Define a sequence of finite sets $S_m \subset \mathbb{N}$ by the initial conditions $$S_1 = \{0\}, \qquad S_2 = \{1\},$$ and the recurrence for $m \ge 3$: $$S_m = \operatorname{Inc}\!\Big(\,\zeta\!\big(\,\zeta(S_{m-1}) \cup \zeta(S_{m-2})\,\big)\Big),$$ where the operations are defined as follows.

Definitions

Characteristic function. Each finite set $A \subset \mathbb{N}$ is identified with its characteristic function $\mathbf{1}_A : \mathbb{N} \to \mathbb{F}_2$, where $\mathbf{1}_A(n) = 1$ if $n \in A$ and $0$ otherwise.

Subset-sum (zeta) transform $\zeta$. Given a finite set $A \subset \mathbb{N}$, define $\zeta(A)$ as the set whose characteristic function is $$\mathbf{1}_{\zeta(A)}(t) \;=\; \sum_{s \,\subseteq\, t} \mathbf{1}_A(s) \pmod{2},$$ where $s \subseteq t$ means that every bit set in the binary representation of $s$ is also set in $t$ (i.e. $s \mathbin{\&} t = s$, the bitwise AND condition). This is the Möbius/zeta transform on the Boolean lattice $2^{\mathbb{N}}$, reduced modulo $2$.

Union $\cup$ (pointwise OR over $\mathbb{F}_2$). Given two finite sets $A, B \subset \mathbb{N}$, the operation $A \cup B$ is simply the set union. Equivalently, in terms of characteristic functions: $$\mathbf{1}_{A \cup B}(n) = \mathbf{1}_A(n) \lor \mathbf{1}_B(n).$$

Increment $\operatorname{Inc}$. $\operatorname{Inc}(A) = \{a + 1 \mid a \in A\}$, i.e. shift every element up by one.

Computed values

Carrying out the recurrence by hand (or by computer), the first several sets are:

$$\begin{aligned} S_1 &= \{0\}, \\ S_2 &= \{1\}, \\ S_3 &= \{1\}, \\ S_4 &= \{2\}, \\ S_5 &= \{2,\, 3,\, 4\}, \\ S_6 &= \{3,\, 5,\, 7\}, \\ S_7 &= \{3,\, 5,\, 8\}, \\ S_8 &= \{4,\, 6,\, 8,\, 9,\, 12,\, 14,\, 16\}. \end{aligned}$$

Question

Is there a closed-form expression for the set $S_m$, or equivalently a closed-form formula for the membership predicate $$P(m,\, r) \;=\; [r \in S_m]$$ as a function of $(m, r)$?

Any structural insight is welcome: a characterisation in terms of binary representations of $m$ and $r$, asymptotic growth of $|S_m|$ or $\max S_m$, etc.