Falsifiable Validation of the Symbolic Irrational Constant ⟦κ⟧

This paper introduces and validates a newly defined irrational constant, symbolized as ⟦κ⟧. Unlike traditional mathematical constants, ⟦κ⟧ is proposed as a symbolic stabilizer—a number that modulates feedback, preserves memory, and balances entropy in symbolic and cognitive systems. The constant emerges from a recursive structure involving the golden ratio and plays a role in modular cognition, symbolic computation, and geometric encoding. To test its validity, we designed five falsifiable experiments. These tests explore how ⟦κ⟧ affects modular loops, memory decay, volatility in symbolic systems, softmax-based selection behavior, and geometric spiral formations. The results consistently show that introducing ⟦κ⟧ changes the behavior of these systems in measurable, non-trivial ways. This paper includes a fully reproducible Python script containing all five validation tests, making the findings verifiable and accessible for further research. The work builds upon the broader KN (Kuro Nova) framework and contributes a symbolic constant with clear experimental grounding.