Empirical Clustering Patterns in Binary Partition Ratios of Parametric Functions: A Large-Scale Computational Study

We present a large-scale computational investigation into the binary partition ratio
R(f ) = μ({x ∈ [0, 1] : f (x) > 0})/μ({x ∈ [0, 1] : f (x) ≤ 0}) and the normalized posi-
tive fraction p(f ) = R(f )/(1 + R(f )) for an ensemble of 684,000 parametrically generated
functions on the unit interval [0,1]. Discrete approximations on fine grids reveal multimodal
clustering of p(f ) around values derived from mathematical constants, including a prominent
mode near p ≈ π/(π + 1) ≈ 0.7585, which lies close to the rational 3/4 (relative difference
≈1.1%).
Supporting indicators for select modes include reduced local variance, delayed hierar-
chical emergence, and differential frequencies. Clustering exhibits marked sensitivity to
generation protocols (coverage ranging from 98.8% in weighted sampling to 35.0% in uncon-
strained random sampling), highlighting representational biases while revealing a persistent
baseline exceeding random expectations.
A key additional finding is approximate logarithmic additivity under function compo-
sition, log R(f ◦ g) ≈ log R(f ) + log R(g) (R2 ≈ 0.92), suggesting valuation-like structure.
Results characterize pre-asymptotic behavior in finite-parametric function spaces, prior to
the high-dimensional concentration toward balance established in the companion paper [1].
Methodological details are provided for full reproducibility, including ablation studies linking
modes to functional classes.