Cognition as Non-Closing Delta-Indexing: In'ei Geometry as Projection-Density Field

This work reformulates cognition, geometry, and chaos under a unified projection-density framework. Instead of treating closure as a primitive, it defines observable structures as local readability phenomena emerging from bounded projection capacity over a non-closing Delta-indexing process. Figures such as circles, cycles, and attractors are interpreted as projection-dependent appearances within a density field rather than intrinsic forms. Chaos is redefined as projection breakdown caused by density overflow, not randomness. The paper introduces local projection density, readability threshold, and the In'ei field as minimal descriptive operators, and positions the In'ei spiral as a generative basis for projection-dependent appearance. Within this framework, Wasan and chaos are treated not as opposites but as distinct phases of a single projection-density continuum.