On the undecidability of an iterative parity-twisted divisor-mapping sequence

I have constructed an arithmetic function that exhibits a chaotic behavior reminiscent of Collatz-like dynamical systems, but with a feedback loop driven by the partitions of parity-shifted divisor geometries. The system appears to inherently embed the Halting Problem within standard multiplicative number theory. I am seeking insights into whether this structure is provably independent of ZFC or if any tools in analytical talteori can crack its asymptotic density.

The Dynamical System Mechanics

Let $n$ be a positive integer. We define an infinite state-machine or sequence of vectors $V_0, V_1, V_2, \dots$ through the following generative process:

1. Initial State ($V_0$): Let the ordered divisors of $n$ be $d_1, d_2, \dots, d_m$. We construct the parity-twisted initial vector $V_0$ of length $m$ by mapping each divisor to: $$v_i = d_i \times (-1)^i$$

2. The Reflection Shift Line (Vector Collapse): For any vector $V_k = [x_1, x_2, \dots, x_L]$ of length $L > 1$, the next intermediate vector $V_{k+1}$ is formed by taking the absolute difference of the elements mirrored from the outside in: $$V_{k+1} = \left[ |x_L - x_1|, |x_{L-1} - x_2|, \dots, |x_{L - \lfloor L/2 \rfloor + 1} - x_{\lfloor L/2 \rfloor}| \right]$$ This operation strictly halves the vector length at each step ($L_{k+1} = \lfloor L_k / 2 \rfloor$).

3. The Feedback Singularity (The Recursive Loop): The vector collapse is performed repeatedly. However, the moment the length of the vector collapses to $L = 1$ (leaving a single scalar integer $S$), we restart the generation:

• We take the absolute value $|S|$.
• If $S = 0$ or $S = 1$, the process terminates (or enters a trivial absorbing state).
• If $S > 1$, we find the ordered divisors of $S$, apply the parity twist, and initiate a brand new "generation" vector $V_{\text{new}}$.

The Function $\Omega_S(n)$

We define our primary function $\Omega_S(n)$ as the total number of full generation restarts (singularities) the system completes before either terminating at $0$ or $1$, or entering a periodic loop. If the sequence of scalar targets $S_1, S_2, S_3, \dots$ grows without bound and the process never halts, we state that: $$\Omega_S(n) = \infty$$

The Ultimate Conjecture

Because the parity-twisted mapping breaks the standard multiplicative bounds of highly composite numbers, predicting whether a given seed $n$ will trigger an infinite expansion or a latent collapse feels fundamentally uncomputable.

Let $\mathcal{S} = \{ n \in \mathbb{N} : \Omega_S(n) = \infty \}$ be the set of all non-halting seeds.

The Transfinite Singularity Conjecture: The set $\mathcal{S}$ is non-empty, and its asymptotic density is strictly non-computable (undecidable). No finite axiomatic framework (such as ZFC) can classify whether an arbitrary integer $n$ belongs to $\mathcal{S}$ without executing the transfinite step-sequence.

Questions for the Community

1. Does this specific method of nesting divisor-differentiation loops inside scalar regeneration steps map onto any existing classes of arithmetic automata?
2. Are there any known criteria in arithmetic dynamics that can prove a sequence of this type is unbounded? Or does the parity twist $(-1)^i$ ensure that the distribution of $S_k$ mimics a deterministic random walk, rendering the divergence problem undecidable?