Lean-Verified Chronos Physics Framework: Temporal Carrier, Source Law, and Formal Calibration — Milestone 3

This record contains the publication-facing verification materials and reproducible Lean 4 certificate packages for the continuing formal validation of the Chronos Navier-Stokes program, extending the original unified clock milestone through certificate versions v1.2–v1.10. The formal development is compiled against OpenAI's public NavierStokesAndEuler repository at the pinned upstream commit using Lean 4.34.0-rc2. It begins from OpenAI's actual selected periodic Navier-Stokes candidate and extends the previously verified Chronos change, arrow, action, and aim clock into a broader temporal-field framework. The formally verified development now includes the closed temporal-exchange balance, the Chronos selector value, an exact selector-defined temporal carrier satisfying, temporal diffusion and relaxation dynamics, a nonnegative temporal-field arrow under the stated sign conditions, an independent change-arrow-aim clock for the temporal carrier, and complete oriented temporal histories including the initial endpoint. The later certificates introduce a general temporal source-law interface and formally analyze a nonlinear radial saturation source law of the form The verified results include its exact energy/action decomposition, nonlinear saturation contribution, nonnegative net arrow under the stated parameter conditions, closed instantaneous and historical clock identities, and an exact necessary-and-sufficient calibration criterion. The calibration theorem separates source identification into zero-state compatibility, directional alignment, and scalar magnitude. The formal development also proves a parameter-free transverse alignment condition and establishes that the radial law depends on the difference between drive and damping rather than identifying those two parameters independently. Taken together, versions v1.2–v1.10 extend the Chronos formal program from a verified Navier-Stokes clock decomposition into a formally connected temporal exchange, selector, carrier, field-dynamics, source-law, and calibration architecture. The uploaded certificate archives contain Lean source modules, reproduction scripts, SHA-256 checksums, build logs, provenance information, and dedicated axiom-audit results. The certificate series was reproduced in the pinned Lean/OpenAI environment with no sorry, admit, or sorryAx dependencies in the audited Chronos declarations.