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

==== 1. Boundary variation: From SbdyS_{\text{bdy}}Sbdy derive JΣμJ^\mu_{\Sigma}JΣμ and the jump conditions for E,B\mathbf{E},\mathbf{B}E,B. ====
# Surface stress tensor: Derive SabS_{ab}Sab and show Israel junction reproduces Newtonian potential outside the shell; extract an effective G(σ,p,rates)G(\sigma,p,\text{rates})G(σ,p,rates).
# Spin-½ on S²: Confirm antiperiodic boundary condition and show how an effective half-radius r0/2r_0/2r0/2 appears in angular momentum quantization.
# Uncertainty link: Model absorption events as a point process on Σ\SigmaΣ and show ΔE Δt≥h/(4π)\Delta E\,\Delta t\ge h/(4\pi)ΔEΔt≥h/(4π) with h/2πh/2\pih/2π explicit.
# Two-boundary force: Compute net momentum transfer between Σ1,Σ2\Sigma_1,\Sigma_2Σ1,Σ2 using EM flux; identify when the sign is attractive and compare magnitude to Gm1m2/r2G m_1 m_2/r^2Gm1m2/r2.
# LNH scaling: Reproduce N∝t2\mathcal{N}\propto t^2N∝t2 and calibrate αm\alpha_mαm.
# Consistency: Check local energy conservation: energy removed from radiation fields equals increase in boundary internal energy (mass + heat + EM).
# Limits: Show third-law compatibility: as T→0T\to 0T→0, boundary event rate → 0, so “time production” rate → 0 (QAT’s arrow slows locally).