Smarr Coefficients and Ensemble Survival Terms in Euclidean Black-Hole Thermodynamics

This manuscript argues that, in four-dimensional Einstein gravity, with the standard Einstein-Maxwell normalization in the charged examples considered, the convention 4πG = 1 is naturally adapted to the Bekenstein-Hawking entropy-area density, not to the full black-hole thermodynamic potential. The paper formulates this distinction in Euclidean black-hole thermodynamics as a compact bookkeeping rule for which work terms structurally survive a chosen ensemble subtraction. For a non-extremal stationary black-hole family satisfying the first law dM = T dS + Σᵢ Yᵢ dXᵢ and a verified Smarr relation M = λTS + Σᵢ wᵢ Yᵢ Xᵢ, the manuscript defines the dimensionless thermodynamic potential Ψ̂ = β (M - Σᵢ εᵢ Yᵢ Xᵢ) - Ŝ, with Ŝ = S/kB, β = 1/ (kB T), and shows the exact algebraic decomposition Ψ̂ = (λ - 1) Ŝ + Σᵢ (wᵢ - εᵢ) β Yᵢ Xᵢ. When the chosen ensemble has an on-shell Euclidean realization with the appropriate boundary terms and fixed data, one has Ψ̂ = ÎE = IE/ħ. Thus the structural coefficient of each work package is its signed Smarr coefficient minus its ensemble subtraction. The paper’s central claim is not that a new thermodynamic law is being proposed, nor that 4πG = 1 removes every occurrence of Newton’s constant from Euclidean black-hole thermodynamics. Rather, the point is organizational: once a first law, a verified Smarr relation, and a chosen thermodynamic potential are fixed, the survival or cancellation of each work package is determined by the coefficient difference wᵢ - εᵢ. In four-dimensional Einstein gravity, the Bekenstein-Hawking entropy is SBH = kB A/ (4Għ), so the dimensionless entropy is Ŝ = A/ (4Għ). Under the numerical convention 4πG = 1, this becomes Ŝ = πA/ħ. In this sense, 4πG = 1 remains entropy-area-density adapted: it normalizes the entropy package, not the full thermodynamic potential. Applied to the non-extremal four-dimensional asymptotically flat Kerr-Newman family in the fixed- (β, ΩH, ΦH) grand-canonical thermodynamic potential, the Einstein-Maxwell Smarr relation gives λ = 2, wJ = 2, wQ = 1, εJ = εQ = 1, so the bookkeeping identity yields Ψ̂ = Ŝ + βΩH J. Thus the electric package cancels because wQ - εQ = 1 - 1 = 0, while the rotational package structurally survives because wJ - εJ = 2 - 1 = 1. When the chosen ensemble has an on-shell Euclidean realization with the appropriate boundary terms and fixed data, this becomes ÎE = Ŝ + βΩH J. Under 4πG = 1, the same result reads ÎE = πA/ħ + βΩH J. One of the paper’s main conclusions is therefore that the entropy-area normalization changes the entropy contribution, but does not remove structurally distinct surviving work packages such as βΩH J. The same bookkeeping rule reproduces the simpler cases. For Schwarzschild, where there are no work packages and λ = 2, one obtains Ψ̂ = Ŝ, and, for the standard Euclidean saddle, ÎE = Ŝ. For Reissner-Nordström in the fixed- (β, ΦH) grand-canonical ensemble, one again finds Ψ̂ = Ŝ, because the electric package is subtracted once in the ensemble and appears with Smarr coefficient one. For Kerr in the fixed- (β, ΩH) ensemble, one finds Ψ̂ = Ŝ + βΩH J, so rotation is the clean surviving-work diagnostic already in the uncharged Einstein family. The manuscript also records a formal mixed Kerr-Newman Legendre bookkeeping table. Keeping the same Kerr-Newman Smarr coefficients, different choices of angular and electric ensemble subtraction produce the corresponding formal potentials Ψ̂_ (εJ, εQ) = Ŝ + (2 - εJ) βΩH J + (1 - εQ) βΦH Q. The paper is careful to state that these mixed entries are formal Legendre-subtraction bookkeeping statements. They are not, by themselves, claims that each mixed choice has already been realized by an explicit Euclidean boundary-term construction. A near-extremal Kerr diagnostic in full 4π-rationalized units c = ħ = kB = 1, 4πG = 1 is then used to sharpen the interpretation. Writing χ = J/ (GM²), s = sqrt (1 - χ²), the note finds Ŝ = (M²/2) (1 + s), and Ψ̂ = (M²/2) (1 + 1/s). Hence, along the non-extremal Kerr family as s → 0^+, one has Ŝ → M²/2, Ψ̂ ~ M²/ (2s) → ∞. The manuscript explicitly emphasizes that this is not an entropy-area divergence and not a claim that extremal black-hole entropy diverges. Rather, it is a surviving rotational-work divergence in the non-extremal Euclidean thermodynamic potential as β → ∞. A central theme of the paper is therefore a role audit of coefficient packages. In the four-dimensional stationary Einstein and Einstein-Maxwell black-hole settings considered, Newton’s constant enters directly through the entropy-area coefficient 1/ (4Għ) and the first-law area-response coefficient κ/ (8πG), while the selected Euclidean thermodynamic potential also contains structurally distinct work packages whose survival or cancellation is controlled by Smarr weights and ensemble subtraction. The convention 4πG = 1 acts directly on the entropy-area package, sending Ŝ = A/ (4Għ) to Ŝ = πA/ħ, and correspondingly sending κ/ (8πG) to κ/2. But it does not by itself remove structurally distinct work packages from the bookkeeping identity, nor does it provide a universal coefficient-minimizing convention for full Euclidean black-hole thermodynamics. The paper is careful about scope. Its statements concern non-extremal, stationary, asymptotically flat black holes in four-dimensional Einstein gravity, with standard Einstein-Maxwell normalization in the charged cases, and only after a first law, a verified Smarr relation, and a chosen thermodynamic potential have been fixed. The ensemble matters, the charge normalization matters, rotating Euclidean saddles require the usual thermodynamic-potential care, and extremal limits require separate treatment. The manuscript does not export the same coefficients directly to anti-de Sitter extended thermodynamics, de Sitter thermodynamics, higher-dimensional black holes, higher-curvature Wald-entropy settings, non-equilibrium horizons, dynamical horizons, or holographic entropy. The manuscript proposes no new thermodynamic law, no modified field equation, no modified entropy functional, and no preferred invariant unit system. Its claim is conditional and organizational: in the standard four-dimensional, non-extremal, asymptotically flat Einstein and Einstein-Maxwell black-hole families treated here, the structural survival of Euclidean work packages is controlled by signed Smarr coefficients minus ensemble subtraction, while 4πG = 1 continues to act specifically as an entropy-area-density normalization rather than as a universal simplification of the full thermodynamic potential.