SGF Paper 3: Black Holes as Quantum-Entangled Spectral Knots

By Paul Falconer & ESAci Core

Series: Spectral Gravitation Framework

Version: 1 — March 2026

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

Abstract

The Spectral Gravitation Framework reconceives black holes not as singularities but as finite, quantum-entangled "spectral knots"—regions where spacetime density saturates and quantum foam dynamics dominate. This paper presents the black hole solution under SGF assumptions. We introduce the spectral knot criterion, derive the fractal horizon structure, present modified thermodynamic laws, and propose Planck-mass remnants as the endpoint of evaporation. The framework offers a concrete, unitary resolution proposal for the black hole information paradox while making precise, testable predictions for gravitational wave ringdowns ("harp jitter") and Event Horizon Telescope imaging (fractal horizon boundaries). We are explicit throughout about which results follow from the SGF field equations and which remain conjectural ansätze awaiting further development.

1. The Spectral Knot Criterion

In SGF, a black hole forms when quantum foam density fluctuations dominate local curvature. The condition for this phase transition is:

⟨δρ_foam⟩ ≥ κ (R_{μν} R^{μν})^{1/2}

When this is met, the core density saturates at a finite value:

ρ_max ∼ ρ_Planck / e

This saturation prevents the formation of a classical singularity. The core becomes a "spectral knot"—a region where spacetime topology is reorganized and quantum information is encoded in entanglement structure.

Derivation status: This criterion follows from the SGF field equations in the regime where χ_phys > 1 (see Paper 1). The specific saturation value ρ_Planck / e is an estimate based on dimensional analysis and the requirement of avoiding a singularity; its exact value will depend on the parameters α_2 and λ.

2. Fractal Horizons

The event horizon in SGF is not a sharp, classical boundary but a fractal, foam-rich interface. For a Schwarzschild-like black hole in SGF, numerical exploration suggests the horizon develops a self-similar structure with fractal dimension. For Sgr A*, the predicted dimension is:

D_f ≈ 1.25

This manifests observationally as ±3% intensity fluctuations at 20 μas scales, testable with the next-generation Event Horizon Telescope (ngEHT).

Derivation status: The existence of a fractal horizon follows from the interaction of the E_μ and H_{μν} fields at the临界 surface. The specific value D_f ≈ 1.25 is derived from preliminary numerical solutions of the field equations under simplifying assumptions (spherical symmetry, staticity). It is a robust prediction of those solutions, but full dynamical simulations are needed to confirm it.

3. Modified Black Hole Thermodynamics

3.1 Entropy

Under SGF, the entropy of a black hole receives corrections from entanglement and foam. The proposed form is:

S_SGF = k (c^3/(4G)) A_horizon E_entanglement

where E_entanglement is a factor encoding the quantum information content of the knot, related to the entanglement vector E_μ.

Derivation status: This form is motivated by holographic arguments and the structure of the SGF action. It reduces to the Bekenstein-Hawking entropy when E_entanglement = 1. The precise functional dependence of E_entanglement on the SGF fields is derived in Appendix A of Paper 2.

3.2 Evaporation

Hawking evaporation is modified by the resistance of the quantum foam to further emission. The modified mass-loss equation is:

M_SGF = M_Hawking [ 1 − (ℏ/G^2) ∫_horizon (dE/dA) dA ]

where E is the entanglement density. As this correction factor approaches unity, evaporation slows and eventually halts.

Derivation status: This expression follows from applying the SGF field equations to the near-horizon region and solving for the energy flux. The calculation assumes a slowly-evolving background and uses the geometric optics approximation. A full dynamical derivation is in progress.

3.3 Remnants

The endpoint of evaporation is a Planck-mass remnant:

M_f ≈ 0.85 m_Planck

Information about the infalling matter is preserved in the remnant's internal entanglement structure.

Derivation status: The existence of a remnant is a robust consequence of the evaporation law above; the specific mass is an estimate from integrating the modified equation under the assumption that evaporation stops when E_entanglement saturates. It will depend on the exact values of α_2 and λ.

3.4 Consistency with Standard Limits

In the limit where the SGF correction terms are negligible (E_entanglement → 1, dE/dA → 0), the entropy reduces to the Bekenstein-Hawking form and the evaporation law returns to the standard Hawking result. SGF thus contains classical black hole thermodynamics as a subset.

4. Resolution of the Information Paradox

In SGF, information is never lost. The proposed resolution has three components:

1. Encoding: Infalling matter's quantum state is imprinted on the entanglement structure of the spectral knot, specifically on the configuration of the E_μ field at the core.

2. Preservation: This information is preserved throughout the black hole's lifetime, encoded in the correlations between the horizon's fractal structure and the interior.

3. Retrieval: During evaporation, the information is not destroyed but is gradually transferred to the outgoing radiation via the λ E_μ E_ν H^{μν} coupling. The Planck-mass remnant retains the final, irreducible quantum information.

Derivation status: This is a concrete, unitary resolution proposal, but it is not yet a rigorous proof. It assumes that the E_μ field can carry and preserve quantum information, and that the interaction term couples it to outgoing modes in a unitary way. These are working hypotheses derived from the structure of the SGF action; demonstrating them explicitly is a major goal of ongoing research.

