Horizon Response Principle (HRP) Addendum: The GW Comparator (Isaacson Flux Normalization)

This preprint is an HRP addendum note (“The GW Comparator”) in the Horizon Response Principle (HRP) suite. It is a triptych-style, constants-explicit normalization note that rewrites the Isaacson short-wave gravitational-wave (GW) energy flux in the HRP slot language kSEG. This addendum is not a horizon sector: it is a non-horizon comparator channel, and no horizon-thermodynamic semantics are asserted. Scope: • 4D Einstein–Hilbert gravity only• Linearized gravitational waves in the Isaacson short-wave (high-frequency) regime on a smooth background• Plane-wave / wave-zone use in a locally inertial background patch (leading order in λGW/L) • Isaacson averaging ⟨·⟩ over scales λGW << ℓₐv << L• All constants explicit (G, c, ħ, kB) • Normalization/comparator record only (no new dynamics or modified field equations) Semantic typing (comparator channel). The left-hand side (LHS) objects are radiative-energy quantities: (i) tGW⏛⏜, the Isaacson effective stress-energy (averaged; gauge-invariant under the short-wave assumptions), (ii) FGW, the GW energy flux (power per area). These are not horizon-sector objects: not the BH Hamiltonian/Noether area variation δH_ξ|ₐrea, not the local-Rindler boost-energy flux δQboost, and not an FLRW projected flux rate dot (Q). Chart pin (length chart vs SI time). The note distinguishes a length chart X⁰ = ct (with ∂/∂X⁰ carrying units 1/m) from SI time t (seconds) with dot () = ∂/∂t, using the explicit map∂₀ = (1/c) ∂ₜ and dX⁰ = c dt. This pin is used to keep all c-factors stable when converting Isaacson expressions to dot-form. Normalization slot and comparator coefficient. HRP packages the Einstein coupling into the reusable slotkSEG: = 4πG / c³. In the Isaacson flux normalization, the standard prefactor becomesc³/ (32πG) = 1/ (8 kSEG), and the note defines the GW comparator coefficientCGW: = 1/ (8 kSEG) = c³/ (32πG). Boxed flux (TT gauge; plane wave; Isaacson regime). For a plane wave in TT gauge, the energy flux is recorded in constants-explicit, HRP-slot form asFGW = (c³/ (32πG) ) ⟨ dot (hTTᵢj) dot (hTTᵢj) ⟩ = (1/ (8 kSEG) ) ⟨ dot (hTTᵢj) dot (hTTᵢj) ⟩, with all quantities understood at leading nontrivial Isaacson order and with the averaging scale pinned as above. Polarization factor lock. With the TT polarization normalization eAᵢj eBᵢj = 2 δ^AB, one has the algebraic contractiondot (hTTᵢj) dot (hTTᵢj) = 2 (dot (h_+) ² + dot (hₓ) ²), so the polarization-resolved flux can also be written asFGW = (c³/ (16πG) ) ⟨ dot (h_+) ² + dot (hₓ) ² ⟩ = (1/ (4 kSEG) ) ⟨ dot (h_+) ² + dot (hₓ) ² ⟩. The note emphasizes that the “1/ (8 kSEG) ” and “1/ (4 kSEG) ” prefactors correspond to two equivalent normalizations (full TT contraction vs polarization sum) linked by this fixed factor-of-2 lock. EF firewall and what is not claimed. No entropy statements (gravitational or entanglement) are made. This addendum is purely radiative-energy bookkeeping in the Isaacson high-frequency approximation. No identification is made between FGW (or tGW⏛⏜) and any triptych LHS quantity; coefficient coincidence does not authorize semantic identification. Within the HRP suite, this addendum provides a radiative-flux comparator face of the same Einstein normalization slot kSEG: it anchors the Isaacson flux prefactor to kSEG without mixing it with the horizon-sector LHS objects of the triptych.