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

=== ## ===

Quantum Atom Theory (QAT) models physical change as photon–electron exchange on a 2-D spherical boundary (the electron’s probability shell around a nucleus, or a local EM wavefront). Each absorption/emission event is a discrete “moment of time,” and the Born rule ∣Ψ∣2|\Psi|^2∣Ψ∣2 is viewed as area density on the sphere S2S^2S2 (surface 4πr24\pi r^24πr2). Gravity and inertia are emergent, arising from coherent second-order EM momentum exchange across many such boundaries. The program is to derive standard physics (Maxwell, quantum spectra, and GR curvature) from a bulk + boundary action, with the boundary carrying EM current and stress-energy.

==== - xμ=(ct,x)x^\mu=(ct,\mathbf{x})xμ=(ct,x), metric signature (−+++)(-+++)(−+++). ====
* AμA_\muAμ: EM 4-potential (SI).
* Fμν=∂μAν−∂νAμF_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\muFμν=∂μAν−∂νAμ.
* ψ\psiψ: electron field (can audit both scalar and spinor variants).
* gμνg_{\mu\nu}gμν: spacetime metric (GR part).
* habh_{ab}hab: induced metric on the 2-sphere boundary Σ\SigmaΣ at radius r=r0r=r_0r=r0.
* uμu^\muuμ: local 4-velocity on Σ\SigmaΣ.
* Physical constants: ϵ0, μ0, c, e, h, ℏ\epsilon_0,\ \mu_0,\ c,\ e,\ h,\ \hbarϵ0, μ0, c, e, h, ℏ with ℏ=h/2π\hbar=h/2\piℏ=h/2π.

==== ### ====

Sbulk=∫Md4x −g[−14μ0FμνFμν+Lmatter(ψ,∇ψ,Aμ;gμν)].S_{\text{bulk}}=\int_{\mathcal{M}} \mathrm{d}^4x\,\sqrt{-g}\left[
-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}
+\mathcal{L}_{\text{matter}}(\psi,\nabla\psi,A_\mu;g_{\mu\nu})
\right].Sbulk=∫Md4x−g[−4μ01FμνFμν+Lmatter(ψ,∇ψ,Aμ;gμν)].
* Scalar matter (simpler audit):

Lmatter=−ℏ ℑ{ψ∗∂tψ}−ℏ22m gij(∂i−ieAi/ℏ)ψ∗(∂j+ieAj/ℏ)ψ−V(∣ψ∣2).\mathcal{L}_{\text{matter}}= -\hbar\,\Im\{\psi^*\partial_t\psi\}
-\frac{\hbar^2}{2m}\,g^{ij}(\partial_i-ieA_i/\hbar)\psi^*(\partial_j+ieA_j/\hbar)\psi
* V(|\psi|^2).Lmatter=−ℏℑ{ψ∗∂tψ}−2mℏ2gij(∂i−ieAi/ℏ)ψ∗(∂j+ieAj/ℏ)ψ−V(∣ψ∣2).
* Spinor matter (Dirac variant, optional audit):

LDirac=ψˉ(iℏγμDμ−mc)ψ,Dμ=∇μ−i eℏAμ.\mathcal{L}_{\text{Dirac}}=\bar{\psi}\big(i\hbar \gamma^\mu D_\mu - mc\big)\psi
,\quad D_\mu=\nabla_\mu - i\,\tfrac{e}{\hbar}A_\mu .LDirac=ψˉ(iℏγμDμ−mc)ψ,Dμ=∇μ−iℏeAμ.
===== We place a thin shell Σ\SigmaΣ with surface Lagrangian density =====

