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

==== Instead of a local shadow, consider that a spherically symmetric mass distribution out to infinity produces a tiny, cumulative attenuation/phase-lag of the ambient EM bath along rays arriving at radius rrr. Let the effective (frequency-averaged) opacity per unit length be κ(r′)\kappa(r')κ(r′) along each incoming ray. The fractional event-rate change at rrr from the exterior material is then, schematically, ====

δRR(r)  ≈  τext(r)  =  ∫r∞ ⁣κ(r′) dr′.\frac{\delta R}{R}(r)\;\approx\;\tau_{\rm ext}(r)\;=\;\int_{r}^{\infty}\!\kappa(r')\,dr'.RδR(r)≈τext(r)=∫r∞κ(r′)dr′.
If the exterior effective opacity falls as κ(r′)∝1/r′2\kappa(r')\propto 1/r'^{2}κ(r′)∝1/r′2 (which happens when the effective cross-section per shell tracks a 1/r′21/r'^{2}1/r′2 mass/area distribution), then

τext(r)  ∝  ∫r∞dr′r′2  =  1r.\tau_{\rm ext}(r)\;\propto\;\int_{r}^{\infty}\frac{dr'}{r'^{2}}\;=\;\frac{1}{r}.τext(r)∝∫r∞r′2dr′=r1.
This immediately gives the right 1/r1/r1/r scaling for ΦQAT\Phi_{\rm QAT}ΦQAT. Physically: not a blunt “shadow,” but an infinitesimal, integrated decohering/phase-lag effect from all shells of exterior matter—exactly the kind of cumulative, weak coupling a QAT EM-exchange picture would expect.