Boundary Language as a Unified Physical Framework:\\ From Scattering Phase, GHY Boundary Term to Modular Flow and Generalized Entropy as a Single Structure

This paper proposes and systematizes the concept of ``boundary language,'' rewriting physical theories as algebraic--geometric structures about ``what is allowed to be exchanged'' on causal cut surfaces, with bulk field theory being merely one realization of this structure. Taking the boundary observable algebra and boundary state as fundamental objects, we unify three seemingly independent theoretical paradigms into the same framework: (1) in scattering theory, the spectral shift function, total scattering phase, and Wigner--Smith group delay; (2) in general relativity, the Gibbons--Hawking--York (GHY) boundary term and Brown--York quasilocal energy; (3) in operator algebras, the Tomita--Takesaki modular flow and relative entropy monotonicity. The core idea is: time is not a parameter of flow within the bulk that is given a priori, but rather a unified translation parameter generated by ``what is allowed in terms of flux balance and information monotonicity'' in the ``boundary language;'' all observable delays, energies, and generalized entropy variations are different projections of the same boundary structure. Mathematically, we formalize this framework as three ``boundary language axioms'': (A1) Conservation and Flux Axiom, viewing the boundary as a balancing interface for energy, charge, and information flux; (A2) Time Generation Axiom, viewing the one-parameter {}^\ast-automorphism group defined on the boundary and its generator as the source of time scale; (A3) Monotonicity and Consistency Axiom, represented by relative entropy monotonicity and its geometric forms (quantum focussing, quantum null energy condition, etc.), excluding supercausality and negative entropy transport. In the scattering realization, we prove: the boundary language satisfying A1--A3 necessarily induces the scale identity on a well-posed short-range scattering system $ \kappa(\omega)=\varphi'(\omega){\pi}=\rho_{rel}(\omega)=1{2\pi}trQ(\omega), where \varphi(\omega) is the total scattering half-phase, \rho_{rel}(\omega) is the relative state density, and Q(\omega)$ is the Wigner--Smith group delay operator. This identity unifies the phase gradient, spectral shift density, and group delay trace as a single boundary object called ``time scale.'' In the gravity realization, we show: the GHY boundary term and the Brown--York quasilocal energy positivity are necessary conditions for the boundary language A1--A2 on the geometric side; thus ADM time, proper time, and Killing time are restated as translations generated by the boundary Hamiltonian. In the operator algebra realization, we provide the canonical model of boundary language through Tomita--Takesaki modular theory and implement A3 through relative entropy monotonicity, thereby characterizing the ``time arrow'' as the monotonic evolution of relative entropy with modular time under a natural class of conditions. Finally, through three model classes---one-dimensional potential scattering, static black hole exterior regions, and Rindler wedges---we demonstrate how the boundary language produces experimentally observable time delays, quasilocal energy, and Unruh temperature, and give several testable spectral--geometric--information theoretical predictions. Detailed appendices provide proofs of key propositions such as the scattering--spectral shift--group delay scale identity, the variational completeness of the GHY term, and relative entropy monotonicity, and introduce the ``error geometry'' framework of finite-order Euler--Maclaurin and Poisson discipline to ensure a controllable mapping from boundary readings to experimental data.