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We present a complete axiomatic framework for New Subquantum Informational Mechanics (NMSI), a fundamental physical theory in which information, rather than energy, constitutes the ontological substrate of reality. The framework is based on ten interdependent axioms describing an eternal, oscillatory, and non-expansive universe, where all physical phenomena emerge from the modulation of informational density within a structured subquantum vacuum. NMSI eliminates spacetime singularities through multi-layer curvature stratification, resolves the Hubble tension (H₀) without ad-hoc parameters, and explains the presence of early mature galaxies observed by the James Webb Space Telescope (JWST) via accelerated structure formation at advanced informational phase. In this framework, dark matter is reinterpreted as a complementary informational phase of baryonic matter, eliminating the need for exotic particles, while dark energy ceases to exist as a physical entity and is replaced by a geometric phase gradient. We provide complete mathematical derivations for all axioms, a rigorous mapping between theoretical quantities and primary observables (spectroscopic redshift, luminosity and angular distances, galaxy rotation curves, gravitational lensing, and gravitational-wave signatures), as well as quantitative, testable predictions tied to specific instruments (DESI, JWST/NIRSpec, ANDES/ELT, LISA). The framework demonstrates full compatibility with local gravity tests (PPN formalism, Mercury perihelion precession, binary pulsars) and includes a detailed parametric sensitivity analysis. Crucially, NMSI is explicitly falsifiable through five independent experimental test classes: (1) temporal redshift variation (dz/dt), (2) direct detection of WIMP-like dark matter particles, (3) variation of the fine-structure constant α(Z), (4) deviations in gravitational-wave waveforms, and (5) anomalous lensing-to-baryonic mass ratios in galaxy clusters. The proposed framework satisfies the criteria of internal logical coherence, experimental falsifiability, and observational relevance, and offers a clearly formulated paradigm shift from energy-based to information-based fundamental physics.