Horizon Thermodynamics as a Dark-Sector Regulator: A Derived Interaction and Late-Time Growth Suppression

This record presents Paper A of a two-paper investigation of the cosmological apparent horizon as a dynamical thermodynamic screen. The paper derives a late-time vacuum-to-cold-dark-matter interaction from the rate at which expansion sweeps vacuum energy across the growing apparent horizon. In a spatially flat matter-plus-vacuum background, the sweep rate is ṀΛ = (9/4) ΩΛ(1 − ΩΛ)c³/G, which is independent of H0. Comparison with the marginal-surface Misner–Sharp rate benchmark c³/(2G) identifies the algebraic crossings ΩΛ = 1/3 and ΩΛ = 2/3 and the dimensionless sweep diagnostic S = (9/2)ΩΛΩm = |dq/d ln a|. The continuously processed branch produces the dark-sector interaction Q = (9/2) β Ieff Hρcrit ΩΛΩm, with Ieff = (1 − q)/2 and the conditional channel benchmark β = 1/12. The source-reduced microscopic scheduler and screening polynomial x(q) = (1 − q)(2 − q)/6 are developed in the companion Paper B. The interaction is implemented in a modified CLASS Boltzmann solver using a CDM-rest-frame transfer and the fiducial smooth-source closure fclust = 0. The matched non-ladder likelihood combines Planck 2018 TT, TE, EE, low-ℓ polarization and lensing, DESI DR2 BAO, Pantheon+ shape information, and DES Y3. No SH0ES or other local-distance-ladder likelihood enters the Paper-A inference. For the fixed interaction, the sampled input normalization is H0,in = 67.626 ± 0.243 km s⁻¹ Mpc⁻¹, with Ωm = 0.325 ± 0.003 and S8 = 0.775 ± 0.006. Direct reconstruction of the integrated endpoint background gives H0,phys ≡ cH(a = 1) = 68.480 ± 0.245 km s⁻¹ Mpc⁻¹. In the matched ΛCDM chain, H0,in and H0,phys coincide at 68.808 ± 0.247 km s⁻¹ Mpc⁻¹, with Ωm = 0.296 ± 0.003 and S8 = 0.790 ± 0.005. Dedicated minimization gives Δχ²min = −8.4 without adding a continuously fitted parameter. As a held-out prediction, removing DES Y3 entirely gives S8 = 0.782 ± 0.008, only 0.33σ from the subsequently compared DES Y3 result. The principal signature is therefore not a large change in the background expansion: the physical endpoint H(0) remains close to its matched ΛCDM value, while the interaction produces broadband suppression of matter clustering and a characteristic density–velocity growth split. The protected over-capacity memory, endpoint comparator, calibration holonomy, and conditional local-ladder readout are deliberately excluded from the Paper-A likelihood. Those questions are developed separately in Paper B, so the background, perturbation, and growth results in this record do not depend on the operational calibration hypothesis.