The black hole information paradox—a fundamental conflict between quantum unitarity and semi-classical gravity—remains unresolved within the framework of quantum gravity. This work builds a conceptual bridge between this modern dilemma and Zeno’s ancient paradox of motion, emphasizing that they all rely on limit processes to reconcile infinite divisibility with finite physical outcomes. Although previous researches have established information conservation in black hole evaporation by copying wormholes, this study transcends mere unitarity verification to answer a deeper question: How do we quantify the statistical evolution and inherent irreversibility of Hawking radiation?

Methods :

i) Gravitational path integral formalism : We compute Rényi entropies S_n = \dfrac11-n \ln \textTr(\rho^n) by analytic continuation ( n \to 1 ), incorporating both disconnected geometries (Hawking saddles) and connected replica wormhole saddles.

ii) JT gravity model : A two-dimensional Jackiw-Teitelboim (JT) gravity coupled to a conformal bath is analyzed, with End-of-the-World (EOW) branes modeling early Hawking radiation states.

iii) Modular thermodynamics : Modular entropy S_m and entanglement capacity C_n are defined as:

　　　　　　　　　　　 S_m = \log\textTr(\rho^n) - n\partial_n\log\textTr(\rho^n), \quad C_n = n^2\partial_n^2\log\textTr(\rho^n),mirroring thermodynamic entropy and heat capacity.

Key results :

i) Page curve restoration : The von Neumann entropy transitions from linear growth (Hawking phase) to zero (wormhole-dominated phase) at the Page time: t_\textPage \propto S_\textBH/(n\kappa c), \quad S_\textBH = A/(4G_N), indicating faster entropy reduction for higher n.

ii) Replica wormhole mechanism : The gravitational path integral reveals a topological transition:

　　　　　　　　　　　 \textTr(\rho^n) \approx \dfrackZ_1^n + k^n Z_n(kZ_1)^n \quad \Rightarrow \quad \textDominance shift at k \sim \mathrme^S_0.

Quantum non-cloning constraints: A reversibility paradox arises: while n-copies -copies of a known state allow full reconstruction, the non-cloning theorem forbids replication of unknown states. This necessitates the a priori existence of multiple replicas to ensure consistency.

iii) Connection to Zeno’s paradox : The resolution of Zeno’s paradox through Newtonian limits ( v = \lim\limits_\Delta t \to 0 \Delta x / \Delta t ) parall) parallels the replica trick’s role in black hole physics:

iv) Infinite-to-finite mapping : Both employ limit operations to compress divergent processes (infinite steps or entropy growth) into finite observables.

Discrete vs. continuous : Challenge classical spacetime continuity, hinting at holographic or discrete quantum spacetime.

Irreversibility and generalized second law : The relative entropy S(\rho \| \sigma) , defined as:

　　　　　　　　　　　　　　　 S(\rho \| \sigma) = \textTr(\rho \log \rho) - \textTr(\rho \log \sigma),monotonically increases during the Hawking-to-wormhole transition, establishing statistical irreversibility in modular space. This generalized second law persists despite microscopic unitarity, analogous to thermodynamic irreversibility in closed systems with the formula: \left(S_n-S_\rho\right)-n\left(\left\langle H\right\rangle_n-\left\langle H\right\rangle_\rho\right)\geqslant 0 .

Conclusion :

Our results address the black hole information paradox and the irreversible problem through three pillars:

i) Unitarity : Replica wormholes enforce unitarity via non-perturbative spacetime topology changes.

ii) Modular thermodynamics : A framework linking entanglement entropy, relative entropy, and irreversibility.

iii) Replica priority principle : Multiple radiation replicas exist a priori, circumventing quantum cloning constraints.

This work transcends mere information conservation, advancing into the second layer of black hole physics: Characterizing the irreversible evolution of Hawking radiation in modular space.