We present Intuitionistic Combinatorial Proofs (ICPs), a concrete geometric semantics of intuitionistic logic based on the principles of the second author's classical Com-binatorial Proofs. An ICP naturally factorizes into a linear fragment, a graphical abstraction of an IMLL proof net (an arena net), and a parallel contraction-weakening fragment (a skew fibration). ICPs relate to game semantics, and can be seen as a strategy in a Hyland-Ong arena, generalized from a tree-like to a dag-like strategy. Our first main result, Polynomial Full Completeness, is that ICPs as a semantics are complexity-aware: the translations to and from sequent calculus are size-preserving (up to a polynomial). By contrast, lambda-calculus and game semantics incur an exponential blowup. Our second main result, Local Canonicity, is that ICPs abstract fully and faithfully over the non-duplicating permutations of the sequent calculus, analogously to the first and second authors' recent result for MALL.