Write \[ E_a(f)=\int_0^1f(x)\,dx-Q_a(f). \] This functional kills affine functions. Twice integrating \(f''\), followed by Fubini, gives \[ E_a(f)=\int_0^1K_a(t)f''(t)\,dt, \quad K_a(t)=\frac{(1-t)^2}{2}-\frac{(a-t)_++(1/2-t)_++(1-a-t)_+}{3}. \] Hence \[ C(a)=\int_0^1|K_a(t)|\,dt. \tag{1} \] The kernel is symmetric about \(1/2\). On the left half, \[ K_a(t)=\begin{cases} t^2/2,&0\leq t\leq a,\\ (3t^2-2t+2a)/6,&a\leq t\leq1/2. \end{cases} \tag{2} \] For \(a\leq1/6\), the second quadratic has roots \[ r_\pm(a)=\frac{1\pm\sqrt{1-6a}}3. \] The upper root crosses \(1/2\) at \(a=1/8\), and the roots meet at \(a=1/6\). Put \(u=\sqrt{1-6a}\). Splitting (1) at these roots gives \[ C(a)=\begin{cases} (1+32u^3-6u^4)/648,&0\leq a\leq1/8,\\ (3-24u^2+64u^3-6u^4)/648,&1/8\leq a\leq1/6,\\ (-8a^2+8a-1)/24,&1/6\leq a\leq1/2. \end{cases} \tag{3} \] The derivatives of the three branches are \[ \frac{u(u-4)}9, \qquad \frac{u^2-8u+2}{9}, \qquad \frac{1-2a}{3}. \tag{4} \] The first branch decreases. In the middle branch the derivative changes from negative to positive at \(u_*=4-\sqrt{14}\). The last branch increases before its right endpoint. The branches agree at \(1/8\) and \(1/6\), so the unique global minimizer and sharp error are \[ a_*=\frac{8\sqrt{14}-29}{6} =0.155543182365255180778331643089\ldots, \] \[ C(a_*)=\frac{3355-896\sqrt{14}}{648} =0.003819415818747713760742493587\ldots. \] At \(a_*\), the left-half kernel zeros are \[ \rho_-=\frac{\sqrt{14}}3-1, \qquad \rho_+=\frac{5-\sqrt{14}}3. \] Equality in the \(L^\infty\)-\(L^1\) bound occurs exactly when \[ f''(t)=\varepsilon\operatorname{sgn}K_{a_*}(t) \quad\text{almost everywhere}, \qquad \varepsilon\in\{-1,1\}. \] Thus every extremizer has the form \[ f(x)=\varepsilon\int_0^x(x-t)\operatorname{sgn}K_{a_*}(t)\,dt+\beta x+\gamma, \qquad \beta,\gamma\in\mathbb R. \] These piecewise-quadratic functions attain the displayed sharp error.

The replay substitutes \(a=(1-u^2)/6\), integrates the two possible negative intervals, differentiates the resulting branches, and checks the minimizing node, sharp constant, and optimal kernel roots with exact arithmetic. Its stable output has SHA-256 digest `e4532299ba7b0a9ad65de1d9c060e8d1f8a8bd81db448521235a9e971da0e692`.