From rough local mass to logarithmic spectral ladders for damped waves

We prove an exact inverse theorem for damped waves on $X=\mathbb{T}\times Y$ with arbitrary bounded measurable transverse damping $a=a(x)\ge 0$. Let ${\mathrm{\Theta }}_a(r)$ be the least average of $a$ on radius-$r$ intervals. Then ${\mathrm{lim\hspace{0.17em}inf}}_{r\downarrow 0}{\mathrm{\Theta }}_a(r)>0$ characterizes exponential stability, while for every $L(S)=\mathrm{\ell }(\log (eS))$ with $\mathrm{\ell }:[1,\mathrm{\infty })\to [1,\mathrm{\infty })$ nondecreasing, unbounded, and eventually doubling, ${\mathrm{\Theta }}_a(r)\gtrsim L(1\mathrm{/}r{)}^{-1}$ is equivalent to stationary- and generator-resolvent bounds $O(\mathrm{\mid }s{\mathrm{\mid }}^{-1}L(\mathrm{\mid }s\mathrm{\mid }))$ and $O(L(\mathrm{\mid }s\mathrm{\mid }))$. One resonant scalar block per dyadic octave already recovers this multiscale mass bound. Every such critical profile occurs for indicator damping on an open dense set of arbitrarily small measure, whereas the stationary resolvent-to-mass implication fails at every sublinear power gauge. For the cusp $a(x)=(\log (e\mathrm{/}\mathrm{\mid }x\mathrm{\mid }){)}^{-A}$, $A>0$, near its isolated damping well, we prove the sharp obstruction is genuine spectrum. Blow-up yields $-{\mathrm{\partial }}_y^2+i\log \mathrm{\mid }y\mathrm{\mid }$, whose spectrum is a simple interlaced vertical ladder. Every fixed finite ladder portion transfers to damped-wave eigenvalues with complete blockwise algebraic count and two-term asymptotics. Combined with the resolvent theorem, these eigenvalues give the sharp regularized decay scale $\exp [-t^{1\mathrm{/}(A+1)}]$.