V2.255: GSL with Nonzero Λ_bare — Second Law Cannot Constrain Λ_bare

Status: COMPLETE — 27/27 unit tests, 10/10 experiment tests

Motivation

Approach D of the RESEARCH_GUIDE asks: does Λ_bare ≠ 0 violate the generalised second law (GSL) at the cosmological horizon? V2.250 briefly checked this and found the GSL is satisfied for all positive Λ. This experiment performs the full analysis: cosmic evolution through radiation → matter → de Sitter eras, entropy accounting in physical Planck units, and a systematic scan over Λ_bare values.

Method

Friedmann dynamics

Standard ΛCDM: H²/H0² = Ω_r/a⁴ + Ω_m/a³ + Ω_Λ_total, where Ω_Λ_total = R + Λ_bare_frac × Ω_Λ_obs. The framework predicts R = |δ|/(6α) ≈ 0.665 (SM) or 0.688 (SM+grav) as the entanglement contribution.

Entropy in Planck units

S_gen = α A_H + δ ln(A_H), where A_H is the physical apparent horizon area in Planck units: A_H = 4π/(H_phys²) with H_phys = H_dimless × H0_Planck.

Physical scale: A_H(today) ≈ 9.0 × 10¹²² l_P².

GSL condition

dS/dt = (α + δ/A_H) × dA_H/dt ≥ 0

This requires either:

• dA/dt ≥ 0 and α + δ/A_H > 0 (standard case), OR
• dA/dt ≤ 0 and α + δ/A_H < 0 (only possible at Planck scale)

Results

1. GSL holds for ALL Λ_bare values

Λ_bare/Λ_obs	Ω_Λ_total	GSL violations	ΔS (l_P²)
-0.9	0.048	0	5.2 × 10¹²⁴
-0.5	0.322	0	7.8 × 10¹²³
0.0	0.665	0	3.8 × 10¹²³
1.0	1.349	0	1.9 × 10¹²³
5.0	4.088	0	6.1 × 10¹²²
10.0	7.512	0	3.3 × 10¹²²

Zero violations across all eras (radiation, matter, transition, de Sitter) for every Λ_bare value tested.

2. Why the GSL is trivially satisfied

The critical area where the log term could compete with the area term is:

A_crit = |δ|/α ≈ 4.0 l_P²

The smallest cosmological horizon in the evolution (at a = 10⁻⁶) is:

A_min ≈ 9.9 × 10¹⁰² l_P²

The ratio A_min/A_crit ≈ 2.5 × 10¹⁰². The log correction contributes 4.4 × 10⁻¹²³ of the area term at the present epoch. The GSL is satisfied by a margin of 10¹²² orders of magnitude.

3. First law vs second law

Law	Constraint on Λ_bare	Strength
First (QNEC)	Λ_bare = 0 exactly	EXACT
Second (GSL)	None	Trivially satisfied

The first law (V2.250) constrains Λ_bare through S”(n) = 8πα - δ/n², which has exactly two terms uniquely fixing G and Λ. The second law provides no constraint because the cosmological horizon is 10¹²² times larger than the critical scale.

4. When could GSL fail?

Two scenarios (neither occurs in ΛCDM):

1. Planck-scale horizons: A_H < |δ|/α ≈ 4 l_P² — never in cosmology
2. Phantom dark energy: w < -1 causes dA/dt < 0 — not in standard model

5. Friedmann consistency (repeated from V2.250)

Model	Ω_Λ predicted	Deviation from obs
SM (R = 0.665)	0.665	2.8σ
SM+grav (R = 0.688)	0.688	0.4σ

Λ_bare needed to match observations:

• SM: Λ_bare = 0.029 × Λ_obs (2.8σ tension)
• SM+grav: Λ_bare = -0.005 × Λ_obs (0.4σ, consistent with zero)

Key Findings

The GSL does not constrain Λ_bare — CONFIRMED

This is a negative result for Approach D: the second law of thermodynamics at the cosmological horizon cannot distinguish Λ_bare = 0 from Λ_bare ≠ 0. The reason is quantitative: the log correction δ that determines Λ is 10⁻¹²³ of the area term α that determines G. The GSL is dominated by the area term and is insensitive to the log correction.

The paradox of the log term

The log correction is:

• Negligible for the second law (10⁻¹²³ of the area term in dS/dt)
• Essential for the first law (determines Λ through S”(n) = 8πα - δ/n²)

This is not a contradiction. The first law (QNEC) probes the functional form of S(n) — specifically its second derivative — which separates the constant and 1/n² contributions cleanly regardless of their relative magnitude. The second law only checks the sign of dS/dt, which is dominated by the area term.

Implications

1. Approach D (GSL) is closed: The GSL cannot derive Λ_bare = 0.
2. Approach D.2 (modular flow) remains possible but needs separate analysis.
3. The constraint hierarchy is established:
   • QNEC (first law): determines G AND Λ, forces Λ_bare = 0
   • GSL (second law): trivially satisfied, no constraint
   • This parallels ordinary thermodynamics where the first law (energy conservation) is more constraining than the second (entropy increase).

Files

• src/gsl_analysis.py: Friedmann dynamics, GSL computation, critical analysis
• tests/test_gsl.py: 27 unit tests (all passing)
• run_experiment.py: 10-test experiment across 9 parts
• results/summary.json: Full numerical results

Significance for the Framework

This experiment closes the GSL branch of Approach D. Together with V2.252 (closing Approach C), the derivation of Λ_bare = 0 now rests firmly on two pillars:

• Approach B (V2.249/251): Modular Hamiltonian double-counting (operator-level)
• Approach D via QNEC (V2.250): Entropy completeness (first law)

The remaining open experiment from the RESEARCH_GUIDE is D.2: Modular flow inconsistency — testing whether Λ_bare ≠ 0 contradicts Bisognano-Wichmann modular flow at the cosmological horizon.