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

=== I think these two will most directly answer your question “Is there a set ratio between couplings and gravity that forms GGG?”: ===

A — Toy heat-kernel (one-loop determinant) for EM + Dirac on a thin spherical shell.
Goal: compute the spectral trace / heat kernel coefficients a0,a1,a2,…a_0,a_1,a_2,\dotsa0,a1,a2,… for fields confined to a thin spherical 2-D shell or with boundary conditions at that shell. From that you extract the induced effective action and identify the coefficient of the emergent Einstein–Hilbert term (the prefactor that becomes 1/G1/G1/G). This will give an explicit κind\kappa_{\rm ind}κind as a function of e, m,e,\ m,e, m, boundary conditions, and cutoff L∗L_*L∗. Why: this is the most direct path to see how α and the spinor content numerically enter the induced GGG. It is standard field theory (heat kernel / zeta regularization), and it will tell us whether realistic EM+spinor content can supply the small prefactor or produce cancellations that push the effective scale to Planck values.

B — Parameter sweep of the QAT formula and checking Dirac ratios.
Goal: numerically map the region in (L∗,nbdy,κind)(L_*, n_{\rm bdy},\kappa_{\rm ind})(L∗,nbdy,κind) space that reproduces observed GGG, then overlay physically plausible sub-regions where κind\kappa_{\rm ind}κind is computed (or estimated) from EM physics. This is excellent for outreach and for checking plausibility quickly. The sweep can use the earlier worksheet formula (as a parametrized guide).