The Lineage Ruler: From Binary Recursion to Quantum Gravity

This study is based on the empirical fact of spatial asymmetry between the atomic nucleus and the electron cloud: an external observer is closer to the electrons and farther from the nucleus. Combined with Coulomb's law, where closer distances yield stronger interactions and farther distances yield weaker ones, this leads to the proposal that macroscopic objects possess an extremely tiny net residual charge difference, on the order of 10 to the minus 52 coulombs. Starting from this point, we establish the integer power relations between the strengths of the four fundamental interactions and the fine-structure constant alpha. We then adopt the binary recursive transformation x goes to 2x+1 and 2x+3 to construct the system of lineage digits. The strengths of the four fundamental interactions are expressed as integer powers of alpha multiplied by rational coefficients: gravity as alpha to the 0, the weak interaction as alpha to the 6, electromagnetism as 4/3 times alpha to the 14, and the strong interaction as 3/4 times alpha to the minus 18. Loop Quantum Gravity still faces several open theoretical difficulties: the area spectrum depends on the free Immirzi parameter, which cannot be determined by the theory itself; the dynamical equations contain no explicit time variable; and there is a lack of directly testable experimental predictions. This paper extends the recursive rule to the Planck scale in quantum gravity, providing a numerical description scheme without adjustable fitting parameters. The layer number of the lineage binary tree can serve as intrinsic discrete time. The microstate counting of binary branches at the punctures of spin networks can replace the traditional SU(2) group representation derivation. The main results include: the area quantization formula A_n equals n squared times the natural log of 16 times the Planck length squared, with n odd, consistent with the Bekenstein-Hawking entropy; the tree layer number can serve as intrinsic time, from which the Schrödinger equation can be derived in the continuum limit; and the gravitational wave echo time interval is predicted as 2 pi G M divided by c cubed times the natural log of 16, with CMB primordial distortion as a second independent prediction. This paper provides an empirical description scheme without adjustable fitting parameters for the parameter freedom, time problem, and experimental testability of loop quantum gravity.