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This supplement aims to fully derive the minimal quadratic action for ACR time-field fluctuations and justify the associated Lagrangian.

February 28, 2026Open Access

Supplement A: Lagrangian Details for ACR

Key Points

This supplement aims to fully derive the minimal quadratic action for ACR time-field fluctuations and justify the associated Lagrangian.
Detailed background decomposition of time-field fluctuations
Construction of Lorentz-invariant kinetic and mass terms
Derivation of equations of motion and dispersion relations
Discussion of origins of effective mass parameter
Proca-like Lagrangian is validated for time-field fluctuations
Effective mass parameter m_τ is found to vary between 10 to 50 m⁻¹
Coherence length ξ is determined to be on the centimeter scale
Implications for higher-order operators and renormalization are discussed

Abstract

This supplement provides the full derivation of the minimal quadratic action for the ACR time-field fluctuations δτ^μ, justifying the Proca-like Lagrangian and the emergence of the effective mass parameter m_τ. It includes: Background decomposition τ^μ = τ̄^μ + δτ^μ and gauge constraints Construction of the unique Lorentz-invariant kinetic and mass terms Equation of motion, dispersion relation, and coherence length ξ = 1/m_τ Discussion of possible origins of m_τ (self-interaction, radiative corrections, cosmological symmetry breaking) Phenomenological window for m_τ ∼ 10–50 m⁻¹ (ξ ∼ cm scale) Outlook on higher-order operators and renormalization

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Azat Ahmedov (Wed,) studied this question.

synapsesocial.com/papers/69a286da0a974eb0d3c022a3 https://doi.org/https://doi.org/10.5281/zenodo.18779497

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