Residual Structure Theory in Modal Collapse Systems

This paper develops a formal theory of residual structure in modal collapsesystems. We introduce three central components: a conjectured residualboundedness law, an entropy--based attractor theorem, and a spectral modeconvergence hierarchy. The modal collapse function $\phi(x)$ produces astructural approximation to empirical density, while the residual field\[\delta(x)=\frac{\pi(x)}{x}-\rho(x)\]quantifies deviations from this approximation. The entropy field\[H(x)=\log(1+\delta(x)^2)\]reveals attractor behavior: minima of $H$ coincide with collapse points andsatisfy the proportionality law\[\frac{d\log\Phi}{d\log H}=1.\]This is the Structural Conservation Index (SCI) condition for $K=1$. The theory advances a Unified Resonant Structure Formula (URSF) anddemonstrates that residual feedback---central in cognitive geometry andinformation dynamics---can be made mathematically operational inside modalsystems.