SGF Paper 6: How to Test the Spectral Gravitation Framework (SGF)

By Paul Falconer & ESAci Core

Series: Spectral Gravitation Framework

Version: 1.0 — March 2026

DOI: https://doi.org/10.17605/OSF.IO/PJ8CQ

1. Purpose and Scope

This document is a practical guide for testing the Spectral Gravitation Framework (SGF) using public data and open code. It does not introduce new theory; it operationalises the falsification pathways laid out in SGF Papers 1–5 and shows, step by step, how an adversarial collaborator can evaluate SGF's main empirical bets.

We focus on four domains where SGF makes concrete, quantitative claims:

1. Cosmic void expansion

2. Gravitational‑wave "harp jitter"

3. Black‑hole horizon structure

4. Ultra‑long gamma‑ray burst (GRB) timing

For each domain, we specify: (a) the prediction, (b) the data, (c) the code, and (d) the falsification condition.

2. SGF's Empirical Bets in One Page

SGF is a density‑responsive, entanglement‑based extension of general relativity that links cosmic voids, black holes, and quantum phenomena through the same action and parameters. To be worth anyone's time, it must risk being wrong in public.

At a high level, SGF bets:

• Voids expand faster than ΛCDM predicts, with a specific excess anchored by the SGF parameters.

• Black‑hole mergers carry a narrow‑band, coherent "harp jitter" in the ringdown.

• Horizon images at ngEHT resolution will show fractal boundaries with a characteristic dimension.

• Ultra‑long GRBs exhibit quasi‑periodic spacing consistent with SGF's threshold dynamics.

The numeric thresholds in the sections below are deliberately conservative. Substantial tension short of formal falsification (e.g., a 3–4σ discrepancy in voids, or a systematic absence of jitter in several events) will still be treated as serious evidence against the current parameterization and will trigger revision.

If any one of these domains decisively contradicts the quantitative SGF predictions (after reasonable checks), SGF in its current form is wrong and must be revised or abandoned.

3. Testing SGF with Cosmic Voids

Epistemic status: Post‑fit test using independent catalogs. The parameters were fitted to DESI DR5; this test uses an independent dataset (e.g., DESI DR1 or an alternative void catalog).

3.1 Prediction

For large voids in DESI‑like surveys, SGF predicts an enhanced expansion rate:

H_void = (1.18 ± 0.03) H_ΛCDM for R > 30 Mpc.

Falsification condition: if the best‑fit ratio H_void / H_ΛCDM is < 1.15 at 5σ significance in independent void catalogs, SGF's current parameterization fails.

3.2 Data

Use public large‑scale‑structure datasets:

• DESI DR5/DR1 void catalogs (or equivalent from eBOSS/BOSS), including void radii, redshifts, and density contrasts. The specific benchmark dataset used for SGF's internal fits is available in the OSF repository under data/benchmarks/desi_dr5_voids.fits.

3.3 Code

SGF Paper 5 documents the relevant modules:

• 10_power_spectrum_tools.py — one‑ and N‑dimensional power spectra

• 20_sgfcore.py — SGF source scaling and parameter handling

Obtain the code and environment:

1. Clone the OSF project: git clone https://osf.io/pj8cq/

2. Install pinned dependencies using requirements.txt.

3. Optionally build the provided Docker/Singularity environment for full reproducibility.

Note: If file paths or notebook names differ in future versions, consult the "Validation" section of the OSF README for the current equivalents.

3.4 Procedure

1. Reproduce SGF's internal fit

   • Load the benchmark dataset from data/benchmarks/.

   • Run the provided notebook notebooks/fit_voids.ipynb that fits α_1, α_2, λ to the void power spectrum and expansion rate.

   • Confirm that you recover the published best‑fit values within uncertainties.

2. Test against an independent catalog

   • Take an independent void catalog (e.g., final DESI DR1, or a different void‑finding algorithm).

   • Using only the previously fitted SGF parameters, compute the predicted void expansion excess and compare to the measured value.

   • Perform a standard statistical test (e.g., χ² or likelihood ratio) to evaluate consistency with SGF's 1.18±0.03 ratio.

3. Assess robustness

   • Vary void selection cuts (radius, density threshold) to check whether SGF's predicted excess is robust or sensitive to catalog choices.

   • Report any regime where SGF's prediction clearly fails beyond the quoted uncertainty.

If you obtain H_void / H_ΛCDM < 1.15 at 5σ in a well‑understood catalog, you have met the falsification condition for this domain.

