Cognitional Mechanics Cosmology: Time, Space and Order Emerging from M3(C) Structure

Cognitional Mechanics Cosmology (CM Cosmology) reconstructs time, space, redshift, and large-scale structure from the unique minimal algebra M₃(ℂ). Rather than postulating a Big Bang singularity, inflation, dark matter particles, or dark energy fields, the framework derives all major cosmological observables from a closed set of algebraic axioms with zero free parameters. Building on this foundation, we introduce Permanent Operational Cosmology (Japanese: Koukyuu Enzan Uchu-ron) as the cosmological realization of CM. It describes a universe governed not by an initial creation event or a future thermal end, but by continuous non-commutative operations whose algebraic structure inherently excludes both absolute beginning and termination. Within this framework, time emerges as the ordering of non-commutative operations, three-dimensional space arises as a necessary projection of the algebra, and cosmological redshift follows from a uniform temporal modulation without requiring spatial expansion. The cosmological constant vanishes as a structural identity, cosmic flatness follows from the algebra's natural metric, and the derived CMB temperature and matter/energy density fractions match observational data to within one percent. Furthermore, information is conserved as the invariant rank of the density matrix, resolving the black-hole information paradox by showing it stems from treating dynamical boundary conditions as static horizons. Apparent cosmological tensions arise when the continuous operational dynamics of the underlying algebra are interpreted through static, snapshot-like models. Permanent Operational Cosmology provides a mathematically complete, zero-parameter alternative to the standard ΛCDM model, grounded in a single algebraic structure that unifies cosmic evolution, thermodynamics, and quantum information.