Self-Reference, M¨obius Iteration, and the 13th Axis: The SL(2,Z) Structure of Recursive Observation in the R12 Rendering Algebra

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Main Author: Lim, Han-Jun
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Published: Zenodo 2026
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author Lim, Han-Jun
author_facet Lim, Han-Jun
contents <p>Abstract</p> <p>The rendering algebra R12 ∼= M12(C) of Array Cosmology uses 12 axes, yet the ratio 13/12 appears universally in physical formulae—the proton mass (144 × 13/2), the baryon knot constant (Kknot = (13/12)Znoise/144), and the proton-toelectron mass ratio (µ = 1836 + (13/12)(π − 3)). We show that this “13th axis” is the mathematical signature of recursive self-observation, governed by the M¨obius transformation f(x) = x/(1 − x) whose matrix M =1 0−1 1 ∈ SL(2,Z). Thirteen<br>A-layer results are proved.</p>
format Recurso digital
id zenodo_https___doi_org_10_5281_zenodo_19153044
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publishDate 2026
publisher Zenodo
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spellingShingle Self-Reference, M¨obius Iteration, and the 13th Axis: The SL(2,Z) Structure of Recursive Observation in the R12 Rendering Algebra
Lim, Han-Jun
<p>Abstract</p> <p>The rendering algebra R12 ∼= M12(C) of Array Cosmology uses 12 axes, yet the ratio 13/12 appears universally in physical formulae—the proton mass (144 × 13/2), the baryon knot constant (Kknot = (13/12)Znoise/144), and the proton-toelectron mass ratio (µ = 1836 + (13/12)(π − 3)). We show that this “13th axis” is the mathematical signature of recursive self-observation, governed by the M¨obius transformation f(x) = x/(1 − x) whose matrix M =1 0−1 1 ∈ SL(2,Z). Thirteen<br>A-layer results are proved.</p>
title Self-Reference, M¨obius Iteration, and the 13th Axis: The SL(2,Z) Structure of Recursive Observation in the R12 Rendering Algebra
url https://doi.org/10.5281/zenodo.19153044