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

==== Short answer: Not directly from geometry alone unless you also predict a physical mass or energy scale. But you can express GGG in terms of fundamental constants and a characteristic mass — and if QAT predicts that mass from geometry, then yes. ====

===== - GGG is a dimensional constant (units m3 kg−1 s−2\mathrm{m^3\,kg^{-1}\,s^{-2}}m3kg−1s−2). To get a number for GGG you need both a geometric relationship and a physical mass (or length/time) scale. Geometry alone (angles, ratios) gives dimensionless results; to get dimensions you must supply a scale (mass, length, time). =====
* Most successful “derivations” of dimensional constants use Planck units (combine c,ℏ,Gc,\hbar,Gc,ℏ,G to define a Planck mass/length/time) — but that treats GGG as fundamental, not derived.

===== If you take the Planck mass mPm_PmP as a defined scale, then =====

  G  =  ℏ cmP2  \boxed{ \; G \;=\; \dfrac{\hbar\, c}{m_P^2} \;}G=mP2ℏc
where ℏ=h2π\hbar=\dfrac{h}{2\pi}ℏ=2πh. This is exact (by definition of Planck mass).

We can show the arithmetic with up-to-date constants:
* ℏ=1.054571817×10−34 J⋅s\hbar = 1.054571817\times10^{-34}\ \mathrm{J\cdot s}ℏ=1.054571817×10−34 J⋅s
* c=299 792 458 m/sc = 299\,792\,458\ \mathrm{m/s}c=299792458 m/s
* mP≈2.176434×10−8 kgm_P \approx 2.176434\times10^{-8}\ \mathrm{kg}mP≈2.176434×10−8 kg (standard Planck mass)

Compute:

G  =  ℏcmP2G \;=\; \frac{\hbar c}{m_P^2}G=mP2ℏc
Numerical evaluation (careful arithmetic) gives:

G≈6.67430×10−11 m3 kg−1 s−2,G \approx 6.67430\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}},G≈6.67430×10−11 m3kg−1s−2,
which matches the measured value to the usual precision.

So if QAT can derive the Planck mass mPm_PmP (or some equivalent mass scale) from its geometry, then QAT would immediately give a value for GGG.

===== If you want GGG from QAT geometry, you must complete one of: =====

Strategy A — predict a fundamental mass from geometry.
* Find a QAT mechanism that produces a natural mass scale MQATM_{\rm QAT}MQAT in terms of h,c,e,αh, c, e, \alphah,c,e,α, and geometry constants (like golden ratio or a characteristic radius r0r_0r0).
* Then set G=ℏc/MQAT2G=\hbar c/M_{\rm QAT}^2G=ℏc/MQAT2. If MQATM_{\rm QAT}MQAT equals the Planck mass (or the theory provides a specific numeric ratio), you recover GGG.

Strategy B — relate GGG to cosmic/global quantities via Mach/Dirac ideas.
* Dirac’s Large Number Hypothesis/relations connect GGG with cosmological scales (age of universe ttt, total mass MUM_UMU, etc.), e.g. constraints like MU∼t2M_U \sim t^2MU∼t2 or G∝1/tG \propto 1/tG∝1/t. If QAT can predict the universe mass/age relation from geometry, you could express GGG in terms of geometry + cosmology. That’s more speculative and requires cosmological dynamics.

===== One way to try to derive GGG inside QAT: =====
# Define an elementary shell energy flux: compute the mean energy per shell event (photon absorbed/emitted) over the electron sphere. Express that in terms of ΔE=hf\Delta E = hfΔE=hf and relevant geometric factors 4πr24\pi r^24πr2.
# Compute an effective energy density ρQAT\rho_{\rm QAT}ρQAT by integrating over typical shell events per volume or per time (requires model of event rate).
# Match this effective energy density to the stress–energy tensor TμνT_{\mu\nu}Tμν term used in Einstein’s equations. In GR, curvature scale R∼κρR\sim \kappa \rhoR∼κρ with κ=8πG/c4\kappa = 8\pi G/c^4κ=8πG/c4. Solve for GGG in terms of ρQAT,ℏ,c\rho_{\rm QAT},\hbar,cρQAT,ℏ,c and geometry constants.
# If QAT supplies a unique ρQAT\rho_{\rm QAT}ρQAT from geometry (no free adjustable scale), that gives a numeric GGG. If not, you’ve reduced the problem to finding the correct QAT energy density.

This approach is essentially what you were already attempting with QAT-adjusted field equations — it just makes explicit the need for a physically computed energy density (or mass scale) to fix GGG.