${\lambda }_{\mathrm{\infty }}$: A New Mathematical Constant from the Spectral Theory of the Prime LCM Matrix

We introduce a new mathematical constant ${\lambda }_{\mathrm{\infty }}=0.674036183193696139936660007576508455780\dots$ (OEIS A396695), defined as the unique solution in $(1\mathrm{/}4,+\mathrm{\infty })$ of $h(x):={\sum }_{p\text{ prime}}1\mathrm{/}(x{p}^2-p+1)=1$. This equation arises as the $n\to \mathrm{\infty }$ limit of the secular equation of rank-$n$ truncations of the infinite prime LCM matrix $\mathcal{L}[p_i,p_j]=1\mathrm{/}\mathrm{l}\mathrm{c}\mathrm{m}(p_i,p_j)$, where $\mathcal{L}=D+v{v}^T$. Viewed as a compact self-adjoint operator on ${\mathrm{\ell }}^2(\mathcal{P})$, $\mathcal{L}$ has spectral radius $\rho (\mathcal{L})={\lambda }_{\mathrm{\infty }}$ (Theorem 5.4). The corresponding integer LCM matrix satisfies ${\lambda }_{\max }(W_N^{\ast })\mathrm{/}N\to {\lambda }_{\mathrm{\infty }}$ (Theorem 5.8), a prime-indexed counterpart to ${\lambda }_{\max }(M_N)\mathrm{/}N\to \zeta (2)={\pi }^2\mathrm{/}6$ for the integer divisor matrix. We prove that $h$ is real-analytic, strictly decreasing and strictly convex on $(1\mathrm{/}4,+\mathrm{\infty })$, ensuring existence and uniqueness of ${\lambda }_{\mathrm{\infty }}$. We compute 500 decimal digits of ${\lambda }_{\mathrm{\infty }}$, certified by rigorous error bounds and independently verified by six computational runs, including a fully independent recomputation (Run E, PARI-GP) and a machine-verifiable Arb interval certificate (Run F) covering 505 digits. Extensive PSLQ and LLL searches find no minimal polynomial of degree $\le 8$ satisfied by ${\lambda }_{\mathrm{\infty }}$ (at 560 decimal digits), and no integer relation against catalogs of up to 31 classical constants. The arithmetic nature of ${\lambda }_{\mathrm{\infty }}$ remains open.