Formal Derivation of the Expansion Equation in the D10Z Model: From 𝐹 = 𝑓 ⋅ 𝑣 ( 𝑍 𝑛 ) F=f⋅v(Z n ) to 𝐻 ( 𝑡 ) = 𝐻 0 𝑒 𝛾 𝑍 𝑛 𝑡 H(t)=H 0 e γZ n t

This manuscript presents the formal derivation and empirical validation of a new cosmic expansion law grounded in the D10Z Mechanics of Infinity. Unlike ΛCDM, which relies on dark energy and inflation, the D10Z model derives expansion directly from nodal vibrational dynamics within a quantum fractal mesh (TTAϵ), the Spiderweb Fabric (SWF).The equation: H(t)=H0⋅eγZntH(t) = H_0 \cdot e^{\gamma Z_n t}H(t)=H0⋅eγZnt is obtained from the D10Z mother law F=f⋅v(Zn)F = f \cdot v(Z_n)F=f⋅v(Zn), interpreting the Hubble parameter as proportional to a fractal nodal activation state ZnZ_nZn. Key results include: A full derivation under the assumption of nodal coherence stability dZn/dt≈0dZ_n/dt \approx 0dZn/dt≈0. Parametrization of Zn(z)=αlog⁡(1+z)+ζ0Z_n(z) = \alpha \log(1 + z) + \zeta_0Zn(z)=αlog(1+z)+ζ0 fitted to real cosmological data. An empirical match to Hubble tension: the model explains H0loc/H0CMB≈1.083H_0^{\text{loc}}/H_0^{\text{CMB}} \approx 1.083H0loc/H0CMB≈1.083 as arising from nodal phase difference alone. Validation metrics: R2=0.9992R^2 = 0.9992R2=0.9992 and better performance than ΛCDM in χ², BIC, and AIC. This framework eliminates the need for external energy components, providing an internal, resonant explanation for cosmic expansion. Keywords:Cosmic Expansion, Hubble Tension, D10Z Mechanics, Manual of Infinity, Nodal Vibration, Spiderweb Fabric, Fractal Geometry, TTAϵ, Z-parameter Cosmology, Exponential Universe, Dark Energy Alternative.