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

=== Below are three compact toy models capturing different possible mechanisms. Each is deliberately simple — they expose the assumptions you’ll want to test. ===

===== 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.

===== Assume an object (or collection of atoms across the Universe) converts incident photon flux into mass (bound energy) at a small efficiency η\etaη. The total rate of mass creation inside the causal sphere is =====

dMudt  =  ηc2∫S(R) ⁣ ⁣F(Ω,t) dA,\frac{dM_u}{dt}\;=\;\frac{\eta}{c^2}\int_{S(R)} \!\!\mathcal{F}(\Omega,t)\, dA,dtdMu=c2η∫S(R)F(Ω,t)dA,
where F\mathcal{F}F is the incoming photon energy flux per unit area on the causal sphere. If the total incoming flux at radius RRR scales like F(t)F(t)F(t) and area ∝ t2t^2t2,

dMudt∼ηc2 F(t) 4πc2t2  =  4πη F(t) t2.\frac{dM_u}{dt}\sim \frac{\eta}{c^2} \, F(t)\,4\pi c^2 t^2 \;=\; 4\pi\eta\, F(t)\, t^2.dtdMu∼c2ηF(t)4πc2t2=4πηF(t)t2.
If F(t)F(t)F(t) decays slowly enough — e.g., F(t)∝1/tF(t)\propto 1/tF(t)∝1/t — then dM/dt∝tdM/dt\propto tdM/dt∝t and integrating gives M∝t2M\propto t^2M∝t2.

One specific sub-case: if photon energy density at the causal surface scales as uγ∝1/R∝1/tu_\gamma \propto 1/R \propto 1/tuγ∝1/R∝1/t, then flux F∼cuγ∝1/tF\sim c u_\gamma\propto 1/tF∼cuγ∝1/t and the model yields M∝t2M\propto t^2M∝t2.

Strengths: Provides a local physical process (photon capture) to create mass.
Weaknesses: Need a source for the photon bath and a mechanism to produce the required slow decline uγ∝1/tu_\gamma\propto 1/tuγ∝1/t; must avoid violating energy conservation observational constraints.

===== Dirac’s large ratio typically written as =====

e24πε0Gmp2∼N1/2.\frac{e^2}{4\pi\varepsilon_0 G m_p^2} \sim N^{1/2}.4πε0Gmp2e2∼N1/2.
If QAT gives a geometric way to compute the effective gravitational coupling GeffG_{\rm eff}Geff from photon–electron surface interactions, one could posit

Geff(t)  ∼  e24πε0mp2 1N(t),G_{\rm eff}(t)\;\sim\;\frac{e^2}{4\pi\varepsilon_0 m_p^2}\,\frac{1}{\mathcal{N}(t)},Geff(t)∼4πε0mp2e2N(t)1,
with N(t)∼(ct/ℓ)2\mathcal{N}(t)\sim (c t/\ell)^2N(t)∼(ct/ℓ)2. Then Geff∝1/t2G_{\rm eff}\propto 1/t^2Geff∝1/t2 or (with different counting) Geff∝1/tG_{\rm eff}\propto 1/tGeff∝1/t depending on the model, producing Dirac-style scalings.

Strengths: Directly ties gravity’s strength to global surface geometry.
Weaknesses: Observations place strong constraints on G˙/G\dot G/GG˙/G; typical allowed variation is very small (current limits ~10⁻¹³–10⁻¹⁴ /yr). Any QAT-derived time-dependence would need to respect those constraints.