Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Leshem, Amir
Institution: Bar-Ilan University
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Israel Science Foundation
Award: G7198679574
OpenAlex ID: W7201854838
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