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 k_SEG 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, k_B)• 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 δQ_boost, and not an FLRW power-like flux rate dot(Q). Normalization slot and membrane coefficients.HRP packages the Einstein coupling into the reusable slotk_SEG := 4πG / c^3.In the standard membrane paradigm for Einstein gravity, the stretched-horizon surface shear viscosity isη_mem = c^3 / (16πG),with the standard formulation also giving a negative surface bulk viscosityζ_mem = − c^3 / (16πG) (teleological boundary-condition origin; convention-dependent sign).This note records the coefficient-level identityη_mem = 1 / (4 k_SEG), i.e. the membrane shear viscosity equals the Einstein–Hilbert prefactor when written in k_SEG-slot form. Connection to the triptych UHRA skeleton (coefficient rewrite only).The triptych isolates the universal reversible area-response coefficient skeletonC_H := (α_H / (2c)) k_SEG^{-1}.Using k_SEG^{-1} = 4 η_mem, this addendum rewrites the same skeleton asC_H = (2 α_H / c) η_mem.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.S_grav 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 isη_mem / (S_grav/A) = ħ / (4π k_B),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 k_SEG: it anchors k_SEG^{-1} to a 2D surface transport coefficient (membrane shear viscosity) without semantic mixing across the triptych sectors. Version note (v2): Added the suite-wide “Interface Contract” (shared across the triptych + addenda) to enforce typed-LHS discipline for this addendum (membrane transport/stress data on a timelike stretched-horizon worldtube \mathcal M), coefficient-only UHRA comparison, and an explicit Entropy Firewall. Clarified operator status: this is not a \delta-form or \dot{}-form horizon-channel statement, and coefficient rewrites here are not maps between triptych operators/LHS objects. Added an explicit suite notation crosswalk (k_{\mathrm{SEG}}, temperature slot T(\alpha_H), and Einstein entropy density S_{\rm grav}/A) inside the pinned conventions. Critical clarification: explicitly distinguishes the Einstein–Hilbert action prefactor c^{3}/(16\pi G) from the field-equation coupling 8\pi G/c^{4} (no conflation), preventing normalization drift when interpreting the membrane-viscosity identity. Strengthened the membrane-coefficient import with reference pins (Thorne–Price–Macdonald; Damour), an explicit renormalization pin (redshift-renormalized horizon-limit coefficients vs location-dependent “bare” stretched-horizon values), and a constitutive-sign convention pin; added a units tripwire for \eta_{\rm mem}=1/(4 k_{\mathrm{SEG}}). Core coefficient identities and the UHRA rewrite are unchanged.