doi: 10.17706/ijapm.2026.16.2.43-51
A Mathematical Framework for Innovation: Cognitive Gauge Theory and Empirical Validation
Jian Gao
- China Development Bank, No. 2, Fuxingmen Nei Street, Beijing 100031, China.
- Manuscript received January 10, 2026
- accepted August 6, 2026
- published October 23, 2026
Abstract
How do transformative ideas emerge from the bounded mind? This paper develops and empirically validates a geometric framework for the cognitive architecture of innovation. We first present the Pyramid Knowledge Gauge Theory (PKGT), which models cognition as a seven-layer hierarchical stack where gauge-theoretic principles govern information processing. From PKGT, we derive Cognitive Gauge Theory (CGT) for innovation, which formalizes idea generation as navigation of a cognitive state space, constrained by a gauge field of attentional costs, curvature producing path-dependence, and a finite discovery budget. Across three experiments, we confirm CGT’s predictions: exploratory search is path-dependent, individuals possess stable discovery budgets, and conceptual recombinations follow predictable geometric patterns. CGT provides cognitive microfoundations for economic theories and formalizes key philosophical ideas concerning knowledge structure and growth.
Keywords
- bounded rationality
- cognitive architecture
- gauge theory
- idea generation
- innovation
- mathematical modeling
How to Cite
Jian Gao, "A Mathematical Framework for Innovation: Cognitive Gauge Theory and Empirical Validation," International Journal of Applied Physics and Mathematics, vol. 16, no. 2, pp. 43-51, 2026. https://doi.org/ 10.17706/ijapm.2026.16.2.43-51
Copyright & License
Copyright © 2026 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).