A Parameter-Free Spectral Framework: Generator Rigidity, Observable Algebra, and Deterministic Predictions from Compact Operator Theory

This work develops a fully deterministic and parameter-free mathematical framework in which all observable quantities arise as structural consequences of a finitely generated observable algebra. Starting from a minimal generator set, we construct the closure algebra of observables and realize it as operators acting on a Hilbert space. Under standard ellipticity and coercivity conditions, the associated operator admits compact resolvent, implying a discrete spectrum. All predictions therefore emerge as eigenvalues determined solely by the underlying algebraic structure, without phenomenological parameters or fitting procedures. The entire derivation is deductive and theorem-based. No probabilistic assumptions or empirical calibration are introduced at any stage. Each step follows from classical functional analysis and spectral theory. A Lean 4 formalization is provided to enable machine verification of the logical core. The project is designed so that all constructions are, in principle, checkable by the Lean proof kernel. Contents include:• full LaTeX source of the paper• compiled PDF• Lean 4 formalization skeleton This repository aims to provide a rigor-maximal, reproducible, and machine-verifiable foundation for deterministic observable prediction.