Part 2 — Figures: Methods & Reproducibility Notes

This page documents how each figure in Part 2 was produced: data sources, equations, assumptions, and minimal code snippets. SPSP–SSC predictions coincide with GR/ΛCDM/QM in validated regimes, so overlays use standard pipelines; falsifiability bounds are shown explicitly.

Clarity statement. All exterior predictions use unmodified GR geodesics and SM/QM dynamics; SPSP–SSC effects appear only through bounded elliptic boundary diagnostics. This preserves validated results and keeps novelty sharply testable.

Figure 1 — Compact Binary GW Phasing (−1PN dipole test)

(Visual in Part 2: frequency-domain phase residuals showing GR curve; SPSP–SSC overlaps GR; a shaded band indicates the −1PN dipole bound.)

What is shown

Fourier-domain phase \(\Psi(f)\) for a compact binary inspiral (restricted PN), contrasted with a hypothetical −1PN dipole term.
SPSP–SSC prediction: no dipole channel → \(\beta_{-2}=0\). Overlay equals GR within the plotted precision.
Falsifiability band: \(|\beta_{-2}|<\beta_{\rm thr}\) mapped to \(|\mathcal F_{\rm dip}|/\mathcal F_{\rm quad}<10^{-3}\).

Equations / model mapping

\[ \Psi(f) \;=\; \Psi_{\rm GR}(f)\;+\;\beta_{-2}\,f^{-7/3}\;+\;\sum_{k\ge 0} \beta_k f^{(k-5)/3}, \quad \text{SPSP–SSC: } \beta_{-2}=0. \]

Here \(\beta_{-2}\) encodes a −1PN dipole term. Because the SPSP–SSC constraint field is elliptic (non-radiative), no scalar dipole flux is allowed, so the GR phasing is recovered identically.

Methods & code snippet

Data LIGO/Virgo public constraints on dipole radiation (O1–O3 analyses); use representative mass pairs to render a GR phasing template (TaylorF2) and a hypothetical −1PN term for the shaded bound.

Method Plot \(\Psi_{\rm GR}(f)\) from a standard PN library; draw a zero-centered band for \(\beta_{-2}\in [-\beta_{\rm thr}, +\beta_{\rm thr}]\). SPSP–SSC curve coincides with GR (central line).

# pseudo-Python (illustrative)
import numpy as np

def psi_GR(f, m1, m2):
    # placeholder for PN phase (TaylorF2-style); use standard library in real code
    # Psi ~ (3/128)(pi M_c f)^(-5/3) * [1 + PN corrections]
    return base_phase(f, m1, m2)

def psi_with_dipole(f, beta_m2):
    return psi_GR(f, m1, m2) + beta_m2 * f**(-7.0/3.0)

f = np.logspace(1, 3, 500)  # 10–1000 Hz
psi_gr = psi_GR(f, m1=30, m2=30)
beta_thr = map_flux_ratio_to_beta_threshold(m1=30, m2=30, ratio=1e-3)
band_lo = psi_with_dipole(f, -beta_thr)
band_hi = psi_with_dipole(f, +beta_thr)
# Plot psi_gr with shaded area [band_lo, band_hi]

Note In real analyses you’d pull exact PN implementations (e.g., LALSuite), but for Part 2 the logic is: GR line = SPSP–SSC line; shaded band = falsifiability window.

Figure 2 — Binary Pulsars: Orbital-Decay Agreement

(Visual in Part 2: x-axis two systems; y-axis ratio \(\dot P_{\rm obs}/\dot P_{\rm GR}\). Unity line; points with error bars for B1913+16 and J0737−3039 near 1.0.)

What is shown

Ratios of observed to GR-predicted orbital period decay for canonical systems.
SPSP–SSC prediction: identical to GR quadrupole → ratios at unity (within error bars).
Model falsification: a persistent excess deficit implying \(|\Delta_{\rm dip}|\ge 10^{-3}\) would rule out SPSP–SSC.

Equations / model mapping

\[ \Delta_{\rm dip}\;\equiv\;\frac{\dot P_{\rm obs}-\dot P_{\rm GR}}{|\dot P_{\rm GR}|},\qquad \boxed{\,|\Delta_{\rm dip}|<10^{-3}\,}\quad\text{(SPSP–SSC hard bound).} \]

Because the constraint field does not radiate, binary energy loss is purely quadrupolar (GR). Hence the ratios cluster at 1.0 subject to measurement uncertainty.

Methods & code snippet

Data Published timing residuals and decay rates for PSR B1913+16 (Hulse–Taylor) and PSR J0737−3039 (Double Pulsar).

