Relational de Sitter State Counting with an SU(3) Clock

Motivated by Maldacena's observer-centric formulation of de~Sitter physics \cite{Maldacena:2024spf}, we develop an observer-dependent state-counting framework in Euclidean de~Sitter space by modeling the observer as a massive equatorial worldline carrying an SU(3) clock. Starting from the gauge-fixed graviton path integral on $S^D$, we trace the one-loop phase $\ii^{D+2}$ to a finite set of scalar and conformal Killing modes and show that, once the worldline is included, the $(D-1)$ transverse negative modes cancel the corresponding $(D-1)$ conformal Killing directions mode by mode. The residual fixed-$\beta$ phase from the global conformal factor and reparametrizations is removed by imposing the Hamiltonian constraint $H_{\text{patch}} - H_{\text{clock}} - \nu = 0$ via a Bromwich inverse Laplace transform, which under explicit complete-monotonicity assumptions yields a real and positive microcanonical density. We stress that this positivity statement is conditional on Assumptions (A1)--(A3) and is established at one loop about the round $S^D$ saddle in the probe regime $G E_{\rm clock}/R\ll 1$; a self-consistent backreacting or higher-loop extension is a natural next step. In earlier work \cite{Ali:2025wld,Ali:2024rnw} we argued that unbroken SU(3) confinement at $T\to 0$ can account for the observed value of the cosmological constant and for the origin of the fundamental constants $(\hbar,G,c)$ as effective couplings fixed by the SU(3) vacuum structure; this makes SU(3) the natural candidate for the internal clock of de~Sitter, whose radius and temperature are themselves set by the same cosmological constant. This idea is implemented with three explicit SU(3) realizations (qutrit, Cartan weight-lattice, and $U(1)^2$ rotor), for which the observer-inclusive density of states factorizes into a universal gravity factor, a universal worldline residue, and a clock-dependent SU(3) weight.