The Geometry of Bitcoin: Superlinear-Feedback Asymptotic Model and Finite-Time Singularity

The Geometry of Bitcoin proposes a superlinear-feedback asymptotic model as an alternative descriptive framework to the Power Law for Bitcoin's long-term price structure. The tangent-based model used here is the closed-form member of a broader family: any system governed by superlinear feedback produces a finite-time singularity. Fitted to the historical macro-bottoms (tangency nodes) of the price series, the tangent-based model reaches a Mean Absolute Centered Error of approximately 2.1%, compared with approximately 6.9% for the best-fit (OLS) Power Law on the same data. The vertical asymptote of the function is interpreted not as a price target, but as a theoretical marker of a structural phase transition in liquid price discovery. The paper also argues, through two structural exclusions, that neither Bitcoin's own supply contraction nor ordinary demand growth can by itself host the self-reinforcing feedback that the observed divergence requires. Scarcity is therefore treated as a necessary precondition rather than the complete mechanism. This suggests a tentative broader reading: an asset with a hard, verifiable supply constraint may serve as a fixed reference point, a candidate instrument for observing the dynamics of an external system indirectly. A substantial part of the paper is devoted to a stochastic simulation in which the feedback exponent is not fixed, but is driven by moving averages of random regime-switching signals on two timescales. The results suggest that: the finite-time singularity is driven by the fluctuations themselves. A noise-free system with the same mean exponent remains stable, while the fluctuating system reached an asymptote in every simulated run, even when the mean exponent was below or exactly at the critical value; more persistent and stronger fluctuations bring the singularity earlier; the mechanism can be expressed as a convexity property of the growth term, consistent with a structural, formal description of antifragility: within this construction, the system gains from noise and shocks rather than being eroded by them. The paper further outlines a conceptual construction of nested functions, in which smaller, faster structures supply the noise of larger, slower ones. It also discusses why a curve-fitting approach is likely to be considerably less accurate over shorter trends than over longer cycles. Beyond the specific case of Bitcoin, the simulation framework may offer a possible direction for further work on systems facing instability or singularity problems, where noise, rather than the average tendency alone, may play a decisive role. This is presented as a possible continuation, not as an established result. Scope: This is a descriptive, exploratory work. It is not a price-prediction tool, not financial advice, and not a claim of physical or economic law. The fitted parameters are empirically calibrated and fragile. The simulation is a deliberately simplified, two-level construction, and its limitations are discussed openly in the paper.