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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 phi within 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 long time goal of AI research. Keeping a model within a coherent state ensures that it produces natural sounding and well constructed sounding language output. When the model drifts into the outer bounds towards infinity it becomes jibberish, incoherent, and is essentially broken. By defining the upper bounds and lower bounds of which a model’s coherency can drift around in, it allows for what could be termed as a ”sanity buffer”, or a Discrete Stability Range using a geometric coupling of 2 discrete equations that produce fractal boundaries that prevent 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 2 converging equations, one for natural drift or upper limit and a baseline constant. 1
For the upper limit, we define a recursive form of Phi. Rephi =ni+ 1 = 1/ni+ 1 This converges naturally to Phi, or the quantized value of 1.618... For the lower baseline, or bedrock, we will be using AlphaPi aka The Kunferman Constant, which converges close to Pi/2 and can be seen as a way of quantizing Pi for computational purposes. alphapi =lim(n−> infinity)[2/n2]∗Sum(j= 1ton −1)[j2/sqrt(n2− j2)] Where:- n is an integer approaching infinity (the total number of discrete divisions)- j is the summation index (chosen to avoid confusion with kappa) This summation converges to approximately 1.5642654391586674, a value remarkably close to pi/2 1.57079632679, but derived without curvature or circular geometry. The remarkably simple implementation then sets the bounds and buffer by stating delta =Rephi −alphapi =approx0.537685496 This creates a quantization buffer, a stabilizing offset that prevents the system from overshooting or collapse into instability. 2
