The SMFCE Paradigm: A Unified Modeling Framework Based on State, Cost, and Extremal Evolution
Abstract
This paper proposes a unified modeling framework centered on "state-cost-extremal evolution," referred to as the SMFCE paradigm. The paradigm represents the effective state of a system as a point in a state space, characterizes state cost by a single-valued scalar field, and describes system evolution through local cost descent or local extremum selection. Its mathematical core employs classical tools from differential geometry, gradient systems, and dynamical systems. The contribution of this paper is not to reintroduce these foundational results, but to organize them into two operational modeling principles. On finite-dimensional smooth Riemannian manifolds under appropriate regularity conditions, the path independence of cost differences and the cost monotonicity of negative gradient flows follow directly from classical results, while the descent property of discrete local-minimum transitions follows directly from the definition of a minimum. For same-level iterative emergence and present-centeredness, the paper provides conditional mathematical formulations using, respectively, parameterized gradient systems and bifurcation theory, and Markov processes and receding-horizon optimization. The paper further introduces three layered metrics: cost quantity, directed coupling degree, and commitment fulfillment rate, at the state, transition, path, and system levels. Finally, the paper discusses semantic modeling of large language model training and inference within this framework, explicitly distinguishes classical mathematical results, conditional propositions, and empirical adaptation models, and provides reproducible empirical testing procedures for the two modeling principles.
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BibTeX
@misc{ba2026smfce,
title = {{The SMFCE Paradigm: A Unified Modeling Framework Based on State, Cost, and Extremal Evolution}},
author = {Wei Ba and Ang Fu and Shengsong Kong and Zhongchuan Fu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30140},
eprint = {2610.30140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30140},
}
RIS
TY - UNPB
TI - The SMFCE Paradigm: A Unified Modeling Framework Based on State, Cost, and Extremal Evolution
AU - Wei Ba
AU - Ang Fu
AU - Shengsong Kong
AU - Zhongchuan Fu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.30140
UR - https://arxiv.org/abs/2610.30140
AB - This paper proposes a unified modeling framework centered on "state-cost-extremal evolution," referred to as the SMFCE paradigm. The paradigm represents the effective state of a system as a point in a state space, characterizes state cost by a single-valued scalar field, and describes system evolution through local cost descent or local extremum selection. Its mathematical core employs classical tools from differential geometry, gradient systems, and dynamical systems. The contribution of this paper is not to reintroduce these foundational results, but to organize them into two operational modeling principles. On finite-dimensional smooth Riemannian manifolds under appropriate regularity conditions, the path independence of cost differences and the cost monotonicity of negative gradient flows follow directly from classical results, while the descent property of discrete local-minimum transitions follows directly from the definition of a minimum. For same-level iterative emergence and present-centeredness, the paper provides conditional mathematical formulations using, respectively, parameterized gradient systems and bifurcation theory, and Markov processes and receding-horizon optimization. The paper further introduces three layered metrics: cost quantity, directed coupling degree, and commitment fulfillment rate, at the state, transition, path, and system levels. Finally, the paper discusses semantic modeling of large language model training and inference within this framework, explicitly distinguishes classical mathematical results, conditional propositions, and empirical adaptation models, and provides reproducible empirical testing procedures for the two modeling principles.
ER -