Lorentzian Regime I of Timeline-Relative Quantum Collapse (TRQC)

Lorentzian Regime I of Timeline-Relative Quantum Collapse develops the continuum realization of TRQC on an effective Lorentzian manifold. The discrete collapse field becomes a real scalar field, structural time becomes a dynamical scalar field, and observers are represented by a unit timelike vector field. A collapse-active domain is defined where structural time is physically valid and collapse roughness and shear satisfy fixed thresholds. The paper introduces a structural time sector Lagrangian, corresponding field equations for structural time evolution, and a covariant collapse equation including observer coupling. The projection domain is expressed in invariant geometric form, and structural aging along observer worldlines reproduces relativistic differential aging as an emergent effect. Regime I bridges the metric-free foundations and relativistic field theory, showing how classical spacetime behavior and observer dynamics arise within collapse-active regions of TRQC. Official project website: https://trqctheory.org