the exact a₄ engine (Maxwell probe, scalar control, massive-vector control, attacks G11–G16), the known-answer tests, the dated gate records and design memos with their retained errors, the exported fit tables, the figure code, and the raw session logs. A verification guide in the record runs everything in an order where each step is checked by the previous ones (Python 3 + numpy; no computer-algebra system). This is a standalone note, not part of the series "A smooth beginning for spacetime." Declarations: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Manuscript drafting, numerical implementation, and verification tooling were carried out with the assistance of a large language model (Claude, Anthropic); every quantitative claim traces to a gate script deposited with this record, and the author takes sole responsibility for the content.