V2.533: Black Hole Entropy Log Correction Spectrum — Charge Dependence from SM Trace Anomaly

Motivation

V2.531 computed γ_BH = −63/10 for Schwarzschild black holes using the SM trace anomaly decomposition into Euler (a) and Weyl (c) central charges. That result applies to neutral BHs. This experiment extends to charged (Reissner-Nordström) black holes, revealing that γ_BH is a function of charge — a qualitative prediction that distinguishes this framework from LQG (which predicts constant γ = −3/2 for all BHs).

Key Physics

The BH entropy log correction γ receives two contributions:

• Euler (a-type): topological, depends only on horizon topology (S² → χ = 2)
• Weyl (c-type): geometric, depends on the Weyl curvature at the horizon

For vacuum BHs (Schwarzschild, Kerr): R_μν = 0, so E₄ = W² pointwise. Both contributions have the same geometric factor, and γ = −4a + 2c is the same for ALL vacuum BHs. Spin-independence is a theorem, not an assumption.

For charged BHs (Reissner-Nordström): R_μν ≠ 0 (electromagnetic stress tensor), E₄ ≠ W². The Weyl integral over the horizon changes with charge. As q = Q/M → 1 (extremal), the Weyl curvature at the horizon vanishes (geometry → AdS₂ × S²), and the c-contribution disappears.

Results

The charge-dependent spectrum

γ_BH(q) = −4 a_total + c_total × 32s²/(1+s)⁴, where s = √(1−q²)

q = Q/M	γ_BH	|γ/γ_LQG|	Physical regime
0	−63/10 = −6.300	4.20	Schwarzschild/Kerr
0.5	−6.363	4.24	Moderate charge
0.7	−6.636	4.42
0.9	−8.042	5.36	High charge
0.95	−9.199	6.13	Near-extremal
0.99	−11.268	7.51
1.0	−149/12 = −12.417	8.28	Extremal

Exact fractions at endpoints

• Schwarzschild: γ = −63/10 (exact, from a = 149/48, c = 367/120)
• Extremal RN: γ = −149/12 (exact, pure a-anomaly, Weyl contribution vanishes)
• Ratio: |γ_extremal/γ_Schwarzschild| = 1.97 (nearly doubles)

Spin-independence for vacuum BHs

Theorem: For any vacuum spacetime (R_μν = 0), the Euler density equals the Weyl invariant pointwise: E₄ = R_μνρσ² = W². Therefore:

• ∫ E₄ = ∫ W² = 64π² (Gauss-Bonnet, topological)
• γ_BH = −4a + 2c for ALL vacuum BHs

Verified numerically: |E₄ − W²| < 10⁻¹⁴ at all radii for Schwarzschild. This means γ = −6.300 holds for Schwarzschild AND Kerr (any spin a*).

Gauss-Bonnet verification

∫ E₄ √g d⁴x = 64π² for BOTH Schwarzschild and RN (topological invariant). Verified numerically to <0.02% for all charges. The Weyl volume integral departs from 64π² for charged BHs: ratio W²/E₄ = 1.0000 (q=0), 1.0056 (q=0.6), 1.1017 (q=0.9).

Per-field decomposition

Schwarzschild (q=0): Gauge vectors dominate at 93.1% of total.

Extremal (q=1): All fields contribute negatively (Weyl “bonus” gone):

• Gauge vectors: 66.6% (down from 93.1%)
• Weyl fermions: 22.1% (up from 7.9%)
• Graviton: 10.9% (up from −0.7%)
• Scalars: 0.4% (flipped sign!)

The extremal limit reveals the true relative importance of each field: without the c-anomaly Weyl contribution that happens to nearly cancel for the SM (c/a = 0.985 ≈ 1), the hierarchy reshuffles dramatically.

BSM predictions in the (Ω_Λ, γ_BH) plane

Scenario	Ω_Λ	σ	γ(q=0)	γ(q=1)
SM + graviton	0.6877	+0.42	−6.300	−12.417
SM + 1 axion	0.6830	−0.23	−6.294	−12.428
SM + sterile ν	0.6805	−0.58	−6.311	−12.478
SM + dark photon	0.7147	+4.11	−6.789	−13.106
MSSM	0.4030	−38.6	−5.956	−14.439

Adding a dark photon shifts BOTH Ω_Λ (by +4.1σ) AND γ_BH (by −0.49) — a correlated, falsifiable signature in two independent observables.

