Emergent_Comparison_Geometry_from_Internal_Consistency_of_Ratio_Fields

We study systems in which only relative comparison under iteration is observ-
able, encoded by ratio fields generated by repeated transformations.
Without assuming topology, metric, algebra, probability, or order, we show
that a single structural requirement—Internal Consistency on decomposition
space—forces an emergent comparison geometry.
This produces: (i) an asymptotic preorder, (ii) multiplicative rigidity, (iii)
representation invariance, and (iv) an obstruction functional measuring failure
of asymptotic compositional closure. This framework can be interpreted as a
general form of coarse-grained dynamical systems in which multiplicative com-
parison corresponds to effective renormalization flow, and internal consistency
enforces asymptotic closure of composition laws