Charge-Anchored Horizon Thermodynamics on Spherical Screens: Dilation Force and Laplace Pressure

A round two-sphere has only one homogeneous deformation: changing its areal radius. In four-dimensional Einstein gravity, any marginal (apparent or trapping) sphere carries a preferred quasilocal energy given by the Misner–Sharp–Hernandez/Kodama charge, which becomes a one-scale quantity proportional to the radius on the marginal branch. Using this charge as an anchor fixes a unique spherical mechanical channel directly from bulk geometry: a radius-conjugate dilation force, an associated isotropic tangential stress, and a screen-normal Laplace pressure. These quantities are determined without reference to boundary observers, subtraction schemes, or any choice of surface-gravity convention.

Applied to FRW and de Sitter apparent horizons, the charge anchor yields an exact mechanical identity relating the horizon Laplace pressure to the cosmic energy density, providing a purely geometric statement of horizon mechanics. Adopting the Einstein area entropy then closes a matched, charge-anchored temperature–acceleration pair. A key clarification follows from treating entropy and radius as independent variables when defining thermodynamic conjugates: the resulting two-channel first law cleanly separates heat-like and work-like contributions. When this exact differential is restricted to the one-parameter marginal family, it reproduces the familiar “two-to-one” bookkeeping relation often written in the literature, showing that it is not an anomaly but a fixed partition enforced by one-scale rigidity and the dilation work channel.

When a Hawking/Unruh temperature assignment is admissible, we introduce a positive Hawking scale and a sector parameter that measures the ratio between the charge-anchored and Hawking accelerations. This motivates a Hawking-scaled energy and force that obey a compact Hawking-packaged first law in constant-sector families. In FRW and de Sitter the sector parameter equals unity, so the charge-anchored and Hawking descriptions coincide. In the Schwarzschild sector the parameter equals two, and the standard factor-of-two Smarr relation is reorganized into a transparent “heat plus force–radius” partition. Finally, propagating the same one-scale closure to a Compton-saturated Planck cell provides a saturated Bekenstein equation of state as a UV benchmark and fixes the associated effective coupling.