Sequential-Innovation Reducibility and the Innovation Spectrum

2026-08-05

AI Summary

This work proposes a novel reduction framework grounded in causal predictability that transcends classical truth-table reducibility. By defining an “innovation spectrum” for individual binary sequences via sequential prediction error, the authors introduce “sequential innovation reducibility,” which captures a finer-grained notion of information extraction between sequences. This reducibility reveals a new geometric structure of the degree-theoretic landscape, composed of a truth-table backbone and a reservoir-immune region. The study establishes that reservoir immunity is invariant under sequential innovation reducibility, and within this immune region, Martin–Löf random degrees form a proper downward-closed substructure. Furthermore, the authors construct a bridging mechanism that connects these two distinct regions, thereby unifying their structural properties within the proposed framework.

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.

Problem

• sequential prediction
• innovation spectrum
• reducibility
• truth-table reducibility
• Martin-Löf randomness

Innovation

• sequential innovation
• innovation spectrum
• truth-table reducibility
• reservoir immunity
• Martin-Löf randomness