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 k_SEG. 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 λ_GW/L)• Isaacson averaging ⟨·⟩ over scales λ_GW << ℓ_av << L• All constants explicit (G, c, ħ, k_B)• 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) t_GWμν, the Isaacson effective stress-energy (averaged; gauge-invariant under the short-wave assumptions),(ii) F_GW, 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 δQ_boost, and not an FLRW projected flux rate dot(Q). Chart pin (length chart vs SI time).The note distinguishes a length chart X^0 = ct (with ∂/∂X^0 carrying units 1/m) from SI time t (seconds) with dot( ) = ∂/∂t, using the explicit map∂_0 = (1/c) ∂_t and dX^0 = 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 slotk_SEG := 4πG / c^3.In the Isaacson flux normalization, the standard prefactor becomesc^3/(32πG) = 1/(8 k_SEG),and the note defines the GW comparator coefficientC_GW := 1/(8 k_SEG) = c^3/(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 asF_GW = (c^3/(32πG)) ⟨ dot(h^TT_ij) dot(h^TT_ij) ⟩ = (1/(8 k_SEG)) ⟨ dot(h^TT_ij) dot(h^TT_ij) ⟩,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 e^A_ij e^B_ij = 2 δAB, one has the algebraic contractiondot(h^TT_ij) dot(h^TT_ij) = 2( dot(h_+)^2 + dot(h_x)^2 ),so the polarization-resolved flux can also be written asF_GW = (c^3/(16πG)) ⟨ dot(h_+)^2 + dot(h_x)^2 ⟩ = (1/(4 k_SEG)) ⟨ dot(h_+)^2 + dot(h_x)^2 ⟩.The note emphasizes that the “1/(8 k_SEG)” and “1/(4 k_SEG)” 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 F_GW (or t_GWμν) 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 k_SEG: it anchors the Isaacson flux prefactor to k_SEG without mixing it with the horizon-sector LHS objects of the triptych.