Local Multifractional Lorentz Spacetime at the Planck Scale: Causal Singularity Resolution and Local Information Conservation Without Holographic Projections

Description / Abstract: This follow-up work systematically resolves the symmetry (Lorentz invariance violation) and causality limitations of the previously proposed Planck-scale discrete fractal spacetime model (Sierpiński topology) by embedding the framework into a continuous, multifractional Lorentz geometry with weighted spacetime measures $d\mu(x)$. Key Findings & Theoretical Highlights: Causality & Lorentz Geometry: By replacing discrete lattice structures with a scale-dependent integration measure, the pseudo-Riemannian signature $(-+++)$ and local causality are strictly preserved. Directional anisotropy is eliminated as classical Lorentz symmetry smoothly transitions into Doubly Special Relativity (DSR) at high energy scales. Dimensional Flow & UV Convergence: The effective spectral dimension continuously runs from $d_S = 4$ at macroscopic scales down to $d_S \approx 2$ at the Planck scale ($l_{Pl}$). This guarantees autonomous ultraviolet (UV) convergence of loop integrals in quantum field theory and strictly caps curvature invariants. Singularity Resolution & Local Information Conservation: At the Planck scale, asymptotic freedom of gravitational coupling prevents gravitational collapse into point singularities. Applying this geometry to evaporating black holes, the local information capacity of the multifractional measure resolves the Hawking information paradox in a strictly unitary and local manner—bypassing the need for global holographic projections. Experimental Testing Protocols: Formulates concrete verification protocols via continuous index-gradient metamaterials in analogue gravity setups and cosmological stochastic gravitational wave background (SGWB) spectral signatures at high frequencies ($f \sim 10^{11}\text{ Hz}$).