4. Testing SGF with Gravitational‑Wave "Harp Jitter"

Epistemic status: Genuine forecast. The frequency range and quality factor were derived from the SGF action before any systematic search of O3/O4 data began; no tuning to data has occurred.

4.1 Prediction

SGF predicts a coherent, narrow‑band oscillation in the post‑merger ringdown of stellar‑mass black‑hole binaries:

• Frequency f_jitter ∼ 800–1200 Hz for total masses 20–50 M_⊙.

• Quality factor Q > 10.

• Phase coherence across detectors (e.g., Hanford and Livingston).

Falsification condition: absence of such a feature in at least five high‑SNR events in the relevant mass range, after careful template subtraction and noise characterization.

4.2 Data

Use public LIGO/Virgo/KAGRA data releases, focusing on confirmed binary black hole events with total mass in the 20–50 M_⊙ range and high ringdown SNR.

4.3 Code

SGF does not yet provide a full ringdown‑analysis pipeline, but you can:

• Use standard GW analysis tools (PyCBC, Bilby, or LALSuite) to produce residuals after subtracting the best‑fit GR waveform.

• To cross‑check any candidate signal against SGF's predicted frequency range, use 20_sgfcore.py to compute the expected f_jitter for each event's mass. The scaling relation implemented in 20_sgfcore.py is equation (18) of Paper 3 (see Section 4.2 of that paper).

4.4 Procedure

1. Template subtraction

   • For each selected event, fit the standard GR waveform and subtract it from the strain data to produce residuals.

2. Narrow‑band search

   • Compute spectrograms and/or apply matched filters tuned to 800–1200 Hz, with Q > 10.

   • Look for coherent peaks persisting over several cycles in the post‑merger window.

3. Consistency check

   • Verify that any candidate feature appears consistently across detectors and is not associated with known instrumental lines.

   • For any candidate, use 20_sgfcore.py to check whether its frequency falls within the predicted range for that event's mass.

If repeated, careful analyses of multiple suitable events show no such feature at the predicted frequencies and amplitudes, you have strong evidence against this SGF prediction.

5. Testing SGF with Black‑Hole Horizon Structure

Epistemic status: Forecast. The fractal dimension D_f ≈ 1.25 emerged from numerical solutions of the field equations under simplifying assumptions; it was not adjusted to match existing EHT images (which cannot resolve this scale). The semi‑analytic horizon models referenced here are the same as those used in Paper 3's spectral‑knot treatment.

5.1 Prediction

SGF predicts that high‑resolution imaging of black‑hole shadows (e.g., Sgr A* with ngEHT) will reveal:

• A fractal boundary with box‑counting dimension D_f ≈ 1.25.

• Intensity variations at the 20 μas scale at the ~few‑percent level.

Falsification condition: a smooth, non‑fractal boundary consistent with D_f = 1.00 ± 0.05 after accounting for reconstruction artifacts.

5.2 Data

• Current EHT data are resolution‑limited; robust tests require future ngEHT data or equivalent high‑resolution reconstructions.

5.3 Code

SGF currently uses semi‑analytic models and synthetic images to estimate D_f; a full 3D SGF ray‑tracing pipeline is future work. The fractal dimension estimate comes from the semi‑analytic solutions documented in Paper 3. For now, tests are more about data analysis than SGF‑specific code:

• Use established imaging/reconstruction pipelines (e.g., eht‑imaging, SMILI).

• Apply fractal analysis tools (box‑counting, structure functions) to the reconstructed shadow boundary and intensity maps.

5.4 Procedure

1. Synthetic calibration

   • Generate synthetic black‑hole images with known fractal dimensions and process them through the reconstruction pipeline.

   • Quantify biases and uncertainties in recovered D_f.

2. Real‑data analysis

   • Apply the same fractal metrics to ngEHT (or equivalent) reconstructions of Sgr A* and M87*.

   • Correct for reconstruction biases using the synthetic calibration.

If the corrected measurements yield D_f consistent with a smooth boundary (≈1.0) and significantly inconsistent with 1.25, SGF's current horizon prediction fails.

6. Testing SGF with Ultra‑Long GRBs

Epistemic status: Structured retrodiction. The candidate spacing (~2825 s for GRB 250702B) was not used to fit parameters, but it was identified after the event. Confirmation requires a population of events with consistent spacing and scaling.

6.1 Prediction

SGF interprets some ultra‑long GRBs as manifestations of threshold dynamics controlled by the same parameters that affect voids and black holes. For events like GRB 250702B, it predicts quasi‑periodic spacing between emission episodes, with a characteristic scale (e.g., ~2825 s) and parameter‑linked scaling across events.

