The Problem Generation Framework of Switch Measure Theory: Five Illustrative Results, 137 Open Problems, Five Deep Problems, and the Impossibility Boundary

This paper establishes the problem generation framework of Switch Measure Theory, and presents five illustrative results and five deep problems of the framework. The framework takes the problem generation operator G as its core tool and the order operators G^k and the graded meta-ization operator Ψ_k as its recursive structure, and systematically generates open problems on the proved theorem basis T = {T_1, …, T_29} (numbering consistent with Tang 2026aw) and the extension direction set D = {D_1, …, D_7}. On the precise meaning of “generation”: the “generation” in this paper refers to structured classification, not algorithmic generation—G is a criticizable classification framework (see Section 2.2 for details). Five illustrative results: (1) the computational complexity upper bound of the switching degree is O(p²) (or O(p^ω) if the precision-matrix form is used), with the exact lower bound left as an open problem; (2) a strict bijection between no-horizontal-step switching sequences and Dyck paths, with the sequence count equal to the Catalan number C_n; (3) via the one-sided inverse Lipschitz condition, the two-step estimator satisfies a minimax lower bound consistent with the boundary information lower bound c=1/8; (4) the switching spectrum satisfies a third-moment Lyapunov condition in high dimensions, and is asymptotically normal; (5) the concrete quadratic form ĥ(τ) produces a pseudo-peak under heterogeneous configurations. Three naming proposals: switching manifold, switching category, regime algebra—these are the author’s convenient names for certain mathematical objects, not claims of terminological priority. Five deep problems (in three levels): (Level One, firmly grounded) D1 non-regular Le Cam theory, D2 the optimal rate spectrum of nonparametric continuous switching; (Level Two, requiring precise construction) D3 the classifying space of the switching category, D4 a sheaf-theoretic interpretation of no false peaks; (Level Three, exploratory) D5 homotopy characterization of the three-valued η. These five problems do not occupy Q numbering or Ω numbering. Order structure: first-order problem set P_0 (100 concrete problems), second-order problem set P_1 (Ω_1–Ω_8), third-order problem set P_2 (Ω_9–Ω_29), three cross-border problem sets with distinct object types (P_cat, P_Gödel, P_term), and the system-level problem set P_sys (Ω_30–Ω_34). This paper proposes a total of 137 open problems: 100 first-order problems, 29 meta-problems, 3 cross-border problems, and 5 system-level problems. Of these, 54 are marked as N-level (first-of-their-kind problems). In addition, this paper proposes five deep problems and eight impossibility conjectures (IMP-01–IMP-08). Together with Tang (2026av, system outline) and Tang (2026aw, mathematical outline), this paper constitutes the complete program of Structural Regime Mathematics. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.