What question did this study set out to answer?

The aim is to derive the internal coefficient lambda in the quartic functional based on stability conditions.

March 13, 2026Open Access

Emergence and Numerical Fixation of the Universal Constant Lambda in the Quartic Variational Framework

Key Points

The aim is to derive the internal coefficient lambda in the quartic functional based on stability conditions.
Utilized the explicit integral formulation of the quartic functional.
Applied the stationarity condition and expanded the static Hessian.
Performed Z3 block decomposition to analyze stability.
Identified a unique value for lambda equal to 3/2.
Confirmed that this value arises solely from internal algebraic structures.
Ensured results are consistent with stationarity and spectral stability.

Abstract

This work presents the complete variational derivation of the internalcoefficient appearing in the quartic functional of the –framework. Using the explicit integral formulation of the functional, thestationarity condition, the fully expanded static Hessian, and its Z3 blockdecomposition, the stability requirement is reduced to a single spectralequation. The internal eigenvalues generated by the destabilizing andstabilizing blocks fix the ratio determining. The procedure yieldsa unique value, with no external parameters, no fitting, and no importedstructures. The result is: λ=32. = 32. λ=23. This value follows solely from the internal algebra of the quartic functionaland represents the only solution consistent with stationarity, spectralstability, and the Z3 cyclic closure.

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