Unconditional Density Bounds for Quadratic Norm-Form Energies via Lorentzian Spectral Weights

Abstract: For a real quadratic field $\mathbb{Q}(\sqrt{d})$, we study the norm-form energy $N = S_ζ² - d \cdot S_L<sup>2$, where $S_ζ$ and $S_L$ are Lorentzian-weighted zero sums with $w(ρ) = 2/(1/4 + γ<sup>2)$. We prove three main results. (1) Spacelike spectral data: $N &lt; 0$ unconditionally for all squarefree $d &gt; 1$, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level $M$, $\mathrm{dens}{N &gt; 0} \leq 2|f_{S_L^{{(M)}}|_\infty} \cdot (W_1(ζ)/\sqrt{d} + ε<em>M)$, established unconditionally via Jacobi--Anger resonance analysis. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice $Λ</em>\infty$ has finite rank (verified for $M \leq 20$, where rank $= 0$), the sharp asymptotic $\mathrm{dens}{N &gt; 0} = C(d)/\sqrt{d} + o(1/\sqrt{d})$ holds. For $d = 5$, $$C(5) = 2\,f_{S_L}(0)\cdot\mathbb{E}[|S_ζ|] = 0.1191$$; the constant depends on $d$ through the zeros of $L(s,χ_d)$, and $C(d) = O(1/\log d)$ as $d \to \infty$.