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

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 $w(\rho )=2\mathrm{/}(\frac{1}{4}+{\gamma }^2)$. 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 $M$, $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\{N>0\}\le 2\mathrm{\parallel }{f}_{S_L^{(M)}}{\mathrm{\parallel }}_{\mathrm{\infty }}\cdot (W_1(\zeta )\mathrm{/}\sqrt{d}+{\epsilon }_M)$, established unconditionally via Jacobi–Anger resonance analysis. At fixed $M$ the bound is nontrivial only for sufficiently large $d$; the $O(1\mathrm{/}\sqrt{d})$ 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 ${\mathrm{\Lambda }}_{\mathrm{\infty }}$ has finite rank (verified to have rank $0$ for $M\le 20$), the sharp asymptotic $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\{N>0\}=C(d)\mathrm{/}\sqrt{d}+o(1\mathrm{/}\sqrt{d})$ holds. For $d=5$, $C(5)=2f_{S_L}(0)\cdot \mathbb{E}[\mathrm{\mid }{S}_{\zeta }\mathrm{\mid }]=0.1193$; the constant depends on $d$ through the zeros of $L(s,{\chi }_d)$, and $C(d)=O(1\mathrm{/}\log d)$ as $d\to \mathrm{\infty }$. 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.