Density of a self-avoiding quadratic sequence with hierarchical "modular" valves

I am exploring a family of integer sequences $(k_n)$ that combine explosive quadratic growth with specific "modular" reduction rules based on the proximity to perfect squares and powers of two. I’ve categorized these reduction mechanisms as "Valves."

The Core Growth Function: Let $f(x) = ax^2 + bx + c$ with $a, b, c, L \in \mathbb{Z}$ and $a > 0$. In the absence of a "valve" trigger, the sequence follows a second-order recurrence: $$k_{n+1} = f(k_n) + L \cdot k_{n-1}$$

The Global Constraint: The sequence is strictly self-avoiding. Any proposed next term $k_{n+1}$ must satisfy $k_{n+1} \notin {k_1, \dots, k_n}$. If a reduction rule proposes a value already in the history, the sequence defaults to the quadratic growth branch.

The Evolution of the Valves:

Valve of the 1st Kind (Square-based): Let $p$ be the largest perfect square such that $p \leq k_n$. Let $gap = (\lfloor\sqrt{k_n}\rfloor + 1)^2 - p$. Given a threshold $\alpha \in (0, 1)$, if $k_n - p < \alpha \cdot gap$ and the target is unvisited, then: $$k_{n+1} = k_n - p$$ Observation: This acts as a "Teleporter," often snapping massive values back to small integers (under $2\sqrt{k_n}$).

Valve of the 2nd Kind (Binary-based): Let $p$ be the largest power of two such that $p \leq k_n$. Let $gap = p$ (since $2^{m+1} - 2^m = 2^m$). Given a threshold $\beta \in (0, 1)$, if $k_n - p < \beta \cdot gap$ and the target is unvisited, then: $$k_{n+1} = k_n - p$$ Observation: This acts as a "Staircase," stripping the most significant bit and reducing the value by at most half.

Valve of the 3rd Kind (The Hybrid): This valve uses "OR" logic between the two previous types, each with its own threshold $\alpha$ (Square) and $\beta$ (Binary). If both are eligible, the rule selects $\min(target_{sq}, target_{bin})$.

If neither reduction target is eligible or both are in the history, the sequence triggers the quadratic growth $f(k_n) + L \cdot k_{n-1}$.

Heuristic Behavior:

With parameters as high as $a=8$ and various $L$ in the one hundreds, I have observed that the Type III valve creates a highly recurrent sequence. Even when $k_n$ spikes to $10^{100}$ or higher, the hierarchical nature of the valves (Binary stripping bits followed by a Square reset) eventually pulls the sequence back to near the origin.

I am interested in the density of the set $K = {k_n : n \in \mathbb{N}}$.

Questions:

1. Divergence of Reciprocals: Under what conditions on $(f, L, \alpha, \beta)$ is the set $K$ "large"? Specifically, is it true that there are $0< \alpha < \beta < 1$,so that the sum $\sum_{k \in K} \frac{1}{k}$ diverges, and also that there are $0< \alpha < \beta < 1$,so that the sum $\sum_{k \in K} \frac{1}{k}$ is converging? How to find out the boundary?

2. My intuition thinks that ordinals can play a role in studying this. But I am a amateur in this area. After some discussion the AI I am working with has proposed the following mapping for the reset of the hierarchy $$\Omega_n = \omega^2 \cdot (\text{Square Falls}) + \omega \cdot (\text{Binary Falls}) + \text{bits}(k_n)$$ Is that a good path for searching for an answer?

But also, while the quadratic growth provides the 'energy' for the sequence to explore high values, the long-term behavior seems dominated by the structure of the descent. Once a spike occurs, the Binary Valve performs a deterministic 'bit-stripping' process. I am observing that these descent paths are significantly longer than the growth phases, suggesting that the sequence's density is determined by how these 'binary ladders' intersect with the set of perfect squares (the Type I Valve) and the set of previously visited integers (the self-avoiding history). But this is, for now at least, above my paygrade it seems.