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

==== 1. Numeric tension exists. The simple estimate shows that with an atomic boundary scale and ordinary atomic densities, the induced-gravity estimate (as written) does not reproduce the observed smallness of G — it predicts gravity many orders of magnitude stronger. To match observed G you must either: - assume an effective boundary scale L_* that is extremely small (much smaller than atomic or nuclear scales), or - assume an effective mode density n_bdy that is astronomically huge, or - include additional suppression factors (the dimensionless kappa_ind would have to be extremely small), or - include cancellation/collective physics (e.g., sign cancellations, negative contributions, or renormalization effects) coming from the full heat-kernel/path integral. ====
# This is exactly the same qualitative issue Sakharov / induced gravity runs into. In induced-gravity derivations the numeric value of G depends on the UV cutoff or micro physics scale (the quantity analogous to your 1/L_''). To produce the tiny measured G you usually need a very high UV cutoff (Planck-scale physics) or a carefully selected set of cancellations. That suggests if QAT is to give the correct numeric G, it must either: - provide a microscopic reason why the effective cutoff scale L_'' is extremely small (Possibility: Planck-scale emergence), or - show a mechanism producing a very small effective kappa_ind, or - identify a huge effective mode count (for example, counting vacuum/field modes up to very high frequency, not just atomic bound states).
# Where Gaussian integrals and Euler identity tie in: the true numeric prefactor kappa_ind comes from the detailed heat-kernel / determinant evaluation (Gaussian integrals and stationary-phase approximations) and can easily supply O(10⁻²–1) factors. But those factors alone cannot bridge the many orders of magnitude gap found above; the scale (L_* or cutoff) is the controlling quantity.