The no-boundary saddle carries the de Sitter first law: entropy in the action, temperature in the cap

We show that a single complex-lapse saddle of the Lorentzian no-boundary path integral carries the complete thermodynamics of the de Sitter horizon, and that its three thermodynamic data occupy three structurally distinct parts of the saddle. The Gibbons–Hawking entropy SdS = A/4G is the real part of the on-shell action, exact and independent of the boundary data.The Gibbons–Hawking temperature T = H/2π is the imaginary part of the Euclidean cap proper time, which we evaluate in closed form and find to equal iπ/2H exactly. The evaluation rests on a clean identity: the no-boundary saddle equation is identical to the smooth-capping condition b2 = −4N2, from which the turning point of the geometry lands precisely on the de Sitter throat q = 1/H2 and the cap proper time collapses to (1/H) ln(±i) = ±iπ/2H. We further isolate the imaginary part of the lapse itself, Im N = H−2p1 − H2q0, and show it is not the temperature but the static-patch redshift structure carried at an area scale. A boxed caution records how a stable, confident, and entirely spurious numerical result for the cap time was caught only by an analytic evaluation. Together with the companion obstruction result [6]—that a microscopic substrate cannot generate these constants—this note shows the constants are present, exactly and in full, on the gravitational side: the no-boundary saddle already knows the de Sitter first law.