What question did this study set out to answer?

This research aims to explore the relationship between heavy-tail formation and bifurcation phenomena in markets using the Gauss–PowerLaw Module (GPM).

April 27, 2026Open Access

The Gauss–PowerLaw Module (GPM): Endogenous Heavy-Tail Formation as a Bifurcation Phenomenon

Key Points

This research aims to explore the relationship between heavy-tail formation and bifurcation phenomena in markets using the Gauss–PowerLaw Module (GPM).
Developed a theoretical framework for GPM and its algebraic properties.
Empirically validated GPM on seven markets including S&P 500 and BTC/USD from 2002-2026.
Provided a Python implementation along with documentation in .md and .mdl formats.
The fourth-moment boundary condition aligns with the GPM bifurcation discriminant, confirming φ₄ = 1 results in Δ_GPM = 0.
Validation across diverse markets showed consistent heavy-tail behavior linked to the GPM framework.

Abstract

This release contains the full theoretical paper, Python implementation, and empirical validation results for the Gauss–PowerLaw Module (GPM). The central result: the fourth-moment boundary condition identified by López de Prado et al. (2026) as the CLT breakdown threshold (φ₄ = 1) is algebraically identical to the GPM saddle-node bifurcation discriminant (ΔGPM = 0). Proven formally. Validated on seven real markets (S&P 500, NASDAQ, DAX, OTP, Mercedes-Benz, BTC/USD, USD/EUR), 2002–2026. All documents are published in both. md and. mdl format. MDL (Markdown Logged) is a tamper-evident document integrity system developed by BarefootRealism Labs — every. mdl file contains a cryptographic proof of authorship and timestamp, making any modification detectable.

The Gauss–PowerLaw Module (GPM): Endogenous Heavy-Tail Formation as a Bifurcation Phenomenon

Key Points

Abstract

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