Base-Unit Realization of the Numerical 4π G = 1 Entropy-Response Convention: Exact M, L, T, Θ Conversion Rules for Einstein Black-Hole Thermodynamics

This manuscript constructs an explicit family of base units realizing the numerical conditions c_χ = 1 and (χ_E)_χ = 1 in four-dimensional Einstein gravity. Together, these conditions imply the reduced numerical shorthand 4π G_χ = 1. The constants-explicit starting point is the derived entropy-response conversion coefficient χ_E := πk_B/(ħη_Ein) = 4πG/c³, where, with G, c, ħ, and k_B held fixed, η_Ein := dS_BH/dA = k_Bc³/(4Għ) is the Bekenstein-Hawking entropy-area density. The quantity χ_E has dimensions of time per mass. It is not introduced as a new independent physical constant, but as a dimensionful conversion coefficient correlating the selected mechanical unit scales. The defining numerical conditions are c_χ = 1 and (χ_E)_χ = 1. For an arbitrary positive reference mass-unit scale 𝓜_χ, these conditions hold if and only if the corresponding time- and length-unit scales are 𝓣_χ = χ_E𝓜_χ and 𝓛_χ = cχ_E𝓜_χ. The construction therefore leaves one freely selectable scale in the mechanical M, L, T sector. If absolute thermodynamic temperature is retained as an independent base dimension, an additional positive temperature-unit scale Θ_χ remains freely selectable. A fully specified member of the basic M, L, T, Θ convention consequently requires the choices of 𝓜_χ and Θ_χ. In these units, Newton’s constant has numerical value G_χ = 1/(4π). This is a statement about the numerical representation of G in the selected unit system. It is not an invariant equality involving a dimensionful quantity, a modified gravitational coupling, or a new physical law. The numerical values of the reduced Planck constant and Boltzmann’s constant remain explicit: ħ_χ = ħ/(c²χ_E𝓜_χ²) and (k_B)_χ = k_BΘ_χ/(c²𝓜_χ). The defining convention does not require either ħ_χ = 1 or (k_B)_χ = 1. Their explicit retention separates the basic entropy-response convention from additional quantum or thermal normalizations. To clarify the status of the factor π, the manuscript embeds the construction in the one-parameter family χ_ν := νk_B/(ħη_Ein) = 4νG/c³, with ν > 0. Units satisfying c_ν = 1 and (χ_ν)_ν = 1 give G_ν = 1/(4ν), while the corresponding Einstein entropy-temperature-area response has coefficient ν/(2π). Within this declared family, the stipulated half-response form T_H,χ δS_BH,χ = (κ_χ/2)δA_χ selects ν = π uniquely. This selection is criterion-relative. The manuscript does not claim that the entropy law or horizon dynamics independently forces the half-response target or the factor π. Rather, π is the member of the declared normalization family that produces the selected reduced coefficient 1/2 when combined with the standard Hawking factor 2π. For the resulting χ convention, the four-dimensional Einstein black-hole thermodynamic ledger becomes 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. Here κ_χ is the numerical inverse-length surface gravity. The displayed stationary-horizon applications assume a consistently normalized future-directed horizon generator and the sign convention κ_geom > 0 on the non-extremal outer event horizons considered. The constants, selected base-unit scales, and generator-normalization prescription are held fixed under the stated solution-space variations. At exact extremality, κ_ν and T_H,ν vanish. Provided the solution-space variations δA_ν and δS_BH,ν remain finite, the evaluated response relation reduces to 0 = 0 for every member of the normalization family. Exact extremality therefore supplies no independent normalization criterion. Extremal configurations may be admitted as continuous limits of non-extremal families where appropriate. The manuscript supplies an exact forward and inverse conversion map for any already-defined physical quantity Y whose dimensions lie in the declared M, L, T, Θ sector. If [Y] = MᵃLᵇTᵈΘᵉ, then its numerical value in the selected units is obtained by dividing by the corresponding powers of 𝓜_χ, 𝓛_χ, 𝓣_χ, and Θ_χ. The inverse map restores the constants-explicit quantity exactly. This reversible map is complete within the declared dimensional sector. Quantities carrying additional independent dimensions, such as electric charge in an SI-like dimensional basis, require corresponding additional base-unit choices. The paper emphasizes that constants should be restored through the exact inverse map. Dimensional analysis alone may not uniquely reconstruct the original combination of G, c, ħ, and k_B from a reduced numerical expression. A Schwarzschild round-trip calculation provides the principal operational check. The numerical Schwarzschild radius, area, surface gravity, entropy, and Hawking temperature are derived in terms of the numerical mass μ := M/𝓜_χ. The resulting expressions satisfy both the thermal area-response relation and the Schwarzschild first law, while the inverse conversion map restores the standard constants-explicit geometric and thermodynamic formulas. The manuscript also records the area-defined mass parameter M_A² := c⁴A/(16πG²), which satisfies the numerical identity (M_A)_χ² = πA_χ. For Kerr-Newman black holes, this area mass coincides with the Christodoulou-Ruffini irreducible mass. The identity is presented as a consistency rewriting of the area sector, not as a new black-hole relation. An optional mechanical completion imposes ħ_χ = 1. This fixes the reference mass-unit scale to 𝓜_χ = m_P/√(4π), with corresponding time- and length-unit scales 𝓣_χ = √(4π)t_P and 𝓛_χ = √(4π)ℓ_P. This 4π-scaled Planck completion is not part of the defining convention. The basic construction leaves 𝓜_χ arbitrary, and the independent thermodynamic-temperature scale Θ_χ remains unfixed unless an additional thermal normalization is imposed. The dimensional reach of the conversion map is distinguished from the physical scope of the displayed black-hole applications. The map applies to already-defined quantities in the M, L, T, Θ sector, but it does not by itself supply field equations, entropy laws, conserved quantities, boundary conditions, horizon-generator normalizations, or variation prescriptions. The displayed entropy-temperature-area-response ledger is restricted to four-dimensional Einstein-Hilbert gravity and to the outer event horizons of the stationary non-extremal black holes considered. It uses the Bekenstein-Hawking entropy, which agrees with the stationary Wald Noether-charge entropy in that theory. No automatic extension to higher-curvature gravity is claimed. The manuscript proposes no new independent physical or fundamental constant, no modified field equation, no new entropy formula, no microscopic derivation of horizon entropy, no uniquely preferred normalization imposed by nature, and no preferred unit system with invariant physical status. Its contribution is operational rather than dynamical: it provides explicit base units, makes the remaining mechanical and thermal scale freedom transparent, and supplies exact reversible conversion rules for the numerical 4π G = 1 entropy-response convention.