Method For each system, compute \(\dot P_{\rm GR}\) from standard GR formulae (including environmental corrections used in the literature), then form the ratio with \(\dot P_{\rm obs}\).

# pseudo-Python (illustrative)
systems = {
  "B1913+16": {"Pdot_obs": -2.4025e-12, "Pdot_GR": -2.4025e-12, "err": 0.2e-3},
  "J0737-3039": {"Pdot_obs": -1.252e-12, "Pdot_GR": -1.252e-12, "err": 1.3e-4},
}
for name, d in systems.items():
    ratio = d["Pdot_obs"]/d["Pdot_GR"]
    delta = (d["Pdot_obs"] - d["Pdot_GR"])/abs(d["Pdot_GR"])
    assert abs(delta) < 1e-3  # SPSP–SSC falsification bound
    # plot 'ratio' with error bar 'err'

Note Numbers above are placeholders for logic illustration; for publication use the exact literature values and uncertainties.

Figure 3 — CMB TT Bandpowers: GR/ΛCDM Overlay

(Visual in Part 2: binned TT bandpowers with error bars; a smooth ΛCDM line runs through the points. SPSP–SSC curve coincides with ΛCDM/GR line.)

What is shown

Observed CMB TT bandpowers with ΛCDM best-fit line.
SPSP–SSC prediction: in validated linear regime, equations reduce to GR/ΛCDM → same TT curve.

Equations / model mapping

Linearized Einstein–Boltzmann system with \(\Phi_g=\Psi\) and standard species (photons, baryons, CDM, neutrinos). SPSP–SSC adds no linear anisotropic stress and no new radiative DOF in this regime; hence CAMB/CLASS spectra apply directly.

Methods & code snippet

Data Planck 2018 TT bandpowers (public likelihood); best-fit ΛCDM parameters.

Method Use standard Boltzmann solver to compute \(C_\ell^{TT}\) and overlay on binned data. SPSP–SSC line = ΛCDM line.

# pseudo-Python (illustrative)
import numpy as np
# e.g., using CAMB/CLASS in a real script:
# params = set_LCDM_params(H0=67.4, ombh2=0.0224, omch2=0.120, As, ns, tau)
# Cl_TT = boltzmann_solver(params).TT

ells = np.arange(2, 2500)
Cl_TT = lcdm_theory_TT(ells)  # placeholder for solver
# Load binned Planck bandpowers (Dl = l(l+1)Cl/2pi)
# Plot points with error bars; overlay Dl from theory.

Note The scientific point is: SPSP–SSC does not alter the validated linear system → no extra freedom → identical TT curve within current precision.

Optional Figure — Weak Lensing & Growth (Poisson Limit)

(Optional visual: matter power spectrum or lensing \(C_\ell^{\phi\phi}\) with ΛCDM line; SPSP–SSC overlapping in the validated window.)

What would be shown

Sub-horizon Poisson-like relation \(-k^2\Phi_g \simeq 4\pi G a^2 \sum_X \rho_X \delta_X\) holds; growth and lensing potentials coincide with GR.
SPSP–SSC line overlaps ΛCDM; deviations, if any, are confined to ultra-large scales (outside current precision) via the projection boundary sector.

Methods & code snippet

Method Compute \(P(k)\) and/or \(C_\ell^{\phi\phi}\) with a standard solver; SPSP–SSC overlay identical in the validated window.

# pseudo-Python (illustrative)
k = np.logspace(-3, 1, 300)  # h/Mpc
P_LCDM = matter_power_LCDM(k, z=0.5)
P_SPSP = P_LCDM  # identical in validated regime
# Plot P(k); note: deviations (if any) pushed to ultra-large scales by design.

Falsifiability Inserts (used in the panels)

Dipole radiation (binaries): \(\displaystyle \boxed{|\Delta_{\rm dip}|<10^{-3}}\) after systematics; else SPSP–SSC falsified.
GW phasing (−1PN): \(\displaystyle \boxed{|\beta_{-2}|<\beta_{\rm thr}}\) with mapping to \(|\mathcal F_{\rm dip}|/\mathcal F_{\rm quad}<10^{-3}\).
Linear cosmology: any significant mismatch with ΛCDM/GR in the validated window falsifies.

Provenance & Reproducibility

Data inputs: community public datasets (pulsar timing literature; LIGO/Virgo public constraints; Planck 2018). Theory overlays: standard GR/ΛCDM pipelines (e.g., LALSuite, CAMB/CLASS). SPSP–SSC mapping: in validated regimes, overlays are identical to GR/ΛCDM curves; shaded regions indicate falsification thresholds, not fitted parameters.