This preprint develops a device- and algorithm-independent theory of multi-scale semantic stability in the geometric framework of observation spaces. Building on previous work on semantic phase transitions in observation geometries, the paper considers metric–measure spaces (X, d, ρ) (X, d, ) (X, d, ρ) satisfying an Ahlfors-type regularity condition and studies how semantic behaviour changes (or remains stable) under coarse-graining of these spaces. The core technical contribution is a class of metric–measure coarse-grainings constructed from scale-controlled partitions. The author proves that, under mild assumptions, these coarse-grained observation geometries preserve both the effective Ahlfors dimension and small-scale regularity in a uniform way. For Poisson Boolean models and random geometric graphs defined on such spaces, the coarse-graining induces a renormalisation map on the effective intensity–radius parameter η=λrD = rDη=λrD that is Lipschitz in logarithmic scale: along each coarse-graining step, ηη can only vary within uniform multiplicative bounds. Combining these geometric and probabilistic results with sharp-threshold assumptions for monotone tasks on Boolean and percolation models, the paper introduces a precise notion of stable semantic phases: regimes in which a semantic task remains reliably solvable across a range of resolutions and under the induced rescalings of the Boolean model. An interior stability theorem shows that phases far enough from critical thresholds are robust under coarse-graining, while a converse result yields necessary conditions: empirical multi-scale stability imposes nontrivial constraints on effective parameter scaling. In particular, the work derives device-independent intensity–resolution trade-offs, implying that any representation system claiming stable semantics across scales must respect universal inequalities relating dimension, density, and semantic capacity. The framework applies abstractly to internal representation manifolds, sensor networks, and other systems whose states admit an observation-geometry description, without reference to specific learning algorithms or architectures.
Takahashi K. (Fri,) studied this question.