Scale Selection in Iterated Function Systems and Banach Contractions via the Susceptibility Peak: A Contraction-Lens Framework on Finite Metric Structures

For a Hutchinson iterated function system (IFS), a Banach contraction on a complete metric space, or a finite-metric dynamical system, a natural question is: at which resolution σ does the contraction's geometric structure (fractal attractor, basin of attraction, periodic part) become optimally visible? We answer this by introducing a scale-selection principle: define the observation scale σ_c := argmax_σ χ(σ), where χ(σ) = |dO(σ)/d log σ| is the susceptibility of a resolution-dependent observable, and prove that σ_c exists under explicit boundary-regularity hypotheses.The framework's main quantitative results are three theorems specialised to Banach contractions and IFS: (i) a geometric scaling identity σ_c = qL for affine Banach contractions with operator norm q and basin scale L, applying directly to Hutchinson IFS with σ_c ∼ q · diam(K⋆); (ii) a discrete Banach theorem on finite metric structures under uniform Lipschitz Lip_d(f) = q < 1, giving an exact collapse-time N⋆ = ⌈log(∆/d_min) / log(1/q)⌉; (iii) a spectral concentration theorem placing σ_c at the inverse log-spectral-gap of the transfer operator at fixed positive noise. A stability lemma for canonical normalisation under smooth windowing and a parametric Banach correspondence observation complete the technical core.The framework is stated explicitly in the non-expansive Lipschitz regime Lip_d(f) ≤ 1 on the metric side, and at fixed positive noise ε ∈ (0, 1) on the spectral side. A four-type classification of operations by injectivity structure organises the broader landscape; cross-domain empirical evidence anchored on a peer-reviewed NISQ-hardware measurement of σ_c is summarised. The middle-thirds Cantor set IFS appears as the principal worked example.