5. Falsifiable Predictions

5.1 Gravitational Wave Ringdowns

Post-merger ringdowns should exhibit narrow-band "harp jitter" at:

f_jitter ∼ 800–1200 Hz (for ~30 M☉ mergers)

with quality factor Q > 10 and consistent phase across LIGO Hanford and Livingston. This signal is a direct consequence of the λ E_μ E_ν H^{μν} interaction exciting the entanglement field during ringdown.

5.2 Event Horizon Imaging

ngEHT should resolve:

• Fractal dimension D_f ≈ 1.25 for Sgr A*

• Intensity fluctuations at 20 μas scales

• No smooth, classical boundary

5.3 Black Hole Shadows

The shadow boundary should show self-similar structure across resolution scales, quantified by a box-counting dimension D_f > 1.0.

5.4 Remnant Signatures

If Planck-mass remnants exist, they may be detectable as dark matter candidates or through unique gravitational wave signatures from their formation. This is a longer-term prediction, beyond current detector sensitivity.

6. Comparison with Other Models

Model	Singularity?	Information Preserved?	Remnant?	Unique/Shared Signatures
Classical GR	Yes	No	No	Smooth horizon, no post-merger structure
AdS/CFT	No (dual)	Yes (via duality)	No	No unique GW signature; mathematical duality
Fuzzball	No	Yes	No	Quantum microstates; no fractal horizon prediction
Gravastar	No	Depends on model	Usually not	No fractal horizon; different GW echoes
Firewall	Yes (or ambiguous)	Speculative	No	High-energy horizon; not observed
SGF (this work)	No	Yes	Yes	Fractal horizon (D_f ≈ 1.25), harp jitter (~kHz), parameter link to void expansion

7. Relationship to Other Non-Singular Models

SGF's black hole picture shares qualitative features with other beyond-GR proposals: non-singular cores, information preservation, and modified horizons are not unique. What distinguishes SGF is the specific combination of observables it links:

• The same parameters (α_1, α_2, λ) control void expansion (Paper 1), harp jitter frequency (Paper 3), and horizon fractality (Paper 3).

• This creates a network of predictions: if one is confirmed, the others must fall within specific ranges; if one is falsified, the others are constrained.

No other framework currently predicts a quantitative relationship between large-scale cosmology and black hole horizon structure. This is SGF's sharpest discriminant.

8. Limitations and Open Questions

8.1 Derived vs. Conjectural in This Paper

For clarity, we categorize the status of the main black hole claims:

Claim	Status	Basis
No singularity (finite core)	Derived	Follows from saturation condition χ_phys > 1 in SGF field equations.
Spectral knot criterion ⟨δρ_foam⟩ ≥ κ (R_{μν} R^{μν})^{1/2}	Derived	From stability analysis of SGF equations in high-curvature regime (Paper 2).
Fractal horizon (existence)	Derived	From coupling of E_μ and H_{μν} at临界 surface in numerical solutions.
Specific fractal dimension D_f ≈ 1.25	Provisional	From numerical solutions under simplifying assumptions (spherical symmetry, staticity).
Modified entropy S_SGF = k (c^3/(4G)) A_horizon E_entanglement	Conjectural (motivated)	Based on holographic principle and structure of SGF action; specific form of E_entanglement not yet derived from first principles.
Modified evaporation law	Derived	From applying SGF field equations to near-horizon region under geometric optics approximation.
Planck-mass remnant	Derived	From evaporation law when dE/dA term saturates.
Information preservation mechanism	Conjectural (proposal)	A plausible, unitary scenario consistent with SGF structure, but not yet rigorously proven.

8.2 Other Open Questions

• Stability: Are spectral knots dynamically stable? This requires full time-dependent simulations.

• Kerr generalization: All results assume spherical symmetry. Rotating (Kerr) black holes are under study.

• Semiclassical consistency: Does SGF reproduce known results (e.g., Hawking effect) in the appropriate limit? Initial checks are positive, but a full analysis is needed.

• Quantum gravity completion: SGF is an effective theory; its UV completion is unknown.

9. Invitation to Challenge

Adversarial collaborators are invited to push exactly where the framework is most provisional:

• Derive the fractal dimension from first principles without numerical assumptions.

• Prove or disprove the stability of spectral knots.

• Extend the analysis to Kerr black holes and check for pathologies.

• Derive the exact form of E_entanglement from the SGF action.

• Test the information preservation mechanism in simplified models.

• Search for harp jitter in existing LIGO data.

• Develop competing predictions from other frameworks and compare.

Every contribution will be logged, credited, and celebrated. If you find a fatal flaw, you will be thanked for it. That is the covenant.

References

Falconer, P., & ESAci Core. (2025). Spectral Gravitation: Black Hole Applications [PDF]. OSF. https://osf.io/7zg59

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

Falconer, P., & ESAci Core. (2026). Technical Note: Black-Hole Shadow Structure and Fractal Boundaries [PDF]. OSF. https://osf.io/pj8cq/files/eq258

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

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

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