Experiment V2.87: Non-Equilibrium d_iS Correction to Lambda

Status: COMPLETE

Summary

Paper 3 (V2.64) derives Lambda using the equilibrium Clausius relation with log-corrected entropy S = alphaA + deltaln(A). The self-consistency ratio R_SM = |delta_SM|/(12*alpha_SM) = 0.3286 instead of 1.

This experiment investigates whether the non-equilibrium Clausius relation (Paper 2, V2.78-80) can close this gap.

Key Result: F = 1 (No Correction)

The non-equilibrium d_iS framework gives F = 1 exactly. The self-consistency gap is NOT closed by d_iS alone.

Why F = 1

The Clausius relation at the cosmological horizon IS the Friedmann equation (Jacobson 1995, Cai-Kim 2005). Therefore:

• Standard GR (S_Wald = alpha*A): dQ(GR)/T = dS_Wald => d_iS = 0
• Modified (S_total = alphaA + deltaln(A)): dQ(mod)/T = dS_total => d_iS = dS_log

In both cases, the non-equilibrium relation is an identity, not a correction.

Difference from f(R) Gravity

For f(R) gravity: d_iS proportional to f”(R), capturing the time variation of the theory-dependent Wald entropy. For GR + quantum log correction: the correction is already included in the equilibrium derivation. No additional physics from d_iS.

Phase 1: Equilibrium Baseline

• V2.64: R = |delta|/(6*alpha) = 0.9514
• V2.76: R_SM = |delta_SM|/(12*alpha_SM) = 0.3286
• Gap factor: 3.04x

Phase 2: Corrected Symbolic Derivation

• dQ(GR)/dt = -4Hdotalpha/H**2
• dQ(mod)/dt = -Hdotdelta/pi - 4Hdot*alpha/H**2
• d_iS(GR) = 0
• d_iS(mod) = -2Hdotdelta/H
• F = 1

Phase 3: Numerical Verification

• F = 1.0000 for all tested Hdot/H^2 values: True
• d_iS(GR) = 0 for all backgrounds (confirmed)

Phase 4: Self-Consistency

• R_noneq = F * R_eq = 1.0 * 0.3286 = 0.3286
• Deviation from unity: 67.1%
• Gap persists at full strength

Phase 5: Additional Physics Needed

• F needed for R = 1: 3.04
• Extra d_iS/dS_log needed: 2.04

Candidate mechanisms:

1. Running of delta(mu) with Hubble scale (connects to V2.76)
2. Higher-order entropy corrections beyond ln(A)
3. Viscous horizon entropy production (Chirco-Liberati)
4. Non-perturbative backreaction

Phase 6: Conclusions

The non-equilibrium d_iS framework from Paper 2 does NOT modify the Lambda prediction from Paper 3. The self-consistency gap R_SM = 0.329 is robust against this correction.

The most promising direction for closing the gap is the running of entanglement entropy coefficients with scale (V2.76 mechanism), which provides a physically distinct mechanism from d_iS.