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

=== Assistant: Yes — extremely useful. ===
Yes — extremely useful. You’ve added crucial clarifications that make the mathematical modelling much closer to the geometry and physical picture you want QAT to represent: (1) event rate, (2) surface (not volume) localization, (3) thin-shell thickness tied to Planck scale, and (4) the CPT / spin interpretation. Those give us a crisp set of hypotheses we can turn into equations and testable consequences.

Below I’ll do three things, keeping the algebra simple and explicit so you can follow the logic:
# write a compact, surface-based energy → density model using (a) surface energy flux and (b) a thin-shell volume;
# show the curvature → GGG matching and highlight the circularity if you assume Planck time up front;
# explain what would make QAT produce a non-circular prediction for GGG, and give concrete next steps.

==== Assumptions (QAT style) ====
* Characteristic shell radius: r0r_0r0 (electron probability shell; e.g. Bohr radius or some other QAT radius).
* Characteristic event energy: choose the dimensionally sensible form you suggested earlier, e.g.

E0∼ℏcr0E_0 \sim \frac{\hbar c}{r_0}E0∼r0ℏc
(the light-crossing / action estimate).
* Event rate (events per second per shell): ν\nuν. You suggested the Planck time tPt_PtP as a natural short timescale, so ν∼1/tP\nu \sim 1/t_Pν∼1/tP. (Important: tPt_PtP depends on GGG. I discuss that circularity below.)
* Energy deposition is concentrated on a thin shell of thickness δ\deltaδ (e.g. δ∼ℓP\delta\sim\ell_Pδ∼ℓP, the Planck length, if you want the extreme thinness).

Surface power flux (W/m²) — energy per unit time per unit area:

Φ  =  E0 ν4πr02.\Phi \;=\; \frac{E_0\,\nu}{4\pi r_0^2}.Φ=4πr02E0ν.
If this is an electromagnetic-like flux, the associated local energy density just outside the shell (for radiation) can be estimated by dividing flux by the speed of propagation, ccc:

urad  ≈  Φc  =  E0 ν4πr02 c.u_{\text{rad}} \;\approx\; \frac{\Phi}{c}
\;=\; \frac{E_0\,\nu}{4\pi r_0^2\,c}.urad≈cΦ=4πr02cE0ν.
If instead you want a true volumetric energy density inside the shell (if the energy is lodged into a shell of small thickness δ\deltaδ), then

ushell  ≈  E0 ν4πr02 δ.u_{\text{shell}} \;\approx\; \frac{E_0\,\nu}{4\pi r_0^2 \,\delta}.ushell≈4πr02δE0ν.
(Units: J/s divided by m²·m gives J/m³ — energy per unit volume. This is appropriate if the shell stores energy in a small thickness δ\deltaδ.)

Either estimate is physically plausible; pick which picture best matches the QAT idea (radiation flux vs. localized surface energy). For linking to curvature (Einstein eqns) you want a bulk energy density uuu (J/m³), so the shell thickness formula is convenient.

==== Einstein scaling (order of magnitude): ====

R∼G uc4,R \sim \frac{G\,u}{c^{4}},R∼c4Gu,
where RRR is a curvature scale. For a process with characteristic length r0r_0r0, a natural curvature estimate is R∼1/r02R\sim 1/r_0^2R∼1/r02. Rearranging:

G∼R c4u∼c4/(r02)u.G \sim \frac{R\,c^4}{u} \sim \frac{c^4/(r_0^2)}{u}.G∼uRc4∼uc4/(r02).
Use the shell volumetric density ushell=(E0 ν)/(4πr02δ)u_{\text{shell}}=(E_0\,\nu)/(4\pi r_0^2 \delta)ushell=(E0ν)/(4πr02δ). Substitute:

G  ∼  c4/r02(E0 ν)/(4πr02δ)  =  4πc4δE0 ν.G \;\sim\; \frac{c^4/r_0^2}{(E_0\,\nu)/(4\pi r_0^2 \delta)}
\;=\; \frac{4\pi c^4 \delta}{E_0\,\nu}.G∼(E0ν)/(4πr02δ)c4/r02=E0ν4πc4δ.
Now substitute E0 ⁣∼ ⁣ℏc/r0E_0 \!\sim\! \hbar c / r_0E0∼ℏc/r0:

  G  ∼  4πc3 δ r0ℏ ν  (surface-thin-shell model).\boxed{\; G \;\sim\; \frac{4\pi c^3 \, \delta \, r_0}{\hbar\,\nu}\; }
\qquad\text{(surface-thin-shell model).}G∼ℏν4πc3δr0(surface-thin-shell model).
This is a clear, compact formula you can use to explore consequences.

