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

Self-reinforcing feedback appears across disciplines, and where it is superlinear it produces not unbounded growth but a finite-time singularity. The classical treatment assumes a fixed feedback exponent: for dy/dx = k(1 + y^m), a singularity occurs if and only if m > 1, at the distance x^* = (1/k)(π/m)/sin(π/m). This paper asks what happens when that exponent is not fixed but fluctuates across the critical value. The question arose from an attempt to determine the exponent of a specific empirical curve, Bitcoin's historical price lows, where a flattening power law, a superlinear trajectory, and a noise-driven one proved difficult to distinguish over the observed range.

The main result is that the mean exponent does not decide the outcome. If the exponent process has a non-vanishing probability of sustained excursions above the critical value, together with a renewal condition, the singularity occurs almost surely whatever the mean may be, including means below the critical value, with the time to the event bounded in expectation and its tail geometric. The proof rests on two elementary facts: the state is non-decreasing, so the level reached is never given back, and the duration of supercritical feedback still required to complete a singularity shrinks as that level rises. Numerical results are given for mean exponents down to 0.80. A complementary result sharpens where the fluctuation must act: a fluctuation entering the rate rather than the exponent cannot produce a singularity at any amplitude, however persistent.

The second half of the paper concerns identification rather than dynamics. Explicit, checkable membership conditions are stated, including the requirement that the fluctuation reach the exponent and that an irreversible state variable be identified. Simulations of nested instances of the same equation, with code included, illustrate a difficulty that applies to every candidate system: an observed quantity may follow none of the levels present, so that an exponent fitted to a record belongs to no part of the system producing it. Two physical domains are examined against these conditions. In tokamak edge instabilities the currently established equation reduces to the same form but with a fixed quadratic exponent and a linear damping, placing them in a distinct class of metastable escape rather than in the one described here; the status of that fixed exponent, which is derived rather than measured, is discussed. In accelerating creep before slope failure, the governing relation used in that field coincides with the equation above, and its exponent is measured above the critical value rather than assumed.

A circuit realizing the equation is described, though not yet built. An oscillator runs between a fixed threshold and a rising one, so that its frequency is inversely proportional to the gap between them, and a second capacitor closes that gap as it accumulates. The exponent such an arrangement produces is not a design choice. It follows from how the gap closes and how the observable reads it, through a relation derived here that accounts for every case considered: charging once per cycle gives one value, charging at a constant rate gives another, and the same relation reproduces the exponents of two mechanical arrangements, one of which coincides with the isothermal and adiabatic compressions of elementary kinetic theory. Every passive arrangement examined proves to be supercritical, and the reason is visible in the relation itself; reaching the critical value requires a limiting case, and going below it requires the charging law to be imposed rather than found. The circuit is a constructed instance and cannot show that the class is non-empty in nature, but it fixes the exponent by geometry rather than by assumption, and it permits the estimator used throughout to be calibrated against a system whose exponent is known.

No prediction is offered for any system, and the conjectures are labelled as such, in particular whether the fluctuations can be generated by nesting rather than imposed, and whether any real system satisfies the conditions. The framework is put forward as a descriptive instrument for recognizing intermittent superlinearity, together with the observations that would exclude it.