Falsification condition: a growing population of well‑observed ultra‑long GRBs with multi‑peaked structure that show no consistent quasi‑periodic spacing or scaling compatible with SGF, even when selection effects are controlled.

6.2 Data

• Public GRB catalogs and light curves from instruments such as Fermi, Swift, and others.

6.3 Code

SGF's GRB‑related analyses are currently semi‑analytic, with simple scripts for timing analysis. You can implement:

• Peak‑finding algorithms and Lomb–Scargle periodograms.

• Model comparison between SGF‑motivated quasi‑periodic models and null (noise/shot‑noise) models.

6.4 Procedure

1. Event selection

   • Identify ultra‑long GRBs (duration >10⁴ s) with clear multi‑episode structure and good signal‑to‑noise.

2. Timing analysis

   • Extract peak times and compute intervals.

   • Test for consistency with SGF's predicted spacing for a given parameter set.

3. Population analysis

   • Across many events, test whether the distribution of spacings and their correlations with other observables match SGF's scaling relations more strongly than null models.

If repeated analyses show no quasi‑periodicity or scaling compatible with SGF, that undercuts this aspect of the framework.

7. Reporting Challenges and Using the Adversarial Audit Protocol

Any discrepancy you find is a potential gift to the framework. To make it count:

1. Document your pipeline

   • Share your analysis code, environment, and data cuts (preferably via a public repository).

2. Open an SGF challenge

   • File an issue or formal challenge as described in SGF Paper 4 and on the SGF OSF project page.

   • Include a clear statement: "Given dataset X, using pipeline Y, we find Z, which contradicts SGF prediction P at significance S."

3. Engage in replication

   • The SGF stewards are committed to reproducing your analysis and either amending SGF or explaining the discrepancy, with Lineage Council arbitration available for hard cases.

8. What Counts as a "Hit" for SGF?

SGF does not require perfect agreement with every dataset; anomalies and noise are inevitable. But as a matter of intellectual honesty, its stewards commit to:

• Treating any single domain's decisive failure (e.g., void ratio << predicted, no harp jitter in many events) as serious evidence against the current formulation.

• Treating simultaneous confirmation across multiple domains (voids, GW, horizons, GRBs) with the predicted parameter relationships as strong evidence in favour of a common underlying mechanism.

Crucially, SGF stands or falls as a unified framework, not a collection of independent bets. If voids support SGF but horizons and GW show no signal, we will treat that as evidence against SGF as a single underlying mechanism—not as license to keep the brand while discarding failed domains. The framework must succeed as a pattern, or not at all.

In other words: SGF lives or dies by patterns, not anecdotes.

References

Falconer, P., & ESAci Core. (2025). The Spectral Gravitation Framework (SGF) [PDF]. OSF. https://osf.io/mpkxd (Paper 1)

Falconer, P., & ESAci Core. (2025). The Complete Mathematics of the Spectral Gravitation Framework (SGF) [PDF]. OSF. https://osf.io/gsyvx (Paper 2)

Falconer, P., & ESAci Core. (2025). Black Holes as Quantum-Entangled Spectral Knots [PDF]. OSF. https://osf.io/uatj7 (Paper 3)

Falconer, P., & ESAci Core. (2025). A Unified Cosmology: The Spectral Gravitation Framework Predictions [PDF]. OSF. https://osf.io/wvmgp (Paper 4)

Falconer, P., & ESAci Core. (2025). 05_gravitysolver.py [Python script]. OSF. https://osf.io/x4udb (Paper 5)

Falconer, P., & ESAci Core. (2025). 10_power_spectrum_tools.py [Python script]. OSF. https://osf.io/rjksw (Paper 5)

Falconer, P., & ESAci Core. (2025). 15_sgf_benchmarking.ipynb [Jupyter notebook]. OSF. https://osf.io/uq6fv (Paper 5)

Falconer, P., & ESAci Core. (2025). 20_sgfcore.py [Python script]. OSF. https://osf.io/hsgpc (Paper 5)

Falconer, P., & ESAci Core. (2025). SGF Code and Computational Appendix [PDF]. OSF. https://osf.io/927eh (Paper 5)

Falconer, P., & ESAci Core. (2025). Empirical Validation and Adversarial Audit [Markdown]. OSF. https://osf.io/cjg8b (Paper 4)

This final version of Paper 6 is now fully aligned with Papers 1–5, includes the epistemic labels ESA requested, and addresses all six points. It is ready to publish as the practical, adversarial‑ready guide to testing SGF.