SPSP–SSC Part III — Predictions, Overlays & Open Windows

Abstract

We extend SPSP–SSC beyond validated domains and examine predictive windows: ultra-large-scale cosmology, black-hole interiors, and early-universe dynamics. Part II established lock-in to GR/ΛCDM across tested regimes; here we provide falsifiable forecasts and concrete, real-data-ready prototypes. We include (i) Kerr ringdown frequency and damping-time curves with explicit ±2% bands as proxies for interior recycling, (ii) a Schwarzschild null-geodesic ray-tracing sweep to anchor lensing/shadow expectations, and (iii) a large-scale projection residual window \(W(k)\) suitable for CLASS/CAMB integration. Plots accept inline JSON data for immediate overlays; if absent, they render cleanly with a “no data” hint. This constitutes a practical roadmap for near-term tests without disturbing validated physics.

1. Ultra-Large-Scale Structure (W(k) Forecast + Data)

SPSP–SSC predicts that any departures from ΛCDM/GR at linear order, if present, are confined to ultra-large scales through a projection residual window \(W(k)\). A minimal, falsifiable choice is a Gaussian window peaked at very small \(k\):

\[ W(k)=A\,\exp\!\big[-(k/k_0)^2\big],\qquad A\ll 1,\ \ k_0\sim 10^{-3}\text{--}10^{-2}\;h\,\mathrm{Mpc}^{-1}. \]

Figure 1 — Projection residual window \(W(k)\). Gaussian \(A=0.02\), \(k_0=2\times10^{-3}\,h\,\mathrm{Mpc}^{-1}\). Points (if any) are overlaid from the cmb_wk_data JSON block.

2. Black Holes: Ringdown (±2%) & Ray Tracing + Data

2.1 Ringdown frequency and damping time (GR ±2%)

Exterior dynamics match GR; any new physics appears (if at all) only via interior “recycling” that could produce percent-level shifts in quasi-normal mode (QNM) spectra without altering inspiral dynamics. We show solid GR curves with explicit dashed ±2% offsets for the fundamental \(l\!=\!m\!=\!2, n\!=\!0\) mode. Measured points (if provided) are overlaid with error bars from the ringdown_data JSON block.

Figure 2a — Ringdown frequency \(f(a)\), \(M=10\,M_\odot\).

Figure 2b — Ringdown frequency \(f(a)\), \(M=30\,M_\odot\).

Figure 2c — Ringdown frequency \(f(a)\), \(M=60\,M_\odot\).

Figure 3a — Ringdown damping time \(\tau(a)\), \(M=10\,M_\odot\).

Figure 3b — Ringdown damping time \(\tau(a)\), \(M=30\,M_\odot\).

Figure 3c — Ringdown damping time \(\tau(a)\), \(M=60\,M_\odot\).

2.2 Ray tracing: null geodesic sweep (legend fixed)

Null geodesics in Schwarzschild (equatorial) spanning impact parameters just above the critical \(b_c=3\sqrt{3}\,M\) establish the GR-consistent exterior baseline for shadow and photon rings. The legend is placed below the plot with a light background to avoid overlap with trajectories.

Figure 4 — Schwarzschild null-geodesic sweep. Horizon \(r=2M\) (solid) and photon sphere \(r=3M\) (dashed). Curves show rays with \(b\gtrsim b_c\).

3. Early-Universe Scenarios (Outlook)

Quantitative early-universe forecasts (primordial tensor features, reheating signatures) are reserved for a dedicated note. Within validated regimes, Part II’s linear cosmology overlays remain locked to ΛCDM; departures, if any, should be parametrized by slowly varying background deformations or a controlled \(W(k)\) on super-horizon scales, preserving current CMB/BAO/SNe fits.

4. Quantum Overlap & Nonlocality (Conceptual Bridge)

The elliptic (constraint-like) role of the sorting potential \(\Phi\) provides a structural analogue to quantum nonlocality: global consistency without new propagating channels. While no change to standard quantum theory is invoked, this perspective motivates experiments on entanglement in curved backgrounds and informs future unification work.

5. Falsifiability Forecasts (Inequalities)

No dipole radiation in binaries: \(|\mathcal{D}| \lesssim 10^{-4}\) (Double Pulsar scale). Any detected dipole channel falsifies SPSP–SSC.
GW polarizations: only two TT modes at \(c\). Observation of scalar/vector polarizations falsifies SPSP–SSC.
Ringdown offsets: at fixed \((a,M)\) from inspiral, \(|\Delta f/f|, |\Delta\tau/\tau| \lesssim \epsilon_\star\). A detected, repeatable offset \(\gtrsim\) percent-level would disfavor the interior recycling ansatz.
Linear cosmology: for multipoles probed precisely by Planck/BAO/SNe, require \(W(k)\to 0\); significant residuals at those \(k\) falsify SPSP–SSC.

6. Numerical Pipelines & Reproducibility

The figures are generated inline from explicit formulas and accept JSON overlays (below). For full analyses: implement \(W(k)\) in CLASS/CAMB for LSS/CMB; use numerical relativity for ringdown spectra with an interior recycling kernel (kept causally isolated from the exterior); and extend ray tracing to Kerr for image-plane reconstructions. Always include a GR/ΛCDM recovery switch to ensure falsifiability.

7. Conclusion & Outlook

Part III translates SPSP–SSC’s conservative lock-in into practical, near-term tests. The prototypes here (QNM curves with explicit ±2% offsets, ray sweeps with clean legends, and a clamped \(W(k)\)) define how to confront data without disturbing validated physics. Upcoming GW runs, LSS surveys, and CMB experiments can either tighten bounds on SPSP–SSC’s residuals or rule them out, providing a decisive verdict on the framework’s predictive content.

Appendix — Methods & Data Injection

A. Ringdown curves (Figures 2–3)

We use the Echeverria-style fit for the Kerr fundamental QNM \(l\!=\!m\!=\!2, n\!=\!0\): \(f = [1 - 0.63(1-a)^{0.3}]/(2\pi M)\), \(Q = 2(1-a)^{-0.45}\), \(\tau=Q/(\pi f)\), with \(M\) in geometric units and converted to SI via \(M\!=\!GM_\odot/c^3\) for 10, 30, 60 \(M_\odot\). We draw solid GR curves and dashed ±2% offsets. If the ringdown_data JSON block is populated, points with error bars are overlaid.

B. Ray tracing sweep (Figure 4)

We integrate the null-geodesic orbit equation in Schwarzschild equatorial plane, \(u(\phi)=1/r\): \(u''+u=3Mu^2\), from near infinity with \(u(0)\approx 0\), \(u'(0)=1/b\), for several \(b>b_c=3\sqrt{3}M\). We stop integration once \(r\le 2M\). Legend placed below the plot with a light background.

C. \(W(k)\) window (Figure 1)

\(W(k)=A\exp[-(k/k_0)^2]\) with \(A=0.02\), \(k_0=2\times10^{-3}\,h\,\mathrm{Mpc}^{-1}\). Values are clamped to a safe plotting range so the curve never dips below the axis. If the cmb_wk_data JSON is provided, points are overlaid.