Nested Formula Framework (NFF): A Hierarchical Framework for Mathematical Discovery Through Recursive Coefficient Nesting

The Nested Formula Framework (NFF) introduces a fundamentally different approach to mathematical modelling. Rather than fitting a single global equation across all variables simultaneously, NFF incorporates variables one at a time through a recursive process in which the coefficients obtained at each level become the dependent quantities modelled at the next. The result is a hierarchy of nested equations that reconstructs complex phenomena progressively, keeping every parameter physically traceable and mathematically justified at every step.

NFF is not a black box. It is a transparent, interpretable framework in which the investigator retains full control over each modelling decision — selecting the functional family that best fits the observed behaviour of the data at each level, from linear and exponential to logarithmic, Gaussian, Gumbel and beyond. This level-by-level inspection is precisely what current machine learning and symbolic regression methods cannot offer.

The framework derives a central structural result — the Fundamental Law N_c = sⁿ — which was not postulated but emerged from systematic derivation across six primary functional families. It further demonstrates that hybrid architectures combining different families across levels generate new functional structures naturally: power laws, double exponentials, iterated logarithms and cascading extreme-value distributions arise without prior assumption.

Applications span seismic attenuation modelling, climate sensitivity analysis, financial yield curves, epidemiological growth dynamics and materials science — anywhere that understanding the structure of a phenomenon matters as much as predicting it.

Researchers working with complex multivariate data are invited to explore NFF as a tool for equation discovery that is both computationally tractable and scientifically meaningful.