PAPER-DCQ3: Neutral Tension Spaces and Categorical Statistics

The first two papers in this series established a geometric foundation for a theory emerging from discrete combinatorics. In Paper DCQ1, a phase-encoded embedding ι : H6 ↩→ Gr(3, 6) created an isometric correspondence between a six-bit space and a complex Grassmannian, endowing the discrete structure with a natural quantum interpretation and a topologically quantized Berry geometry Paper DCQ1.Paper DCQ2 demonstrated that this embedding induces a dynamical substructure: the discrete states reside as non-degenerate minima of a Morse function on a 6-dimensional phase-orbit submanifold N ≃ (CP1)3, with the Berry curvature further constraining the effective, physical degrees of freedom to an emergent four-dimensional arena Paper DCQ2. This geometric picture, however, leaves two fundamental gaps that the present work aims to fill: From kinematics to dynamics. Paper DCQ1 constructed the state space, andPaper DCQ2 identified the classical stable points (Morse minima) and geometricconstraints on that space. Yet a quantum principle to define transition amplitudesbetween these states, and a variational principle to derive equations of motion,were still missing. Paper DCQ3 introduces the “neutral tension space” and theassociated path integral to provide precisely this dynamical principle. From implicit to explicit statistics. The 20+4 decomposition in Paper DCQ1hinted at a bosonfermion statistical split, but the statistical rules appeared thereas a consequence of representation theory, not as actively implemented principles.A central goal of Paper DCQ3 is to elevate statistics itself symmetry vs. antisymmetrytoan architectural, firstprinciple feature encoded by category theory andtopology (see Section 2 and Appendix D).