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

=== From our worksheet the schematic induced formula was ===

GQAT  ≈  c3κind ℏ  L∗2nbdy(Equation A)G_{\rm QAT}\;\approx\;\frac{c^3}{\kappa_{\rm ind}\,\hbar}\;\frac{L_*^{2}}{n_{\rm bdy}} \qquad\text{(Equation A)}GQAT≈κindℏc3nbdyL∗2(Equation A)
where
'' L∗L_''L∗ = microscopic boundary length (cutoff),
* nbdyn_{\rm bdy}nbdy = effective mode density (modes per m3^33),
* κind\kappa_{\rm ind}κind = dimensionless spectral prefactor from one-loop determinants / heat-kernel integrals (Gaussian integrals, phases etc.).

This expression has correct units because c3ℏ\dfrac{c^3}{\hbar}ℏc3 has units m3⋅kg−1⋅s−2\text{m}^3\cdot\text{kg}^{-1}\cdot\text{s}^{-2}m3⋅kg−1⋅s−2 when combined with a dimensionless factor and L∗2/nbdyL_*^2/n_{\rm bdy}L∗2/nbdy (m2^22/m−3^ {-3}−3 = m5^55 ??? — note: this particular dimensional form was a simple scaling heuristic earlier; we must be careful to keep dimensions consistent when we refine the derivation. The numerical worksheet used it as a parametric scaling. In a rigorous heat-kernel derivation the exact geometric factors will be different but of the same spirit: GGG controlled by a UV scale and spectral density.)

Where α can enter:
α\alphaα will enter the prefactor κind\kappa_{\rm ind}κind and the effective mode density nbdyn_{\rm bdy}nbdy through the electromagnetic boundary physics:
* The energy per event (photon exchange) scales like ℏω\hbar \omegaℏω; spectral weight of modes depends on the EM spectrum and selection rules (that spectrum is weighted by eee and α via interaction matrix elements).
* Material/plasma response functions, cross sections, polarizabilities etc. contain eee, mmm, ε0\varepsilon_0ε0 and thus produce α\alphaα-dependent factors in the effective action tracing.
* Boundary conditions for spinor fields (antiperiodic vs periodic) change heat-kernel coefficients; spin contributions bring factors of 2 and sign changes (and spinor determinants bring factors proportional to the number of fermionic degrees of freedom).

So the path to derive GGG numerically from e, ℏ, ce,\ \hbar,\ ce, ℏ, c is: show how EM+Dirac boundary spectrum (which depends on e and α) produces a specific κind\kappa_{\rm ind}κind and nbdyn_{\rm bdy}nbdy; plug into the induced scale formula and you then get GGG.