Applicability Domain of the Finite-Observation-Window Law, Cosmic Resource Barrier, and Microscopic Origin of Topological Decoherence—TRFT IV

The Topological Relational Field Theory (TRFT) series [1–3] previously elevated the finite observation window to a thermodynamic coordinate and, in TRFT III, proposed a candidate fourth law based on the time-conjugate field \( Φ \). In this paper, within the frameworks of standard quantum estimation and open-system theory, we perform a rigorous independent test and boundary delimitation of this candidate law, and complete three substantive developments of cross-scale significance. The core conclusions are fully testable independently of the framework (dual-track independence). First, we establish the precise domains of kinematics and thermodynamics. By introducing the Scaled Reference Point Relation (SRPR) and the quantum Cramér–Rao bound (CRB), we prove that in the macroscopic reality regime (\( r \) > 1), the CRB strictly takes over statistical precision, and the candidate law degenerates into a conditional constraint on bandwidth; whereas in the extremely short, few-body attosecond limit (\( r \) ≪ 1), the original law again becomes an absolutely tight upper bound, thereby precisely delimiting the valid territory of the law. Second, we reveal the "third class of irreversibility" of the arrow of time (the cosmic resource barrier). The derivation shows that the number of independent measurements required for reverse recovery of decoherence explodes exponentially as \( e^{2\varGamma t} \). Combined with the upper bound on baryonic matter in the observable universe, this gives a resource barrier \( Γt≈91.8 \). Even approaching the Bremermann limit, the barrier is delayed only logarithmically to about 139. This establishes an extremely hard physical limit based on the total capacity of the universe, opening a new paradigm independent of statistical probability and local energy consumption. Third, we reveal the microscopic origin of topological decoherence and the prerequisite for an engineering breakthrough. Within the SSH + Lindblad model, we rigorously prove that uniform environment coupling (\( P=I \)) locks the loss rate due to wavefunction normalization; only by "screening the edges and coupling solely to the bulk region" (\( P ≠ I \)) can the topological decoherence rate be exponentially suppressed by wavefunction overlap (\( {\varGamma }_{\mathrm{topo}}\propto e^{-cN} \)). This mechanism connects with the macroscopic barrier and establishes the core spatial-geometric prerequisite that must be satisfied for topological quantum device design in the NISQ era.