Geometric Origin of Algebraic Solvability in Stochastic Gene Expression Cascades

摘要: The exact solvability of linear stochastic cascades is often treated as an algorithmic coincidence of reaction topology, leaving open the structural question of why non-equilibrium generators admit finite-dimensional algebraic reduction. We demonstrate that tractability originates not from linearity alone, but from a unique geometric representation that aligns Hilbert-space adjoints with algebraic grading. Using the classical two-stage gene expression model as a minimal testbed, we prove that a unique Poisson-weighted inner product converts the intrinsically non-Hermitian stochastic generator into a quasi-Hermitian, block-upper-triangular form. This metric-induced alignment unlocks the underlying Lie superalgebraic skeleton, yielding three exact dynamical consequences: (i) transient evolution reduces to a sequentially solvable filtration without truncation; (ii) the moment hierarchy closes rigorously at the expectation functional level; and (iii) the steady state emerges as a stationary zero-mode with an orthogonal Charlier expansion. Beyond explicit solutions, we derive a Reconstruction Criterion that delineates the boundary of finite-dimensional integrability and introduce a four-level solvability hierarchy, showing how rational feedback naturally descends to localized Weyl algebras while transcendental regulation forces analytic breakdown. All constructions are rigorously defined on a common invariant core, with physical convergence guaranteed in $\ell^1$ by Markov ergodicity. By shifting the analysis from case-by-case computation to representational classification, this work provides an \emph{a priori} diagnostic protocol for algebraic tractability in arbitrary birth--death networks.