Horizon Response Principle (HRP) Addendum: The Membrane Face (Stretched-Horizon Transport)

This preprint is an HRP addendum note (“The Membrane Face”) in the Horizon Response Principle (HRP) suite. It is a triptych-style, constants-explicit bookkeeping note that relates the HRP normalization slot kSEG to standard membrane-paradigm stretched-horizon transport coefficients in 4D Einstein–Hilbert gravity. This addendum is not a fourth horizon sector: it concerns a regime-limited, timelike surrogate boundary (“stretched horizon”) rather than a null horizon sector statement. Scope: • 4D Einstein–Hilbert gravity only• Membrane paradigm in the stretched-horizon (timelike worldtube) regime• Near-horizon, long-wavelength, near-equilibrium linear-response transport coefficients (imported standard values) • All constants explicit (G, c, ħ, kB) • No new dynamics or modified field equations Semantic typing (surrogate-boundary addendum). The LHS objects here are membrane surface stress/transport coefficients on a timelike stretched-horizon worldtube M (a surrogate boundary). They are not identified with any triptych LHS object: not the BH Hamiltonian/Noether area variation δH_ξ|ₐrea, not the local-Rindler boost-energy flux δQboost, and not an FLRW power-like flux rate dot (Q). Normalization slot and membrane coefficients. HRP packages the Einstein coupling into the reusable slotkSEG: = 4πG / c³. In the standard membrane paradigm for Einstein gravity, the stretched-horizon surface shear viscosity isηₘem = c³ / (16πG), with the standard formulation also giving a negative surface bulk viscosityζₘem = − c³ / (16πG) (teleological boundary-condition origin; convention-dependent sign). This note records the coefficient-level identityηₘem = 1 / (4 kSEG), i. e. the membrane shear viscosity equals the Einstein–Hilbert prefactor when written in kSEG-slot form. Connection to the triptych UHRA skeleton (coefficient rewrite only). The triptych isolates the universal reversible area-response coefficient skeletonCH: = (αH / (2c) ) kSEG^-1. Using kSEG^-1 = 4 ηₘem, this addendum rewrites the same skeleton asCH = (2 αH / c) ηₘem. This is coefficient bookkeeping only and does not convert a membrane stress into a horizon heat flux or identify LHS objects across sectors. EF firewall and optional comparator. Sgrav denotes Wald/Bekenstein–Hawking gravitational entropy in Einstein gravity; no identification with entanglement/generalized entropy is made. An optional constants-explicit comparator recorded in the note isηₘem / (Sgrav/A) = ħ / (4π kB), stated purely as an Einstein-gravity normalization identity (no AdS/CFT or KSS-bound universality claim is made here). What is not claimed. • No new HRP horizon sector (timelike surrogate boundary, not a null-horizon sector) • No Clausius/first-law statement for the membrane stress tensor• No entropy-production or irreversible channel derived• No identification of gravitational entropy with entanglement/generalized entropy• No cross-sector identification of distinct LHS objects Within the HRP suite, this addendum provides a “materials/constitutive” face of the same Einstein normalization slot kSEG: it anchors kSEG^-1 to a 2D surface transport coefficient (membrane shear viscosity) without semantic mixing across the triptych sectors.