Exact Critical-Scale Second Moments for Maaß Symmetric-Square L-Functions

We prove exact critical-scale second-moment bounds for Maaß symmetric-square L-functions at the spectral width G = T1/5, uniformly for microscopic vertical shifts U ≤ (log T)B. More generally, for fixed B, δ > 0, ∑ⱼ αⱼ hT,G(tⱼ) L( 1/2 + iU, sym² uⱼ ) ² B,δ TG(log T)C uniformly throughout T1/5 ≤ G ≤ T1-δ, U ≤ (log T)B. Gaussian majorization yields the corresponding harmonic sharp-window bound at the same exact power scale, and standard harmonic-weight removal gives an unweighted estimate of size T1+ε G. The exponent $1/5$, the shifted half-integral-weight Voronoi formula, the relevant saddle-point mechanisms, and the underlying quadratic second-moment inputs are not new individually. The contribution is the simultaneous closure of the known one-piece approximate-functional-equation / twisted-first-moment architecture at the exact critical width, without the positive-power enlargement present in the published endpoint hypotheses. The proof retains the full decay of the shifted approximate functional equation, routes the positive-discriminant range down to the first integer gap ($q = n - 2m = 1$), treats the microscopic hypergeometric transition directly from the exact Euler representation, replaces generic power-loss arithmetic estimates by logarithmic quadratic-family and Zagier-series interfaces, and closes the resulting dependencies through a one-way logarithmic bookkeeping scheme. Thus the paper establishes an exact endpoint theorem within the existing symmetric-square second-moment framework rather than claiming a new origin for the critical exponent itself. Scope This release should be read as an exact endpoint synthesis of established analytic ingredients. It does not claim that the exponent (1/5), the half-integral-weight Voronoi formula, saddle-point analysis, or quadratic Dirichlet second moments are new individually. The new point is that the complete collection of analytic and arithmetic interfaces can be closed simultaneously at the exact power width (G=T1/5) in the microscopic-shift regime. Citation Please cite the Zenodo record and DOI associated with Version v1.0 once assigned. Suggested bibliographic form: Lee, Byoungwoo. Exact Critical-Scale Second Moments for Maaß Symmetric-Square L-Functions. Version v1.0, Zenodo, 2026. DOI: 10.5281/zenodo.22160710. Files The principal file in this release is the PDF manuscript: Exact_Critical_Scale_Second_Moments_for_Maass_Symmetric_Square_L_Functions.pdf