Charge-Unit Extension of the Numerical 4π G = 1 Entropy-Response Convention: Exact M, L, T, Θ, 𝖰 Conversion Rules for Einstein–Maxwell Black-Hole Thermodynamics

This manuscript extends the numerical 4π G = 1 entropy-response convention from the mechanical and thermodynamic-temperature sector to a dimensional basis that also includes electric charge. The inherited gravitational convention is defined through the constants-explicit entropy-response conversion coefficient χ_E := 4πG/c³, together with numerical units satisfying c_χ = 1 and (χ_E)_χ = 1. For an arbitrary positive reference mass-unit scale 𝓜_χ, the corresponding time- and length-unit scales are 𝓣_χ = χ_E𝓜_χ and 𝓛_χ = cχ_E𝓜_χ. In these units, Newton’s constant has numerical value G_χ = 1/(4π). The preceding base-unit realization supplied an exact forward and inverse conversion framework in the M, L, T, Θ dimensional sector but deliberately left quantities carrying an independent electric-charge dimension outside the declared map. The purpose of the present manuscript is to determine the additional charge-unit scale required to extend that framework consistently to rationalized Einstein-Maxwell quantities. The dimensional basis is enlarged to M, L, T, Θ, 𝖰, where 𝖰 denotes the independent electric-charge dimension and Q denotes physical electric charge. The additional numerical target is stipulated to be (ε₀)_χ = 1. This condition is not implied by c_χ = 1 or (χ_E)_χ = 1. It is an independent rationalized electromagnetic normalization within the extended unit system. The manuscript does not claim that nature uniquely selects this target or that it defines a physically preferred charge unit. Using the charge-based dimensional form of the vacuum permittivity, the manuscript shows that the three numerical conditions c_χ = 1, (χ_E)_χ = 1, and (ε₀)_χ = 1 hold if and only if the positive charge-unit scale is 𝓠_χ = √(4πε₀G)𝓜_χ. The reference mass-unit scale 𝓜_χ remains freely selectable. If thermodynamic temperature is retained as an independent base dimension, the temperature-unit scale Θ_χ also remains freely selectable. The additional electromagnetic target fixes 𝓠_χ, so no independent charge-scale freedom remains after the normalization has been imposed. The resulting electromagnetic numerical values are (ε₀)_χ = 1, (μ₀)_χ = 1, and (k_e)_χ = 1/(4π), where k_e = 1/(4πε₀) is the Coulomb constant. Combining this result with the inherited gravitational normalization gives G_χ = (k_e)_χ = 1/(4π). This is an equality of numerical values in the selected units only. Newton’s constant and Coulomb’s constant retain different physical dimensions and govern different interactions. The manuscript proposes no physical identification of the two constants and no gravity-electromagnetism unification. The paper also distinguishes the selected unit normalization from the status of ε₀ and μ₀ in the post-2019 SI. Their SI numerical values are not independently exact, while the vacuum relation ε₀μ₀c² = 1 remains exact. The condition (ε₀)_χ = 1 therefore changes the numerical representation of the vacuum permittivity; it does not assign a new physical value to it. For an already-defined physical quantity Y with dimensions in the extended sector, [Y] = MᵃLᵇTᵈΘᵉ𝖰ᶠ, the manuscript supplies an exact forward and inverse conversion map using the base-unit scales 𝓜_χ, 𝓛_χ, 𝓣_χ, Θ_χ, and 𝓠_χ. The map is exact and reversible within the declared M, L, T, Θ, 𝖰 sector. Quantities carrying further independent dimensions require corresponding additional base-unit choices. Constants should be restored through the inverse map, since dimensional analysis alone may not uniquely identify the original constants-explicit combination. Frequently used charge-sector conversions are derived for electric charge, electric current, electric potential, charge density, angular momentum, angular velocity, and surface gravity. In particular, the dimensional map gives the electric-potential conversion required for the product ΦδQ to transform exactly as an energy. The angular-momentum and angular-velocity conversions similarly preserve the work term ΩδJ. The four-dimensional Einstein entropy-temperature-area-response ledger inherited from the preceding base-unit realization remains S_BH,χ = [π(k_B)_χ/ħ_χ]A_χ, T_H,χ = [ħ_χκ_χ]/[2π(k_B)_χ], and T_H,χ δS_BH,χ = (κ_χ/2)δA_χ. For asymptotically flat black holes with ADM energy E = Mc², the numerical mass-energy variable is defined by μ := M/𝓜_χ = E_χ. For the stationary non-extremal charged black holes considered, the numerical first law becomes δμ = (κ_χ/2)δA_χ + Ω_χδJ_χ + Φ_χδQ_χ. The constants G, c, ħ, k_B, ε₀, and μ₀, the selected base-unit scales, the normalization prescription for the horizon generator, and the electromagnetic gauge prescription are held fixed under the stated solution-space variations. The horizon generator is future-directed and consistently normalized, with the sign convention κ_geom > 0 on the non-extremal outer event horizons considered. The electric potential is referenced to spatial infinity and represented in the gauge for which it vanishes there. A Reissner-Nordström round-trip calculation provides the principal operational check of the charge-unit extension. In the positive-mass non-extremal sector, the numerical charge-to-mass ratio is Q_χ/μ, and the horizon condition becomes |Q_χ| < μ. The two radial roots, the outer-horizon area, inverse-length surface gravity, and horizon electric potential are converted into compact expressions in terms of μ and Q_χ. For Q_χ ≠ 0, the roots are the outer and inner horizons; in the neutral limit, the inner root degenerates to r_- = 0. Direct differentiation of the numerical outer-horizon area verifies the charged first law δμ = (κ_χ/2)δA_χ + Φ_χδQ_χ, while the corresponding Reissner-Nordström Smarr relation becomes μ = κ_χA_χ + Φ_χQ_χ. The entropy-temperature sector independently reproduces the same area contribution through T_H,χ δS_BH,χ = (κ_χ/2)δA_χ. Applying the inverse conversion map restores the standard rationalized-SI Reissner-Nordström horizon radii, horizon electric potential, and constants-explicit first law, δ(Mc²) = [c⁴κ_geom/(8πG)]δA + Φ_HδQ. The extremal Reissner-Nordström condition becomes |Q_χ| = μ, with vanishing surface gravity and unit-magnitude numerical horizon potential. Exact extremality is treated as a continuous limit of non-extremal solutions where appropriate. A supplementary Kerr-Newman calculation provides a rotating charged consistency check. The manuscript introduces dimensionless spin and charge variables corresponding to cJ/(GM²) and Q/[M√(4πε₀G)], and gives a compact numerical ledger for the outer-horizon area, surface gravity, angular velocity, electric potential, angular momentum, and charge. These quantities satisfy the numerical Kerr-Newman first law δμ = (κ_χ/2)δA_χ + Ω_χδJ_χ + Φ_χδQ_χ, together with the Smarr relation μ = κ_χA_χ + 2Ω_χJ_χ + Φ_χQ_χ. The Kerr, Reissner-Nordström, and Schwarzschild sectors are recovered as the corresponding limits. The Schwarzschild reduction agrees with the preceding uncharged base-unit realization, providing a direct consistency check between the mechanical and charge-extended frameworks. The manuscript also rewrites the established Christodoulou-Ruffini mass relation using variables constructed from the numerical horizon area, angular momentum, and charge. In this form, the area variable equals the squared numerical irreducible mass. The result is presented as a consistency rewriting of the known mass formula, not as a new black-hole relation. The Kerr-Newman ledger further yields the normalized horizon-data identity (κ_χA_χ/μ)² + (Ω_χA_χ/μ)² + (Q_χ/μ)² = 1. Restoring constants gives [c²κ_geomA/(4πGM)]² + [cΩ_HA/(4πGM)]² + [Q/(M√(4πε₀G))]² = 1. This identity is an exact compression of standard Kerr-Newman relations. It is not proposed as an independent dynamical equation or a new physical law. The dimensional reach of the extended conversion map is distinguished from the physical scope of the displayed black-hole applications. The map applies to any already-defined quantity whose dimensions lie in the declared M, L, T, Θ, 𝖰 sector. It does not by itself supply a field equation, electromagnetic action, entropy law, conserved energy, gauge or boundary prescription, horizon normalization, or variation prescription. The displayed thermodynamic applications are restricted to four-dimensional, asymptotically flat Einstein-Maxwell theory; rationalized SI-normalized electric charge; stationary non-extremal outer event horizons; consistently normalized future-directed horizon generators; and an electric potential referenced to spatial infinity. Extremal configurations may be admitted as continuous limits where appropriate. An action-level electromagnetic field dictionary lies outside the scope of the manuscript. The dimensional charge extension does not by itself determine field-level normalization, component, or action conventions. The manuscript proposes no new physical or fundamental constant, no new field equation, no modified Maxwell theory, no new black-hole solution, no new entropy law, no microscopic entropy derivation, no gravity-electromagnetism unification, no physical equality of gravitational and electric interaction strengths, and no preferred charge unit imposed by nature. It also makes no automatic extension of the displayed asymptotically flat Einstein-Maxwell ledger to magnetic charge, dyons, anti-de Sitter backgrounds, nonlinear electrodynamics, charged matter, or higher-curvature gravity. Its contribution is operational rather than dynamical: the independent target (ε₀)_χ = 1 fixes the charge-unit scale to 𝓠_χ = √(4πε₀G)𝓜_χ, thereby extending the underlying entropy-response convention to an exact reversible M, L, T, Θ, 𝖰 conversion framework for rationalized Einstein-Maxwell black-hole thermodynamics.