ABE: A Physics-Based Mathematical Framework for Modeling and Forecasting Ordered Numerical Sequences

Description

Acceleration Balance Equation (ABE): A Unified Mathematical Framework for Reconstructing, Modeling, and Forecasting the Dynamics of Ordered Numerical Sequences

This comprehensive research manuscript introduces and systematically develops the Acceleration Balance Equation (ABE), an interdisciplinary mathematical framework designed to reconstruct effective dynamical structures from ordered numerical sequences and investigate their forecasting capabilities across economics, finance, engineering, and other complex systems.

Drawing on concepts from continuum mechanics, nonlinear dynamical systems, numerical analysis, optimization, and differential geometry, ABE establishes a unified theoretical architecture connecting observed numerical sequences with mathematically defined representations of informational density, velocity, pressure, acceleration, and viscosity.

The research develops five interconnected mathematical components:

1. 1ABE — Fundamental Dynamical Model: Establishes the mathematical foundations of intrinsic sequence dynamics, including the fundamental acceleration balance, Real and Physical ABE viscosities, critical-viscosity conditions, directional stability, and operational procedures for predicting subsequent interval means.

2. 2ABE and nABE — Coupled Dynamical Systems: Extend the fundamental model to interacting numerical sequences through directed cross-sequence relationships, interaction matrices, and cross-viscosity structures. These formulations provide a framework for studying collective dynamics and testing whether information from additional sequences improves predictions for predefined targets.

3. LABE and nLABE — Variational Extensions: Introduce residual-based action functionals and minimum-action principles for evaluating dynamically consistent future continuations, with conditional probabilistic extensions.

4. GABE and nGABE — Geometric Extensions: Develop geometric representations induced by ABE dynamical residuals, establishing conditions under which metric structures, preferred trajectories, and collective dynamical geometry can be investigated.

5. Empirical Validation and Reproducible Forecasting: Present an exploratory evaluation covering 88 numerical series and five forecasting-weight configurations per series, together with procedures for chronological evaluation, causal information management, benchmark comparisons, uncertainty diagnostics, and reproducibility.

Under the balanced 50/50 evaluation criterion, the complete empirical cohort reports a mean FINAL directional hit rate of 53.72%. A retrospectively selected subset of 70 series reports 59.71%; this latter result is descriptive and does not constitute independent confirmation of predictive superiority. The manuscript explicitly distinguishes observed historical performance from prospective validation and identifies the experimental requirements necessary to establish incremental forecasting skill.

The central scientific contribution is the construction of an integrated, mathematically explicit and falsifiable sequence-to-dynamics architecture that connects intrinsic modeling, multicomponent interactions, variational optimization, geometric interpretation, and operational forecasting within a common framework.

Rather than assuming that observed numerical sequences obey literal fluid-mechanical laws, the research formulates ABE as an effective mathematical representation whose properties depend on stated structural and constitutive assumptions. Mathematical consistency, empirical reliability, parameter identifiability, and predictive superiority are treated as distinct scientific questions.

Particular emphasis is placed on nABE as a central direction for future research, with the objective of establishing whether appropriately identified and calibrated interactions among numerical sequences can provide demonstrable predictive improvements over isolated 1ABE models and conventional benchmarks.

This integrated Version 1 consolidates the theoretical, mathematical, computational, and experimental foundations of the ABE research programme. It constitutes a publicly documented stage of an evolving scientific investigation, with further mathematical refinements, methodological corrections, independent empirical validation, and reproducibility improvements planned for subsequent versions.

The broader ambition of ABE is to contribute a transparent and testable mathematical framework for investigating dynamical organization, interdependence, uncertainty, and predictability in complex numerical systems.