A Global Rigid Proper-Scale Clock from an Adelic Scalar

Abstract | A single spacetime-dependent scalar π(x^μ) taking values in the restricted adeles ℝ × ℤ₂ × ℤ₃ × ℤ₅ × ℤ₇ × ℤ₁₁ with six axioms defines a global rigid proper-scale clock π_scale that is exactly synchronized for every observer in the universe. The same construction derives the gravitational constant G (from residue at pointer 122) and dark-matter abundance ratio (DM ratio ≈ 5.3) without tuning. The prime bundle (5,7,11) defines the next transcendent irrational after π: the minimal quadratic core τ = √33 in [5,6) and maximal unit ε = 6 + √33 (gap shift δ = 6). This reproduces the observed DM ratio and G digits via Diophantine ansätze derived from dimensional indexing and continued fraction convergents. U(1)_Y coupling and the Einstein–Hilbert coefficient remain measured inputs; hypercharges are derived in Paper III.