A Coefficient-Role Audit of Entropy Functionals Beyond Einstein Gravity

This manuscript extends the coefficient-role diagnostic developed for four-dimensional Einstein black-hole thermodynamics to settings in which the relevant gravitational entropy is no longer given by the bare Einstein area law. In Einstein gravity, the Bekenstein-Hawking entropy identifies a clean entropy-area density, η_Ein := dS_BH/dA = k_B/(4Għ). In that restricted setting, a convention such as 4πG = 1 can be read as foregrounding the Einstein entropy-area package rather than the source coupling or action normalization. The present manuscript asks how that diagnostic changes once the relevant entropy is no longer simply k_B A/(4Għ). Its central answer is that one should not begin by asking what replaces G. One should begin by asking what entropy package is actually present. The manuscript’s claim is therefore organizational rather than dynamical. It does not propose a new entropy law, a modified field equation, a universal effective Newton constant, a universal convention of the form 4πG_ent = 1, or an invariantly preferred unit system. Its claim is that beyond Einstein gravity the coefficient-role question must be reformulated as an audit of the relevant entropy functional. Here “entropy package” means the coefficient-bearing structure inside the relevant entropy prescription, not a new fundamental coupling. The manuscript’s central methodological rule is: first identify the relevant gravitational or generalized entropy functional, and only then ask whether it admits a clean entropy coupling at all. The paper formulates this as a four-class entropy-package audit. Given a stationary black-hole sector or a semiclassical generalized-entropy setting, the audit has four possible outputs: 1. a clean global or sectoral entropy coupling; 2. a local entropy density only; 3. a local geometrical entropy functional not generically reducible to area over one coupling; 4. a mixed generalized entropy containing distinct gravitational and non-gravitational packages. These four classes classify entropy packages in specified sectors. The manuscript’s core claim is therefore that the Einstein coefficient-role diagnostic generalizes not as a universal replacement G → G_ent, but as a classification of when such an entropy-coupling compression is legitimate and when it is not. The manuscript is best read as a classification note; its main deliverable is the final four-class audit table rather than any new entropy formula. The paper begins by restating the Einstein baseline. In ordinary four-dimensional Einstein gravity, S_BH = k_B A/(4Għ), η_Ein = k_B/(4Għ), and therefore, after c = 1 and the relevant gravitational unit scale has been fixed, 4πG = 1 implies η_Ein = πk_B/ħ. But this Einstein result is not exported as a universal template. Outside the Einstein area-law setting, the relevant entropy may instead be a Wald entropy, a local entropy density, a curvature-dependent functional, or a generalized entropy. The manuscript’s disciplined extension is therefore: first identify the entropy functional; only then ask whether it defines a clean entropy coupling. The manuscript then introduces the Wald entropy as the relevant target for a general diffeomorphism-invariant gravitational theory. For a scalar Lagrangian L, including the appropriate coupling normalization, define E_Rᵃᵇᶜᵈ := δL/δR_abcd. For a stationary black hole with bifurcate Killing horizon and horizon cross-section B, the Wald entropy is written schematically as S_Wald = -(2πk_B/ħ) ∫_B E_Rᵃᵇᶜᵈ ε_ab ε_cd √h dD-2x, with ε_ab ε^ab = -2. When the integrand can be read as a density relative to the area element dA := √h dD-2x, the manuscript defines the local Wald entropy density η_Wald(x) := -(2πk_B/ħ) E_Rᵃᵇᶜᵈ ε_ab ε_cd. A central interpretive point is that η_Wald(x) is first an integrand density on a horizon cross-section with respect to the area element dA, not automatically a thermodynamic derivative dS/dA. In the Einstein case the entropy density is globally constant and may be read as an area derivative. In the general Wald case, a clean entropy-area coupling arises only in special sectors where that density is constant or otherwise compressible. The paper also makes a novelty-boundary clarification here. It does not claim that effective-coupling interpretations of Wald entropy are new. Prior work already interprets Wald entropy in effective-coupling language. The narrower claim made in this manuscript is classificatory: when an entropy-coupling interpretation is legitimate, when only a local density exists, when the entropy is genuinely functional, and when the full entropy is mixed. The first two substantive examples are metric f(R) gravity and Jordan-frame scalar-tensor gravity. These are chosen because they sit near the boundary between the Einstein area law and genuinely functional entropy. For metric f(R) gravity, with action normalization I = (1/(16πG_D)) ∫ √-g