The finite sectional reaction vector field is locally Lipschitz

Define the extension $F$ on all real augmented states by evaluating the same finite event polynomials, without the algorithm's population-validity test. On nonnegative states it agrees with the algorithm's right-hand side; the extension need not describe physical events off that orthant. On a ball $\mathrm{\parallel }n{\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{\parallel }\tilde{n}{\mathrm{\parallel }}_{\mathrm{\infty }}\le R$, every quadratic monomial obeys $\mathrm{\mid }{n}_i{n}_j-{\tilde{n}}_i{\tilde{n}}_j\mathrm{\mid }\le R\mathrm{\mid }{n}_i-{\tilde{n}}_i\mathrm{\mid }+R\mathrm{\mid }{n}_j-{\tilde{n}}_j\mathrm{\mid }$ by adding and subtracting $n_i{\tilde{n}}_j$. Linear monomials obey the same bound with coefficient $1$.

Each coordinate of $F(n,L)-F(\tilde{n},\tilde{L})$ is a finite weighted sum of these differences; the reservoir coordinate is independent of $L$. Summing the finitely many coefficient bounds gives a finite constant $C_R$ with $\mathrm{\parallel }F(n,L)-F(\tilde{n},\tilde{L}){\mathrm{\parallel }}_2\le C_R\mathrm{\parallel }(n,L)-(\tilde{n},\tilde{L}){\mathrm{\parallel }}_2$ on the ball.