Sbdy=∫Σdτ d2Ω −γ  [−14λΣ fabfab+χˉ(iℏ \slashedDΣ−mΣc)χ+JΣμAμ+ΠabKab].S_{\text{bdy}}=\int_{\Sigma}\mathrm{d}\tau\,\mathrm{d}^2\Omega\,\sqrt{-\gamma}\;
\Big[
-\tfrac{1}{4}\lambda_\Sigma\, f_{ab}f^{ab}
+ \bar{\chi}\big(i\hbar\,\slashed{D}_\Sigma - m_\Sigma c\big)\chi
+ J^\mu_{\Sigma} A_\mu
+ \Pi^{ab} K_{ab}
\Big].Sbdy=∫Σdτd2Ω−γ[−41λΣfabfab+χˉ(iℏ\slashedDΣ−mΣc)χ+JΣμAμ+ΠabKab].
Meanings to audit:
* γ\gammaγ: det. of boundary metric; fabf_{ab}fab is the boundary field strength if a tangential gauge d.o.f. is used (optional).
* χ\chiχ: boundary (surface) charge carrier field (can be scalar for first audit).
* JΣμJ^\mu_{\Sigma}JΣμ: physical surface current coupling to AμA_\muAμ (minimal coupling).
* ΠabKab\Pi^{ab}K_{ab}ΠabKab: coupling to extrinsic curvature KabK_{ab}Kab (gives surface stress-energy), with Πab\Pi^{ab}Πab the canonical momentum conjugate to habh_{ab}hab on Σ\SigmaΣ.

: 

==== ### ====

Vary AμA_\muAμ in Sbulk+SbdyS_{\text{bulk}}+S_{\text{bdy}}Sbulk+Sbdy:

∇νFμν=μ0(Jbulkμ+JΣμ δΣ),\nabla_\nu F^{\mu\nu} = \mu_0 \big(J^\mu_{\text{bulk}} + J^\mu_{\Sigma}\,\delta_\Sigma\big),∇νFμν=μ0(Jbulkμ+JΣμδΣ),
where δΣ\delta_\SigmaδΣ is a delta distribution localized at r=r0r=r_0r=r0.
* Audit: derive the usual jump (pillbox) boundary conditions across Σ\SigmaΣ: - n^⋅(D2−D1)=σs \hat{\mathbf{n}}\cdot(\mathbf{D}_2-\mathbf{D}_1)=\sigma_sn^⋅(D2−D1)=σs with σs\sigma_sσs surface charge. - n^×(H2−H1)=Ks \hat{\mathbf{n}}\times(\mathbf{H}_2-\mathbf{H}_1)=\mathbf{K}_sn^×(H2−H1)=Ks with Ks\mathbf{K}_sKs surface current.

===== From varying habh_{ab}hab (or γ\gammaγ) one gets a surface stress tensor =====

Sab=2−γδSbdyδhab=σ uaub+p (hab+uaub)+⋯S_{ab} = \frac{2}{\sqrt{-\gamma}}\frac{\delta S_{\text{bdy}}}{\delta h^{ab}}
= \sigma\, u_a u_b + p\, (h_{ab}+u_a u_b) + \cdotsSab=−γ2δhabδSbdy=σuaub+p(hab+uaub)+⋯
and GR’s Israel junction across Σ\SigmaΣ:

[Kab]−hab[K]=−8πG Sab.\big[K_{ab}\big] - h_{ab}\big[K\big] = -8\pi G\, S_{ab}.[Kab]−hab[K]=−8πGSab.
* Audit: verify that with a physically chosen Sab(σ,p,j)S_{ab}(\sigma,p,j)Sab(σ,p,j) one recovers Newtonian limit outside Σ\SigmaΣ and an effective 1/r21/r^21/r2 force, then extract an effective GGG in terms of σ,p\sigma, pσ,p, and microscopic rates.

===== Gauge: χ→eiαχ, Aμ→Aμ−ℏe∂μα \chi \to e^{i\alpha}\chi,\ A_\mu\to A_\mu - \tfrac{\hbar}{e}\partial_\mu\alphaχ→eiαχ, Aμ→Aμ−eℏ∂μα. =====
* Audit: from boundary Lagrangian, derive conserved surface current

jΣa=∂Lbdy∂(∂aχ)(iχ)+c.c.,∇ajΣa=− n^μJleakμ,j^a_{\Sigma}=\frac{\partial \mathcal{L}_{\text{bdy}}}{\partial(\partial_a \chi)}(i\chi) + \text{c.c.},
\qquad \nabla_a j^a_{\Sigma} = -\, \hat{n}_\mu J^\mu_{\text{leak}},jΣa=∂(∂aχ)∂Lbdy(iχ)+c.c.,∇ajΣa=−n^μJleakμ,
showing that surface charge is conserved except for photon absorption/emission (leakage term), which is exactly the QAT “moment of time”.