f(R) d^Dx + … , the stationary Wald entropy can be written as S_Wald = (k_B/(4G_D ħ)) ∫_B f’(R) dA. If f’(R) is constant on the horizon cross-section, so that f’(R)|_B = f’(R_H), then the entropy compresses to S_Wald = k_B f’(R_H) A/(4G_D ħ), with entropy-area density η_grav = k_B f’(R_H)/(4G_D ħ). In that stationary sector, the manuscript identifies the clean sectoral entropy coupling G_ent,H = G_D / f’(R_H), so that η_grav = k_B/(4G_ent,H ħ). This is classified as a Class A output: a clean sectoral entropy coupling. If instead f’(R) varies over the horizon cross-section, the manuscript does not promote the result to a single horizon-wide coupling. The appropriate object is only the local entropy density η_grav(x) = k_B f’(R(x))/(4G_D ħ). This is classified as a Class B output: local entropy density only. The same logic is then applied to Jordan-frame scalar-tensor gravity written as I = (1/(16πG)) ∫ √-g [F(φ)R - Z(φ)(∇φ)^2 - 2U(φ)] d^4x + … . The relevant stationary Wald entropy is S_Wald = (k_B/(4Għ)) ∫_B F(φ) dA. If F(φ) is constant on the horizon cross-section, so that F(φ)|_B = F(φ_H), then the entropy compresses to S_Wald = k_B F(φ_H) A/(4Għ), with entropy-area density η_grav = k_B F(φ_H)/(4Għ), and corresponding clean sectoral entropy coupling G_ent,H = G / F(φ_H). This is again a Class A output. If instead F(φ) varies across the horizon cross-section, the correct object is only the local entropy density η_grav(x) = k_B F(φ(x))/(4Għ), not a single horizon-wide coupling. This is again Class B. The manuscript is careful about the assumptions here. The clean Class A reading requires finite, positive, horizon-constant data such as f’(R_H) or F(φ_H). It also stresses that these identifications are normalization-sensitive and formulation-sensitive. Metric and Palatini f(R) are not silently identified, and Jordan-frame scalar-tensor coefficients should not be compared across papers without checking whether factors of G, 16π, or scalar normalization have been absorbed into F(φ). Where used, G_ent is only diagnostic shorthand for a compressible entropy package. It is not automatically the source coupling, the measured Newton constant, or the action coupling. The next step in the audit is the first genuinely non-compressible case: Lovelock gravity, including standard higher-dimensional Einstein-Gauss-Bonnet gravity as its lowest nontrivial higher-curvature example. For a stationary black hole in Lovelock gravity, the entropy may be written schematically as S_L = (k_B/ħ) Σ_m 4πm c_m ∫_B √h̃ Lₘ₋₁(B) dD-2x, where the higher-order terms depend on intrinsic curvature invariants of the horizon cross-section itself. The manuscript’s structural point is that these non-Einstein terms are not merely horizon-dependent scalars multiplying the area element. They are intrinsic-curvature contributions. Therefore the generic Lovelock entropy package is not, in general, reducible to k_B A/(4G_ent ħ) for a single global G_ent. This is classified as a Class C output: local geometrical entropy functional. The paper notes an important caveat: in highly symmetric sectors, the intrinsic-curvature terms may reduce to constants over the horizon cross-section, so the full functional may compress to area times a sector-dependent factor. But this is treated as a special compression of a genuinely functional entropy, not as a general theory-level replacement of G by a single effective coupling. The manuscript also explicitly restricts the scope of this discussion. Standard Gauss-Bonnet gravity is dynamically nontrivial as a Lovelock term in D > 4. In ordinary four dimensions it is topological in the bulk, and nonstandard four-dimensional Einstein-Gauss-Bonnet constructions are not taken as part of the baseline audit. The final substantive case is generalized entropy and quantum extremal surfaces. Here the relevant entropy object is not purely gravitational. In the simplest semiclassical Einstein-bulk notation, the generalized entropy is written schematically as S_gen = k_B ⟨A⟩/(4G_N ħ) + k_B S_bulk^vN + S_ct + … . A quantum extremal surface is selected by extremizing S_gen, not by extremizing the area term alone. This is where the audit becomes mixed rather than merely functional. Even if the gravitational term were a clean area term, the entropy being extremized contains additional non-geometrical contributions. Conversely, in a higher-derivative bulk theory, the gravitational term may already be a Wald/Jacobson-Myers term on stationary horizons or a Dong-Camps-type functional for holographic entanglement surfaces before the bulk von Neumann entropy is added. The manuscript’s coefficient-role point is that the area or higher-derivative gravitational term is only one package inside the full generalized entropy. The audit should therefore be applied to the renormalized generalized-entropy package, not to a regulator-dependent bare area term in isolation. Therefore a convention such as 4πG_N = 1 can at most adapt the gravita