No three-equation throat candidate found within b₀/ℓ ∈ [1, 10³] on the grids of fixed Morris–Thorne backgrounds examined, for a massless conformally coupled scalar in the static vacuum

On the fixed Morris–Thorne backgrounds examined here, when the only source is the Hadamard-renormalized stress tensor of N massless, conformally coupled scalar fields in the static (Boulware-type) vacuum, with the renormalization length ℓ fixed to an absolute length and the R2 counterterm set to zero, no candidate satisfying the three throat components of the semiclassical Einstein equations simultaneously is found within our grid and the window b0/ℓ ∈ [1, 103]; whether the radial equation admits an upper bound (b0 < ℓ/x0 < ℓ) or only a lower bound (b0 > ℓ/x0, N ≥ Nmin) is decided by the sign of the throat logarithmic coefficient Brr, a closed-form expression in b′(b0) and b0 f′/2f. On the Brr > 0 side (21 backgrounds, all with trr(1) > 0) the tt equation changes sign only near one point of the a = 1/2 family of our grid (b0 ≈ 0.05 ℓ; the θθ equation is not satisfied there), and for the six outside-family backgrounds only the radial equation was examined; on the Brr < 0 side (13 points on the rule-c* face, all with negative asymptotic mass; the seven control backgrounds have trr(1) > 0 and only a lower bound) the tt equation fails throughout the window at the 12 points whose sign is certain, through the rule identity Rtt = C/trr(x) and the computed C < 0. The numbers are those of the production settings (ω ≤ 40/b0, l ≤ 60), at which the near-throat numerical part does not satisfy the conservation law; 24 backgrounds recomputed at converged settings keep the signs of all throat components, while four of them move beyond the stage-1 uncertainty, so the conservation reservation is kept. We do not solve the self-consistent radial profile, we do not claim novelty, and we make no statement on stability or physical realizability. AI assistants (Anthropic Claude Code; Claude Opus 5.5 for planning and independent review, Claude Fable 5.1 for execution up to draft v0.7 and Claude Opus 5.5 from draft v0.8) were used for computation, documentation and drafting under the author's direction; every claim was accepted by an independent AI review instance before the author froze it, and the author takes full responsibility for the content.