Cross-Scale Constraints on Higher-Dimensional Gravity: From Sub-Millimeter Screening to Emergent Spectral Networks

This preprint presents a comprehensive theoretical analysis of higher-dimensional gravitational mechanics across four scale regimes spanning fifty orders of magnitude (\bm{10^{-35}\text{ m}} to \bm{10^{20}\text{ m}}). By testing the mathematical stability and observational constraints of extra spatial dimensions under uncompactified power-law leakage, Kaluza-Klein screening, Randall-Sundrum warping, and discrete quantum spectral networks, we establish a scale-dependent boundary framework for physical space.

Key results include:

Derivation of a strict upper bound on uncompactified \bm{4\text{D}} leakage from Mercury's orbital precession (\bm{\epsilon \le 3.09 \times 10^{-8}\text{ AU} \approx 4.62\text{ km}}).

Evaluation of sub-millimeter torsion-balance bounds (\bm{R \le 30\text{--}50\,\mu\text{m}}).

Proof that \bm{5\text{D}} warped bulk dynamics (\bm{1/r^4} force decay) cannot account for galactic dark matter.

Demonstration that discrete spectral networks exhibit dimensional reduction (\bm{d_s = 2 \to 3}), curing short-range gravitational singularities while restoring macroscopic classical gravity.