What question did this study set out to answer?

The aim is to explore the geometric and dynamical implications of a phase-encoded embedding in quantum systems.

March 8, 2026Open Access

PAPER-DCQ2-Emergent Dimensionality, Phase–Orbit Geometry, and Morse–Thimble Convergence

Key Points

The aim is to explore the geometric and dynamical implications of a phase-encoded embedding in quantum systems.
Developed a geometric framework connecting discrete and continuous quantum states.
Analyzed emergent dimensional reduction to a 4-dimensional Hausdorff space.
Investigated phase-orbit geometry related to quantized Berry curvature and Chern numbers.
Constructed a Morse function characterizing the structure of discrete code states.
Identified an effective reduction from an 18-dimensional space to a 4-dimensional space.
Described phase-orbit geometry with distinct quantized Berry curvature and Chern numbers.
Established a Morse-theoretic structure with 64 minima corresponding to discrete code states.
Provided a foundational framework for understanding structural equations of motion.

Abstract

Paper DCQ1 established a discrete–continuous–quantum correspondence between the six-bit binary configuration space H6 = ±16, the Grassmannian Gr (3, 6), and the fermionic Fock space Λ3 (C6), via an isometric phase-encoded embedding. This second paper develops the geometric and dynamical consequences of thatembedding: • Dimensional harmonization: Emergent reduction from the 18-dimensionalGr (3, 6) to an effective 4-dimensional Hausdorff configuration space Cδ. • Phase-orbit geometry: The submanifold N ≃ (CP1) 3 carrying quantizedBerry curvature with Chern numbers (1, 1, 1). • Morse-theoretic structure: A natural Morse function on N with exactly 64 non-degenerate minima at the discrete code states, leading to a convergent thimble decomposition. These results provide a pre-dynamical foundation linking discrete states to continuousgeometric dynamics and setting the stage for structural equations of motion.

PAPER-DCQ2-Emergent Dimensionality, Phase–Orbit Geometry, and Morse–Thimble Convergence

Key Points

Abstract

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