Closed form for a set sequence defined via the mod-2 zeta transform ?

Questions
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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.

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