Da Vinci's Law of Fluid Dynamic Unity: An Operational Cross-Medium Invariance Law with Canonical Test Protocols

This work formalizes Da Vinci’s Law of Fluid Dynamic Unity (DL-FDU) as an operational and falsifiable framework for cross-medium fluid dynamic similarity. Classical similarity theory establishes that flows governed by the incompressible Navier Stokes equations exhibit identical dimensionless behavior when control parameters and boundary conditions are matched. DL-FDU extends this mathematical principle into an empirical validation structure by specifying explicit control parameter vectors (Π), observable sets (O), tolerance maps (δ), and a quantitative falsification rule. Three canonical test cases are developed in full detail: the circular cylinder wake at Re = 10⁴, the sphere drag crisis across 10⁴ < Re < 10⁶, and NACA 0012 airfoil lift drag polars for 3×10⁵ < Re_c < 10⁶. For each case, geometry admissibility conditions, matched dimensionless control parameters, operational definitions of observables, uncertainty propagation procedures, and tolerance thresholds are defined. DL-FDU is falsified if a cross-medium difference exceeds both the declared tolerance and the combined measurement uncertainty (Δ_i > δ_i and Δ_i − U_i > 0). A systematic literature execution attempt reveals a structural reporting barrier: while extensive benchmark data exist for both air and water experiments, published studies rarely provide observable level uncertainty budgets with sufficiently detailed control parameter matching to enable formal cross medium validation. The resulting limitation arises from reporting conventions rather than from any identified breakdown of classical similarity. DL-FDU therefore introduces an operational metrological layer bridging mathematical similarity and empirical verification. The canonical specifications are complete and experimentally executable. Empirical validation requires purpose built replicate based experiments under matched control parameters and declared uncertainty reconciliation. Version 1.0 — Canonical Specification.