ora-0127

The conventional online-learning protocol, static regret, and its convex specialization are developed systematically by Shalev-Shwartz [ 2012 ] and Hazan [ 2016 ] ; the expanded second edition of Hazan’s tutorial supplies the milestone sequence used later in this book [ Hazan , 2023 ] . These accounts provide the classical-chain case recovered in 9.21. The separation made here among progressive revelation, comparator information, aggregation, and the comparison observer is the UODL reformulation rather than a claim made by those sources.

For nerves, simplicial sets, inner horns, and quasi-categories, consult Richter [ 2020 ] and Riehl and Verity [ 2022 ] . Riehl [ 2014 ] develops homotopy-coherent diagrams and derived universal constructions at the level needed to replace strict online histories by coherent ones. Those texts supply the homotopical machinery; the use of skeletal growth as online revelation, horn fillers as decision repairs, and semantic localization as persistence is specific to the present framework.

The information-relative account begins with Witsenhausen’s analysis of information structures, feedback, and causality [ Witsenhausen , 1971 ] and his intrinsic formulation of discrete stochastic control [ Witsenhausen , 1975 ] . A categorical extension of the intrinsic model appears in Mahadevan [ 2021 ] . Together these works motivate the passage from one global clock to an information category, and from replaying realized observations to evaluating each policy on its own induced closed-loop history.

Finally, Helbling et al. [ 2026 ] supplies the learned recurrent mechanism in the Sudoku example and the Fixed-Point Forcing procedure discussed in 9.16. The crossword and Sudoku limit constructions, the admission map from recurrent fixed points to the universal solution object, and the separation of dynamical from constraint defect are introduced here. They should therefore be read as proposed UODL semantics for constraint reasoning, not as results attributed to the flow-reasoning paper.