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 $N = S_\zeta^2 - d \cdot S_L^2$, where $S_\zeta$ and $S_L$ are Lorentzian-weighted zero sums with $w(\rho) = 2/(\tfrac{1}{4} + \gamma^2)$. We prove three main results. (1) Spacelike spectral data: $N < 0$ unconditionally for all squarefree $d > 1$, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level $M$, $\mathrm{dens}\{N > 0\} \leq 2\|f_{S_L^{(M)}}\|_\infty \cdot (W_1(\zeta)/\sqrt{d} + \epsilon_M)$, established unconditionally via Jacobi--Anger resonance analysis. At fixed $M$ the bound is nontrivial only for sufficiently large $d$; the $O(1/\sqrt{d})$ rate requires $M$ to grow with $d$, which in turn requires a uniform density bound that we establish under a computationally verified finite-rank condition on the resonance lattice. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice $\Lambda_\infty$ has finite rank (verified to have rank $0$ for $M \leq 20$), the sharp asymptotic $\mathrm{dens}\{N > 0\} = C(d)/\sqrt{d} + o(1/\sqrt{d})$ holds. For $d = 5$, $C(5) = 2\,f_{S_L}(0)\cdot\mathbb{E}[|S_\zeta|] = 0.1193$; the constant depends on $d$ through the zeros of $L(s,\chi_d)$, and $C(d) = O(1/\log d)$ as $d \to \infty$. Appendix F tabulates between 1004 and 1044 zeros at 70 decimal places for $L(s,\chi_2)$, $L(s,\chi_3)$, $L(s,\chi_5)$, $L(s,\chi_6)$, $L(s,\chi_7)$, $L(s,\chi_{10})$, $L(s,\chi_{11})$, and $L(s,\chi_{13})$, all rigorously certified by ARB interval arithmetic.