Symmetric-Power Stability for GL(2) in the Grid Theory Framework (Part I: 2 ≤ 𝑚 ≤ 4 )

# Overview This record releases **“Symmetric-Power Stability for GL(2) in the Grid Theory Framework (Part I: 2 ≤ m ≤ 4)”** as a PDF-only preprint. The paper packages the low-degree symmetric-power operations\(\{\mathrm{Sym}^m : 2 \le m \le 4\}\) as an **importable contraction module** inside theGrid Theory operadic layer, with a parameter-box constant policy. # Main statement (module form) For non-CM holomorphic \(\mathrm{GL}(2)\) newforms \(\pi\) in a balanced admissible regime,and for each \(m \in \{2,3,4\}\), the paper proves a contraction inequality for theexplicit-formula global energy:\[ E_{\mathrm{glob}}(\mathrm{Sym}^m\pi) \le \lambda_m^2\, E_{\mathrm{glob}}(\pi), \qquad 0 < \lambda_m < 1,\]where \(\lambda_m\) depends only on the fixed parameter box PB and on \(m\).Equivalently, for the windowed mean-square deviation statistic \(\mathrm{MSD}_w\),\[ \mathrm{MSD}_w(L(s,\mathrm{Sym}^m\pi);T)\le \lambda_m\, \mathrm{MSD}_w(L(s,\pi);T).\] # Structural ingredients - **Ideal finite-place rate:** an \(\mathrm{SU}(2)\) character \(L^2\)-identity yields the sharp ratio \(4/(m+1)^2\) for normalized traces.- **Window transfer:** a weighted Sato–Tate proximity condition on a finite polynomial packet (formulated as an input module / hypothesis, and amenable to direct numerical validation).- **Calibration:** a Gram/norm-equivalence alignment yielding an explicit constant \(C_{\mathrm{eq}}(\mathrm{PB})\) that enters \(\lambda_m\).- **Archimedean attenuation:** a multiplicative attenuation factor \(\theta_\infty\) in balanced windows. # Scope & limitations - The low-degree range \(2 \le m \le 4\) uses **known functorial closure** (Gelbart–Jacquet; Kim–Shahidi; Kim).- Any “exact GUE / MUGS” linkage appears only as an **optional plugin** and is not required for the main theorem. # Citation Byoungwoo Lee, “Symmetric-Power Stability for GL(2) in the Grid Theory Framework, Part I: 2 ≤ m ≤ 4”, version v2.3.