The Lineage Ruler: From Binary Recursion to the Periodic Table

The electron shell capacities (2, 8, 18, 32, 50...), periodic pairing structure (2, 8, 8, 18, 18, 32, 32...), subshell capacities (s=2, p=6, d=10, f=14...), group column widths (2, 6, 10, 14...), and nuclear magic numbers (2, 8, 20, 28, 50, 82, 126...) have long been regarded as mutually independent experimental data, separately explained by different theoretical frameworks such as quantum mechanics and the nuclear shell model. This paper is based on the empirical fact of spatial asymmetry between the atomic nucleus and the electron cloud, and proposes that macroscopic objects possess an extremely tiny net residual charge difference, on the order of 10 to the minus 52 coulombs. From this starting point, we establish the integer power relations between the strengths of the four fundamental interactions and the fine-structure constant alpha: 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. The powers 0, 6, 14, and 18 trace back to the lineage digits {0,3,7,9}, from which the binary recursive rule x goes to 2x+1 and 2x+3 emerges. This paper demonstrates that all of the above numerical features can be uniformly represented by the same simple binary recursive rule. This rule generates all odd integers, and through simple operations such as natural pairing summation, left-right branch mapping, and multiplication by 2, all of the above sequences can be obtained. This paper also identifies a purely numerical cross-domain coincidence: the total number of nodes at depth 7 of the recursive tree is 2 to the 7th power, which equals 128; adding the lineage digit 9 yields the number 137, which coincides with the integer part of the reciprocal of the fine-structure constant, approximately 137.036. The number 137 is also the theoretical upper limit of the nuclear critical charge in the Dirac equation. This paper also presents two testable predictions concerning the position of new magic numbers in superheavy nuclei and the truncation of the 8th period of the periodic table. This study offers only a parallel mathematical perspective and does not construct a new dynamical theory.