Classical Mechanics from a Resource Constraint: Newton's Laws, Gravitation, and the Lagrangian Formulation from the Budget Equation and a Cubic-Lattice Substrate

The standard structures of classical mechanics --- Newton I/II/III, conservation laws, the Lagrangian and Hamiltonian formalisms, and Newtonian gravitation --- are derived from the budget equation $I^2+R^2+C^2=1$ and the cubic-lattice substrate of the foundation paper~\citep{wittwer2026lattice-v1.0}, given explicit semiclassical, symmetry-lifting, and weak-field assumptions, without introducing the Newtonian axioms as independent postulates. The derivation yields the classical Lagrangian $L=T-V$ through the chain dissonance functional $\to$ master equation $\to$ Schr"odinger equation $\to$ Hamilton--Jacobi in the semiclassical eikonal limit; Links 1 and 2 are imported substrate-level results from companion work, and the present paper develops the resulting mechanical consequences. Newton's three laws then follow from the Euler--Lagrange equation, while the conservation laws follow by Noether inversion from polycrystalline substrate symmetries lifted to continuous $SO(3)$ by Haar averaging, with corrections of order $(L_g/L)^2 \lesssim 10^{-56}$. Newtonian gravitation is reconstructed from a Gauss-type argument for a conserved scalar mass source, with $G \propto c^3 a^2/\hbar$ obtained as a Buckingham-$\pi$ closure when the substrate length is identified with the Planck length. The scalar gravitational potential is assigned to the $A_{1g}$ sector of the substrate dissonance and sourced by universal trace-coupling to the stress-energy tensor; this source coupling is a working ansatz, not a microphysical derivation. Buckingham's $\pi$-theorem precludes a numerical prediction of $G$ from dimensional analysis alone, but a microdynamical derivation of the coupling prefactor plus one independent dimensional anchor would close the framework numerically. Classical mechanics thus appears as the macroscopic-soliton semiclassical limit of the lattice dynamics established in the companion papers.