m_d

m_d — down quark

Down is the isospin partner of up. The ratio $m_d / m_u$ comes from isospin rotation on the geometric cascade:

$$ m_d = m_u \cdot \frac{\sin^2\theta_5(q_-)}{\sin^2\theta_3(q_-)} \cdot R_d, $$

with $R_d \approx 2.415$ (negative-isospin u → d correction, App. P §C8).

Computation

$$ m_d = 2.156 \cdot \frac{0.11181}{0.12489} \cdot 2.415 = 4.656\ \text{MeV}. $$

PT: 4.656 MeV vs PDG: 4.67 ± 0.07 MeV. Gap: 0.29%.

The ratio m_d / m_u

$m_d / m_u = 2.160$. Dimensionless, directly comparable: PDG gives ~2.16, PT 2.160. This ratio is a clear signature of the q_- geometric branch.