Does every order $r\geq 2$ admit an additive basis with $\sum_{n\leq x}f_r(n)^2\ll x$? (Erdős #1192)

For $A\subset \mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\cdots+a_r$ with $a_i\in A$. Call $A$ a basis of order $r$ if $f_r(n)>0$ for all sufficiently large $n$. Does there exist, for every $r\geq 2$, a basis $A$ of order $r$ such that $$\sum_{n\leq x}f_r(n)^2 \ll x$$ for all $x$? Note this $L^2$ bound is optimal up to constants: by Cauchy–Schwarz, any basis of order $r$ satisfies $\sum_{n\leq x}f_r(n)^2\geq (\sum_{n\leq x}f_r(n))^2/x\gg x$, so the question asks for bases whose representation counts are as evenly spread as possible.

Acceptance. FULLY RESOLVES: for every $r\geq 3$ (the case $r=2$ being known, per background), construct a basis $A$ of order $r$ — explicit or probabilistic — with complete proofs of both properties: (i) $f_r(n)>0$ for all sufficiently large $n$, and (ii) $\sum_{n\leq x}f_r(n)^2\ll x$ for all $x$; machine-checkable (Lean/Coq) preferred, else a full written proof. OR a proof that for some $r\geq 3$ no such basis exists. ADVANCES: (a) the case $r=3$ alone, with full proof; (b) any single $r\geq 3$; (c) a basis of order $r\geq 3$ with a proven mean-square bound weaker than $\ll x$ but strictly stronger than anything stated as known in the background (state the bound precisely, with proof); (d) a proof that the Erdős–Rényi random construction can be repaired into a genuine basis for some $r\geq 3$ at bounded cost to the $L^2$ bound. Deliver the construction + proof file, plus (where a construction is explicit) code computing $f_r$ on an initial segment as a sanity certificate.