Hacking the Colebrook-White equation: The Tolentino-𝒁𝒁 method via algebraic collapse and its bit-to-bit equivalence in 64-bit architectures

The resolution of the Colebrook-White equation has historically been a stability challenge in Computational Fluid Dynamics (CFD) due to its transcendental and implicit nature. This work presents the Tolentino-Z method (𝑇𝑇𝑧𝑧), an explicit sequential resolution architecture based on an algebraic collapse that eliminates the need for conventional iterative schemes. Unlike iterative methods that depend on stopping criteria, the 𝑇𝑇𝑧𝑧 method is structured as a fractional sequential equation with a closed two-stage cycle (𝑍𝑍1, 𝑍𝑍2), composed of three concatenated stages: generation of calibrated explicit logarithmic seeds 𝑆𝑆𝑘𝑘; execution of the Z-refinement operator (Z-Engine), which is an explicit rational operator derived from the algebraic collapse of the Newton-Raphson iteration; and direct recovery of the friction factor 𝑓𝑓. The parameters b = 2.25 and n = 0.89 of the base seed were calibrated using a systematic grid search in Python 3.12. The results demonstrate that the 𝑍𝑍2 configuration with 𝑆𝑆2 seed, over the range of relative roughness from 0.0 to 0.05 and Reynolds number from 4×10³ to 1×10⁸, achieves a maximum relative error of 2.74×10⁻¹⁷, attaining bit-to-bit equivalence under the IEEE 754 standard. This precision exhausts the 52-bit mantissa in 64-bit architectures, guaranteeing absolute stability and asymptotic convergence over the entire domain of Reynolds number and relative roughness. The Tolentino-Z method constitutes a deterministic, explicit alternative for high-reliability engineering applications, digital twins, and industrial software with high energy efficiency.