Calculus-Generated Fiber Topology for Organ Genesis (CGFT): Differential Geometry and Integral Density Functions as the Mathematical Foundation of Clone-Fiber Organ Construction

A fiber is a line on a coordinate axis. A line is a parametric curve. A parametric curve is generated by calculus. This paper establishes the mathematical foundation of the Clone-Fiber Organ Genesis (CFOG) series using differential geometry and integral density functions. Core framework: Single fiber = parametric space curve r(t) = (x(t), y(t), z(t)); Tangent vector r'(t) determines cell alignment direction (Contact Guidance, mathematical form); Fiber network = volume integral of density function rho(x,y,z); N in V=N/D = triple integral of rho over organ domain; Density gradient nabla-rho = direction of Qi flow (mathematical identity with GMFT meridian theory). Organ-specific generation equations: Heart — Frenet-Serret helical curves with variable omega/R/p per layer; Liver — hexagonal tiling density function replicating hepatic lobule geometry; Spinal cord — parallel linear fiber bundles with identical parametric extension from both severed ends; Kidney — closed-loop curves with stereoscopic distortion coefficient for glomerular filtration membrane. V=N/D in calculus: V = (integral of rho dV) / (count of discontinuity points). V-maximization conditions: maximize rho (dense fiber design) + eliminate discontinuities (continuous fiber network). Mathematical unification with Eastern medicine: meridians = integral curves of nabla-rho field; acupoints = local maxima of rho; acupuncture = external field applied at rho maximum = gradient correction = Qi reconnection. Two thousand years of empirical medicine described by partial differential equations. This is the seventh and mathematically foundational layer of the unified invention series: ECCE, CPMH, SIBM, GMFT, FICI, CFOG, CGFT. Theoretical basis: V = N / D (Tendo Economics, Katayama 2026). Published as open knowledge base for mathematical biologists, computational anatomists, and bioprinting engineers. The equations are here. Compute the organ.