The Intrinsic Operational Gradient Theorem

We formalize a structural principle implicit throughout mathematics but rarely stated explicitly: composable operations induce intrinsic gradients of difficulty. Forward construction and reverse reconstruction are generically asymmetric, even in purely abstract settings. This asymmetry does not arise from physical time, probability, or specific computational models, but from the combinatorics of operations themselves. We present the Intrinsic Operational Gradient Theorem (IOGT), prove it under minimal assumptions, relate it to established mathematical structures (notably Morse theory and Kolmogorov complexity), and explain why its foundational role has historically remained implicit. The theorem clarifies why notions such as difficulty, irreversibility, and attractors are unavoidable across mathematical practice.