We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every
p ≥ 2 p\geq 2 p ≥ 2 , assuming that the
( p − 1 ) (p-1) ( p − 1 ) th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a
p p p th-order method that finds a point with tangent residual at most
ε \varepsilon ε in
O ~ ( ε − 2 / ( 3 p − 1 ) ) \widetilde{O}(\varepsilon^{-2/(3p-1)}) O ( ε − 2/ ( 3 p − 1 ) ) oracle calls. This improves all prior upper bounds for
p p p th-order methods: in particular, it improves the previous best-known
O ~ ( ε − 1 / p ) \widetilde{O}(\varepsilon^{-1/p}) O ( ε − 1/ p ) tangent-residual complexity as well as the classical
O ( ε − 2 / ( p + 1 ) ) O(\varepsilon^{-2/(p+1)}) O ( ε − 2/ ( p + 1 ) ) bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of
Ω ( ε − 2 / ( 3 p − 1 ) ) Ω(\varepsilon^{-2/(3p-1)}) Ω ( ε − 2/ ( 3 p − 1 ) ) for every deterministic algorithm in the
p p p th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on
ε \varepsilon ε , up to logarithmic factors, for all
p ≥ 2 p\geq2 p ≥ 2 .