Quantum-Computational Methods of the Ψ-Model: Algorithms, Derivations, Advantages, and Verification Protocols

Abstract

This preprint presents a self-contained scientific account of twenty quantum-computational algorithms and algorithmic directions arising from the Ψ-model. The central organizing principle is representation-first computation: whenever a task contains deterministic, quasi-deterministic, or geometrically structured contributions, those contributions are extracted from brute-force optimization and transferred to the representation layer where they become simplest. The paper develops χ-aware frame control for deterministic phase-drift compensation, ψ-preconditioning as an exact operator rewrite, ψ-QLSA and ψ-QSP/QSVT as solver-level consequences of that rewrite, projector-native low-band methods, transport between equivalent low-band carriers through an intertwiner J, topology-driven classification procedures, anomaly- and holonomy-based selectors, and detector-aware measurement formalisms. Exact components are accompanied by full derivations; benchmark and synthetic components are paired with explicit falsifiers, anti-tautological metrics, and frozen verification protocols. The result is a coherent representation-aware stack for quantum computation rather than one isolated algorithm. This record contains both the English and Ukrainian PDF versions of the manuscript.