Poincarè Symmetries, Gravitoelectromagnetic Coupling, and Emergent Conservation Laws from Worldline Non-Injectivity

This paper derives two sets of results from the principle of worldline non-injectivity within the TPST–DGQ framework. A timelike worldline $X^\mu(\tau)$ with Lorentz factor $\gamma > \gamma_{\rm crit}$ intersects a constant-time hypersurface $\Sigma_t$ in $N > 1$ distinct spatial points, generating a multi-sheet structure of spacetime with topological phase offsets $\Phi_n = \gamma^2 v(\tau_n - \tau_1)$.

The first result concerns the Poincaré symmetries of classical physics. We prove that all four conservation laws — energy, momentum, angular momentum, and Lorentz boost invariance — emerge from the invariance of the topological average $\mathcal{W} = N^{-1}\sum_n \int \mathcal{L}^{(n)},dt$ under the corresponding transformations. The inter-sheet corrections to each conservation law involve $\langle\Delta_n\rangle = 0$ or $\langle\tilde{\Delta}_n\rangle = 0$ and vanish exactly by the symmetry of the fold distribution. The conservation laws are therefore derived consequences of the multi-sheet geometry, not postulated symmetries of a background spacetime.

The second result concerns the coupling between gravitation and electromagnetism. A gravitational field breaks the time-translation symmetry of the topological average by producing a non-zero inter-sheet phase gradient: $\partial_t\Phi_n|_{\rm grav} = GM\ell_0/(c^2 r^2)$. In the presence of an external magnetic field $B_z$, this phase gradient induces a fluctuation of the local electric field: $\sigma(\delta E_y) = GM B_z/(c^2 r^2)$. This gravitoelectromagnetic effect is of first order in Newton's constant $G$, in contrast to standard general-relativistic couplings between gravity and electromagnetism which arise at higher order. For terrestrial experiments the signal is below current sensitivity. For neutron stars with surface magnetic fields $B \sim 10^{12}$ T, the induced electric field reaches $\sim 21$ MV/m, a scale potentially relevant for pulsar magnetosphere electrodynamics.

Both results follow from the universal cancellation identity $N(\epsilon)\cdot\epsilon^{d-2} = O(1)$, which now operates at nine levels spanning holography, classical electromagnetism, quantum mechanics, thermodynamics, electromagnetic fields, gravity, quantum statistics, noncommutative spacetime geometry, and Poincaré symmetries. The paper is fully self-contained.

This manuscript is current in Official Peer Review.

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