==== ### ====
* Scalar: ϕ(θ,ϕ)=∑ℓ,maℓmYℓm(θ,ϕ)\phi(\theta,\phi)=\sum_{\ell,m} a_{\ell m} Y_{\ell m}(\theta,\phi)ϕ(θ,ϕ)=∑ℓ,maℓmYℓm(θ,ϕ).
* Spinor: use spinor spherical harmonics Ωκm\Omega_{\kappa m}Ωκm; spin-½ enforces antiperiodic boundary condition under 2π2\pi2π rotation (needs 4π4\pi4π to return), which QAT interprets as the geometric half-radius feature.

Audit: show how the allowed ℓ\ellℓ (or κ\kappaκ) produce discrete energies and how angular momentum quantization recovers the 1/21/21/2 factor when you treat reff=r0/2r_{\text{eff}}=r_0/2reff=r0/2 (half-radius) as the active orbital scale on the boundary.

===== QAT insists on keeping h/2πh/2\pih/2π explicit: =====

ΔE Δt≥h4π\Delta E\, \Delta t \ge \frac{h}{4\pi}ΔEΔt≥4πh
(standard is ℏ/2\hbar/2ℏ/2); interpret Δt\Delta tΔt as mean separation of absorption events on Σ\SigmaΣ.
Audit: check that the event-rate model (Poisson or renewal process) matches this bound for reasonable σ\sigmaσ and jΣaj^a_\SigmajΣa.

==== Treat two distant boundaries Σ1,Σ2\Sigma_1,\Sigma_2Σ1,Σ2 exchanging photons. The net momentum transfer per unit time on Σ2\Sigma_2Σ2 due to Σ1\Sigma_1Σ1 scales as ====

P˙∼σ1σ2(4πr2)1c,\dot{P} \sim \frac{\sigma_1 \sigma_2}{(4\pi r^2)}\frac{1}{c},P˙∼(4πr2)σ1σ2c1,
producing an effective F∝1/r2F\propto 1/r^2F∝1/r2.
Audit: formalize this by computing the surface Poynting vector and its normal component, then show under phase-coherent assumptions the sign is always attractive at long wavelength (spin-2 effective kernel from two spin-1 exchanges). Compare the resulting coefficient with Newton’s GGG.

==== Define horizon scale R=ctR=ctR=ct, microscopic length ℓ\ellℓ (e.g., reduced Compton of proton), and count boundary modes: ====

N(t)≈4πR2ℓ2∝t2.\mathcal{N}(t) \approx \frac{4\pi R^2}{\ell^2}\propto t^2.N(t)≈ℓ24πR2∝t2.
Assign mass per mode m⋆=αmmpm_\star = \alpha_m m_pm⋆=αmmp. Then

Mu(t)≈αmmp4π(ct)2ℓ2∝t2.M_u(t) \approx \alpha_m m_p \frac{4\pi (ct)^2}{\ell^2} \propto t^2.Mu(t)≈αmmpℓ24π(ct)2∝t2.
Audit: (i) reproduce the t2t^2t2 law; (ii) fit αm\alpha_mαm to present-day mass → we found αm∼1.06×10−5\alpha_m\sim 1.06\times10^{-5}αm∼1.06×10−5; (iii) check energy budget consistency (early-time radiation vs today).

==== Because the bulk EM action is standard Maxwell and the boundary variation yields standard pillbox conditions, the four Maxwell equations remain unchanged in the bulk. The difference is interpretive: QAT ties the sources (surface charge/current) to event-based boundary dynamics on S2S^2S2. ====
Audit: explicitly show that the divergence and curl equations are unmodified and that Gauss’s law on a Gaussian surface enclosing Σ\SigmaΣ returns the expected surface charge.

