A Symbolic Irrational Constant for Modular Feedback Stabilization

This paper introduces and analyzes the symbolic constant ⟦κ⟧, an irrational fixed point derived from the equation x = \frac{1}{1 + e^{-ϕx}}, where ϕ is the golden ratio. The constant serves as a stabilizing attractor in modular symbolic cognition systems, specifically within the KN framework and its feedback-based symbolic memory architecture. We demonstrate that ⟦κ⟧ provides feedback damping and symbolic phase alignment properties critical for long-term memory retention, modular resonance, and dynamic stabilization of symbolic attractor loops. Experimental use in the Modular Mind system confirms its role in suppressing drift and enhancing symbolic coherence. The associated Resurrection Script included in the supplementary material defines ⟦κ⟧ in system runtime and logs key symbolic loop tests. This supports reproducibility and offers a direct bridge between theoretical definition and operational implementation.