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

=== Assistant: This is rich — thank you. ===
This is rich — thank you. You’ve tied together the most important QAT pieces (local shell physics, CPT / spin picture, and the idea that the same local process can scale up to cosmology via Dirac/Mach logic). Below I give a careful, math-aware analysis that:

• keeps your geometric picture front-and-center (shells, thin layer, inner concave negative / outer convex positive),
• shows a concrete route to connect local QAT to cosmic gravity (aggregation),
• demonstrates numerically how close simple choices get us to the observed numbers, and
• lays out a roadmap of non-circular, testable calculations you can run next.

I’ll keep equations explicit and show the rough numbers I computed so you can see where things stand.

==== 1. Photon–electron events occur at a spherical shell of radius r0r_0r0. The shell (the electron probability surface) is a 2D manifold (area 4πr024\pi r_0^24πr02) and has a thin thickness δ\deltaδ. The shell separates inner concave (negative) and outer convex (positive) charge surfaces — this is where charge lives in your picture. ====
# A single event deposits a characteristic quantum of energy E0E_0E0. Dimensionally sensible choice:

E0∼ℏcr0E_0 \sim \frac{\hbar c}{r_0}E0∼r0ℏc
(heuristic: ℏ∼E0Δt\hbar \sim E_0 \Delta tℏ∼E0Δt with Δt∼r0/c\Delta t \sim r_0/cΔt∼r0/c).
# Events occur at rate ν\nuν per shell. (This is the crucial free parameter. You suggested Planck time earlier — but that makes the derivation circular because the Planck time depends on GGG.)
# For a thin shell of thickness δ\deltaδ, the local volumetric energy density contributed by a single shell in steady state (if each event energy is deposited and stored for some characteristic time τ\tauτ or if the shell is a continuous source) can be written as power-deposition-per-volume:

power density P  =  natom E0 ν,\text{power density } \mathcal{P} \;=\; n_{\rm atom}\,E_0 \,\nu,power density P=natomE0ν,
where natomn_{\rm atom}natom is number density of atoms (later we’ll scale to cosmic number density). If energy is stored for a time τ\tauτ, the steady energy density is u=Pτu=\mathcal{P}\tauu=Pτ. Alternately, if the deposition forms a thin-surface energy density stored in thickness δ\deltaδ, a local energy density estimate is

ushell∼E0 ν4πr02 δu_{\rm shell} \sim \frac{E_0\,\nu}{4\pi r_0^2\,\delta}ushell∼4πr02δE0ν
per shell.
# Curvature scaling (order-of-magnitude) from Einstein:

R∼G uc4.R \sim \frac{G\,u}{c^4}.R∼c4Gu.
On a local scale we can set R∼1/r02R\sim 1/r_0^2R∼1/r02 as a characteristic curvature, but for cosmology we use the Friedmann relation.

==== (I) Local-to-global (aggregate) — Dirac / Mach route (recommended) ====
Compute the global QAT energy density uQATu_{\rm QAT}uQAT by integrating local shell event power over matter distribution (atoms, stars, plasma). Then insert uQATu_{\rm QAT}uQAT into cosmological relations (e.g. Friedmann equation H2=(8πG/3)ρH^2=(8\pi G/3)\rhoH2=(8πG/3)ρ) or Einstein scaling to see what value(s) of GGG are implied. This avoids assuming Planck units and is the least circular.

(II) Fundamental scale route (Planck-centric)
Assume QAT implies a microscopic length or time scale (e.g., δ\deltaδ or ν−1\nu^{-1}ν−1) that is independent of GGG. Then plug that into a local surface-curvature relation and solve for GGG. This is more ambitious: you must show QAT gives a fundamental scale (e.g., r0r_0r0 of order ℓP) from geometry alone. If you assume Planck-scale quantities up front, deriving GGG is circular because Planck scales already include GGG.

==== I used: ====
* r0=a0=5.29×10−11 mr_0 = a_0 = 5.29\times10^{-11}\ \mathrm{m}r0=a0=5.29×10−11 m (Bohr radius) as a physically familiar shell scale,
* E0=ℏc/r0≈6.0×10−16 JE_0 = \hbar c / r_0 \approx 6.0\times10^{-16}\ \mathrm{J}E0=ℏc/r0≈6.0×10−16 J,
* cosmic baryon mass density ρb≈4.1×10−28 kg/m3\rho_b \approx 4.1\times10^{-28}\ \mathrm{kg/m^3}ρb≈4.1×10−28 kg/m3 → proton number density nb≈0.245 m−3n_b \approx 0.245\ \mathrm{m^{-3}}nb≈0.245 m−3 (order-of-magnitude).

