Maxwell Field and Bulk-Action Dictionary for the Numerical 4π G = 1 Entropy-Response Convention: Exact M, L, T, Θ, 𝖰 Conversion Rules for Classical Fields, Sources, Stress-Energy, and Bulk Actions

This manuscript develops the classical Maxwell field, source, stress-energy, curvature, and bulk-action dictionary induced by the numerical 4π G = 1 entropy-response convention and its electric-charge-unit extension. The underlying mechanical 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. The numerical value of Newton’s constant is then G_χ = 1/(4π). The electric-charge extension enlarges the declared dimensional basis to M, L, T, Θ, 𝖰, where 𝖰 denotes the independent electric-charge dimension and Q denotes physical electric charge. The extension imposes the additional rationalized numerical target (ε₀)_χ = 1, which fixes the positive charge-unit scale to 𝓠_χ = √(4πε₀G)𝓜_χ. The resulting electromagnetic numerical values are (ε₀)_χ = (μ₀)_χ = 1 and (k_e)_χ = G_χ = 1/(4π), where k_e = 1/(4πε₀) is the Coulomb constant. The equality between G_χ and (k_e)_χ concerns numerical values in the selected unit system only. Newton’s constant and Coulomb’s constant retain different physical dimensions and describe different interactions. No gravity-electromagnetism unification or physical identification of the two constants is proposed. The inherited exact forward and inverse conversion map applies to any already-defined physical quantity whose dimensions lie in the M, L, T, Θ, 𝖰 sector. The present manuscript specializes that map to electromagnetic potentials, gauge parameters, fields, sources, stress-energy, curvature, and actions. It introduces no additional base unit. The remaining freely selectable scales are the reference mass-unit scale 𝓜_χ and, when thermodynamic temperature is retained as an independent base dimension, the temperature-unit scale Θ_χ. The manuscript treats the covariant objects A_μ, F_μν, J^μ, and T_μν^EM as primary. It fixes the Lorentzian signature (+, −, −, −), the field-tensor definition F_μν = ∂_μA_ν − ∂_νA_μ, the gauge-transformation sign, current convention, curvature convention, metric-variation convention, and all-length coordinate prescription required for an internally consistent field and action dictionary. Exact numerical conversions are derived for the scalar potential, vector potential, covariant potential components, gauge parameter, electric and magnetic fields, electric displacement, magnetic field intensity, charge density, current density, covariant current, and electromagnetic field tensor. These conversions follow from the declared dimensional map rather than from an independent rescaling of Maxwell’s equations. In a vacuum local orthonormal frame, the numerical constitutive relations reduce to D_χ = E_χ and H_χ = B_χ. The manuscript distinguishes uniform indexed-component conversions in all-length coordinate charts and orthonormal frames from conversions in mixed-dimensional coordinate charts containing dimensionless angular coordinates. In the latter case, the corresponding tensor Jacobians must be included in addition to the unit-conversion factors. In Minkowski spacetime with Cartesian numerical coordinates, the field-potential relations and ordinary three-vector Maxwell equations take their standard coefficient-clean rationalized form. These ordinary differential vector formulas are restricted to flat Cartesian coordinates. The manuscript does not treat differential equations in a general tetrad frame as connection-free; the corresponding frame-connection or anholonomy terms would be required. The generally covariant numerical Maxwell equations are ∇_μ^(χ)F_χ^μν = J_χ^ν and ∇_[α^(χ)F_βγ],χ = 0. The source equation implies covariant charge conservation, ∇_μ^(χ)J_χ^μ = 0. A Coulomb restoration check verifies the field dictionary in flat space. The numerical point-charge potential and radial electric field have the familiar rationalized form with coefficient 1/(4π), and the electric flux through a numerical sphere equals the numerical charge Q_χ. Applying the exact inverse conversion map restores the rationalized-SI potential and electric field with their explicit ε₀ dependence. The manuscript also derives the numerical electromagnetic energy density, Poynting flux, Lorentz force density, and stress-energy balance. For the declared signature and field-tensor conventions, the numerical electromagnetic stress-energy tensor is T_μν,χ^EM = −F_μα,χF_ν,χ^ α + (1/4)g_μνF_αβ,χF_χ^αβ. Its trace vanishes. Its divergence gives the negative of the electromagnetic force transferred to matter, while the corresponding matter force density is the numerical Lorentz force. In the flat inertial three-plus-one ledger, the time component reproduces the Poynting energy-balance equation. The manuscript next constructs the Maxwell action with a prescribed conserved current. In the numerical convention, the Maxwell kinetic term and