This paper presents a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is believed to be relevant to the strong coupling limit of the $E_8 \times E_8$ heterotic string. The authors address the gauge and gravitational anomalies that arise at early stages and propose a refinement of the standard Green-Schwarz mechanism for their cancellation. This leads to the conclusion that the gauge group is a copy of $E_8$ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and provides new tests of the hypothesis that there is a consistent quantum $M$-theory with eleven-dimensional supergravity as its low-energy limit. The paper also discusses the classical and quantum consistency of the theory, highlighting the need for a cutoff to handle divergences and the peculiarities in constructing the classical theory. The authors provide a detailed derivation of the gauge transformation law for the three-form $C$ and the behavior of $G$ near the boundary, and use these to analyze gauge and gravitational anomalies. They predict the structure of the anomaly polynomial $\widehat{I}_8$ and verify two out of three predictions. The paper concludes with a construction of the locally supersymmetric Lagrangian, demonstrating the existence of the supersymmetric coupling of the vector multiplet on the boundary to the supergravity multiplet in the bulk and showing the limits of the classical construction by identifying infinities that arise at higher orders in $\kappa$.This paper presents a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is believed to be relevant to the strong coupling limit of the $E_8 \times E_8$ heterotic string. The authors address the gauge and gravitational anomalies that arise at early stages and propose a refinement of the standard Green-Schwarz mechanism for their cancellation. This leads to the conclusion that the gauge group is a copy of $E_8$ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and provides new tests of the hypothesis that there is a consistent quantum $M$-theory with eleven-dimensional supergravity as its low-energy limit. The paper also discusses the classical and quantum consistency of the theory, highlighting the need for a cutoff to handle divergences and the peculiarities in constructing the classical theory. The authors provide a detailed derivation of the gauge transformation law for the three-form $C$ and the behavior of $G$ near the boundary, and use these to analyze gauge and gravitational anomalies. They predict the structure of the anomaly polynomial $\widehat{I}_8$ and verify two out of three predictions. The paper concludes with a construction of the locally supersymmetric Lagrangian, demonstrating the existence of the supersymmetric coupling of the vector multiplet on the boundary to the supergravity multiplet in the bulk and showing the limits of the classical construction by identifying infinities that arise at higher orders in $\kappa$.