Response slopes (per added field)

Field	dΩ_Λ/dn	dγ/dn (q=0)	dγ/dΩ (the “slope”)
Scalar	−0.005	+0.006	−1.2
Weyl fermion	−0.007	−0.011	+1.5
Vector	+0.027	−0.489	−18.1

Vectors create a steep “cliff” in the (Ω_Λ, γ_BH) plane: adding one vector simultaneously pushes Ω_Λ up by +4σ and γ_BH down by −0.49. This correlated response is the framework’s strongest discriminating feature.

Comparison with other quantum gravity approaches

| Approach | γ_BH | Field-dep? | Charge-dep? | |Δ| from framework | |----------|------|-----------|-------------|-------------------| | This framework | −6.30 to −12.42 | YES | YES | — | | LQG (Kaul-Majumdar) | −1.50 | NO | NO | 4.80 | | LQG (grand canonical) | −0.50 | NO | NO | 5.80 | | Euclidean QG | −4.98 | YES | YES | 1.32 | | Induced gravity | −3.33 | YES | NO | 2.97 | | String theory (BPS) | −4.00 | NO | NO | 2.30 | | Asymptotic safety | −5.0 | YES | YES | 1.30 |

Key discriminator: The framework vs LQG gap GROWS with charge — from 4.2× (neutral) to 8.3× (extremal). This is because LQG’s γ = −3/2 is universal (from SU(2) Chern-Simons theory on the horizon), while the framework’s γ depends on the trace anomaly which responds to the background geometry.

Honest assessment

Strengths

1. Genuine new prediction: The charge-dependent spectrum γ(q) has not been computed for the full SM before. The endpoints −63/10 and −149/12 are exact rational numbers.
2. Qualitative discriminator: LQG predicts charge-independence; this framework predicts charge-dependence. This is a YES/NO distinction, not a numerical one.
3. Spin-independence proved: E₄ = W² for vacuum spacetimes → γ identical for Schwarzschild and Kerr. A clean theorem.
4. Correlated with Λ: Both γ_BH and Ω_Λ come from the same (a, c) coefficients. No other framework links these observables.
5. Per-field decomposition shifts dramatically: The extremal limit exposes the “true” field hierarchy without the Weyl bonus.

Weaknesses and caveats

1. Not directly measurable: The log correction to BH entropy is a sub-leading quantum effect. No current or planned experiment can measure γ_BH for astrophysical BHs.
2. Formula assumption: The formula γ = −4a + 2c × (Weyl factor) assumes the log correction comes from the heat kernel on the BH background. The “Weyl factor” P(q) = 64s²/(1+s)⁴ comes from the surface integral of W² over the horizon. This is the natural entanglement entropy interpretation, but the volume integral over the Euclidean manifold gives a slightly different answer (W²/E₄ ratio reaches 1.10 at q=0.9). The surface and volume approaches should agree for the log coefficient, but a rigorous derivation for non-vacuum BHs requires the full conical singularity calculation.
3. Charged BHs are astrophysically rare: Real BHs are electrically neutral to excellent approximation. The Kerr spectrum (rotating, neutral) is physically more relevant — but it’s flat (γ = −6.30 for all spins).
4. The c-coefficients are from standard QFT literature, not independently verified by the framework’s lattice methods. The lattice verifies only the a-anomaly (δ = −1/90 for a scalar, corresponding to −4a).
5. Near-extremal limit: As q → 1, T_H → 0 and quantum corrections may diverge. The simple formula may break down near extremality.

What this means for the science

The charge-dependent BH entropy spectrum is the framework’s strongest theoretical discriminator against LQG, even though it’s not experimentally accessible today. The key insight is physical:

• This framework: γ_BH arises from quantum fields’ entanglement across the horizon. The entanglement depends on what the fields “see” — which includes the background Weyl curvature. For charged BHs, the Weyl curvature at the horizon is suppressed, so the c-contribution shrinks.
• LQG: γ_BH arises from the quantum geometry of the horizon itself (punctures, SU(2) representations). The horizon quantum state is universal — it doesn’t care about matter content or background curvature.

These are fundamentally different physical pictures, and the charge-dependent spectrum distinguishes them cleanly.

Tests

46/46 tests passing, covering:

• Exact trace anomaly coefficients (a = 149/48, c = 367/120)
• Weyl suppression factor P(q) analytically and numerically
• γ_BH at Schwarzschild, extremal, and intermediate charges
• Spin-independence theorem (E₄ = W² for vacuum)
• Gauss-Bonnet topological integral (∫E₄ = 64π²)
• BSM scenario predictions and response functions
• QG comparison discriminating power

Files

• src/bh_entropy_spectrum.py: Core computation (anomaly coefficients, γ(q), BSM scenarios, curvature invariants)
• tests/test_bh_entropy_spectrum.py: 46 tests
• run_experiment.py: Full 12-section analysis
• results.json: Machine-readable results