The Contraction Lens: Observation Scales and Non-Injective Operations Across Mathematics and Physics

Non-injective maps on finite structures—maps where distinct inputs can share
an output—contract their image under iteration. We introduce the observation
scale σc, the resolution at which a non-injective map’s contraction geometry is opti-
mally visible, defined via a susceptibility peak in a resolution-dependent observable.
We prove that σc exists for every non-injective map on a finite structure and show
that the scale has been detected across five physical domains spanning twelve or-
ders of magnitude, with statistically significant peaks (p < 0.02) in each case. As
a secondary contribution, we propose a four-type classification of mathematical op-
erations by injectivity structure: contraction (Type D), oversaturation (Type O),
symmetry constraint (Type S), and preservation (Type R). The companion paper [1]
develops the core theory for Type D; here we develop σc, identify physical instances
of contraction, and apply the classification to illustrative examples including Gold-
bach’s conjecture (Type O) and the Riemann hypothesis (Type S), without claiming
resolution of either.