Some memos on Stable Symplectic Structured Space

In these memos, we define a pregeometry ${\mathcal{T}}_{\mathbb{S}}^{alg}$ and a geometry ${\mathcal{G}}_{\mathbb{S}}^{alg}$ which integrate symplectic manifolds with $E_{\mathrm{\infty }}$-ring sheaves, enabling the construction of ${\mathcal{G}}_{\mathbb{S}}^{alg}$-schemes as structured $\mathrm{\infty }$-topoi. Our framework and results establish a profound connection between algebraic invariants and homological properties, opening new pathways for exploring symplectic phenomena through the lens of higher category theory and derived geometry.