This paper presents a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is relevant to the strong coupling limit of the $ E_8 \times E_8 $ heterotic string. Gauge and gravitational anomalies enter early and require a refinement of the Green-Schwarz mechanism for their cancellation. This uniquely determines the gauge group to be $ E_8 $ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and leads to new tests of the hypothesis that there is a consistent quantum M-theory with eleven-dimensional supergravity as its low energy limit. The paper explores the coupling of ten-dimensional vector multiplets on the boundary of an eleven-manifold to the eleven-dimensional supergravity multiplet in the bulk. The supergravity action in the bulk is given by $ -\frac{1}{2\kappa^{2}}\int_{M^{11}}d^{11}x\sqrt{g}R $, while the action on the boundary is $ -\frac{1}{4\lambda^{2}}\int_{M^{10}}d^{10}x\sqrt{g}\mathrm{t r}F^{2} $. The dimensionless number $ \eta = \lambda^{6}/\kappa^{4} $ is determined by analyzing gravitational and gauge anomalies, leading to $ \eta = 128\pi^{5} $ and $ \lambda^{2} = 2\pi(4\pi\kappa^{2})^{2/3} $. Anomalies in ten dimensions are described by a twelve-form $ I_{12}(R, F_1, F_2) $, which is a sixth-order polynomial in the Riemann tensor and field strengths. The Green-Schwarz mechanism in ten dimensions involves a two-form field B whose field strength H obeys $ dH = I_4 $. In eleven dimensions, the analogous field is a three-form C, and the Bianchi identity is modified to include a delta function term at the boundary. This leads to a correction to the Bianchi identity and a new interaction term that cancels anomalies. The paper also discusses the structure of the Lagrangian and the supersymmetry variations of the fields, showing that the theory is consistent at the quantum level. The analysis reveals that the gauge coupling is related to the gravitational coupling by $ \lambda^{2} = 2\pi(4\pi\kappa^{2})^{2/3} $. The paper concludes that the eleven-dimensional supergravity on a manifold with boundary is consistent with the ten-dimensional $ E_8 \times E_8 $ heterotic string theory.This paper presents a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is relevant to the strong coupling limit of the $ E_8 \times E_8 $ heterotic string. Gauge and gravitational anomalies enter early and require a refinement of the Green-Schwarz mechanism for their cancellation. This uniquely determines the gauge group to be $ E_8 $ for each boundary component, fixes the gauge coupling constant in terms of the gravitational constant, and leads to new tests of the hypothesis that there is a consistent quantum M-theory with eleven-dimensional supergravity as its low energy limit. The paper explores the coupling of ten-dimensional vector multiplets on the boundary of an eleven-manifold to the eleven-dimensional supergravity multiplet in the bulk. The supergravity action in the bulk is given by $ -\frac{1}{2\kappa^{2}}\int_{M^{11}}d^{11}x\sqrt{g}R $, while the action on the boundary is $ -\frac{1}{4\lambda^{2}}\int_{M^{10}}d^{10}x\sqrt{g}\mathrm{t r}F^{2} $. The dimensionless number $ \eta = \lambda^{6}/\kappa^{4} $ is determined by analyzing gravitational and gauge anomalies, leading to $ \eta = 128\pi^{5} $ and $ \lambda^{2} = 2\pi(4\pi\kappa^{2})^{2/3} $. Anomalies in ten dimensions are described by a twelve-form $ I_{12}(R, F_1, F_2) $, which is a sixth-order polynomial in the Riemann tensor and field strengths. The Green-Schwarz mechanism in ten dimensions involves a two-form field B whose field strength H obeys $ dH = I_4 $. In eleven dimensions, the analogous field is a three-form C, and the Bianchi identity is modified to include a delta function term at the boundary. This leads to a correction to the Bianchi identity and a new interaction term that cancels anomalies. The paper also discusses the structure of the Lagrangian and the supersymmetry variations of the fields, showing that the theory is consistent at the quantum level. The analysis reveals that the gauge coupling is related to the gravitational coupling by $ \lambda^{2} = 2\pi(4\pi\kappa^{2})^{2/3} $. The paper concludes that the eleven-dimensional supergravity on a manifold with boundary is consistent with the ten-dimensional $ E_8 \times E_8 $ heterotic string theory.