The Irreversibility Field (I-Field):A Classical Framework for Fundamental Irreversibility in Physics

 We present the I-field: a classical scalar field minimally coupled to matter whose equation of motion contains an explicit time-asymmetric dissipation term, derived from the Euler-Lagrange-Rayleigh (ELR) formalism [@rayleigh1877]. The field does not modify the gravitational sector: Einstein's field equations are unchanged, and the total stress-energy tensor of matter plus I-field is covariantly conserved. In the limit $\gamma \to 0$ the theory reduces exactly to standard classical field theory.  The dissipation term $\gamma u^\mu \partial_\mu \mathcal{I}$, where $u^\mu$ is the four-velocity of the cosmological rest frame and $\gamma > 0$ is a coupling constant, is odd under time reversal while every term derived from a Lagrangian is even. This explicit breaking of time-reversal symmetry at the level of the field equation --- rather than through boundary conditions or statistical postulates --- has three consequences derived as theorems within the framework:   1. The I-field carries a covariant entropy production density   $\sigma_{\mathcal{I}} = \gamma\dot{\mathcal{I}}^2 \geq 0$   pointwise, establishing the second law of thermodynamics as   a field-theoretic identity rather than a postulate.     2. The energy transferred from matter to the I-field is strictly   non-negative, providing a microscopic account of dissipation   without invoking a heat bath or environment.     3. The preferred time direction is globally well-defined,   identified with the cosmological rest frame in which the   cosmic microwave background is isotropic [@fixsen2009].     The theory is self-contained and makes no modifications to the gravitational sector. The framework provides a minimal, classical extension of standard field theory in which irreversibility is fundamental rather than