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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19212911 |
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- <p>This paper develops a finite-level gain theory for Recursive Interval Geometry (RIG). Working in the finite-level extraction formalism over $<span><span>\mathbb Z[\Omega^{-1}]$</span></span>, it establishes an exact channel decomposition of the full-depth leaf energy, derives the sharp extraction gain law for every word operator <span><span>ω\omega</span><span><span><span>ω</span></span></span></span>, characterizes the extremal channel-supported states, determines the attainable gain set, and proves the pointwise composition law for successive extractions.</p> <p>The main mathematical point is that the gain is governed exactly by the structural <span><span>$</span><span><span><span><span>\len$</span></span></span></span></span>-count <span><span>$N_{\len}(\omega)$</span></span>. This yields a concrete and self-contained finite-level framework in which extraction, localization, and composition can be analyzed explicitly.</p> <p>Beyond the present results, the paper is intended as a preparatory step toward several later directions in the RIG program. In particular, the orthogonal channel decomposition and gain formalism are designed to support future defect bounds and quantitative off-channel estimates, and to serve as part of the mathematical foundation for later work on RIG information theory and RIG dynamical systems.</p> <p>This Zenodo record is posted to provide a stable public version of the manuscript and to support transparent citation and dissemination.</p> <p> </p>