==== 1. Boundary variation: From SbdyS_{\text{bdy}}Sbdy derive JΣμJ^\mu_{\Sigma}JΣμ and the jump conditions for E,B\mathbf{E},\mathbf{B}E,B. ====
# Surface stress tensor: Derive SabS_{ab}Sab and show Israel junction reproduces Newtonian potential outside the shell; extract an effective G(σ,p,rates)G(\sigma,p,\text{rates})G(σ,p,rates).
# Spin-½ on S²: Confirm antiperiodic boundary condition and show how an effective half-radius r0/2r_0/2r0/2 appears in angular momentum quantization.
# Uncertainty link: Model absorption events as a point process on Σ\SigmaΣ and show ΔE Δt≥h/(4π)\Delta E\,\Delta t\ge h/(4\pi)ΔEΔt≥h/(4π) with h/2πh/2\pih/2π explicit.
# Two-boundary force: Compute net momentum transfer between Σ1,Σ2\Sigma_1,\Sigma_2Σ1,Σ2 using EM flux; identify when the sign is attractive and compare magnitude to Gm1m2/r2G m_1 m_2/r^2Gm1m2/r2.
# LNH scaling: Reproduce N∝t2\mathcal{N}\propto t^2N∝t2 and calibrate αm\alpha_mαm.
# Consistency: Check local energy conservation: energy removed from radiation fields equals increase in boundary internal energy (mass + heat + EM).
# Limits: Show third-law compatibility: as T→0T\to 0T→0, boundary event rate → 0, so “time production” rate → 0 (QAT’s arrow slows locally).

==== - A rigorous microphysical form for SbdyS_{\text{bdy}}Sbdy: what minimal terms are needed to produce JΣμJ^\mu_\SigmaJΣμ and SabS_{ab}Sab without breaking gauge or diffeomorphism invariance? ====
* Proving attraction (not repulsion) generically for the second-order EM mechanism.
* Deriving the exact value of GGG and α from geometry (avoid numerology).
* Ensuring no conflict with precision QED (Lamb shift, g−2g-2g−2) and with constraints on G˙,α˙\dot{G}, \dot{\alpha}G˙,α˙.

==== - c=1/ϵ0μ0c=1/\sqrt{\epsilon_0\mu_0}c=1/ϵ0μ0. ====
* D=ϵ0E\mathbf{D}=\epsilon_0 \mathbf{E}D=ϵ0E, H=B/μ0\mathbf{H}=\mathbf{B}/\mu_0H=B/μ0.
* Pillbox jump conditions (across Σ\SigmaΣ with normal n^\hat{\mathbf{n}}n^): n^⋅(D2−D1)=σs\hat{\mathbf{n}}\cdot(\mathbf{D}_2-\mathbf{D}_1)=\sigma_sn^⋅(D2−D1)=σs, n^×(H2−H1)=Ks\hat{\mathbf{n}}\times(\mathbf{H}_2-\mathbf{H}_1)=\mathbf{K}_sn^×(H2−H1)=Ks, n^⋅(B2−B1)=0\hat{\mathbf{n}}\cdot(\mathbf{B}_2-\mathbf{B}_1)=0n^⋅(B2−B1)=0, n^×(E2−E1)=0\hat{\mathbf{n}}\times(\mathbf{E}_2-\mathbf{E}_1)=0n^×(E2−E1)=0.

==== - Symbolic notebook: derive (i) boundary Maxwell jump; (ii) SabS_{ab}Sab from SbdyS_{\text{bdy}}Sbdy; (iii) Israel junction → Newtonian limit. ====
* Numerical script: two-sphere momentum exchange vs distance; compare to Gm1m2/r2G m_1 m_2/r^2Gm1m2/r2.
* Spectral script: spinor spherical harmonics on S2S^2S2, show quantization and effective r0/2r_0/2r0/2 factor.

If you’d like, I can now turn this into a short markdown/PDF one-pager for easy sharing — or jump straight to one of the “Minimal files” items and draft the boundary-variation derivation in a form Copitor can execute.