The Anti-Arrow of Time: Reversibility, Depth, and the Paradox of Robustness

In this work, the origin of the Arrow of Time is explored within the framework of Timeless Correlation Theory (TCT), based on the relational Page–Wootters formalism. In this operational approach, time is represented by a finite “System–Clock” setup, and irreversibility is treated as a property of concrete time-reversal protocols (Loschmidt echoes) in the presence of noise. It is shown that the breakdown of quantum reversibility is driven not only by thermodynamic entropy, but also by dynamic control fragility — the accumulation of small systematic errors in a finite number of control operations. Key results: A new rule for quantum control (The N–rule). The reversibility of the Loschmidt echo is comparatively robust to incoherent (thermodynamic) noise, yet catastrophically sensitive to systematic control errors. This fragility is governed primarily by the number of operations N, rather than by the total evolution time τ, which gives the Arrow of Time a fundamentally operational character. Dynamic Robustness Mode. A paradoxical regime is identified in which coarsening the clock step (larger Δt at fixed N) slows down the decay of the Loschmidt echo and makes the reversal protocol more robust. This suggests a nontrivial strategy for error mitigation in quantum control and quantum computing. Temporal budget concept. The work introduces the notion of a finite “temporal budget” for noisy clocks — a limited entropic resource of System–Clock correlations. Within TCT, thermodynamic irreversibility arises when this budget is exhausted due to accumulated decoherence. Taken together, the results unify thermodynamic and dynamic aspects of time into a single relational picture: thermodynamic noise sets the size of the available temporal budget, while dynamic control fragility determines the rate at which it is spent. The work should be viewed as a preliminary theoretical framework and remains under active development.---Version notes (v2): Minor revision.