Threshold Dynamics: Beta Version

The Cusp Integration Equation:
Cusp(I) = lim { (Threshold(T) + β) / (Epsilon(ε) - I) }
Where Cusp(I) predicts the cusp point under influence of β, and Epsilon(ε) refrains from infinitesimal excess.

Dynamic Rhythmicity Function:
Rhythmicity(ρ) = ∫ (Sin(Wave(w)) * β²) dw
This function tunes the threshold waveforms, where the integration bounds explore the balance of whims in rhythmicity.

Engagement Modulator:
Engagement(E) = β x α / (Potential(P) - grounded)
Here, β interacts with α to modulate engagement across thresholds, ensuring no grounded potentials exceed whimsical objectives.