A smooth beginning for spacetime I: The thermodynamic bridge

Paper I of the trilogy "A smooth beginning for spacetime." A single complex-lapse saddle of the Lorentzian no-boundary path integral is shown to carry the complete de Sitter thermodynamics — the Gibbons–Hawking entropy in the real part of the on-shell action, the temperature in the imaginary cap proper time (exactly iπ/2H), and the observable local thermodynamics (Tolman temperature and Brown–York quasi-local energy) in the imaginary lapse. The organising fact is the capping identity b² = −4N²: the condition for a smooth beginning and the condition for a saddle are one equation. The first law dE = T dS = V dP and the Smarr relation close on saddle data alone. All three ledgers are read at the de Sitter throat, the surface the companion paper promotes to the fluctuation data boundary. The record includes the preprint, all figures, and six self-contained Python verification scripts with their logs; every quantitative claim is gated against analytic or independently computed values (tolerances 1e-9 to 1e-16). Drafting and computation were assisted by a large language model (Claude, Anthropic); the author takes sole responsibility for the content. No specific funding was received.