V2.651: BH Entropy Log Correction Staircase — Mass-Dependent Species Activation

Status: COMPLETED

Question

How does the black hole entropy log correction coefficient γ depend on black hole mass? Is this mass-dependence a qualitative discriminant against Loop Quantum Gravity?

The Physics

In the entanglement entropy framework:

$S_{BH} = \frac{A}{4G} + \gamma \ln\left(\frac{A}{\ell_P^2}\right) + O(1)$

where γ = Σ δ_i summed over all fields whose Compton wavelength fits inside the horizon. A field with mass m contributes when m·n ≪ 1, where n ~ 2M/M_Pl is the “lattice size” of the black hole. The activation threshold for each field is:

$M_{\text{threshold}} = \frac{M_{\text{Pl}}^2}{2m}$

As a BH evaporates (M decreases, T_H increases), massive fields transition from decoupled (m·n ≫ 1) to active (m·n ≪ 1). This creates a species-activation staircase in γ(M).

LQG predicts γ = -3/2 regardless of BH mass. This is a qualitative, not merely quantitative, difference.

What’s New

Previous experiments (V2.261, V2.507, V2.628) computed γ for fixed field content. V2.648/V2.650 established the three-regime mass-independence of δ (UV regime: δ constant; crossover at m·n ~ 1; decoupled at m·n ≫ 1). This experiment is the first to combine these results into a continuous γ(M) function through all 15 SM mass thresholds, revealing the staircase structure as a unique prediction.

Key Results

1. Asymptotic Limits

Regime	Active fields	γ	Ratio to LQG
Full SM + graviton (PBH limit, M ≪ 10¹² g)	All 62 fields	-12.417	8.28×
Massless only (stellar BH, M ≫ 10²³ g)	γ, g, G (3 fields)	-7.556	5.04×
LQG (all masses)	—	-1.500	1.00×
String theory (N=2 extremal)	—	-2.000	1.33×
Asymptotic Safety	—	0.000	—

The framework exceeds LQG by a factor of 5–8× depending on BH mass. This is not a subtle correction — it’s an order of magnitude.

2. The Staircase: 15 Steps

As BH mass decreases (reading top to bottom), fields activate:

Step	Field	δ_step	γ_cumulative	M_threshold (g)
0	(massless: γ, 8g, G)	—	-7.556	∞
1	ν₁	-0.061	-7.617	1.55 × 10²⁵
2	ν₂	-0.061	-7.678	1.33 × 10²⁵
3	ν₃	-0.061	-7.739	2.66 × 10²⁴
4	electron	-0.122	-7.861	2.60 × 10¹⁷
5	up	-0.367	-8.228	6.15 × 10¹⁶
6	down	-0.367	-8.594	2.85 × 10¹⁶
7	strange	-0.367	-8.961	1.43 × 10¹⁵
8	muon	-0.122	-9.083	1.26 × 10¹⁵
9	charm	-0.367	-9.450	1.05 × 10¹⁴
10	tau	-0.122	-9.572	7.48 × 10¹³
11	bottom	-0.367	-9.939	3.18 × 10¹³
12	W± bosons	-1.378	-11.317	1.65 × 10¹²
13	Z boson	-0.689	-12.006	1.46 × 10¹²
14	Higgs (4 dof)	-0.044	-12.050	1.06 × 10¹²
15	top quark	-0.367	-12.417	7.68 × 10¹¹

The dynamic range is 39.1% — γ changes by 4.86 from the massless to the full-SM limit. The largest single step is the W± bosons (δ = -1.378), reflecting the large trace anomaly of vector fields.

3. Remnant Mass Predictions

Approach	M_rem / M_Pl	M_rem (grams)
Framework (full SM)	0.994	2.17 × 10⁻⁵
Framework (massless)	0.775	1.69 × 10⁻⁵
LQG	0.345	7.53 × 10⁻⁶
String (N=2)	0.399	8.70 × 10⁻⁶

The framework predicts a near-Planck-mass remnant (M_rem ≈ M_Pl to 0.6%!), while LQG predicts a remnant 2.9× lighter. This affects:

• PBH remnant abundance as dark matter candidates
• The endpoint spectrum of Hawking evaporation

4. BSM Sensitivity

Scenario	γ (PBH)	Δγ from SM	M_rem/M_Pl
SM + graviton	-12.417	0	0.994
+1 axion	-12.428	-0.011	0.995
+1 sterile ν	-12.478	-0.061	0.997
+1 dark photon	-13.106	-0.689	1.021
MSSM (light)	-13.939	-1.522	1.053

A dark photon adds -0.689 to γ (vectors have large trace anomaly). MSSM shifts γ by -1.522, nearly equal to LQG’s entire prediction! Every BSM particle shifts both Ω_Λ (V2.649) and γ_BH simultaneously — an over-determined system.

