Exact order versus order of additive bases: evaluate $\lim_r h(r)/r^2$, and determine $h(4)$ (Erdős #336)

For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (meaning every sufficiently large integer is the sum of at most $r$ integers from $A$) whose exact order is $k$ (meaning every sufficiently large integer is the sum of exactly $k$ integers from $A$). Find the value of $$\lim_{r\to\infty} \frac{h(r)}{r^2}.$$ A simple example separating the two notions: $A=\bigcup_{k\geq 0}(2^{2k},2^{2k+1}]$ has order $2$ but exact order $3$.

Acceptance. FULLY RESOLVES: the exact value of $\lim_r h(r)/r^2$ (including a proof that the limit exists, if not already established en route): a family of constructions attaining the value asymptotically, with complete proofs of both the order and the exact order of each basis, together with a matching asymptotic upper bound. ADVANCES: (a) a proven improvement of either the lower bound $1/3$ or the upper bound $1/2$ stated in the background; (b) determination of $h(4)$: either a basis of order $4$ with exact order $11$ (explicit construction plus complete proofs of both properties) or a proof that every order-$4$ basis with an exact order has exact order at most $10$; (c) new exact values $h(5), h(6), \ldots$ with proofs; (d) a reproducible computer search over structured (finite-pattern / eventually periodic) bases that certifies a new lower bound for a specific $h(r)$, with code and the proof completing the certification. Deliver the proofs; for construction-based claims, the explicit basis description plus its certification; for search-based claims, the code + certificate.