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

Abstract

Lorentz transformations are linear on Minkowski spacetime but induce Möbius maps on velocity, null-direction, celestial, and spinor ratios. Starting from this correspondence, we introduce the active model lift B = λA ∈ GL(2,ℂ), with A ∈ SL(2,ℂ) and κ = |λ|. The scalar factor does not alter the projective Möbius action generated by A, while, under the momentum-spinor postulate, Hermitian momentum bilinears acquire the factor κ². For a fixed domain with uniform κ and Lorentz map and homogeneous reference momentum density, this yields PΩ,r = Cᵣκ²VΩ; equivalent effective cells give Pₙ = nP₁. Timelike states connected by the lift define a conditional microscopic estimator of κ from invariant-mass ratios without Hubble-like or Modular-Weierstrass input. A companion directional model supplies a distinct macroscopic estimator from a Hubble-like benchmark, a Planck-duration impulse, and modular closure. Under the stated benchmark choices, both routes give κ ∼ 10¹⁴. Their non-exact agreement is reported as conditional cross-scale consistency, not experimental validation. Kinematically, the scalar lift preserves nullness and adds no projective light-direction change beyond the Lorentz sector.