A Möbius-Scale Spinor Lift of Lorentz Kinematics: The Contained-Volume Law for Four-Momentum

Lorentz transformations are linear on Minkowski spacetime but induce Möbius maps on velocity ratios, null directions, celestial coordinates, and spinor ratios. We develop this relation into a minimal Möbius-scale lift by replacing the Lorentz spinor matrix A ∈ SL(2,C) with B = λA ∈ GL(2,C), where λ = κe^(iθ). The projective Möbius action is unchanged, while Hermitian bilinears, including four-momentum, scale as |λ|² = κ². For a fixed domain in which the scale factor and Lorentz map are approximately uniform and the reference momentum density is approximately homogeneous, this yields the conditional law PΩ ∝ κ²VΩ. A possible discrete-space extension is also identified: if the contained volume is composed of identical elementary cells, VΩ = nV₀, the same law yields Pₙ = nP₁ in the uniform-cell limit, defining a discrete collective momentum ladder without claiming quantum-mechanical quantization in the present kinematics. A Modular–Weierstrass closure determines κ through elliptic invariants and produces inverse-square amplification near the square lattice. The scalar lift preserves the null cone and projective light directions, making the isotropic construction kinematically compatible with an unchanged local null structure, while any direction-dependent corrections remain subject to experimental Lorentz-isotropy bounds.