Signature Failure and Minimum Spatial Quanta: Shear-Induced Quantum Boundaries for Gravitational Path Integrals

Abstract

We investigate whether quantum metric fluctuations can destroy the differentiable Lorentzian support required for spacetime. We conjecture that non-Lorentzian metric elements are absent from the physical manifold and restrict admissible histories to a single continuing, causally connected four-dimensional geometry. To explore this proposal, we extrapolate the Gaussian transverse-traceless graviton vacuum to an additive metric ensemble with physical Fourier cutoff $k_{\max }=1/q$. A lattice model with local repair and selection of periodically winding differentiable-state components gives a minimum surviving scale $q_{\min }\simeq 1.9{\ell }_{\mathrm{P}}$ over the simulated duration. We then examine two complementary curvature-induced deformations. For the Schwarzschild electric Weyl field, the smallest boundary radius at the center of a black hole, occurring when $q\simeq 2.5{\ell }_{\mathrm{P}}$, is $r_{\mathrm{qb}}\simeq 2.62C_W^{1/3}(r_g{\ell }_{\mathrm{P}}^2{)}^{1/3}\simeq 1.91\times 10^{-22}{C}_W^{1/3}{\left(M/{\mathrm{M}}_{\odot }\right)}^{1/3}\mathrm{m}$, where $r_g=GM/c^2$ and $C_W$ parameterizes the finite-$q$ Weyl response. For homogeneous radiation, a supplementary Gaussian four-metric ensemble subject to linearized de Donder conditions gives ${\rho }_{\mathrm{qb}}/{\rho }_{\mathrm{P}}\simeq 1.31\times 10^{-2}/C_R$ at $q=2.5{\ell }_{\mathrm{P}}$, where $C_R$ is the corresponding Ricci-response coefficient. The local homogeneous criterion is unchanged under reversal of the expansion rate. These model-dependent thresholds motivate a possible replacement of classical singular continuations by finite-curvature quantum boundaries.

12 pages, 3 figures, 2 Python scripts