Thermodynamic Split Conjecture (TSC)

• The Thermodynamic Split Conjecture (TSC) is a principle differentiating the microstate counting and thermodynamic derivations between black hole and cosmological horizons in UV-complete quantum gravity.
• It employs the BKE framework by evaluating asymptotic charges, the presence of a global timelike Killing vector, and a universal near-horizon decoupling region essential for black hole entropy.
• TSC provides practical insights in statistical mechanics and observational cosmology by predicting deviations from the standard Bekenstein–Hawking area law in cosmological settings.

The Thermodynamic Split Conjecture (TSC) is a formal principle that articulates the generic inequivalence between the microscopic and thermodynamic structures underpinning black hole event horizons and cosmological event horizons in UV-complete quantum gravitational theories. It crystallizes distinctions in the operational meaning and derivation of horizon entropy for black holes versus cosmological spacetimes, with implications spanning statistical mechanics, non-equilibrium states, and observational cosmology.

1. Definition and Formal Statement

The TSC posits that in any UV-complete theory of quantum gravity, the microstate counting and thermodynamic properties giving rise to the Bekenstein–Hawking area law for black holes are not generically mirrored for cosmological horizons. This assertion follows from the absence, in cosmological spacetimes, of several critical structural ingredients: well-defined asymptotic charges, a globally defined timelike Killing vector, and a universal near-horizon decoupling region. As a consequence, entropy for cosmological horizons cannot be constructed from a universal microcanonical ensemble with degeneracy

$$S_{\text{micro}} = \log \Omega(E,\{Q_i\}) = \frac{A}{4 G_N} + O(1)$$

whereas this construction is standard for black holes.

2. Structural Criteria: The BKE Framework

To formalize this difference, the BKE criterion (Trivedi, 17 Sep 2025) encapsulates three binary conditions fundamental for black hole entropy derivation:

Condition	Criterion	Physical Meaning
B (Boundary/Charges)	1 if asymptotic region allows conserved charges via Gauss law	Existence of superselection sectors labeled by {Q_i}; crucial for microstate counting
K (Killing/Gibbs Structure)	1 if a global timelike Killing vector with horizon regularity, constant surface gravity, and global KMS state exist	Enables equilibrium thermodynamics and Gibbs ensembles
E (Near-Horizon Control)	1 if a universal near-horizon decoupling region exists for well-defined entropy functional	Controls path integral contributions and allows regulator independence

Only when $B \cdot K \cdot E = 1$ is there a controlled, observer-independent microscopic counting leading to the Bekenstein–Hawking formula. Cosmological horizons typically fail all three ($B = K = E = 0$), precluding this construction.

3. Thermodynamic Splitting in Statistical Mechanics

The spirit of TSC is also manifest in statistical mechanics for reparametrization-invariant systems (Chirco et al., 2015). In such frameworks, the absence of a preferred time requires the system to be split into subsystems: a clock and one or more interacting components.

• Two-part split allows definition of a conserved generalized energy $I$ for the system ($C = C^a + C^b = 0$), where $C$ is the total constraint, and enables construction of statistical states via clock averaging:

  $$\overline{f}(\theta, \gamma) = \frac{\int_\gamma f\,\theta}{\int_\gamma \theta}$$

  using a physical clock one-form $\theta$.

• Three-part split is necessary and sufficient for equilibrium. The total constraint is

  $C = C^a + C^b + C^c + V^{bc} = 0,$

  with $I = I^b + I^c$ additive and the interaction $V^{bc}$ weak. Thermodynamic quantities (entropy, generalized temperature) emerge only when energy is equi-partitioned between subsystems.

This decomposition is essential for the formulation of thermal time and for stat-mechanical constructions in generally covariant systems, underscoring the necessity of a physical "split" for meaningful thermodynamics.

4. Thermodynamic Splitting in Non-Equilibrium Steady States

In conformal field theories and hydrodynamic regimes, TSC is closely connected to the universality of steady-state currents between equilibrium baths (Chang et al., 2013). When two baths with fixed thermodynamic boundary conditions are connected, the system evolves into a steady-state split region ("central plateau") characterized by universal heat current expressions:

• 1+1D CFTs: The steady state current depends only on the asymptotic data:

  $T^{tx} = \frac{c\pi}{12} (T_L^2 - T_R^2)$

  independent of microscopic details.

• Higher dimensions: Splitting and matching conditions yield

  $J = \frac{2\Delta P}{v_L + v_R}$

  in terms of bath pressures and front velocities.

This universal behavior supports the notion that far-from-equilibrium steady states are determined solely by the thermodynamic properties of the system's splits. The late-time plateau region, and its associated currents, are fully set by asymptotic equilibrium parameters—a manifestation of the TSC.

5. Thermodynamic Split in Irregularly Ordered Ground States

The TSC nomenclature is adopted in the context of lattice gas models with irregularly ordered ground states (Sasa, 2010). In these models, highly non-trivial ground-state structures (generated by one-dimensional automata, e.g. Wolfram’s Rule 102) admit a thermodynamic transition marked by a self-duality line in the system's parameter space:

$(\sinh(J/T)) \cdot (\sinh(\mu/T)) = -1$

This duality and the associated discontinuous transitions in density—verified through Monte Carlo simulations and fluctuation analysis—are referred to as a "split" of the thermodynamic phase, separating a vacuum-like regime from an irregular, fractal-ordered phase with a continuously distributed set of pure states.

6. Observational Verdict: Tests of the Area Law in Cosmology

To empirically assess the TSC in cosmology, the observational scaling test (Trivedi, 17 Sep 2025) compares data-defined entropy proxies to the area scaling predicted by the Bekenstein–Hawking law:

$S(z) \stackrel{?}{\propto} \frac{1}{H(z)^2}$

using as proxies:

• Information entropy from Fourier modes
• Shannon or compression-based pixel entropies

Deviation from the predicted scaling exponent ($\beta = 2$) in $S_{\text{LHS}}(z) \propto H(z)^{-\beta}$ would indicate a breakdown of the black hole paradigm, consistent with the TSC’s assertion of inequivalence for cosmological horizons. The result of such tests provides a route to falsifying or reinforcing the conjecture.

7. Significance and Future Directions

The TSC refines the conditions under which the area law is valid, demarcating controlled (black hole) cases from cosmological scenarios that lack the requisite structure for a microstate counting derivation. Its implications encompass:

• The necessity for “splitting” both in fundamental quantum gravitation and in statistical mechanics for defining equilibrium and thermodynamic ensembles.
• A roadmap for developing UV-complete descriptions of cosmic horizon entropy, potentially with observer- or patch-dependent constructs replacing the area law.
• Testable predictions using cosmological data, providing empirical access to foundational questions of horizon thermodynamics.

The TSC thus unifies theoretical developments in gravity, quantum statistical mechanics, and field theory, and sets a program for future research into the microphysical origins and observability of horizon entropy in cosmology and beyond.