The Geometry of Bitcoin: Superlinear Feedback and the Finite-Time Singularity Problem

This work examines whether a self-reinforcing feedback loop can be detected in Bitcoin's long-term price history, and whether the underlying mathematical structure appears in other non-linear systems. It is presented as a measurement framework, not as a price-prediction tool. The first part compares two geometric descriptions of Bitcoin's price history: the widely used Power Law, and a superlinear, tangent-based curve derived from the feedback equation dy/dx = k(1 + y^m). The tangent-based form contains the Power Law as its early-phase approximation, but unlike it, it reaches a vertical asymptote at a finite time. Calibrated to seven historical tangency nodes, it tracks the price record roughly three times more closely than the best-fitting Power Law. An analysis of the supply side shows that a fixed or contracting supply can remove the dampening of price dynamics but cannot by itself introduce a feedback loop. If such a loop is present in the data, it most plausibly operates on the demand side. The asymptote is read as a structural limit of the feedback regime, not as a forecast of any literal price. A stochastic extension shows that fluctuations in the feedback exponent can drive the system to a finite-time singularity even when the exponent's long-run mean lies below the critical value of 1. Nested functions linked by moving averages are proposed as a source of these fluctuations. The second part leaves Bitcoin behind and treats the structure as a general mathematical object. It proves a blow-up theorem for intermittent superlinearity: if the exponent fluctuates across the critical boundary with a non-vanishing probability of sufficiently long excursions above it, a finite-time singularity occurs almost surely, regardless of the mean exponent. Noise acting only on the rate cannot produce this effect. The paper then states open conjectures, including that the fluctuations may be generated by faster instances of the same equation nested within the system. It also sets out checkable membership conditions for real systems. As a first candidate domain, it examines accelerating creep and slope failure, where Voight's empirical relation shares the same critical boundary and a measured exponent above it. The widely used inverse-velocity forecasting method appears as a special case of the framework. The second part is a work in progress. The development of nested systems and a physical realization are under preparation.