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

==== A direct QAT toy derivation (worked numerically earlier): ====
* Let causal radius R(t)=ctR(t) = c tR(t)=ct.
* Count angular modes up to cutoff length ℓ\ellℓ (choose microscopic length scale, e.g., proton reduced-Compton ℓ=ℏ/(mpc)\ell=\hbar/(m_p c)ℓ=ℏ/(mpc)):

N(t)∼4πR(t)2ℓ2∝t2.\mathcal{N}(t) \sim \frac{4\pi R(t)^2}{\ell^2} \propto t^2.N(t)∼ℓ24πR(t)2∝t2.
* Assign a small mass per mode m⋆=αmmpm_\star = \alpha_m m_pm⋆=αmmp. Then

Mu(t)∼m⋆ N(t)∼αmmp 4π(ct/ℓ)2.M_u(t) \sim m_\star \, \mathcal{N}(t) \sim \alpha_m m_p \, 4\pi (c t/\ell)^2.Mu(t)∼m⋆N(t)∼αmmp4π(ct/ℓ)2.
Numerical evaluation (done earlier) gave αm≈1.06×10−5\alpha_m \approx 1.06\times10^{-5}αm≈1.06×10−5 if you match observed cosmic mass today. So QAT can reproduce Dirac’s Mu∝t2M_u \propto t^2Mu∝t2 scaling with a modest dimensionless factor.

Energy budget check (crucial): converting today’s photon backgrounds into present mass is impossible: CMB + starlight energy is short by factors ∼6×103\sim 6\times10^3∼6×103–2×1042\times10^42×104. That forces either:
* mass creation occurred mainly in early epochs when photon energy densities were enormous, or
* the "mass per mode" is bookkeeping for previously converted energy, not continuous conversion now.

Status: scaling fits; energy budget and observational constraints (BBN, CMB) must be checked carefully for early-time conversions.