Derived Stratified-Microlocal Framework and Moduli Space Resolution for the Cheeger-Goresky-Macpherson Conjecture

Abstract

In this paper, We define the stratified metric $\infty$-category ${\mathbf{StratMet}}_{\infty }$ and the middle perversity moduli stack ${\mathcal{M}}^{\mathrm{mid}}$. We construct a universal truncation complex $\hat{I}{\text{c◯}}_{X,\mathrm{FS}}^{\bullet ,\mathrm{univ}}$ for a projective variety $X\subseteq {\mathbb{P}}^N$. By introducing the stratified singular characteristic variety ${\mathrm{SSH}}_{\mathrm{strat}}$, we establish a microlocal correspondence between metric asymptotic behavior and topology, proving the natural isomorphism $$H_2^{\ast }(X_{\mathrm{reg}},d{s}_{\mathrm{FS}}^2)\cong I{H}^{\ast }(X,\mathbb{C}).$$ This framework transcends transverse singularity constraints, achieves moduli space parametrized duality, and develops new paradigms for high-codimension singular topology, quantum singularity theory, and $p$-adic Hodge theory.