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

=== Below are short mathematical sketches showing how QAT maps to known frameworks and where work remains. ===

==== Spherical geometry gives the inverse square law trivially: for flux Φ\PhiΦ radiated from a point, ====

Φ(r)=P4πr2.\Phi(r) = \frac{P}{4\pi r^2}.Φ(r)=4πr2P.
If a test body intercepts an effective cross-section σ\sigmaσ of the outward photon momentum flux, the force is

F∼σΦ(r)c∝1r2.F \sim \sigma \frac{\Phi(r)}{c} \propto \frac{1}{r^2}.F∼σcΦ(r)∝r21.
QAT reading: outgoing photon momentum normal to the sphere (Huygens wavefront) produces an effective 1/r² geometry; an emergent gravitational attraction can arise from net inward-directed EM momentum transfer when absorption/emission patterns are coherent and asymmetric across many atoms.

Status: geometrical origin of 1/r² is robust; turning that into a universal attractive force with the precise magnitude GGG is the nontrivial step.

==== - Standard QM: Born’s rule ρ(x)=∣Ψ(x)∣2\rho(x) = |\Psi(x)|^2ρ(x)=∣Ψ(x)∣2. ====
* QAT: the electron probability cloud is literally an area density on a spherical shell (4πr²). Many features follow naturally: - Squaring (Ψ²) because amplitude → power/energy density over area. - Quantization (discrete energy levels) from boundary conditions on the spherical shell (standing waves on S² → spherical harmonics YℓmY_{\ell m}Yℓm). - Electron spin-½ emerges naturally when the correct spinor boundary conditions are used (Dirac spinors on S² require 4π rotation).
* Formal step needed: replace hand-waving with a boundary quantum field theory: derive boundary Dirac/Schrödinger operators on S², couple to Aμ, enforce energy quantization.

Key equation targets to derive:
* Dirac equation on S² with minimal coupling (iγμ(∇μ−ieAμ)−m)ψ=0(i\gamma^\mu(\nabla_\mu - ieA_\mu) - m)\psi=0(iγμ(∇μ−ieAμ)−m)ψ=0.
* Show boundary eigenmodes produce observed atomic spectra and Heisenberg uncertainty ∆x∆p ≥ ℏ/2 arises from finite mode support.

==== Two steps give a rigorous link: ====

(A) Build a surface stress–energy SabS_{ab}Sab from photon–electron boundary interactions (energy per unit area σ, tangential pressure p, and flow j):

Sab=σuaub+p hab+jaub+jbua,S_{ab} = \sigma u_a u_b + p \, h_{ab} + j_{a}u_{b} + j_{b}u_{a},Sab=σuaub+phab+jaub+jbua,
where habh_{ab}hab is the induced metric on the 2-surface and uau^aua the local 4-velocity.

(B) Use the Israel junction condition to relate a thin shell’s surface stress tensor to the jump in extrinsic curvature KabK_{ab}Kab and thus to spacetime curvature:

[Kab]−hab [K]=−8πG Sab.\big[ K_{ab} \big] - h_{ab}\, \big[ K\big] = -8\pi G \, S_{ab}.[Kab]−hab[K]=−8πGSab.
This is a standard GR result for surface layers. Therefore if QAT yields a well-defined SabS_{ab}Sab coming from area-based photon energy, it will generate curvature (gravity) exactly as GR prescribes for thin shells. The effective long-range metric (Schwarzschild-like) can then be found by solving Einstein’s equations outside the shell.

How to connect to standard stress-energy Tμν: smear the surface into a thin volume with thickness δ and define

Teffμν(x)≈σ(θ,ϕ) δ(r−r0)δ uμuν+⋯ ,T^{\mu\nu}_{\rm eff}(x) \approx \sigma(\theta,\phi)\, \frac{\delta(r-r_0)}{\delta}\, u^\mu u^\nu + \cdots,Teffμν(x)≈σ(θ,ϕ)δδ(r−r0)uμuν+⋯,
then plug into Rμν−12Rgμν=8πGTμνR_{\mu\nu}-\tfrac12 R g_{\mu\nu}=8\pi G T_{\mu\nu}Rμν−21Rgμν=8πGTμν.

Status: the mathematical toolset is standard (Israel junctions); QAT's job is to produce explicit SabS_{ab}Sab from microphysics (photon exchange rates → σ and p). That is the critical technical step left undone.

==== 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.