De Sitter Horizon Freeze-Out and the Physical Threshold of Cosmic Einselection: From Planetary Radar to Post-Biological Horizon Saturation

Paper IV in the Time-Symmetric Quantum Darwinism (TSQD) Series. In the preceding papers of this series (doi:10.5281/zenodo.23572180, doi:10.5281/zenodo.23677536, and doi:10.5281/zenodo.23818446), we formulated Time-Symmetric Quantum Darwinism (TSQD), showing that a self-selecting final boundary condition <Φ*| = S[<Φ*|] on the Future Conformal Boundary (I+) prunes unamplified quantum branches, resolves the Fermi Paradox via the inflationary cavity resonance λ = N ptotal ≈ 1, and preserves local unitarity and the Born rule via the Aharonov–Bergmann–Lebowitz (ABL) Degeneracy Theorem. A central open question remains: what is the exact physical broadcast threshold Rcrit required for a macroscopic pointer state to survive cosmological expansion and lock in <Φ*| on I+? Moreover, how does such a record evade the Dyson–Kleban–Susskind thermalization barrier in asymptotically de Sitter (ΛCDM) spacetime? Here we calculate the radiative transfer, intergalactic Thomson optical depth (τIGM ≈ 8.1 × 10−4), plasma frequency evolution, and Gibbons–Hawking horizon freeze-out (TdS = ℏH∞ / 2πkB ≈ 2.2 × 10−30 K) from first principles. We prove the Two-Stage Horizon Selection Theorem (the “Trunk-and-Crown” Theorem), demonstrating that S enforces a bifurcated physical threshold across a hierarchically branched Hilbert space: Stage I Trunk Ignition (Pcrit(I) ∼ 106–107 W): Narrowband planetary radar escapes the galactic ionosphere and prevents local single-mode quantum erasure, selecting a single planetary trunk among N ∼ 1022 stars. Stage II Crown Saturation (Pcrit(II) ∼ 1026 W ≈ 1 L☉): De Sitter fast-scrambling (t* ∼ H∞−1 ln SdS) suppresses partially illuminated sub-branches unless an amplifier illuminates the entire AH = 4πRH2 ≈ 3.2 × 1053 m2 cosmic event horizon with ≥ 1 coherent photon per square meter per second. Finally, we derive the Biological Bootloader Corollary and evaluate the thermodynamically and geometrically capped von Neumann disassembly kinetics, showing that while planetary irradiance and Stefan–Boltzmann waste-heat limits throttle pure exponential replication into a linear mass-ejection regime of Δτswarm ∼ 103–104 yr, this transition still spans a vanishingly brief cosmological interval (Δτ / H∞−1 ∼ 10−6), confining retro-causal anthropic shielding of a terrestrial biosphere to the pre-autonomy window (τ < τaut). Complete Series Links: Paper I: Cosmic Fixed-Point Einselection: A Time-Symmetric Quantum Darwinist Resolution of the Measurement Problem — 10.5281/zenodo.23572180 Paper II: The Cosmic Return Stroke: Resolving the Fermi Paradox via Time-Symmetric Quantum Amplification — 10.5281/zenodo.23677536 Paper III: Foundational Consistency of Time-Symmetric Quantum Darwinism: No-Signaling, Relativistic Covariance, and Born-Rule Recovery — 10.5281/zenodo.23818446 Paper IV: De Sitter Horizon Freeze-Out and the Physical Threshold of Cosmic Einselection: From Planetary Radar to Post-Biological Horizon Saturation (Present Work)