5. The Critical PBH (Evaporating Now)

For a PBH with M ~ 5 × 10¹⁴ g (T_H ~ 21 MeV, evaporating on cosmological timescales):

• 11 of 18 fields active (all except charm, tau, bottom, W, Z, Higgs, top)
• γ = -9.042 (6.0× LQG)
• At this mass, the strange quark and muon are partially activated (86%, 83%)

This is the mass range where PBH evaporation could be observed via gamma-ray bursts (Fermi-LAT). The Hawking spectrum near this mass carries the staircase fingerprint.

Discriminating Tests

Test 1: Absolute γ value (any BH)

• Framework: -12.4 to -7.6 depending on mass
• LQG: -1.5 (always)
• Separation: 5–8× — qualitative, not marginal

Test 2: Mass dependence (QUALITATIVE)

• Only the entanglement approach predicts γ(M) is mass-dependent
• LQG, string theory, and asymptotic safety all predict constant γ
• This is the most powerful discriminant: it’s a YES/NO test, not a precision measurement

Test 3: Analog BH species scaling (LAB-TESTABLE)

• 1-component BEC sonic horizon: γ = -1/90 = -0.0111
• 2-component BEC: γ = -2/90 = -0.0222
• Framework: γ(2)/γ(1) = 2.000; LQG: γ(2)/γ(1) = 1.000
• Required precision: ~50% (even a crude measurement distinguishes)
• Steinhauer’s group (Technion) has the experimental capability NOW

Test 4: PBH remnant mass

• Framework: M_rem ≈ M_Pl (within 0.6%)
• LQG: M_rem = 0.35 M_Pl
• Ratio: 2.9× — affects PBH dark matter abundance and evaporation endpoint

Honest Assessment

Strengths

1. The staircase is a unique prediction — no other QG approach predicts mass-dependent γ
2. The species-scaling test is lab-accessible with existing analog BH technology
3. The factor 8.3× separation from LQG makes this robust against systematic uncertainties
4. Self-consistent: same trace anomaly coefficients predict both Ω_Λ and γ_BH

Weaknesses and Caveats

1. Method ambiguity persists: The entanglement entropy (Solodukhin) formula gives γ_BH including Weyl curvature terms for actual Schwarzschild BH. The full Solodukhin result is γ_BH = -(4a + 2c/3) ≈ -14.46, not -12.42. The flat-space δ_total = -149/12 applies to the cosmological horizon (zero Weyl curvature). For Schwarzschild, the Weyl contribution adds ~2 to |γ|. The QUALITATIVE prediction (staircase, species-dependence) is robust to this ambiguity; the exact numerical value depends on which formula is correct.

2. Smooth vs sharp transition: The activation function near m·n ~ 1 is modeled as a smooth tanh. The actual transition depends on details of how entanglement entropy responds to mass at finite lattice size (V2.648). The step locations are robust; the step shapes have O(1) uncertainty.

3. Confinement: Gluons are counted as massless fundamental fields. Below T_QCD ~ 150 MeV, they’re confined into hadrons. The framework’s convention (count fundamental fields, V2.499) is theoretically motivated but not independently verified for BH entropy.

4. Not directly testable for astrophysical BHs: The log correction is subdominant (γ·ln(A/l_P²) ~ 200 vs A/4G ~ 10⁷⁷ for solar-mass BH). Direct measurement requires either analog BH experiments or primordial BH evaporation observations.

Conclusion

The BH entropy log correction staircase is the most powerful qualitative discriminant between the entanglement entropy approach and Loop Quantum Gravity. It requires no precision measurement — even a crude analog BH experiment distinguishing γ(2 fields) from γ(1 field) at 50% precision would constitute a decisive test. The prediction is unique, falsifiable, and accessible with existing experimental technology.

The staircase pattern — 15 steps spanning a 39% dynamic range — constitutes a “particle physics fingerprint” embedded in black hole thermodynamics. It connects the SM field content to quantum gravity observables in a way no other approach achieves.