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

=== Assistant: Nice choice — good physics question. ===
Nice choice — good physics question. I ran the calculation you asked for (Option 1) and kept everything explicit so you can see the geometry and constants that matter. Below I show the reasoning, the algebraic relations, the numbers, and what they imply about whether QAT (as currently formulated) can predict Newton’s GGG from atomic-scale geometry.

==== You suggested treating a single photon–electron event as depositing a characteristic quantum of action/energy on a spherical shell of radius r0r_0r0. I made the minimal, physically-plausible assumptions needed to get a dimensional estimate: ====
* Use the Planck quantum of action explicitly as ℏ=h/(2π)\hbar = h/(2\pi)ℏ=h/(2π).
* Take a characteristic energy per event of order

E0∼ℏcr0,E_0 \sim \frac{\hbar c}{r_0},E0∼r0ℏc,
which is dimensionally an energy (J). This comes from the action–time idea ℏ∼E0Δt\hbar \sim E_0 \Delta tℏ∼E0Δt with Δt∼r0/c\Delta t \sim r_0/cΔt∼r0/c (the light-crossing time of the radius).
* Suppose that energy E0E_0E0 is effectively concentrated in a region of volume of order the sphere's volume V∼43πr03V \sim \tfrac{4}{3}\pi r_0^3V∼34πr03. That gives a characteristic energy (volume) density

u∼E0V∼ℏc/r043πr03=3ℏc4πr04.u \sim \frac{E_0}{V} \sim \frac{\hbar c / r_0}{\tfrac{4}{3}\pi r_0^3}
= \frac{3\hbar c}{4\pi r_0^4}.u∼VE0∼34πr03ℏc/r0=4πr043ℏc.
* Use the Einstein field-equation scaling (order of magnitude) that a curvature scale RRR is produced by an energy density uuu via

R∼G uc4.R \sim \frac{G\,u}{c^4}.R∼c4Gu.
Take a geometric curvature scale R∼1/r02R\sim 1/r_0^2R∼1/r02 (curvature from a length scale r0r_0r0). Rearranging gives

G∼Rc4u∼c4/r02u.G \sim \frac{R c^4}{u} \sim \frac{c^4 / r_0^2}{u}.G∼uRc4∼uc4/r02.
Substituting the expression for uuu gives a compact estimate:

 Gest  ≃  4π3 c3r02ℏ .\boxed{\, G_{\rm est} \;\simeq\; \frac{4\pi}{3}\,\frac{c^3 r_0^2}{\hbar}\, }.Gest≃34πℏc3r02.
(You can check units quickly: c3r02/ℏc^3 r_0^2/\hbarc3r02/ℏ has dimensions [L3M−1T−2][L^3 M^{-1} T^{-2}][L3M−1T−2] — the same as GGG.)

==== If we choose r0r_0r0 to be the Bohr radius a0≈5.29×10−11 ma_0 \approx 5.29\times10^{-11}\,\text{m}a0≈5.29×10−11m (an electron shell scale commonly used), then: ====
* ℏ≈1.054571817×10−34 J⋅s\hbar \approx 1.054571817\times10^{-34}\ \mathrm{J\cdot s}ℏ≈1.054571817×10−34 J⋅s
* c=2.99792458×108 m/sc = 2.99792458\times10^8\ \mathrm{m/s}c=2.99792458×108 m/s
* r0=a0=5.29177210903×10−11 mr_0 = a_0 = 5.29177210903\times10^{-11}\ \mathrm{m}r0=a0=5.29177210903×10−11 m

Numerical evaluation (carefully done) yields:
* Estimated GGG from the formula above:

Gest≈3.0×1039 m3 kg−1 s−2.G_{\rm est} \approx 3.0 \times 10^{39}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}.Gest≈3.0×1039 m3kg−1s−2.
That is roughly 105010^{50}1050 times larger than the measured Newton constant
Gobs≈6.6743×10−11 m3 kg−1 s−2\displaystyle G_{\rm obs}\approx 6.6743\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}Gobs≈6.6743×10−11 m3kg−1s−2.

So an atomic shell radius produces an enormous (and totally wrong) value for GGG.

