Appendix C Lemma 1: Laplacian Clarification

• Definition 4/Appendix C Lemma 1’s global strict inequality → a weak inequality globally, strict only on overlap. For $f(x)=e^{-\lambda|x|}$, $\lambda\geq0$, and $a>b,c>d$, $$f(a-c)f(b-d)\geq f(a-d)f(b-c).$$ It is strict when $\lambda>0$, $b<c$, and $d<a$, so $(b,a)$ and $(d,c)$ overlap. Source: Algorithmic Monoculture and Social Welfare.
• Evidence: at $\lambda=1$ and $(a,b,c,d)=(11,10,2,1)$, both products equal $e^{-18}$. Separated intervals lie on linear absolute-value pieces, whose cross difference cancels; overlap makes that difference strict. Density normalization cancels from the comparison. Integrating the weak global inequality with strictness on a positive-mass overlap region supplies the downstream strict Laplace ranking comparison.