The Bondi k-Factor as Quantum Operator: A Two-Sector Decomposition of the Lorentz Transformation

We promote the Bondi k-factor to an operator k̂ = (Ĥ_cm + p̂c)/(mc²) and prove that the Lorentz transformation decomposes into two sectors with structurally distinct quantum behaviour: a time dilation sector k̂ + k̂⁻¹ = 2Ê/(mc²), nonlinear in momentum, and a simultaneity sector k̂ − k̂⁻¹ = 2p̂c/(mc²), linear in momentum.

The asymmetry is proved via Jensen's inequality. Rest mass m > 0 bounds the energy spectrum below, making E(p) strictly convex. Convexity simultaneously produces Jensen corrections, Pikovski decoherence, and the Pauli obstruction — all confined to the time dilation sector. The simultaneity sector is protected exactly by linearity. The decomposition is universal across physically distinct clock types (light-bounce, radioactive decay, gravitational pendulum): which sector contributes corrections and which is protected is independent of the clock mechanism.

The decomposition provides a structural explanation for the angular separation of the quantum corrections computed by Grochowski et al.: the time dilation sector and the simultaneity sector map onto their "quantum time dilation" and "quantum Doppler shift" contributions, separable by angular detection geometry. The framework does not predict these corrections but organises them under a single structural cause. All corrections vanish in the classical limit σ_p → 0, recovering the Bondi derivation of Part I exactly. The axiomatic and categorical foundations are developed in Part II.