32. Geometric Subsystem Quantisation programme: Rigidity, Decidability, and a Linearised Spectral Bridge

We present four self-contained rigorous theorems and one finite-to-continuum spectral convergence result in the Geometric Subsystem Quantisation programme. The first theorem establishes rigidity of the Weyl-polynomial quantisation of the reduced harmonic oscillator under spectral matching. The second gives an exact criterion for separation of finite algebraic quantisation corrections by parent observables. The third classifies global sector labels under finite monodromy. The fourth provides an obligation-reduction engine for polynomial quantisation constraints. All proofs are complete for the stated classes. Standard foundational results from real algebraic geometry and computational algebra are cited rather than reproved. No result claims physical QFT uniqueness beyond the explicitly declared finite-dimensional algebraic setting.