Functional Accumulation of Energy–Momentum Trajectories Precludes Background Spacetime

This paper demonstrates that the standard spacetime-first formulation of relativistic physics contains a fundamental structural circularity. Using Dirac’s reparametrization-invariant first-order Hamiltonian formalism, we show that spacetime coordinates arise as accumulated functionals of the energy–momentum trajectory and therefore cannot serve as an independent geometric background. Proper time and worldline geometry emerge only after gauge choice and are not primitive geometric inputs.

This dependency inversion clarifies long-standing tensions involving background dependence, the problem of time, and generally covariant observables. It establishes energy and momentum as the logically prior variables and positions spacetime as a derived relational construct. The paper concludes by indicating how this viewpoint naturally extends to a background-free geometric framework defined on the Spectral Manifold, where Lorentzian structure appears directly in (E, p).

This work serves as a conceptual bridge between classical relativistic mechanics and emerging energy-first approaches to spacetime and geometry.