CNRS-Multi: Scale-Derivative Exactness and Structural Cross-Scale Coupling in Multi-Scale Diffusion: A Progressive Place-Value Framework

We introduce a representational framework for multi-scale functions in which the derivative with respect to the scale variable is an exact structural operation—a digit shift on the coefficient sequence—rather than an approximation. The framework, the progressive place-value system (PPS), stores a smooth function of scale as a list of Taylor coefficients; differentiation with respect to scale is realised by dropping the leading coefficient and shifting the remaining ones, with zero truncation error by construction (Theorem 2.4). For multi-scale partial differential equations with scale-dependent coefficients, the PPS representation yields two further structural benefits. Multiplication of two PPS-represented functions is a Cauchy convolution of their coefficient sequences (Theorem 2.7), which automatically generates all cross-scale coupling terms without any ad hoc specification. The PPS data structure holds all scale levels simultaneously, eliminating the stiffness that arises when a monolithic multi-scale solver must use the fastest time scale everywhere. We apply the framework to a one-dimensional cross-scale diffusion problem with an exponentially scale-dependent diffusion coefficient, representing the diffusion of a signalling molecule across cell, tissue, and organ scales. Numerical comparison against a classical explicit solver demonstrates: (i) exact scale derivatives versus O(h2) finite-difference error; (ii) automatic cross-scale coupling in the PPS convolution versus absent coupling in the classical solver; (iii) the scale gradient of the concentration field as a new native variable with no classical analogue. The Python implementation (cnrs diffusion.py) is provided as supplementary material; all results are reproducible.