$$m_e={2R_{\infty}h\over c\alpha^2}$$

$$m_e{\alpha^2\over R_{\infty}}={2h\over c}$$

$$m_e{\alpha^2\over \pi R_{\infty}}={2h\over \pi c}$$

Let $r_e={\alpha^2\over\pi R_{\infty}}$, then:

$$m_er_e={2h\over\pi c}$$

Now, because for every action there is an equal and opposite reaction, for every force there is an equal and opposite force, for every torque, there is an equal and opposite torque, equate $m_pr_p$ to $m_er_e$ to balance torque/spin between proton and electron:

$$m_er_e={2h\over\pi c}=m_pr_p$$

$$\therefore {m_p\over m_e}={r_e\over r_p}={\alpha^2\over\pi r_pR_{\infty}}=1836.15267$$

Where:

$$m_pr_p={2h\over\pi c}=4\ell m_{\ell}$$

$$r_p=0.841235640294664\;fm$$

$m_p=$ proton mass

$m_e=$ electron mass

$r_p=$ proton radius

$r_e=$ effective torque arm radius for electron

$\alpha=$ fine-structure constant

$h=$ Planck constant

$c=$ speed of light

$R_{\infty}=$ Rydberg constant

$\ell=$ Planck length

$m_{\ell}=$ Planck mass

QED.