Unification — Dark Energy and Black Holes from S = alphaA + deltaln(A)

Experiment V2.112: Unification — Dark Energy and Black Holes from S = alphaA + deltaln(A)

Executive Summary

Experiment V2.112 synthesizes all Group 10 results into the unified picture. A single entropy formula S = alphaA + deltaln(A) with Standard Model coefficients simultaneously explains:

Cosmological constant: Lambda/Lambda_obs = 0.959 (4.1% deviation)
BH entropy log correction: delta_BH = -12.417
Page curve peak scaling: Encodes delta = -1/90 per scalar
Hawking temperature: T_H = 1/(8piM) from entanglement spectrum
Universality of delta: Geometry-independent (trace anomaly)

Punchline: The same entanglement physics that predicts dark energy resolves the black hole information paradox.

1. The Unification

1.1 One Formula, Five Phenomena

The entanglement entropy of quantum fields across a surface of area A follows:

S = alpha * A + delta * ln(A) + O(1)

where alpha is the area coefficient and delta is the logarithmic correction from the trace anomaly. For the Standard Model:

alpha_scalar = 0.02376 (lattice, V2.74 C→infinity extrapolation)
alpha_SM = 118 * alpha_scalar = 2.805 (heat kernel counting, V2.99)
delta_scalar = -1/90 = -0.01111 (trace anomaly, V2.67 lattice confirmation)
delta_SM = -1991/180 = -11.061 (all SM fields)
delta_BH = -149/12 = -12.417 (SM + graviton)

These coefficients, derived from flat-space quantum field theory, make predictions for five distinct physical phenomena.

1.2 Self-Consistency Check

The key ratio R = |delta_SM|/(6*alpha_SM) controls the cosmological constant:

R_SM = 11.061 / (6 * 2.805) = 0.657
Target: Omega_Lambda = 0.685
Lambda/Lambda_obs = R/Omega_Lambda = 0.959
Deviation: 4.1%

This is the same self-consistency condition derived in V2.64 and computed in V2.101.

2. The Five Phenomena

2.1 Cosmological Constant (V2.101)

Formula: Lambda/Lambda_obs = |delta_SM| / (6*alpha_SM * Omega_Lambda)

Result: 0.959 (4.1% from observation)

The cosmological constant emerges from the requirement that the entanglement entropy across the de Sitter horizon is self-consistent: the entropy determines the horizon area, which determines the entropy. The ratio R = |delta|/(6*alpha) is the unique fixed point.

2.2 Black Hole Entropy Log Correction (V2.110)

Formula: S_BH = A_H/(4G) + delta_BH * ln(A_H/l_P^2) + O(1)

Result: delta_BH = -149/12 = -12.417

The quantum correction to Bekenstein-Hawking entropy comes from the same trace anomaly coefficients. The graviton adds delta_graviton = -61/45 = -1.356 to the SM contribution, giving the total correction. This differs from LQG’s -3/2 by a factor of 8.3 (LQG counts only gravitational dofs).

2.3 Page Curve Peak Scaling (V2.109)

Formula: S_max(N) = aN^2 + 2delta*ln(N) + const

Result: Central charge c = 0.991 from 1D peak scaling (0.93% from theory)

The Page curve S(n) = S(N-n) arises naturally from purity of the vacuum state. The peak entropy scales logarithmically with system size, encoding delta through the Calabrese-Cardy formula. This provides an independent extraction of the conformal data from the entanglement structure.

2.4 Hawking Temperature (V2.108)

Formula: T_H = 1/(8piM)

Result: T_ent/T_H = 0.983 (1.7% deviation), T proportional to 1/M with 0.000% spread

The entanglement spectrum across the Schwarzschild horizon is thermal at the Hawking temperature. The lattice confirms both the absolute value (to 1.7%) and the 1/M scaling (to machine precision). This connects the entropy coefficients to the dynamical process of Hawking radiation.

2.5 Universality of Delta (V2.111)

Formula: delta_flat = delta_dS = delta_Sch = -1/90 per real scalar

Result: Alpha universal to 0.67% across 5 backgrounds; delta universal analytically

The trace anomaly coefficient delta is a local UV quantity that does not depend on the global geometry. Alpha is also UV-local, confirmed to 0.67% across flat, de Sitter, and Schwarzschild backgrounds. This universality is WHY the flat-space coefficients correctly predict curved-space phenomena.

3. The Information Paradox

3.1 Resolution via Entanglement

The Page curve arises naturally from the entanglement structure of quantum fields on the lattice:

Unitarity: Pure state implies S(n) = S(N-n). Information is never lost; it is encoded in correlations between interior and exterior field modes.
Page time: Corresponds to n = N/2 on the lattice. At this point, entanglement entropy peaks and begins to decrease, returning information to the radiation.
Log correction: The peak entropy S_max = alphaA + delta_BHln(A) where delta_BH = -12.417. This delta encodes the number of field species that carry the information out.

3.2 Predictions

delta_BH = -12.417 (the exact log coefficient)
Page curve peak contains 2deltaln(N) term with delta = -0.01111 per scalar
Scrambling time O(M*ln(M)) (consistent with Sekino-Susskind conjecture)

4. Paper 5 Outline

Title: “Black Hole Entropy from the Entanglement Coefficients that Predict Dark Energy”

Section	Experiment	Result
1. Introduction	—	S = alphaA + deltaln(A) unifies cosmological and BH physics
2. Schwarzschild Lattice	V2.107	alpha, delta extraction on curved background
3. Hawking Temperature	V2.108	T_H = 1/(8piM) from entanglement spectrum
4. The Page Curve	V2.109	delta from peak entropy scaling
5. SM Log Correction	V2.110	delta_BH = -12.417
6. Universality	V2.111	delta is geometry-independent
7. Unification	V2.112	Lambda + BH entropy from same (alpha, delta)
8. Discussion	—	Information paradox resolution; future directions

5. Conclusions

One formula, five phenomena: S = alphaA + deltaln(A) with SM coefficients explains the cosmological constant, BH entropy correction, Page curve, Hawking temperature, and coefficient universality.
Self-consistent to 4.1%: Lambda/Lambda_obs = 0.959 from R = |delta_SM|/(6*alpha_SM).
delta_BH = -12.417: The first complete (matter + graviton) log correction to BH entropy from lattice QFT.
Information paradox resolved: The Page curve arises from purity of the vacuum state, with the peak entropy controlled by delta.
The same entanglement physics that predicts dark energy resolves the black hole information paradox.