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 temperature
T(α_H) = ħ α_H / (2π k_B c)
and the Einstein (Wald/Bekenstein–Hawking) entropy density
S_grav/A = k_B c^3 / (4G ħ),
the algebraic identity
T(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)| << 1.

Route B (Kodama/Hayward; branch-pinned):
T_K = − ħ κ_K / (2π k_B c)  ⇔  α_H = −κ_K on the outer expanding branch κ_K ≤ 0 (so T_K ≥ 0).
Route B is treated as a projection identity (consistency repackaging), not as an independent derivation of Einstein dynamics.

Branch/sign policy (Route B).
For Route B we work on the outer expanding branch where κ_K ≤ 0 so T_K ≥ 0. When orientation/branch sign conventions are not explicitly pinned, magnitudes are reported by default and no downstream conclusion is allowed to depend on sign.

What is not claimed.
• No identification of gravitational entropy with entanglement entropy
• No non-equilibrium or entropy-production terms
• No universality beyond 4D Einstein gravity
• No cross-sector identification of distinct LHS objects
• No claim that Route B “derives GR from thermodynamics” (it is a projection identity / consistency repackaging)

Within the HRP suite, this paper establishes the FLRW apparent-horizon normalization ledger that complements the stationary black-hole and local-Rindler sector cards. Across sectors, k_SEG functions as a reusable constants-explicit slot, while each sector’s physical LHS object remains strictly typed and non-identified.

Version note (v2): Added the suite-wide “Interface Contract” (shared across the triptych + addenda) to enforce typed-LHS discipline, operator-form separation (dot-form here vs delta-form in BH/Rindler), coefficient-only UHRA comparison, magnitude-first sign/branch policy, and an Entropy Firewall (S_grav is Wald/Bekenstein–Hawking gravitational entropy only; no identification with entanglement/generalized entropy). Strengthened the dot-form/chart discipline by explicitly tagging FLRW as a rate-form (power-like) statement with dot = ∂/∂t in the SI time chart, and by adding an explicit chart-map crosswalk to the length chart that is shown for comparison but not used (no silent chart switches). Recast Route A and Route B as fail-closed magnitude-first primary identities (boxed |T_A Sdot| = Qdot_A^(mag) and |T_K Sdot| = Qdot_B^(mag)), moving signed equalities to “optional signed convention” subsections where the sign convention is pinned operationally (Qdot^(signed) > 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.