In any joint law with these second moments, \[\|\phi _0-\phi _3\|_{L^2}=\sqrt {19/5}, \qquad \|\phi _i-\phi _{i+1}\|_{L^2}=\sqrt {1/5} \quad (i=0,1,2).\] But the first difference is the sum of the three consecutive differences. The triangle inequality would give \(\sqrt {19/5}\le 3\sqrt {1/5}\), contradicting \(19>9\). Any proposed positive-semidefinite covariance is a Gram matrix of real vectors and obeys the same norm inequality. Thus the obstruction already rules out semidefinite completion, without a nondegeneracy assumption.

The angular description gives another view of the same obstruction. Unit Gram vectors have consecutive angles \(\alpha =\arccos (9/10)\) and endpoint angle \(\pi -\alpha \). For unit vectors with successive angles \(\beta ,\gamma \) and \(\beta +\gamma \le \pi \), orthogonal decomposition along the middle vector and Cauchy–Schwarz give \[\langle u,w\rangle \ge \cos \beta \cos \gamma -\sin \beta \sin \gamma =\cos (\beta +\gamma ).\] Since arccosine decreases, the endpoint angle is at most \(\beta +\gamma \). If the sum exceeds \(\pi \), that bound is automatic. Iteration gives endpoint angle at most \(3\alpha \). But \(\alpha <\pi/4\), because \(9/10>1/\sqrt 2\); hence \(3\alpha <\pi -\alpha \), again a contradiction.