A common simplifying assumption in geometric analysis is that intrinsic geometric growth determines the scaling behavior of diffusion operators defined on a space. In this work, we show that shell growth alone does not, in general, determine short-time heat-kernel scaling unless admissible propagation is fixed. We introduce a notion of recursive shell geometry based purely on equidistance relations, independent of topology, coordinates, or the triangle inequality. On such structures, admissible propagation constraints define canonical Dirichlet forms and Laplacian operators in the sense of Dirichlet-form theory. Our principal result establishes a non-identifiability phenomenon: even for spaces with fixed, homogeneous shell growth of dimension n, including continuous limit spaces, the associated Laplacians may exhibit distinct short-time scaling exponents. We prove a dual-geometry scaling law in which the effective spectral dimension governing diffusion at short times is an invariant of admissible propagation not determined by intrinsic shell geometry alone, with a crossover determined by admissible escape to the intrinsic shell dimension at longer times. The result is proved under explicit regularity hypotheses and is illustrated by a finite, computable example. Classical manifolds and metric geometries arise as special cases in which admissible propagation aligns isotropically with shell geometry, recovering the standard single-exponent behavior in the intermediate time regime. The contribution is therefore not the discovery of multiscale diffusion per se, but a geometry-first analytic framework that separates intrinsic shell geometry from propagation constraints and clarifies when geometric and dynamical notions of dimension coincide—and when they provably do not.
Singh et al. (Sun,) studied this question.