Experience Fields: Path-Dependent Algebraic Structures for Learning Systems

We introduce the Experience Field \((\mathcal{X}, +, \cdot)\), an algebraic structure where elements carry computational history in the form of \((v, a, \lambda)\) triples — value, age, and learning rate. Unlike classical number systems where arithmetic depends only on operand values, Experience Field arithmetic incorporates the accumulated experience of each operand. We prove fundamental structural theorems: age monotonicity under learning (Theorem 1), stability bounds in \((0,1]\) (Theorems 3–4), momentum non-negativity (Theorem 5), and value additivity (Theorem 9). The learning rate \(\lambda\) governs adaptation dynamics: \(\lambda \to 0\) yields ossification, constant \(\lambda\) produces eternal adaptivity, and growing \(\lambda\) signals instability. All results are formally verified in Lean 4 with 12 theorems, 12 hypotheses, and 8 axioms. The framework provides a foundation for modeling path-dependent phenomena in AI training, material fatigue, evolutionary dynamics, and financial crisis propagation.