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

=== Assistant: Done — I ran the Planck-scale A worksheet and show the results plus a short interpretation. ===
Done — I ran the Planck-scale (A) worksheet and show the results plus a short interpretation.

===== (Computed with the worksheet code.) =====

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==== 1. With Planck-length boundary scale, G_QAT becomes tiny (far smaller than the observed gravitational constant) for the same assumed n_bdy. Numerically: with L_* = ℓ_P and n_bdy = 10²⁸ m⁻³, the induced G is ~10⁻³⁷–10⁻³⁹ (depending on κ), whereas observed G ≈ 6.674×10⁻¹¹. ====
# But the required mode density to reproduce the observed G is now small and reasonable: - For κ = 1.0, n_required ≈ 1 m⁻³ (i.e., one effective mode per cubic metre) — meaning if each “mode” in the induced-gravity formula already counts all the right microscopic degrees of freedom up to the Planck scale, you don’t need an enormous number density to get the right G. - For κ = 0.1 → n_required ≈ 10 m⁻³. - For κ = 0.01 → n_required ≈ 100 m⁻³. These required densities are tiny (compared with atomic densities), showing that a Planck-scale cutoff can naturally give the tiny observed G with plausible prefactors.
# Interpretation (big picture): - If QAT provides a reason that the effective boundary micro-scale that controls the induced-gravity integral is of order the Planck length, then the induced-gravity mechanism can produce the right order of magnitude for Newton’s G without extreme fine-tuning of mode counts. - Conversely, if the relevant micro-scale is atomic (Bohr-scale) then you need implausibly large mode counts or suppression to match G (we saw that earlier). - So the central question becomes: can QAT justify why the effective boundary thickness/UV cutoff is Planck-scale (or similarly small)? If yes, the induced-gravity route is promising numerically.
# Where Gaussian integrals & Euler identity fit: - The heat-kernel/one-loop determinants that generate the induced Einstein–Hilbert term produce the 1/L_''² scaling (Gaussian integrals) and the numeric prefactor κ (which includes phase/oscillatory factors where Euler’s identity shows up in stationary-phase evaluations). Those prefactors move results by factors O(10⁻²–1), but the dominant control is the cutoff scale L∗L_''L∗.

==== - (i) I can sweep <code>(L_*, n_bdy, κ)</code> and produce a small contour plot showing the region where G_QAT ≈ G_obs — useful to visualize what scales/densities are needed. ====
'' (ii) I can write a short note (1–2 paragraphs) arguing physical reasons why QAT might naturally select L_'' ≈ ℓ_P (or another small scale), or list candidate mechanisms in QAT that could set the cutoff.
* (iii) I can draft the toy heat-kernel calculation for a scalar/EM field on a thin spherical shell to estimate κ more concretely (this is more technical but gives a concrete number for κ).

Which would you like next?