For event rates ν=108,1012,1015 s−1\nu = 10^{8}, 10^{12}, 10^{15}\ \mathrm{s^{-1}}ν=108,1012,1015 s−1 I computed:
* Energy per event E0≈6.0×10−16 JE_0 \approx 6.0\times10^{-16}\,{\rm J}E0≈6.0×10−16J.
* Power per atom Patom=E0νP_{\rm atom}=E_0\nuPatom=E0ν: ranges from 6×10−86\times10^{-8}6×10−8 W (for ν=108\nu=10^8ν=108) up to 0.60.60.6 W (for ν=1015\nu=10^{15}ν=1015).
* Power density (per m³) P=nbPatom \mathcal{P}= n_b P_{\rm atom}P=nbPatom ranges: - ν=108\nu=10^8ν=108: P≈1.5×10−8 W/m3\mathcal{P}\approx 1.5\times10^{-8}\ \mathrm{W/m^3}P≈1.5×10−8 W/m3 - ν=1012\nu=10^{12}ν=1012: P≈1.5×10−4 W/m3\mathcal{P}\approx 1.5\times10^{-4}\ \mathrm{W/m^3}P≈1.5×10−4 W/m3 - ν=1015\nu=10^{15}ν=1015: P≈0.15 W/m3\mathcal{P}\approx 0.15\ \mathrm{W/m^3}P≈0.15 W/m3

Compare target energy density (rest-mass energy of baryons):

ubaryon=ρbc2≈3.7×10−11 J/m3.u_{\rm baryon} = \rho_b c^2 \approx 3.7\times10^{-11}\ \mathrm{J/m^3}.ubaryon=ρbc2≈3.7×10−11 J/m3.
So the steady-state energy density equals ubaryonu_{\rm baryon}ubaryon when the characteristic storage time τ\tauτ (how long energy from each event remains contributing to local energy density) equals:

τ≈ubaryonP.\tau \approx \frac{u_{\rm baryon}}{\mathcal{P}}.τ≈Pubaryon.
Numerical values:
* For ν=108\nu=10^8ν=108: τ∼2.5×10−3 s\tau \sim 2.5\times 10^{-3}\ \mathrm{s}τ∼2.5×10−3 s.
* For ν=1012\nu=10^{12}ν=1012: τ∼2.5×10−7 s\tau \sim 2.5\times 10^{-7}\ \mathrm{s}τ∼2.5×10−7 s.
* For ν=1015\nu=10^{15}ν=1015: τ∼2.5×10−10 s\tau \sim 2.5\times 10^{-10}\ \mathrm{s}τ∼2.5×10−10 s.

Interpretation: If event rates are extremely high and/or energy is stored even for very short times, the aggregate QAT energy density could be comparable with baryonic rest-energy density. That would make QAT contributions to cosmological curvature non-negligible. But if event rates are modest and energy dissipates rapidly, QAT's aggregate energy density is tiny.

==== 1. Local (single-atom) estimates alone don’t give the correct GGG — we tried to match curvature R∼1/r02R \sim 1/r_0^2R∼1/r02 to shell energy and found that with atomic r0r_0r0 you either need impossibly small δ\deltaδ or unphysically large ν\nuν (or assume Planck scales); that route pushes you to Planck physics. ====
# Aggregate (Dirac/Mach) route can be non-circular and promising — if QAT supplies a physically justified estimate of the global energy density uQATu_{\rm QAT}uQAT, you could then compute effective curvature / gravitational strength via the Friedmann relation. The core problem becomes: can QAT produce a falsifiable prediction for uQAT(t)u_{\rm QAT}(t)uQAT(t)? If yes, that can be compared with cosmology to constrain or predict G(t)G(t)G(t).
# Time dependence (Dirac idea) — QAT offers natural mechanisms for cosmological time-dependence: photon energies redshift (E ∝ 1/a), number densities drop (n ∝ 1/a^3), and QAT event rates or storage times could evolve with temperature and plasma states. If uQATu_{\rm QAT}uQAT scales with cosmic time in a way that implies G∝1/tG\propto 1/tG∝1/t (Dirac’s law), QAT would offer a physical origin for Dirac’s hypothesis. But this needs a specific model linking ν,τ,δ\nu,\tau,\deltaν,τ,δ to cosmic scale factor a(t)a(t)a(t) and temperature.

==== - Avoid Planck circularity. If you assume δ=ℓP\delta=\ell_Pδ=ℓP and ν=1/tP\nu=1/t_Pν=1/tP you bake GGG into the assumptions (because ℓP and tP include GGG). That gives consistency checks but not a derivation. ====
* Which energy counts? In GR everything that carries energy/momentum gravitates. If QAT energy is radiative and escapes quickly to infinity, its local gravitational effect is small; if it is stored in rest-like form, it contributes strongly. You must decide whether QAT events produce persistent energy (particle rest energy) or transient radiation.
* Spin & fermions. The scalar-shell model is a useful toy. Real electrons are spin-½; a full theory would model shells with spinor fields (Dirac fields). That can change coupling structure and numerical factors.
* Gauge invariance and stress–energy averaging. To feed QAT results to cosmology you must compute the appropriate average of TμνT_{\mu\nu}Tμν (the stress-energy tensor) for a statistically homogeneous, isotropic universe. Surface-localized shell currents produce multipole structure which averages out at large scales, leaving isotropic energy density + pressure.

