The Bitcoin Realized Power-Law Envelope (RPLE): An Empirical Framework Based on Realized Price and Power-Law Scaling

This paper presents the Bitcoin Realized Power-Law Envelope (RPLE), a reduced-form framework that reconciles on-chain economic anchoring with a non-linear decay in the admissible realized-price-normalized overextension. By utilizing the Realized Price as a baseline for aggregate cost-basis, the RPLE defines a dynamic valuation corridor where the upper boundary scales according to a power-law function relative to an on-chain proxy of aggregate cost basis (Realized Price), with an empirically calibrated time-decay in the admissible premium. This approach reconciles on-chain fundamentals with the diminishing marginal utility of speculation. The framework departs from traditional time-regression models by anchoring the envelope to an on-chain cost-basis proxy (Realized Price) rather than fitting price directly as a function of elapsed time. This formulation captures the compression of speculative multiples as the network achieves market maturation. While the framework presupposes a structural decay in speculative multiples, the governing parameters are currently phenomenological, derived via historical calibration. Empirical validation over a sixteen-year horizon (2010–2026) demonstrates that price action is constrained within the envelope with high empirical regularity, suggesting that excursions are transient deviations from a long-term structural mean. Unlike cyclical models, the RPLE provides a scale-invariant metric for assessing market overextension. These results suggest that Bitcoin’s valuation dynamics can be effectively modeled as a power-law compression of speculative premiums. The RPLE thus provides a consistent quantitative reference for identifying market regimes and supporting rule-based regime classification and risk constraint design.