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

===== Assume the Universe can be modelled as causal spheres of radius R(t)=c tR(t)=c\,tR(t)=ct and that each independent angular quantum (mode) on the boundary contributes a fixed mass unit m⋆m_\starm⋆ (emergent mass per mode). The number of modes up to angular resolution set by a microscopic length scale ℓ\ellℓ (for example, proton Compton length λp=h/(mpc)\lambda_{p}=h/(m_p c)λp=h/(mpc)) is =====

Nmodes(t)  ∼  4πR(t)2ℓ2  ⇒  Mu(t)  ∼  m⋆ 4πc2t2ℓ2.\mathcal{N}_{\rm modes}(t)\;\sim\; \frac{4\pi R(t)^2}{\ell^2} \;\Rightarrow\; M_u(t)\;\sim\; m_\star\,\frac{4\pi c^2 t^2}{\ell^2}.Nmodes(t)∼ℓ24πR(t)2⇒Mu(t)∼m⋆ℓ24πc2t2.
This gives Mu∝t2M_u\propto t^2Mu∝t2 directly.
* Choose ℓ\ellℓ as a fixed microscopic length (Compton wavelength), then

Mu(t)mp∼4πm⋆mp(ctℓ)2.\frac{M_u(t)}{m_p}\sim \frac{4\pi m_\star}{m_p}\left(\frac{c t}{\ell}\right)^2.mpMu(t)∼mp4πm⋆(ℓct)2.
If m⋆∼mpm_\star\sim m_pm⋆∼mp up to constants, you recover the Dirac large-number squares N∼(ct/ℓ)2N\sim (ct/\ell)^2N∼(ct/ℓ)2.

Strengths: Simple, geometric, matches Dirac scaling exactly.
Weaknesses: Why should mass be assigned per angular mode? Conservation issues: where does the energy come from? Must specify m⋆m_\starm⋆ mechanism and explain local matter-energy conservation.