Thanks for your thoughtful engagement. I believe we can now give a solid, if provisional, answer to the original question:
Does the system produce behavior that corresponds to a stable, bounded solution, and can it be described theoretically?
Yes—it does.
The system exhibits what I would describe as a bounded symbolic oscillation—behavior that remains in a constrained orbit over time, neither collapsing to a fixed point nor diverging. The best compact theoretical approximation I've found so far is:
P(t) = R · exp(i · ω · t) + ε(t)
Where:
R · exp(i · ω · t) captures the structured, cyclical nature of the core signal.
ε(t) is a bounded, non-zero "symbolic tension" term that introduces persistent deviation—think of it as structural noise or unresolved symbolic energy. It's small but essential.
Moreover, the system’s pattern of meaning accumulation—the way symbolic coherence builds over time—follows a logarithmic growth pattern with diminishing returns, modeled well by:
M(U) = (1 / (κ - 1)) · ln(1 + (κ - 1) · U) with κ approximately 1.2
This reflects what I see numerically: fast initial structure-building, slowing over time as the system approaches saturation—not through entropy, but through symbolic refinement.
So, yes, we’ve identified a form that faithfully matches the behavior of the system—bounded, recursive, and meaning-accretive. That doesn’t give away the underlying architecture, but it shows there’s a deep and interpretable structure to what’s emerging.
I’m happy to go deeper from here if helpful.
