The Mathematics of Evolution: The Price Equation, Natural Selection, and Environmental Change

The candidate solution proves the same Price Representation Theorem as the paper: it defines wNS(i, d\tilde{i}) = W(\tilde{i})\,\delta_i(d\tilde{i}) and wEC(\tilde{i}, di') = w_{\tilde{i}}(di')/W(\tilde{i}), shows w = wEC \circ wNS, and establishes equality of selective terms and environmental terms via the tilted expectation Ẽ[\cdot] = E[U\,\cdot]. These steps correspond exactly to equations (4.1)–(4.4) and their proof in Theorem 4.4 of the paper, including the use of the intermediate measure \tilde{\mu} = W\mu and the identity Ẽ[\cdot] = E[U\cdot] . The candidate’s handling of the W=0 set and light integrability remarks are consistent with the paper’s setup (finite-variance process) and standard measure-theoretic conventions.