Horizon Response Principle (HRP) Sector III: FLRW Apparent Horizons

This preprint is Sector III (FLRW Apparent Horizons) of the Horizon Response Principle (HRP) triptych (BH / Local Rindler / FLRW). It provides a constants-explicit, sector-typed normalization card for 4D Einstein–Hilbert gravity in the FLRW apparent-horizon setting. Scope:• 4D Einstein–Hilbert gravity only• FLRW spacetimes and their apparent horizons• Reversible (near-equilibrium) area channel only• All constants explicit (G, c, ħ, k_B)• No new dynamics or modified field equations Sector typing (FLRW sector).In this sector the natural left-hand side (LHS) object is a rate-form projected flux (power-like), not a BH Hamiltonian/Noether variation and not a local-Rindler boost-energy heat flux. The sector is therefore reported in dot-form (SI-time rates), not in delta-form. Chart pin (this matters here).This paper uses the SI time chart x^0 = t with overdot meaning ∂/∂t. The apparent-horizon radius R_A is a physical length. No silent chart switches. Normalization backbone.Using the acceleration temperatureT(α_H) = ħ α_H / (2π k_B c)and the Einstein (Wald/Bekenstein–Hawking) entropy densityS_grav/A = k_B c^3 / (4G ħ),the algebraic identityT(α_H)(S_grav/A) = α_H c^2 / (8πG)exposes a classical coefficient skeleton that HRP packages via k_SEG := 4πG / c^3. FLRW bindings (two standard routes).This sector records two standard bindings of the abstract acceleration scale α_H to the apparent horizon, and shows that each yields a reversible apparent-horizon rate identity within its stated scope: Route A (quasi-static temperature):T_A = ħ c / (2π k_B R_A) ⇔ α_H = c^2 / R_A,used together with the quasi-static regime pin |dot(R_A)/(2 H R_A)| 0 defined as net energy entering R ≤ R_A; Route B pinned to the outer branch with kappa_K ≤ 0 so T_K ≥ 0). Made the Route A acceleration binding alpha_H = c^2/R_A explicit at first use (in addition to the later UHRA-binding summary) and clarified the provenance of the projected flux rate from the unified-first-law / Cai–Kim apparent-horizon projection, fixing c-placement with rho treated as energy density (J m^-3). Strengthened zero-division and vanishing-factor cautions (H = 0, f = 0) with explicit cancel-before-specialize logic and an explicit continuity pin. Added a Planck-length convention and a compact rate-form UHRA skeleton summary; the core coefficient skeleton and the Einstein evolution equation are unchanged.