Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

==== Pick one clean QAT hypothesis, for example Model 1 (surface-counting mass). Compute: ====
* Choose ℓ\ellℓ = proton reduced Compton wavelength λˉp=ℏ/(mpc)≈1.32×10−15\bar\lambda_p=\hbar/(m_p c)\approx 1.32\times10^{-15}λˉp=ℏ/(mpc)≈1.32×10−15 m.
* Let m⋆=αm mpm_\star = \alpha_m\, m_pm⋆=αmmp (some fraction αm\alpha_mαm of proton mass per mode).
* Evaluate Mu(t0)=4παmmp(ct0/ℓ)2M_u(t_0)=4\pi \alpha_m m_p (c t_0/\ell)^2Mu(t0)=4παmmp(ct0/ℓ)2; compare with measured mass in horizon Mobs∼ρc×43πRu3M_{\rm obs}\sim \rho_c \times \tfrac{4}{3}\pi R_u^3Mobs∼ρc×34πRu3.

Solve for αm\alpha_mαm: if αm\alpha_mαm is O(1) the model is parsimonious; if αm\alpha_mαm is astronomically small or large that tells you whether the idea is plausible.

I can do this computation now and show numbers (it’s quick).