arXiv:2611.12967v1 Announce Type: cross Abstract: We present a constrained mixed variational formulation for incremental irreversible thermodynamics. A single augmented action combines a linearized contact-increment penalty, equilibrium and dissipative Fenchel--Young defects, a storage--transport functional, and multipliers enforcing the constitutive increments. Under frozen-coefficient and boundary assumptions, stationarity yields the linearized contact condition, the Tisza relation, the Onsager flux--force law, and the implicit balance equation. An exact-elimination theorem shows that the contact and multiplier fields can be reconstructed algebraically from a solution of the reduced weak balance problem. We also prove off-shell invariance of the equilibrium--storage--multiplier sector under an equilibrium-sector partial Legendre transformation; invariance of the full augmented action is not claimed. For Fourier heat conduction, we derive a weighted drift identity and a per-step energy-drift bound for the energy-representation three-term incremental functional. In one spatial dimension, a small-time-step estimate yields a uniform $H^1$ bound and a discrete $L^2(0,T_f;H^2)$ bound, leading to cumulative $O(\delta t)$ drift over fixed time intervals. A conservative mixed discontinuous Galerkin (DG0)--Raviart--Thomas (RT0) discretization with fully implicit conductivity $\kappa(T)=1+T^2$ preserves discrete energy to roundoff. A separate lumped-mass piecewise-linear benchmark reproduces the predicted finite-step drift.