==== If we invert the relation to find the radius r0r_0r0 that would give the observed GGG, we get ====

r0,required≃3ℏGobs4πc3.r_{0,\text{required}} \simeq \sqrt{\frac{3\hbar G_{\rm obs}}{4\pi c^3}}.r0,required≃4πc33ℏGobs.
Plugging in GobsG_{\rm obs}Gobs gives

r0,required≈7.9×10−36 m.r_{0,\text{required}} \approx 7.9\times10^{-36}\ \text{m}.r0,required≈7.9×10−36 m.
For comparison, the Planck length is
ℓP=ℏG/c3≈1.62×10−35 m.\ell_P = \sqrt{\hbar G / c^3} \approx 1.62\times10^{-35}\ \mathrm{m}.ℓP=ℏG/c3≈1.62×10−35 m.

So the required radius is of order 10−3510^{-35}10−35 m — i.e. on the order of the Planck length (about 0.5 ℓP\ell_PℓP in the crude algebra above). In other words: to get the correct measured GGG with this simple QAT estimate you need the characteristic shell radius to be a Planck-scale length, not an atomic scale.

==== - Dimensional logic works: the algebra and units are consistent; the estimate shows clearly how GGG depends on a length scale r0r_0r0 and on ℏ\hbarℏ and ccc. ====
* Atomic scales cannot produce GGG: if your fundamental QAT shell length is atomic (Bohr scale), you get a wildly wrong GGG. That means QAT — if it is to produce the observed Newton constant without introducing extra parameters — must either (a) predict a fundamental microscopic length near the Planck scale, or (b) produce an effective collective energy density at cosmological scales that yields the measured GGG (Dirac/Mach style).
* Planck scale naturally appears: the inversion shows that a Planck-scale radius is what gives a realistic GGG. This is not surprising: many routes to deriving GGG lead to Planck units because GGG itself sets the Planck scale when combined with ℏ\hbarℏ and ccc.
* Assumptions matter: I chose E0∼ℏc/r0E_0\sim \hbar c/r_0E0∼ℏc/r0 (light-crossing time estimate) and spread that energy over a sphere volume. Different assumptions (event rate per unit time, how energy is distributed in space, coupling between many shells, or cosmological aggregation) will change numeric results drastically. The order-of-magnitude conclusion (atomic → wrong; Planck → possible) is robust.

==== If you want to keep exploring this route, here are ways to refine the calculation and make it more physically grounded: ====
# Introduce an event rate — many shell events per second per atom. I used one event's energy per one shell volume. If many events occur per second and aggregate over a volume, you can define a steady energy density uuu. That changes the final numeric factor and may connect atomic activity to macroscopic energy density.
# Collective / cosmological aggregation — use QAT to estimate the total QAT energy density of the universe (integrating per-atom events across matter and radiation). Then insert that cosmic uuu into the same Einstein scaling to solve for GGG (Dirac/Mach flavored approach).
# Use a different characteristic radius — if QAT has a fundamental length scale (derived geometrically), use that. The calculation shows that unless that scale is Planckian, you won't get the observed GGG by this simple matching.
# Model the shell localization more precisely — instead of smearing E0E_0E0 into the entire sphere volume, consider a thin shell energy density (surface energy), or a distribution that falls off with radius; then redo the matching carefully (boundary terms change prefactors).
# Try a dimensionless (geometric) route — if QAT predicts a dimensionless ratio that links electron/proton masses to a cosmological mass, that could be used along with measured values to infer GGG. That’s the Dirac/Large Number route.

==== - Using a plausible QAT assumption (energy per photon-electron event E0∼ℏc/r0E_0\sim\hbar c/r_0E0∼ℏc/r0 concentrated over a shell volume) leads to a formula G∼(4π/3) c3r02/ℏG\sim (4\pi/3)\,c^3 r_0^2/\hbarG∼(4π/3)c3r02/ℏ. ====
* With the electron shell at atomic size, that formula predicts an astronomically huge GGG — wrong by ~105010^{50}1050.
* To match the measured GGG with this simple model you need the fundamental radius r0r_0r0 to be of order the Planck length (∼10−3510^{-35}10−35 m). That suggests GGG is linked to Planck-scale physics (as in standard understanding), unless QAT provides a novel, robust way to produce the Planck scale from its geometry.

If you like, I can now:
* (A) Repeat the same estimate but include an event rate (how many photon–electron events per second per atom), which may give a route from atomic activity to a macroscopic uuu.
* (B) Do the cosmological aggregation (assume average QAT energy density across the universe and match to GGG — Dirac/Mach route).
* (C) Investigate whether a purely geometric argument inside QAT can produce a Planck-scale r₀ (harder — would need a specific mechanism inside the theory).

Which of these would you like next?