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January 22, 2026Open Access

Companion A: Effective Lagrangian Derivation — From 5D Junction Dynamics to Sₑffq

Key Points

This work aims to provide a detailed derivation of the effective collective-coordinate action S_eff[q] using the NJSR framework.
Derivation starts from a 5D brane-bulk setup.
Explicitly defines junction and boundary conditions.
Identifies M(q) and V(q) through assumptions and gauge choices.
Documents intermediate steps in the derivation process.
Successfully derives the reduced 1D effective action S_eff[q]=∫dt(½M(q)dot q² - V(q)).
Clarifies the relationships between M(q) and V(q) based on the initial 5D framework.

Abstract

This companion provides the technical derivation of the effective collective-coordinate action Sₑffq used by the NJSR framework. Starting from a 5D brane–bulk setup with explicit junction/boundary conditions, it derives the reduced 1D formSₑffq= dt (12 M (q) q² - V (q) ), and documents the assumptions, gauge choices, and intermediate steps required to identify M (q) and V (q). This document is intended to be cited by the Paper 3 journal version as the canonical source for the effective-action reduction.

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