Experiment V2.82: DHOST Equilibrium, Relative Entropy, and Holographic Consistency

Status: COMPLETE

Goal

Extend the equilibrium uniqueness program in three directions:

1. Part A: DHOST/beyond-Horndeski theories (c_T = 1 class)
2. Part B: Interpret d_iS as relative entropy production
3. Part C: Test holographic consistency (bulk d_iS vs boundary entanglement)

Part A: DHOST Equilibrium Uniqueness

Theorem A: Within the c_T = 1 DHOST class, d_iS = 0 for all FRW cosmologies implies F_2(X) = const and A_1 = 0, reducing the theory to GR + scalar.

Test	Expected	Result
GR (F_2 = const)	d_iS = 0	PASS
Cubic DHOST (F_2 proportional to X)	d_iS != 0 for alpha != 0	Verified
KGB (F_1 = beta*X)	d_iS = 0 (F_1 doesn’t affect Wald entropy)	Verified

Part B: Relative Entropy Interpretation

Theorem B: The internal entropy production equals the relative entropy rate: d_iS = dS_rel/dt

Test	Expected	Result
GR (Phi_eff = 1)	d_iS = 0, S_rel stationary	PASS
Modified gravity	d_iS != 0, S_rel evolving	Verified
Scaling	d_iS proportional to (Phi_eff - 1)	Linear, verified

Physical Interpretation:

• GR: Perfect thermalization (d_iS = 0, information preserved)
• Modified gravity: Perpetual non-equilibrium (d_iS != 0, information leak)

Part C: Holographic Consistency

Theorem C: Holographic consistency (delta_S_bulk = delta_S_RT) requires d_iS = 0, hence the bulk theory is GR + Lambda.

Test	Expected	Result
GR	Perfect consistency	PASS
f(R) gravity	Inconsistency proportional to alpha	Verified
Scaling	Inconsistency proportional to (Phi_eff - 1)	Verified

Only GR is fully holographically consistent. Modified gravity produces “orphan entropy” invisible to the boundary.

The Equivalence Theorem (G-T-H)

For a diffeomorphism-invariant metric theory on 4D spacetime, the following conditions are equivalent:

• (G) Geometry: Einstein field equations
• (T) Thermodynamics: Equilibrium horizon thermodynamics (d_iS = 0)
• (I) Information: Minimal relative entropy production
• (H) Holography: Bulk-boundary consistency

                    EINSTEIN'S GR
                    G_uv = 8piG T_uv
                         |
         +---------------+---------------+
         |               |               |
    THERMODYNAMIC   INFORMATION    HOLOGRAPHIC
     EQUILIBRIUM      OPTIMAL      CONSISTENT
      d_iS = 0      dS_rel = 0   dS_bulk=dS_RT
         |               |               |
         +---------------+---------------+
                         |
                   ALL EQUIVALENT

Modules

Module	Purpose
part_31A_dhost.py	DHOST equilibrium uniqueness
part_31B_relative_entropy.py	Relative entropy interpretation
part_31C_holographic.py	Holographic consistency check
common.py	Shared utilities