Collinear columns identify a fitted mean, not its coefficients

Both coefficient vectors are distinct, but $X_n{\theta }_1={\mathbf{1}}_n=X_n{\theta }_2$. Thus the asserted injectivity fails, and the observation distributions under ${\theta }_1$ and ${\theta }_2$ are identical because the additive noise law is common.

Let $T(Y)\in {\mathbb{R}}^2$ be any measurable coefficient estimator. For every $a\in {\mathbb{R}}^2$, completing squares gives $\mathrm{\parallel }a-{\theta }_1{\mathrm{\parallel }}_2^2+\mathrm{\parallel }a-{\theta }_2{\mathrm{\parallel }}_2^2=2{\parallel a-(\frac{1}{2},\frac{1}{2})\parallel }_2^2+1\ge 1.$ If either coefficient risk is infinite, their maximum is already at least $1\mathrm{/}2$. Otherwise both squared losses are integrable under the common observation law. Their sum is integrable by [F3], so linearity and monotonicity in [F3] show that the sum of the two risks is at least $1$. Thus their maximum is at least $1\mathrm{/}2$ for every $n$.

The common fitted mean $\mu ={\mathbf{1}}_n$ lies in $\mathrm{col}(X_n)$, which has rank $1$. By [F2], its projected least-squares fit has expected squared fitted-value risk ${\sigma }^2$, or ${\sigma }^2\mathrm{/}n$ per observation by [F4]. This vanishes as $n$ grows while the coefficient-risk lower bound in step 1.2 stays at least $1\mathrm{/}2$.