Graded Branch Unpredictability Near Nonlinear Transitions: A Decision-Scale Framework and Held-Out Energy-Balance Demonstration

Abstract

Forecast laws near nonlinear transitions may be poorly concentrated even when they are not cleanly bimodal. For a conditional law P(t,h) on a metric state space, we study Uδ(t,h) = 1 − supₐ P(t,h){d(Y,a) ≤ δ}: the minimum probability that a point forecast misses the realization by more than tolerance δ. The quantity is the complement of the classical concentration function; the contribution is its assembly into a decision-scale framework with excess-error severity, temporal exposure, information gain, and mechanism ambiguity. For K resolved equal-probability branches, Uδ = 1 − 1/K, while Kδ^(∞) = (1 − Uδ)⁻¹ is an exponentiated local min-entropy. Unlike relative-entropy predictability, which measures information relative to climatology, Uδ measures residual point-action risk at a declared physical resolution. Additional observations reduce its expected value under Blackwell ordering, and historically indistinguishable mechanisms with future separation Δ impose a two-point squared-risk bound e₀Δ²/2. The principal numerical test uses stochastic zero-dimensional energy-balance models with two radiation closures. A prospectively frozen high-resolution holdout at previously unused feedback softness m = 3 comprises 48 parameter cases: two radiation laws, two horizons, four transition timings, and three diffusion levels. Every case contains a non-bimodal episode with U₁K ≥ 0.35 and U₈K < 0.20. Median exposure above U₁K = 0.25 is 0.549 and 0.409 of the forecast interval, increases of 0.432 and 0.263 over fixed sharp-feedback baselines. Both primary scientific endpoints and every numerical endpoint pass; two stronger secondary hypotheses do not generalize across both radiation laws. Exploratory sharp-transition scaling and a historical hidden-Markov pilot remain nonconfirmatory. The demonstrated result is therefore specific but affirmative: moderate decision-relevant unpredictability can persist much longer than extreme branch ambiguity within the declared nonlinear model family. Version 2.0 is deposited as the fixed public baseline for subsequent prospectively preregistered extensions and additional calculations. The record contains the main paper, supplementary material, complete reproducibility package, and SHA-256 checksum file. Archive integrity — SHA-256 of the complete reproducibility ZIP: 25edd0c4c3b21fb0f5aa6e0a7de3505ae3fe212c574a5d12f6a83c6f776fcb11