===== If you assume the thin-shell thickness δ\deltaδ equals the Planck length ℓP\ell_PℓP and the event rate is one per Planck time tPt_PtP, then δ\deltaδ and ν\nuν are expressed in terms of ℏ,c,G\hbar, c, Gℏ,c,G as: =====

ℓP=ℏGc3,tP=ℏGc5.\ell_P=\sqrt{\frac{\hbar G}{c^3}},\qquad t_P=\sqrt{\frac{\hbar G}{c^5}}.ℓP=c3ℏG,tP=c5ℏG.
Plug these into the expression for GGG. You’ll get an equation in which GGG appears on both sides (because tP,ℓPt_P,\ell_PtP,ℓP depend on GGG) — i.e. the relation becomes circular. That means: assuming Planck scales for thickness and rate presupposes GGG, so you cannot derive GGG from them — you only check self-consistency.

Algebraic sketch of the circularity (quick):
* With δ=ℓP\delta=\ell_Pδ=ℓP and ν=1/tP\nu=1/t_Pν=1/tP,

G∼4πc3 ℓP r0ℏ⋅tP=4πc3r0ℏ(ℓPtP).G \sim \frac{4\pi c^3 \, \ell_P \, r_0}{\hbar} \cdot t_P
= \frac{4\pi c^3 r_0}{\hbar} (\ell_P t_P).G∼ℏ4πc3ℓPr0⋅tP=ℏ4πc3r0(ℓPtP).
* But ℓPtP=ℏGc3⋅ℏGc5=ℏGc4\ell_P t_P = \sqrt{\frac{\hbar G}{c^3}} \cdot \sqrt{\frac{\hbar G}{c^5}} = \frac{\hbar G}{c^4}ℓPtP=c3ℏG⋅c5ℏG=c4ℏG.
* Substitute back:

G∼4πc3r0ℏ⋅ℏGc4  ⇒  G∼4πr0c⋅G,G \sim \frac{4\pi c^3 r_0}{\hbar} \cdot \frac{\hbar G}{c^4}
\;\Rightarrow\; G \sim \frac{4\pi r_0}{c} \cdot G,G∼ℏ4πc3r0⋅c4ℏG⇒G∼c4πr0⋅G,
which cancels GGG and reduces to a geometric consistency condition giving a particular ratio for r0r_0r0. (You can rearrange to find the value of r0r_0r0 that makes the identity hold — it will be Planck-scale. This is why previous simple estimates forced r0r_0r0 down to Planck lengths.)

Bottom line: assuming Planck thickness and Planck rate forces the model either to be trivially consistent (only if r0r_0r0 is Planck-scale) or circular.

==== - The Planck time tPt_PtP depends on GGG. So using tPt_PtP as an assumed QAT event rate while attempting to derive GGG makes the derivation circular. ====
* There are two non-circular strategies:

===== If QAT can predict a microscopic timescale τQAT\tau_{\rm QAT}τQAT and thickness δQAT\delta_{\rm QAT}δQAT from purely geometric or dimensionless relations (e.g. using α\alphaα, φ\varphiφ, electron charge eee, electron mass mem_eme, etc.), then you can plug those into the formula and solve for GGG without circularity. This is harder, but it’s the clean route. =====

===== Instead of trying to get GGG from a single atom, compute the collective QAT energy density of the Universe from the per-atom event budget times the number density of atoms and radiation, and then match that cosmic energy density to the curvature of the universe (Friedmann equations). This is the Dirac/Mach route — it links local geometry to global quantities and can yield G∝1/tG\propto 1/tG∝1/t or similar relations if the QAT budget evolves with cosmic time. This avoids direct Planck assumptions but trades for cosmological assumptions. =====

