Validation against Solar-System PPN, binary pulsars, gravitational waves, and linear cosmology (with inline figures). Stylesheet and MathJax configuration merged from Part 1.
We validate the Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC) against precision tests of gravity and cosmology. In SPSP–SSC, a projection geometry with an elliptic sorting constraint reproduces general relativity (GR) locally and adds no propagating degrees of freedom. In validated regimes, the full post-Newtonian (PPN) catalogue equals GR; scalar dipole emission is absent; and only two transverse–traceless tensor modes propagate at luminal speed. Binary pulsar orbital decay and gravitational-wave propagation therefore coincide with GR predictions. On cosmological scales, SPSP–SSC reduces to the ΛCDM linear perturbation system, reproducing the CMB acoustic structure and lensing at measured multipoles. We state explicit kill-switch inequalities: any detection of dipole radiation, non-GR polarizations, non-luminal GW speed, or ΛCDM-inconsistent linear spectra would rule out the model.
Alternative theories of gravity often add fields that conflict with precision tests, or they remain too flexible to be falsified. Part 1 of SPSP–SSC introduced a single-sphere projection geometry with a sorting potential \( \Phi \) that acts via an elliptic constraint rather than a wave equation. The propagating sector thus remains Einsteinian: two tensor polarizations only, with no scalar/vector radiation channels.
This Part 2 provides a systematic validation. We show: (i) exact PPN lock-in for Solar-System tests; (ii) binary pulsar decay with no dipole channel; (iii) luminal gravitational-wave speed and tensor-only polarizations; and (iv) ΛCDM-consistent linear perturbations. We include self-contained figures for PPN, binaries, and a CMB TT overlay (schematic subset) to demonstrate agreement and clarify falsifiability.
We take \( Z(\theta)=Z_0>0 \), \( f(\theta)=f_0>0 \), a locally frozen cycle \( \partial\theta_0=0 \), and screening \( \delta\mathcal P\to0 \). The sorting field is elliptic and enforces the Poisson relation \[ \nabla^2 \Phi = \frac{1}{Z_0}\,\rho \quad \Leftrightarrow \quad 4\pi G = \frac{1}{Z_0}. \] There are no preferred-frame forces or extra long-range fields; post-Newtonian corrections arise solely from Einstein–Hilbert nonlinearity, as in GR.
Hence SPSP–SSC is indistinguishable from GR in all standard Solar-System tests in the validated regime.
| Parameter | GR | SPSP–SSC | Status |
|---|---|---|---|
| γ | 1 | 1 | Matches GR |
| β | 1 | 1 | Matches GR |
| ξ | 0 | 0 | Matches GR |
| α1 | 0 | 0 | Matches GR |
| α2 | 0 | 0 | Matches GR |
| α3 | 0 | 0 | Matches GR |
| ζ1..4 | 0 | 0 | Matches GR |
With \( \Phi \) elliptic, there is no scalar radiation channel; only two transverse–traceless tensor polarizations \( (h_+,h_\times) \) propagate. The propagation is luminal \( v_{\rm gw}=c \).
SPSP–SSC reproduces the GR quadrupole formula and hence the observed orbital decay of well-measured pulsar binaries.
Only tensor \( (+,\times) \) polarizations are allowed in validated regimes; no scalar/vector GW modes. Propagation is luminal, consistent with multimessenger constraints; any robust deviation would falsify the model.
In a spatially flat FRW metric \( ds^2=a^2(\tau)[-d\tau^2+\delta_{ij}dx^i dx^j] \), SPSP–SSC reduces to an effective background expansion \[ H_{\rm eff}(\tau)\approx H_{\Lambda{\rm CDM}}(\tau) \] within validated regimes. Screening ensures \( \delta\mathcal P\to0 \) at epochs relevant to the CMB acoustic peaks.
In Newtonian gauge, \( ds^2=a^2(\tau)[-(1+2\Psi)d\tau^2+(1-2\Phi_g)\delta_{ij}dx^i dx^j] \). The elliptic constraint removes linear anisotropic stress in validated regimes, giving \( \Phi_g=\Psi \). Vector modes decay; tensors satisfy \[ h''_{ij}+2\mathcal H h'_{ij}+k^2 h_{ij}=0 \] with two TT polarizations at speed \( c \).
With \( \Phi_g=\Psi \), the linear system equals GR/ΛCDM; photon/neutrino multipole hierarchies and transfer functions are unchanged.
The plot below shows a compact, illustrative subset of CMB TT bandpowers spanning the first acoustic peaks and the damping tail, overlaid with a smooth GR/SPSP–SSC curve. Replace or extend the inline array with your exact bandpowers as needed.
Kill switches: (i) any non-GR PPN parameter; (ii) binary dipole radiation; (iii) extra GW polarizations or non-luminal propagation; (iv) ΛCDM-inconsistent CMB/LSS spectra at validated scales.
SPSP–SSC is conservative—locked to GR where tested—and innovative in geometry: a single-sphere projection with an elliptic constraint that adds no propagating modes. Unlike many modified-gravity scenarios, it avoids dipole radiation and extra polarizations by construction while remaining decisively falsifiable at frontiers: very low-ℓ cosmology, horizon-scale black holes, and high-precision GW polarimetry.
If correct, SPSP–SSC reframes ultra-large-scale cosmology and black-hole interiors and points toward a geometric underpinning for nonlocal correlations, all while leaving local physics intact. Upcoming surveys and GW observatories offer a clean path to verification or exclusion.
We have provided a comprehensive, data-facing validation of SPSP–SSC across Solar-System PPN tests, binary pulsar decay, gravitational-wave polarizations and speed, and linear cosmology. In every validated regime, SPSP–SSC reproduces GR/ΛCDM predictions exactly. The theory is therefore conservative where precision demands it, yet falsifiable by specific observations. The next generation of cosmology and GW measurements will determine whether SPSP–SSC survives or is decisively ruled out.
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