Abstract | The same single scalar π(x^μ) ∈ ℝ × ℤ₂ × ℤ₃ × ℤ₅ × ℤ₇ × ℤ₁₁ that defines the global proper-time clock in Paper I and the global proper-scale clock in Paper II now defines a third rigid clock: the global proper-color clock. The parity sequence of the natural numbers — odd/even — evaluated under the adelic norm defined by the five primes {2,3,5,7,11} generates a bounded, oscillating function whose generation-indexed assignment fixes the exact rational hypercharges of the Standard Model. No new axioms or free parameters are introduced. U(1)_Y hypercharges are derived exactly; simulations yield coupling normalization emerges from rigidity and renormalization-group evolution (RGE) flow, reproducing the measured value. U(1)_Y charges and coupling are no longer measured inputs — they are derived from the same arithmetic structure that yields the CKM matrix and the numerical value of G.

This is Paper III of a series of VII introducing the Residue Bootstrap Operator (RBO) and Four Clocks unification framework. Paper I: https://doi.org/10.5281/zenodo.18209655 Paper II: https://doi.org/10.5281/zenodo.18300361

License note: Initially published under CC-BY-NC-ND 4.0. License will be upgraded to CC-BY 4.0 (full open access) from version 2 onwards.