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

=== Below are practical, progressive steps. I recommend building a small chain of calculations and checks you (and collaborators or I) can run. ===

==== 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).

==== Construct a simple integral for photon energy incident on causal sphere, approximate efficiency η\etaη of converting that energy to rest mass, then compute required η\etaη to produce Mu∼M_u \simMu∼ observed. Check whether integrated photon backgrounds (CMB, starlight, cosmic background) can supply such energy. ====

I can run this numerical estimate next if you want.

==== From the model’s predicted Geff(t)G_{\rm eff}(t)Geff(t) or Mu(t)M_u(t)Mu(t) compute G˙/G\dot G/GG˙/G and compare with observational limits. If the model predicts too-large variation, it’s ruled out or needs modification (e.g., time-variation only early on, or suppressed by conformal factors). ====

==== Try to derive α or α⁻¹ from an explicit geometrical construction on the sphere (as you’ve been doing for golden angles). This is harder, but the aim is to show α arises from surface geometry counting or angular packing arguments once the microscopic scale is set by ℓ\ellℓ. ====