Bound R-Squared for Individual and Combined OLS Regressions

What range can R-squared take in ordinary least squares, and why? On the same response and observations, suppose the regressions `y ~ x1` and `y ~ x2` have R-squared values 0.1 and 0.2. What is the possible range for `y ~ x1 + x2`? ### Constraints & Assumptions For the main question, use in-sample OLS with an intercept, a nonconstant response, and the usual centered definition `R² = 1 - SSE/SST`. Compare models fitted on the same rows. Also explain what changes without an intercept or when scoring on different data. ### Clarifying Questions Is an intercept included? Are the data and response identical across fits? Is this ordinary R-squared or adjusted R-squared? Is evaluation in sample? ### What a Strong Answer Covers Projection or nested-model reasoning for the bounds, the role of predictor correlation, and an explanation of why the two individual scores do not determine the combined score. ### Follow-up Questions Can the joint model explain all variation even when each predictor has a small individual score? Can adding a predictor reduce ordinary training R-squared? Under what conditions can R-squared be negative or undefined?