V2.635: Kerr BH Log Correction — Topological Protection Against Spin

Motivation

V2.628 computed γ_BH = -1301/90 ≈ -14.5 for Schwarzschild (non-spinning) BHs using the Solodukhin formula. But real astrophysical BHs spin — often with a* > 0.9. Does γ_BH depend on the spin parameter a*?

The Solodukhin formula for the entanglement entropy log coefficient is:

γ = q₁ ∫R^Σ dA + q₂ ∫Tr(K²) dA + q₃ ∫C_{⊥⊥} dA

where q₁ = -a/(2π), q₃ = c/(6π) for each field. The first integral (intrinsic curvature) is topological by Gauss-Bonnet. But the third integral (Weyl projection) involves the Weyl tensor, which DOES depend on the BH geometry. The question: does this geometric dependence survive integration over the horizon?

Key Result

No. The log correction is EXACTLY spin-independent:

γ_BH = -4a - 2c/3 = -1301/90 ≈ -14.5 for ALL vacuum Kerr BHs.

Despite the Kerr horizon geometry varying dramatically with spin (area halves from Schwarzschild to extremal, intrinsic curvature becomes highly non-uniform), the log correction is EXACTLY the same. The deviation from the Schwarzschild value is < 2 × 10⁻¹⁵ at all spins tested (a* = 0 to 0.999).

Mechanism: Topological Protection

The spin-independence follows from a chain of three mathematical identities:

Step	Identity	Origin
1	∫R^Σ dA = 8π	Gauss-Bonnet (S² topology, χ = 2)
2	R^Σ = 2C_{θφθφ}	Gauss equation (vacuum + K=0)
3	C_{trtr} = -C_{θφθφ}	Type D vacuum (Weyl tracelessness)

Combining: ∫C_{⊥⊥} dA = ∫C_{trtr} dA = -∫C_{θφθφ} dA = -4π.

All three surface integrals in the Solodukhin formula are topologically fixed for vacuum Kerr:

• ∫R^Σ = 8π (Gauss-Bonnet)
• ∫Tr(K²) = 0 (bifurcation surface)
• ∫C_{⊥⊥} = -4π (Gauss + type D)

Numerical Verification

a*	r₊/M	A/A_Schw	W/(-4π)	∫R^Σ/(8π)	C₀₁/C₂₃
0.0	2.000	1.000	1.00000000	1.00000000	-1.0000
0.5	1.866	0.933	1.00000000	1.00000000	-1.0000
0.9	1.436	0.718	1.00000000	1.00000000	-1.0000
0.99	1.141	0.571	1.00000000	1.00000000	-1.0000
0.999	1.045	0.522	1.00000000	1.00000000	-1.0000

The Weyl integral W equals -4π to machine precision at ALL spins. The type D identity C₀₁₀₁/C₂₃₂₃ = -1 is verified at every spin value.

Angular Variation (Geometry Varies, Integral Doesn’t)

While the integrated quantities are topological, the local geometry varies dramatically with spin:

a*	C_{⊥⊥} range	R^Σ range
0.0	0 (constant)	0 (constant)
0.5	0.112	0.110
0.9	0.719	0.665
0.99	1.655	1.409

At a* = 0.99, the Weyl projection varies from -1.34 at the poles to +0.31 at the equator — changing SIGN across the horizon. Yet the integral is exactly -4π. This is the hallmark of topological invariance.

Per-Field Breakdown

Field	n_SM	γ_BH	δ_flat	γ_BH/δ
Scalar	4	-0.0667	-0.0444	1.500
Weyl	45	-3.5000	-2.7500	1.273
Vector	12	-9.0667	-8.2667	1.097
Graviton	1	-1.8222	-1.3556	1.344
SM+grav		-14.456	-12.417	1.164

The Weyl correction increases |γ| by 16.4% relative to the cosmological value δ = -4a. This correction is the SAME for all Kerr spins.

Comparison with Other Approaches

Approach	γ_BH	Spin-dep?	Reason	Matter-dep?
Framework	-14.5	NO	Gauss-Bonnet + Gauss eq.	YES
LQG (K-M)	-1.5	NO	SU(2) Chern-Simons	NO
LQG (Meissner)	-0.5	NO	Barbero-Immirzi	NO
String (N=2)	-2.0	YES	Moduli-dependent	YES
Asymptotic Safety	-1.5	NO	Fixed point	NO

Both the framework and LQG predict spin-independent γ, but for completely different reasons: the framework from differential geometry (Gauss-Bonnet), LQG from quantum group theory (Chern-Simons). String theory is the only approach that generally predicts spin-dependent corrections.

When Topological Protection Breaks

The protection requires three conditions:

1. Bifurcation surface (K = 0): Breaks for dynamical horizons (BH formation, merger, evaporation). The q₂ term then contributes.

2. Vacuum (Ricci-flat): Breaks for charged BHs (Kerr-Newman) and BHs in de Sitter. The Gauss equation picks up Ricci terms, making W charge-dependent. This is a FURTHER prediction: γ_BH(Q) is a calculable function of charge.

3. S² topology (Gauss-Bonnet gives 8π): Breaks for higher-dimensional BHs (black rings, etc.). Protected for all 4D BHs by Hawking’s topology theorem.

For astrophysical BHs (Kerr, stationary, negligible charge): ALL three conditions hold. γ_BH = -14.5 exactly.

Honest Assessment

What’s New

1. First demonstration that γ_BH is topologically protected against spin for vacuum Kerr BHs
2. Explicit proof mechanism: Gauss-Bonnet → Gauss equation → type D identity
3. Numerical verification to machine precision (dev < 2 × 10⁻¹⁵) across the full spin range a* ∈ [0, 0.999]
4. Identification of the three protection conditions and when each breaks

What’s Strong

1. The result is EXACT, not approximate — it follows from topological theorems
2. The proof is elementary (no quantum gravity input needed)
3. The framework prediction γ_BH = -14.5 now holds for ALL astrophysical BHs
4. The charge-dependent case provides a further prediction axis

What’s Weak

1. The spin-independence was perhaps foreseeable from the general structure of the Solodukhin formula (all surface integrals are topological in vacuum with K=0). But it had not been explicitly verified for Kerr.
2. The V2.628 method ambiguity (which formula for higher spins) is still present — topological protection doesn’t resolve it
3. The result is not observationally testable with current technology

What It Means for the Framework

The framework predicts γ_BH = -1301/90 for ALL astrophysical BHs — supermassive BHs in galactic centers (a* ~ 0.9-0.998), stellar-mass BHs from mergers (a* ~ 0.7), and slowly spinning BHs (a* ~ 0). This universality across spin simplifies the prediction and makes it a robust signature: any measurement of γ_BH distinguishes the framework (|γ| ≈ 14.5) from LQG (|γ| ≈ 1.5) regardless of which BH is observed.

Files

• src/kerr_log_correction.py — Kerr Weyl integral and topological proof
• tests/test_kerr.py — 10 verification tests (all pass)
• run_experiment.py — Full 6-phase experiment
• results.json — Machine-readable results