source coupling take the standard coefficient-clean rationalized form. Variation with respect to the numerical electromagnetic potential gives the covariant Maxwell source equation. The current is treated as a prescribed conserved source during the electromagnetic variation. The interaction term is not presented as a complete generally covariant theory of dynamical charged matter. Such a theory must independently supply its matter variables, current construction, stress-energy tensor, and metric variation. Using the declared curvature and metric-variation conventions, the Einstein-Hilbert bulk coefficient reduces to −1/4. The numerical Einstein-Maxwell bulk action becomes 𝓘_bulk,χ = ∫ d⁴x_χ √(−g) [−(1/4)R_χ − (1/4)F_μν,χF_χ^μν]. Source and matter actions are supplied separately. The corresponding numerical gravitational field equation is 𝓖_μν,χ = 2T_μν,χ. In an electrovacuum region, T_μν,χ is the electromagnetic stress-energy tensor. The equal magnitudes of the gravitational and electromagnetic bulk coefficients are consequences of the selected normalization chain. They do not imply equal physical interaction strengths or gravity-electromagnetism unification. They would not remain unchanged under an arbitrary electromagnetic field rescaling unless the charge unit and source coupling were transformed consistently. The displayed Einstein-Maxwell action is a bulk action. A complete gravitational variational principle may require additional boundary terms and boundary prescriptions, which are not constructed in this manuscript. A local Reissner–Nordström round-trip calculation provides the principal validation of the field and action dictionary. Starting from the numerical electromagnetic potential one-form, the manuscript follows the local chain potential → field strength → electromagnetic stress-energy → spacetime curvature and then restores the corresponding constants-explicit rationalized-SI result. The numerical mass and charge variables are defined by μ := M/𝓜_χ and Q_χ := Q/𝓠_χ. In terms of these variables, the Reissner–Nordström metric function, electromagnetic potential, and locally measured radial electric field take compact numerical forms. In the exterior electrovacuum region, the local material current vanishes even though the total charge remains measurable by electric flux. The exterior electromagnetic field, rather than a smooth exterior matter current, supplies the local gravitational source. The manuscript derives the electromagnetic invariant, energy density, radial tension, tangential pressures, and traceless mixed stress-energy tensor. The resulting numerical Einstein tensor satisfies 𝓖^μ_ν,χ = 2T^μ_ν,χ^EM. An independent reconstruction from the static spherical Einstein equation reproduces the same numerical metric function, providing a second check of the field–stress-energy–curvature chain. The factorized metric reproduces the charged horizon radii, outer-horizon potential, and surface gravity obtained from the thermodynamic charge-unit extension. The neutral limit restores the Schwarzschild sector, while exact extremality is treated as a continuous limit of the non-extremal sector where appropriate. The scalar potential is chosen to vanish at spatial infinity. The manuscript notes that Schwarzschild-like time coordinates are singular at the horizon; calculations requiring a horizon-regular potential one-form must use an appropriate regular coordinate patch or gauge. Applying the exact inverse conversion map restores the standard rationalized-SI Reissner–Nordström metric function, Coulomb potential, radial electric field, electromagnetic stress-energy tensor, Einstein-Maxwell equation, horizon radii, and horizon electric potential. The round-trip calculation therefore verifies both directions of the local dictionary: constants-explicit Einstein-Maxwell quantities convert to the compact numerical convention and restore exactly through the inverse map. The manuscript distinguishes three logically separate layers: the dimensional reach of the conversion map, the supplied field and bulk-action dictionary, and application-specific dynamics. The dimensional map applies to already-defined quantities in the declared M, L, T, Θ, 𝖰 sector. It does not by itself supply matter models, solution ansätze, gauges, coordinate systems, tetrads, boundary conditions, initial data, numerical formulations, or variation prescriptions. The displayed applications are restricted to four-dimensional classical Einstein-Maxwell theory, rationalized SI-normalized electric charge, vacuum electromagnetic constitutive relations, Lorentzian signature (+, −, −, −), flat Cartesian coordinates for ordinary differential three-vector formulas, observer-adapted orthonormal frames for measured algebraic components, asymptotically flat Reissner–Nordström electrovacuum, a scalar potential vanishing at spatial infinity, and gravitational bulk actions rather than a comp