Notes on slalom prediction
Takashi Yamazoe
TL;DR
This work develops a unified framework for slalom prediction and localization by introducing global, local, and infinite localization systems $\mathbf{GLc}(b,h)$, $\mathbf{LLc}(b,h)$, and $\mathbf{ILc}(b,h)$ alongside predictor-based systems $\mathbf{GPr}(b,h)$, $\mathbf{LPr}(b,h)$, $\mathbf{IPr}(b,h)$, and investigates their ZFC-provable relationships. Through Tukey connections, it derives monotonicity, limit behavior as width sequences $h$ grow, and explicit formulas for key cardinals such as $\mathfrak{b}$ and $\mathfrak{d}$ in the various localization regimes, including the $\,\omega$-base cases $\mathfrak{b}^\mathrm{GLc}_{\omega,h}$ and $\mathfrak{d}^\mathrm{GLc}_{\omega,h}$. The paper also analyzes how width-augmentation can invert the relative difficulty of predictions versus localizations, giving conditions under which $\mathfrak{e}^\mathrm{G}_{b,h}$ and $\mathfrak{pr}^\mathrm{G}_{b,h}$ attain their limit values and when $h\to\infty$ yields equality with related localization invariants. By relating these invariants to ideals on the reals (e.g., $\mathcal{N}$, $\mathcal{M}$, $\mathcal{E}$), it clarifies how slalom-based properties interact with additivity/non-additivity and uniformity, including specific results like $\mathop{\mathrm{add}}(\mathcal{NA})=\mathop{\mathrm{non}}(\mathcal{NA})=\minGLc$ and model-dependent separations for $\mathop{\mathrm{non}}(\mathcal{MA})$.
Abstract
We study a concept of evasion and prediction associated with slaloms, called slalom prediction. This article collects ZFC-provable properties on the slalom prediction.