Alpha Pi and and Recursive Phi as safety equations preventing infinite poles in AI Coherency

Defining Quantization Buffer for Coherency Using Phi Recursion and AlphaPi C.R. Kunferman December 20, 2025 1 Introduction In terms of AI coherence, there currently exists a large number of safety guidelines and arbitrary methods to keep the coherence of models from drifting into infinite complexity. When diverging into such realms of chaos, the information becomes heavy and expensive. In this paper the author presents a dynamical system with hard safety constraints that still allow for the dynamic adaptability needed for natural communication, while keeping bounded limits that prevent diverging into infinity. This method uses AlphaPi and a recursive form that converges to φwithin 10 steps to create a quantization buffer, a safety margin to allow more focus and targeted coherence that never diverges. This creates a bounded domain for calculations where standard physics and AI models currently break due to infinity. 2 Abstract The need and importance of an AI system’s ability to remain coherent has been a longtime goal of AI research. Keeping a model within a coherent state ensures that it produces natural-sounding and well-constructed language output. When the model drifts into the outer bounds toward infinity it becomes gibberish, incoherent, and is essentially broken. By defining the upper bounds and lower bounds within which a model’s coherency can drift, it allows for what could be termed a “sanity buffer,” or a Discrete Stability Range using a geometric coupling of two discrete equations that produce fractal boundaries preventing a discrete system from collapsing into the infinite pole found in current approximation and analytical continuation solutions. 3 Methods To start, we will be using two converging equations: one for natural drift or upper limit, and one baseline constant. 1 Upper Limit: Recursive Phi For the upper limit, we define a recursive form of φ: ni+1 =1 ni + 1. This converges naturally to φ, the quantized value of approximately 1.618 . . . Lower Baseline: AlphaPi (Kunferman Constant) For the lower baseline, or bedrock, we use AlphaPi, also known as the Kunferman Constant, which converges close to π/2 and can be seen as a way of quantizing πfor computational purposes: απ = lim n→∞ "2 n2 n−1 X j=1 j2 pn2−j2#. Here: •nis an integer approaching infinity (the total number of discrete divisions), •jis the summation index (chosen to avoid confusion with κ). This summation converges to approximately 1.5642654391586674, a value remarkably close to π/2≈1.57079632679, but derived without curvature or circular geometry. Quantization Buffer The implementation sets the bounds and buffer by defining δ=φ−απ ≈0.0537685496. This creates a quantization buffer: a stabilizing offset that prevents the system from overshooting or collapsing into instability. 4 Conclusion By coupling a rapidly convergent recursive form of φwith the computationally derived constant απ, this work introduces a simple yet effective quantization buffer that prevents divergence in nonlinear iterative systems. The buffer δ=φ−απ defines a narrow but robust stability corridor that constrains the system away from its singularities while preserving natural dynamical flexibility. 2 This framework provides a mathematically grounded method for maintaining coherency in discrete systems, including AI models whose internal states risk drifting toward unstable or infinite regimes. Because the mechanism relies only on rational operations and bounded recursion, it offers a computationally efficient alternative to floating-point or trigonometric stabilization methods. The resulting bounded domain suggests potential applications in neural architectures, fixed-point computation, and any system requiring guaranteed non-divergent behavior under iterative updates. The quantization buffer formalism demonstrates that coherence can be preserved not through arbitrary constraints, but through intrinsic geometric relationships between convergent constants. 3