Sequential-Innovation Reducibility and the Innovation Spectrum

Amir Leshem

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arXiv · 2608.05443

Sequential-Innovation Reducibility and the Innovation Spectrum

Amir Leshem

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Abstract

Sequential prediction naturally induces an innovation sequence consisting of the prediction errors produced by a causal predictor. We use the collection of all such innovation sequences to define the \emph{innovation spectrum} of an individual binary sequence and, from it, a new reducibility based on sequential information extraction. We show that this reducibility refines truth-table reducibility while exhibiting a fundamentally different geometry. The degree structure decomposes into two canonical regions: a truth-table spine, whose induced order is isomorphic to the truth-table degrees, and a complementary reservoir-immune region, consisting of sequences from which no infinite computable predictable reservoir can be extracted. We establish bridge constructions connecting the two regions, prove that reservoir immunity is preserved under sequential innovation, and show that the Martin--Löf-random degrees form a proper downward-closed substructure inside the reservoir-immune region. These results reveal a new geometric organization of individual sequences based on causal predictability rather than classical oracle computation.

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Amir Leshem. 2026-08-05. Sequential-Innovation Reducibility and the Innovation Spectrum. https://arxiv.org/abs/2608.05443

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