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

=== Below is a prioritized roadmap (mathematical + numerical + verification tasks). I’ll label tasks as (Math), (Numerics), (Check) so you can delegate or run them later. ===

A. Lagrangian & boundary formulation — make the 2-D spherical boundary the primary object. (Math)
* Write a bulk+boundary action Sbulk[A,ψ,… ]+Sshell[Φ,… ]S_{\rm bulk}[A,\psi,\dots]+S_{\rm shell}[\Phi,\dots]Sbulk[A,ψ,…]+Sshell[Φ,…]. The boundary Lagrangian should include a U(1) field coupling (surface current) and matter degrees of freedom confined to the 2-D shell.
* Example boundary coupling: surface current jsurfμj^\mu_{\rm surf}jsurfμ couples to the 3+1 AμA_\muAμ pulled back to the shell: Sint=∫shelld3σ  jμ(σ)Aμ(σ).S_{\rm int} = \int_{\text{shell}} d^3\sigma \; j^\mu(\sigma) A_\mu(\sigma).Sint=∫shelld3σjμ(σ)Aμ(σ). Use explicit coordinates on the sphere (θ,φ,t) and the embedding map into spacetime.
* Derive boundary Euler-Lagrange equations and exact boundary conditions for bulk fields (this gives the jump conditions like a pillbox across the shell). That produces the quantized boundary constraints.

B. Noether currents, U(1) symmetry and charge quantization. (Math)
* Use global U(1) to derive conserved surface current JμJ^\muJμ via Noether’s theorem. Show how the integral of J0J^0J0 over the shell gives discrete charge units — connect to “surface occupation” and why e emerges as a quantum of surface charge.
* Show how minimal coupling yields a gauge holonomy condition ∮A⋅dl=2πq/ℏ\oint A\cdot dl = 2\pi q/\hbar∮A⋅dl=2πq/ℏ or related boundary quantization. Use h/2πh/2\pih/2π explicitly rather than ħ if you want the 2π geometry visible.

C. Derive boundary Maxwell conditions (pillbox) and link to Gauss / shell theorem. (Math)
* Integrate Maxwell across thin shell (pillbox) to obtain exact normal field discontinuity: n^⋅(Eout−Ein)=σ/ε0\hat n\cdot(\mathbf{E}_{\rm out}-\mathbf{E}_{\rm in})=\sigma/\varepsilon_0n^⋅(Eout−Ein)=σ/ε0. This ties surface charge density σ to the shell geometry.
* Similarly derive tangential component conditions for A and E and relate them to the surface current and the boundary Lagrangian.

D. Relate surface charge density to e and geometric constants. (Math→Numeric)
* Propose a model: the minimal surface charge per surface cell (unit area patch) is an eigenvalue coming from the boundary eigenproblem (e.g., allowed normal modes on shell). Solve quantization of boundary modes → discrete σ_n. The smallest nonzero σ would be identified with e / (patch area).
* Estimate patch area using the action quantum: h/2πh/2\pih/2π sets a minimal angular phase cell on the sphere, area ∼ (h/2π)/(... geometry dependent), build algebraic relation to produce e.

E. Produce a closed expression for α (attempt). (Math)
* α is dimensionless: α=e24πε0ℏc\alpha = \dfrac{e^2}{4\pi\varepsilon_0 \hbar c}α=4πε0ℏce2. Map each symbol: - e2e^2e2 ↔ minimal surface charge squared from the shell quantization. - ℏ=h/2π\hbar = h/2\piℏ=h/2π ↔ minimal action from antiperiodic boundary / phase cell; explicitly keep h/2πh/2\pih/2π visible so geometry with 2π appears. - ccc ↔ process speed (light speed) — in QAT this is the speed of the generating spherical wavefronts. - ε0\varepsilon_0ε0 ↔ permittivity from the embedding medium, or effective geometric constant from how field lines emerge from the shell (use Gauss law on curved surface).
* Algebraically attempt to eliminate dimensional prefactors to produce a pure number. This is the key hard step: show the combination of geometric factors (4π from surface area, 2π from diameter, patch area from action quantization) reduces to a pure number ≈1/137.

F. Numerical experiments (high precision). (Numerics)
* Using the concrete boundary model, compute the predicted α numerically. If that fails, compute the ratio expression and compare to measured α.
* Independently: re-run the spectral/rainbow test with: 1. high-precision solar spectrum data, 2. high-precision refractive index tables (Hale & Querry / refractiveindex.info), 3. atomic emission line sets (Hα, Hβ, Na D), and 4. selected plasma lines. See how the weighted mean scattering angle moves; map it to a “geometric angle” and compare to the golden-angle and 1/α.

G. Plasma / finite-temperature generalization. (Math & Numerics)
* Extend the boundary model to include temperature/ionization: at high T the shell broadens into plasma double layers — solve MHD/EM boundary conditions on the shell and compute how effective surface charge density and fields change. This will test your intuition about plasma playing a dominant role at large scales.

H. Independent checks and publication. (Check)
* Ask an expert (GR/QFT/plasma) to review the Lagrangian & boundary derivation. Provide them the math, the assumptions, and the numerical output.
* Run the algebra + numerics with multiple symbolic engines or double-precision libraries and compare. Also run a sensitivity analysis so you can show how robust any derived number is to small changes in model assumptions.