V2.369 - Black Hole Remnant Mass from Entanglement Entropy
V2.369: Black Hole Remnant Mass from Entanglement Entropy
Status: SUCCESS (26/26 tests pass) Date: 2026-03-10 Category: Black Hole Entropy — Remnant Prediction
Headline
The framework predicts M_remnant = 0.994 M_Pl — the SM field content produces a BH remnant within 0.6% of the Planck mass. This arises from the remarkable near-coincidence |δ_total|/(4π) = 0.988 ≈ 1. LQG predicts 0.35 M_Pl (2.9× lighter). String theory predictions vary from 0.28 to 0.40 M_Pl, or no remnant at all for BPS states.
Scientific Question
If the BH entropy has a log correction S = A/(4l_P²) + δ·ln(A/l_P²) + O(1), what happens at the endpoint of Hawking evaporation? The log term modifies the thermodynamics: when δ < 0, there exists a minimum BH area where dS/dA = 0, creating a stable remnant. What mass does the framework predict?
The Computation
Setting dS/dA = 0:
dS/dA = 1/(4l_P²) + δ/A = 0
A_crit = 4|δ| l_P²Converting to mass via A = 16πG²M²/c⁴ = 16π l_P² (M/M_Pl)²:
M_remnant = M_Pl × √(|δ|/(4π))For δ = -149/12 (SM + graviton):
M_rem = M_Pl × √(149/(48π)) = M_Pl × √(0.98809) = **0.9940 M_Pl**Key Results
1. The Remnant Mass
| Quantity | Value |
|---|---|
| δ_total (exact) | -149/12 |
| |δ|/(4π) | 0.98809 |
| A_crit | 49.7 l_P² |
| M_remnant | 0.994 M_Pl |
| M_remnant (kg) | 2.16 × 10⁻⁸ kg |
| M_remnant (GeV) | 1.21 × 10¹⁹ GeV |
The SM field content (4 scalars + 45 Weyl + 12 vectors + graviton) produces |δ| = 149/12 ≈ 12.42, which is within 1.2% of 4π ≈ 12.57. This near-coincidence makes M_rem ≈ M_Pl.
2. Comparison Across Quantum Gravity Approaches
| Approach | δ | M_rem/M_Pl | Species-dep? |
|---|---|---|---|
| This framework (SM+grav) | -149/12 | 0.994 | YES |
| This framework (SM only) | -1991/180 | 0.938 | YES |
| LQG (Kaul-Majumdar) | -3/2 | 0.346 | NO |
| String theory (N=2, 4D) | -2 | 0.399 | NO |
| String theory (N=4, 5D) | -1 | 0.282 | NO |
| String theory (BPS extremal) | 0 | ∞ (no remnant) | NO |
The framework’s remnant is 2.9× heavier than LQG’s. This is a sharp, unambiguous discriminator.
3. Gravitational Wave Echoes
For a 30 M_☉ black hole:
- Echo delay (framework): 26.8 ms
- Echo delay (LQG): 27.1 ms
- Difference: 0.31 ms (ratio 0.99)
Both predictions are in the LIGO band (~ms). However, the echo delays are nearly identical (the difference is only 1.2%) because the delay depends on ln(r_s/r_crit), and the factor of 2.9 in r_crit translates to only ln(2.9) ≈ 1.06 out of a total ln(r_s/l_P) ≈ 87.5. GW echoes cannot distinguish the two frameworks unless BH mass is known to sub-percent precision.
4. Species-Dependent Remnant
| Scenario | δ | M_rem/M_Pl | Deviation from M_Pl |
|---|---|---|---|
| SM + graviton | -12.417 | 0.994 | 0.60% |
| + 1 axion | -12.428 | 0.995 | 0.55% |
| + 1 sterile ν | -12.478 | 0.997 | 0.35% |
| + 3 sterile ν (seesaw) | -12.600 | 1.001 | 0.13% |
| + 1 dark photon | -13.106 | 1.021 | 2.12% |
| MSSM + graviton | -14.439 | 1.072 | 7.19% |
The remnant mass depends on particle physics. LQG predicts 0.346 M_Pl for ALL of these scenarios. If remnants could be observed (e.g., through PBH evaporation endpoints), the mass would tell us about the particle spectrum.
5. The |δ|/(4π) ≈ 1 Near-Coincidence
For M_rem = M_Pl exactly, we would need δ = -4π ≈ -12.566. The actual SM value is -149/12 ≈ -12.417, short by 0.15. This gap equals ~13 real scalars or ~2.4 Weyl fermions.
If the seesaw mechanism adds 3 right-handed neutrinos (Majorana), δ shifts by 3 × (-11/180) = -0.183, which would give M_rem = 1.001 M_Pl — even closer to M_Pl.
Is |δ|/(4π) ≈ 1 a coincidence, or does it point to an underlying principle? An open question.
Honest Limitations
1. Negative Entropy at the Remnant
The formula S = A/4 + δ·ln(A) gives S(A_crit) = |δ|(1 - ln(4|δ|)) ≈ -36 nats. This is negative, which is unphysical. The resolution: the O(1) constant C₀ in S = A/4 + δ·ln(A) + C₀ is physically important and must be chosen so that S(A_crit) ≥ 0. This doesn’t affect the remnant mass (determined by dS/dA = 0), but it means the constant term carries physical content that we haven’t computed from first principles.
2. GW Echoes Are Not Discriminating
The echo delay differs by only 1.2% between the framework and LQG. This is a log effect — the factor of 2.9 in remnant radius translates to a sub-percent fractional change in echo timing. GW echoes cannot distinguish the frameworks.
3. Remnant Existence is Not Unique
Most quantum gravity approaches (LQG, string theory with non-zero δ, this framework) predict some form of remnant. The MASS differs, but the qualitative prediction (remnant exists) is shared. The unique feature is the species-dependence and the near-Planck mass.
4. No Foreseeable Measurement
Black hole remnants at ~M_Pl (2×10⁻⁸ kg) are far below any conceivable detection threshold. PBH evaporation endpoints would require PBHs in the right mass range and close enough to observe the final evaporation stages.
What This Means
The Unique Prediction
The framework predicts M_rem = 0.994 M_Pl — distinguishable from LQG (0.35 M_Pl) and string theory (0.28–0.40 M_Pl). The remnant mass is species-dependent, changing with BSM physics. This is the first prediction that links the BH remnant mass to the SM field content.
The Surprising Near-Coincidence
|δ_SM+grav| ≈ 4π to 1.2%. The entire SM spectrum (61 distinct quantum fields) conspires to produce a trace anomaly sum near 4π, making the remnant mass near the Planck mass. Whether this is a coincidence or a deep principle connecting the SM to Planck-scale physics is an open question.
Connection to Λ
The SAME δ_total = -149/12 that determines Λ also determines M_rem. This is a consistency requirement: the trace anomaly sum must simultaneously produce:
- Ω_Λ = 0.688 (cosmological prediction)
- M_rem = 0.994 M_Pl (BH remnant prediction)
- γ_BH = -149/12 (log correction to BH entropy)
All three come from one number. No free parameters.
Files
src/bh_remnant.py: Core module (remnant mass, evaporation curve, GW echoes, species dependence)tests/test_bh_remnant.py: 26 tests, all passingrun_experiment.py: Full experiment with 8 analysis sectionsresults.json: Machine-readable output