Spectral Admissibility of Real Observers in Euclidean de Sitter Gravity

Abstract: The Euclidean de Sitter path integral contains the familiar phase associated with conformal negative modes. Maldacena's construction shows that a suitably included real observer can reorganize the refined state-counting problem. This paper does not rederive that cancellation. It addresses the prior semiclassical admissibility question: which observer sectors couple to the de Sitter saddle as genuine metric observers without becoming spectators or producing infrared-singular backreaction? On $S^D$, after gauge fixing and zero-mode projection, the observer's quadratic influence is governed by a Schur complement. We formulate a form-domain criterion: if the observer kernel is positive and the mixed metric-observer source is bounded after applying $Δ{ΦΦ}^{-1/2}$, the induced metric correction is a bounded quadratic-form perturbation on the chosen channel. In the gapped case, $Δ{ΦΦ}\geq m_^2\mathbf{1}$ gives $|K^\dagger Δ{ΦΦ}^{-1} K|{\rm op} \leq |K|{\rm op}^2/m^2$; metric-coupled soft modes produce corrections growing as $1/\varepsilon$. We prove a sufficiency theorem: on any stable channel with coercive form $Q_{gg}^P \geq δP |h|^2$, the Gaussian saddle remains controlled whenever $|Δ{ΦΦ}^{-1/2} \mathfrak{j}P|{\rm op}^2 < δ_P$. We construct a localized gapped clock-detector with internal oscillators on a smeared worldline that satisfies the criterion with a computable bound and gives explicit $S^4$ benchmark versus the round-sphere TT scale. The conformal channel is treated only as an indefinite or contour-defined sector; boundedness does not imply positivity. The criterion identifies the semiclassically admissible observer class. Phase cancellation follows only when this class overlaps the relevant conformal or negative-mode sector and is combined with an independent contour or state-counting prescription.