Solar-System Validation from SPSP–SSC

Executive statement. In SPSP–SSC the sorting potential \(\Phi\) is an elliptic constraint ( no kinetic term ), and projection terms are screened in high-density/near-source environments. Hence in the Solar System the field equations reduce to Einstein’s equations with metric coupling only. The Parametrized Post-Newtonian (PPN) parameters are therefore \(\gamma=\beta=1\) and all others vanish, so every classic GR test (perihelion advance, light-deflection, Shapiro delay, redshift, geodetic/frame-dragging) is recovered exactly.

1. From the SPSP–SSC action to the exterior field equations

Start from the validated-regime SPSP–SSC action (your Part 1/Section 4 form):

\[ S=\int d^4x\,\sqrt{-g}\,\Big[\tfrac{M_P^2}{2}R+\mathcal L_{\rm SM} -\,\Phi\,(\rho-\varepsilon)\Big]\;+\;S_{\rm proj}^{\rm (screened)}. \]

Elliptic constraint: variation w.r.t. \(\Phi\) gives \(\nabla^2\Phi=4\pi G\,\rho\) (in the Newtonian limit), and in vacuum outside bodies (\(\rho=\varepsilon=0\)) this reduces to \(\nabla^2\Phi=0\). With regular boundary conditions, \(\Phi=\)const. and drops out of the exterior dynamics.
Screening: the projection functional is weakly coupled and exponentially suppressed locally (your \(\sigma(\theta)\) factor), so \(S_{\rm proj}^{\rm (screened)}\to 0\) in Solar-System conditions.
Result: the metric sector is governed solely by the Einstein–Hilbert term with minimally coupled matter \(\Rightarrow\) Einstein equations in vacuum: \(G_{\mu\nu}=0\).

1.1 PPN catalogue from the action

Expanding the metric in PPN gauge to \(\mathcal O(v^4)\) under these assumptions gives (as in your Appendix F):

\[ \boxed{\;\gamma=1,\quad\beta=1,\quad \xi=\alpha_1=\alpha_2=\alpha_3=\zeta_1=\zeta_2=\zeta_3=\zeta_4=0\;}. \]

This alone fixes the Solar-System phenomenology to GR values, because the classic observables depend only on \(\gamma\) and \(\beta\) (at the orders of interest): perihelion \(\propto (2+2\gamma-\beta)\), light bending \(\propto (1+\gamma)\), Shapiro delay \(\propto (1+\gamma)\).

2. Spherically symmetric exterior metric

With \(G_{\mu\nu}=0\) outside the Sun, the unique static, spherically symmetric vacuum solution is Schwarzschild:

\[ ds^2=-\Big(1-\frac{2GM_\odot}{rc^2}\Big)c^2dt^2 +\Big(1-\frac{2GM_\odot}{rc^2}\Big)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2). \]

All Solar-System predictions below follow directly from geodesics of this metric (or, equivalently, from PPN with \(\gamma=\beta=1\)).

3. Perihelion advance (derivation)

Restrict to the orbital plane \(\theta=\pi/2\). Using specific angular momentum \(h=r^2\dot\phi\) and \(u\equiv 1/r\), the geodesic equations reduce to the relativistic Binet equation (standard GR result, now justified by the reduction above):

\[ \frac{d^2u}{d\phi^2}+u=\frac{GM_\odot}{h^2}+\frac{3GM_\odot}{c^2}u^2. \]

Treat the \(u^2\) term as a small perturbation to the Newtonian solution \(u_0=\frac{GM_\odot}{h^2}(1+e\cos\phi)\). Solving to first order in \(GM_\odot/(ac^2)\) gives a small advance of the perihelion per orbit:

\[ \boxed{\;\Delta\varpi_{\rm per\,orbit}=\frac{6\pi GM_\odot}{a(1-e^2)c^2}\;}. \]

Converting to arcseconds per century with Mercury’s \(a,e,P\) yields \(\approx 43''/\text{century}\), matching observation once Newtonian planetary perturbations are subtracted. (Your numerical table in the Solar-System page reproduces Mercury, Venus, Earth, Mars.)

4. Light deflection (derivation)

In the weak-field limit, the PPN result for a light ray with impact parameter \(b\) is

\[ \Delta\phi=\frac{1+\gamma}{2}\,\frac{4GM_\odot}{b\,c^2}\;\;\xrightarrow{\ \gamma=1\ }\;\frac{4GM_\odot}{b\,c^2}. \]

At the solar limb (\(b\simeq R_\odot\)), this yields \(\Delta\phi\simeq 1.751''\), the classic Eddington value. Because SPSP–SSC fixes \(\gamma=1\), the prediction is exactly GR’s.

5. Shapiro time delay (derivation)

For emitter at \(r_E\) and receiver at \(r_R\) with closest approach \(b\), the PPN time delay along a null geodesic is

\[ \Delta t=\frac{1+\gamma}{2}\,\frac{2GM_\odot}{c^3} \ln\!\left(\frac{4\,r_E r_R}{b^2}\right)\;\;\xrightarrow{\ \gamma=1\ }\;\frac{2GM_\odot}{c^3} \ln\!\left(\frac{4\,r_E r_R}{b^2}\right). \]

With \(r_E\simeq 1\,\text{AU}\), \(r_R\simeq 1.5\,\text{AU}\), \(b\simeq R_\odot\), one finds \(\Delta t\sim 10^2\,\mu\text{s}\), consistent with Cassini-level confirmations. Again SPSP–SSC gives the GR value because \(\gamma=1\).

6. Why SPSP–SSC must agree with GR here

No new exterior DOF: \(\Phi\) is elliptic and non-radiating; in vacuum \(\nabla^2\Phi=0\Rightarrow\Phi=\)const., so it does not modify the metric outside sources.
Screening: projection terms are suppressed in high-density/near-source regimes; they do not enter the exterior PPN expansion.
PPN identities: the metric is purely Einsteinian with minimal coupling \(\Rightarrow \gamma=\beta=1\), others 0.

Remarks on scope & falsifiability

The derivations use only the SPSP–SSC reduction to vacuum Einstein equations plus standard geodesic motion. Any measured \(\gamma\neq 1\) or \(\beta\neq 1\) in the Solar System would immediately falsify SPSP–SSC.
Strong-field/binary tests (pulsar decay, GW phasing) likewise coincide with GR in validated regimes because the constraint field cannot radiate (no scalar dipole channel). Those are covered in your Part 2/Radiation section.