The Mathematics of Physical Change: Calibrated Clocks and Lean-Checked Bridges Across Gravity, Quantum Dynamics, Fluid Mechanics, and Computation

Can temporal change be represented by a structured field or observable from which calibrated clocks and geometric relationships can be constructed? This paper investigates that question through a mathematical synthesis of Matthew J. Hall’s Chronos developments and their archived Lean 4 verification evidence. The central construction extracts a geometric Change clock from selected solutions of a 55-component nonlinear Fourier evolution system. Its normalized readout has zero initial value, unit initial right derivative, and arbitrarily accurate local first-order agreement with elapsed evolution time. Further results establish cross-solution first-order agreement, ordering of scale-separated events, quantitative finite-schedule separation, and robust event reconstruction under bounded perturbations. The paper examines connections to Lorentzian geometry, Einstein–scalar field equations, gravitational constraints, scalar stress-energy conservation, independent geometric Bianchi identities, and endpoint curvature. It also reviews radial time-field gravity, Navier–Stokes energy accounting and Aimed Action clocks, Wheeler–DeWitt Frozen-to-Flow bridges, quantum–classical shared-time correspondences, and computational Change clocks with explicitly imported complexity bounds. A unified mathematical interpretation identifies shared structures of observable change, accumulation, correspondence, and calibration while preserving each construction’s hypotheses and physical meaning. Two additional synthesis propositions receive conventional mathematical proofs. A detailed theorem inventory distinguishes archived Lean-checked results, source and audit evidence, mathematical deductions, and proposed physical interpretations. The accompanying source supplement provides theorem inventories, source excerpts, provenance records, verification evidence, and validated Zenodo references. The paper supports a mathematical program for constructing clocks within evolving systems without assuming a universal χ constant or claiming a general resolution of the Exponential Time Hypothesis.