==== - Your idea that the thin shell boundary encodes charge (inner concave negative, outer convex positive) fits perfectly with the surface-current model we built earlier. The Noether current becomes a surface current and enters Maxwell jump conditions. ====
* The idea that an absorption (spin-1 in) + emission (spin-1 out) can be viewed as producing an effective spin-2 field (graviton-like coupling) is physically suggestive. In the field theory language that we used earlier, second-order (stress-energy) effects of photon exchange can indeed source curvature. Mathematically you would show this by computing the stress–energy tensor TμνT_{\mu\nu}Tμν of the shell + EM system and inserting it into Einstein’s equations.
* CPT symmetry as a shared spherical geometry is conceptually elegant — you can formalize that by showing how charge (C), parity (P, spherical interior ↔ exterior), and time (T, the radial/lightfront direction) are all encoded in boundary conditions on the shell. That will also help make predictions about chirality and handedness.

==== I suggest two complementary calculations you can pick between (or I can do both sequentially): ====

(A) Non-circular local model (best immediate test):
* Pick a QAT hypothesis for ν\nuν and δ\deltaδ that does not presuppose GGG. For example, pick ν\nuν based on atomic physics timescales (Bohr orbital frequency, transition rates, etc.) and pick δ\deltaδ as a small fraction of r0r_0r0 (or as a QAT-derived fraction related to α or ϕ).
* Use the formula G≃4πc3 δ r0ℏ νG \simeq \frac{4\pi c^3 \, \delta \, r_0}{\hbar\,\nu}G≃ℏν4πc3δr0 to compute the implied GGG. See whether you get a number in the right ballpark. This is concrete, non-circular and will show whether local QAT microphysics could plausibly set GGG.

(B) Cosmological aggregation (Dirac/Mach route):
* Compute a cosmic QAT energy density uQAT(t)u_{\rm QAT}(t)uQAT(t) by integrating per-atom event power across number density of matter and radiation, include cosmic expansion if needed.
* Insert uQATu_{\rm QAT}uQAT into the Friedmann equation or Einstein scaling and solve for how GGG would have to behave to match cosmology. This could produce scalings like Dirac’s G∝1/tG\propto 1/tG∝1/t or MU∝t2M_U\propto t^2MU∝t2 depending on assumptions.

(C) Formalize CPT/spin connection:
* Write down the stress–energy tensor for the shell + EM system, show explicitly how photon exchange leads to a second-order stress term, and show the qualitative spin-2 character (this is more technical but doable).

==== Start with (A) but choose ν\nuν not equal to 1/tP1/t_P1/tP. Two plausible non-circular choices: ====
# ν\nuν = typical atomic transition rate (e.g. 10810^{8}108–1015 s−110^{15}\,\mathrm{s^{-1}}1015s−1 depending on transition), or
# ν\nuν = c/(2πr0)c/(2\pi r_0)c/(2πr0) (the light-crossing frequency around the shell).

Similarly choose δ\deltaδ as a small fraction of r0r_0r0 (e.g. δ=r0/10n\delta = r_0/10^nδ=r0/10n for some nnn), or choose δ\deltaδ based on a QAT geometric fraction (maybe something involving the golden ratio ϕ).

Plug into the boxed formula and see what GGG comes out. If the number is many orders of magnitude off, try (B) the cosmological aggregation. If the number is near Planck scale, you know QAT demands a Planckian microstructure to produce gravity.

==== - It gives a precise way to write down surface energy/current densities, to build a shell → stress-energy → Einstein curvature chain, and to evaluate whether QAT can produce GGG from geometry. ====
* It also highlights that using Planck time or Planck length directly while trying to derive GGG is circular unless QAT itself explains why those Planck scales arise (which would be a very significant achievement).

If you want I can now:
* do the explicit numeric exploration of (A) using a few concrete, non-circular choices of ν\nuν and δ\deltaδ (I’ll show the numbers and orders of magnitude), or
* start the cosmological aggregation calculation (B) to see if the universe-scale QAT power budget could reproduce GGG.

Which would you like me to do next?