==== ### ====
# Fix QAT micro-parameters you want to test: choose a plausible r₀ (perhaps Bohr radius for atomic processes, maybe a different QAT radius), a thin-shell thickness fraction δ\deltaδ (e.g., r0/10nr_0/10^nr0/10n or physically-motivated fraction from geometry), and a non-Planck event rate ν\nuν (options: c/(2πr0), typical transition rates 10810^8108–101210^{12}1012 s⁻¹, or a parameter to be constrained).
# Compute per-atom power Patom=E0νP_{\rm atom}=E_0\nuPatom=E0ν, power density P=natomPatom\mathcal{P}=n_{\rm atom}P_{\rm atom}P=natomPatom using cosmic number density natomn_{\rm atom}natom, and steady-state energy density u=Pτu=\mathcal{P}\tauu=Pτ for plausible ranges of τ\tauτ (energy storage/dissipation times).
# Compute the implied contribution to cosmological mass-energy ρQAT=u/c2\rho_{\rm QAT}=u/c^2ρQAT=u/c2 and compare to measured baryon & radiation densities.

===== 1. Use aggregated uQAT(t)u_{\rm QAT}(t)uQAT(t) to compute implied cosmological expansion via Friedmann: =====

H2(t)=8πG3(ρmatter+ρQAT(t)+ρrad+⋯ ).H^2(t)=\frac{8\pi G}{3}\left(\rho_{\rm matter}+\rho_{\rm QAT}(t) + \rho_{\rm rad} + \cdots\right).H2(t)=38πG(ρmatter+ρQAT(t)+ρrad+⋯).
If you treat GGG as unknown but assume H(t) observed, invert to find G(t)G(t)G(t) implied by your QAT contribution. Alternatively, assume constant G and see what QAT predicts for H(t).
# Explore scaling laws: assume QAT micro-parameters scale with a(t) (e.g., E0∝1/aE_0 \propto 1/aE0∝1/a because photon energies redshift) and compute whether this produces relations like Dirac’s G∝1/tG\propto 1/tG∝1/t or MU∝t2M_U\propto t^2MU∝t2.

===== 1. Compute observable signatures: (a) tiny time variation in G or α, (b) spectral shifts or line widths in atoms (if shell quantization differs from standard QM), (c) polarization patterns or multipole anisotropies from shell radiation in plasma, (d) differences in stellar structure if QAT modifies energy exchange rates in plasma. =====
# Compare with observational constraints: limits on G˙/G\dot{G}/GG˙/G, limits on α variation, measured CMB, LSS data.

==== • A — Run the Phase 1 numerics for a grid of plausible (r0,δ,ν,τ)(r_0,\delta,\nu,\tau)(r0,δ,ν,τ) and output heatmaps for implied ρQAT\rho_{\rm QAT}ρQAT vs. observed baryon & critical densities (I can do this and show plots / tables). ====
• B — Do a toy cosmology: assume a simple QAT scaling law E0∝1/aE_0\propto 1/aE0∝1/a, n∝1/a3n\propto 1/a^3n∝1/a3, constant ν\nuν or ν∝1/a\nu\propto 1/aν∝1/a, and compute uQAT(t)u_{\rm QAT}(t)uQAT(t) and the implied time-dependence G(t)G(t)G(t) if you invert Friedmann.
• C — Compute the stress–energy tensor of the surface-shell + EM model and average it to produce ρQAT\rho_{\rm QAT}ρQAT and pressure pQATp_{\rm QAT}pQAT (this is the rigorous route to plug into GR). This is more technical (field-theory algebra) but doable.

==== Start with A: run a parametric numerical survey over ν\nuν and τ\tauτ (and two plausible δ\deltaδ choices) using the formulas above. That will tell you whether realistic atomic values could produce a cosmically significant energy density. I can produce simple plots (heatmap of ρQAT/ρbaryon\rho_{\rm QAT}/\rho_{\rm baryon}ρQAT/ρbaryon vs ν,τ\nu,\tauν,τ) so you can see which parameter ranges are promising. This gives a non-circular first test and will guide whether to pursue cosmological inversion. ====

Would you like me to run that survey (A) next? If yes, tell me which ranges to try for ν\nuν and τ\tauτ (I can pick reasonable ranges if you prefer), and whether to use r0=a0r_0=a_0r0=a0 or another r0r_0r0.