At the Fujita exponent $p=1+2/N$ a strictly positive mild solution of the semilinear heat equation can be built whose initial trace in $L^1$ is zero. The construction uses a logarithmic‑modulated self‑similar ansatz that produces an ancient orbit in Gaussian weighted spaces.

The resulting solution satisfies \[ \|U(t)\|_1\sim C_N(\log(1/t))^{-N/2},\qquad \|U(t)\|_\infty\sim c_Nt^{-N/2}(\log(1/t))^{-N/2}, \] with explicit constants $C_N$ and $c_N$. These rates show that the solution lies outside every stronger logarithmic uniqueness class, proving that uniqueness fails in the critical $L^1$ setting.

Moreover, the construction shows that the positivity property is lost for every bounded nonnegative datum – even for the constant initial condition $u_0\equiv1$. This answers long‑standing open problems posed by Brezis, Cazenave and Miyamoto.

The phenomenon is specific to the critical exponent. In subcritical ($p<p_F$) or supercritical ($p>p_F$) cases, proofs of existence, uniqueness and positivity in the corresponding critical spaces remain valid.

These results clarify the delicate nature of the doubly critical case and demonstrate that standard semigroup arguments break down precisely at the Fujita exponent.