The coefficient of Ω_amΩ_am in the Dirichlet a₅ heat-kernel coefficient: −270, not +360

The boundary heat-kernel coefficient a₅ (a5/2 in the half-integer convention) for a Laplace-type operator with Dirichlet conditions contains a term in the normal–tangential components of the bundle curvature, tr(Ω_amΩ_am). Two values for its coefficient are in the literature, and both appear in the same paper: Branson, Gilkey, Kirsten and Vassilevich (Nucl. Phys. B 563 (1999) 603; hep-th/9906144) quote +360 in their Lemmas 2.2–2.3, taken over from Branson–Gilkey–Vassilevich (1997, hep-th/9504029, Theorem 1.1 — where the derivation of that coefficient is explicitly omitted) and Kirsten (1998, hep-th/9708081, eq. 4), while the pure-Dirichlet specialization of their own Theorem 2.4 for mixed boundary conditions (w₈ + w₉χ + w₁₀ at χ = −1, with w₈ = w₁₀ = −45, w₉ = 180 fixed by the local index theorem on S¹×[0,1] and the conformal-variation relations) gives −270. The Neumann value, +90, is common to both. The paper's Lemma 3.4 does not list the relations that would have confronted the two, and Vassilevich's 2003 review restates only a₀–a₄, so the discrepancy has, to the author's knowledge, not been resolved in print. This note settles it numerically. The arbiter is the Dirichlet ζ-determinant of −∇² + E + λ² on the hemisphere for the charge-(±s) monopole bundles, s = 1 and s = 2, with λ² = ℓ(ℓ+1), ℓ = 2…40 — the spin blocks of the gauge-fixed graviton and ghost operators of the Nariai instanton at its equator, computed in the course of a larger programme (the two papers below) but independent of that origin. The totally geodesic equator tests Lemma 2.2 (the totally geodesic case of BGV 1997) directly. In the large-λ expansion the λ⁻³ coefficient is c₁ = (π/11520)[45 n_c + 720 tr E² + 360 tr E − c_Ω tr Ω_amΩ_am], with tr Ω_amΩ_am = −2s²; the three non-Ω terms are fixed by theorems (the 720 and 180 by the exact constant-mass shift, the 45 by the exact scalar tower, which the same instrument reproduces to 5×10⁻⁴), so c_Ω is the only unknown. The determinants are Gel'fand–Yaglom/Dunne–Kirsten mode sums with the subtracted asymptotics resummed analytically (precision ~10⁻⁹), certified by two gates: the s = 0 tower reproduced to 10⁻¹¹ by the exact hemisphere determinant, and the spectral doubling identity (Dirichlet + Neumann hemisphere = closed sphere) holding row by row. Result: c_Ω = −269.7 ± 2.4 (s = 1, 39 rows) and −269.95 ± 0.37 (s = 2, 27 rows), against the alternatives −270 and +360 — the +360 would more than triple c₁ for s = 2. A mixed-set check (Neumann on one component, Dirichlet on the other) confirms the structure w₁ = −w₂, w₈ = w₁₀ of Theorem 2.4 as well. The correction is: replace (90Π₊ + 360Π₋) by (90Π₊ − 270Π₋) in the Ω_amΩ_am term of the pure-boundary-condition a₅. It affects large-mass expansions and ζ(−½)-type quantities for charged or spinning fields with Dirichlet conditions whenever the bundle curvature has a normal component at the boundary. What the note does not do is trace how the +360 arose; the structure of Theorem 2.4 shows it is not a simple sign. Scope, stated plainly: one geometry (the hemisphere) and two spins; the mixed-set check fixes w₁ = −w₂ and w₈ = w₁₀ but not w₉ on its own — that is the content of the two Dirichlet rows. A second, independent arbiter (the unit disc in a uniform magnetic field) was attempted and set aside unfinished; it is recorded as an open item in the working record. The author would welcome a derivation or a correction from the original authors. 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.