Dynamic Complex Geometry: Unifying Euclidean Space to Calabi-Yau Manifolds from the Phase Coupling Equation
- 1Independent Researcher, Chongqing, China
Abstract
Classical differential geometry treats space as an a priori existing object, built upon the real number field—real numbers can only describe “how large” but not “how to turn”. This paper proposes Dynamic Complex Geometry, with the core proposition: All mathematical spaces are steady-state configurations of the phase field. A spiral curve is uniquely determined by its pitch function κ = dβ/dγ, with curvature and torsion as derived quantities; the topological charge I = (1/2π)∮ κ dγ reduces abstract invariants to measurable geometric displacements. Complex analysis, differential geometry, and topology are shown to be different-dimensional projections of the same spiral curve. Euclidean space, Riemann surfaces, and Calabi-Yau manifolds are steady-state solutions of the same equation under three different boundary conditions. On this basis, the paper reveals the generative origin of gravity: the field equations of General Relativity are not independent postulates but inevitable derivations from phase closure and topological conservation under emergent continuity; flat space is not the geometric ground state but a high-energy constrained state; gravity is not a force but the natural tendency of space to return to a low-energy curved state. Dynamic Complex Geometry further yields testable predictions in the strong-field limit—including vacuum dispersion, the emergent approximateness of the equivalence principle, the nature of black holes as dead states, and the emergent nature of the cosmological constant—these effects are necessary consequences rigorously derived from first principles, rather than phenomenological assumptions of traditional quantum gravity programs. This paper is the geometric chapter of the L1 layer of the Generativism series, with Dynamic Complex Analysis as its mathematical foundation and Dynamic Topology as its invariant support. Keywords: dynamic complex geometry; pitch function; steady-state configurations of the phase field; Calabi-Yau manifold; origin of gravity; testable predictions
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BibTeX
@misc{jun2026dynamic,
title = {{Dynamic Complex Geometry: Unifying Euclidean Space to Calabi-Yau Manifolds from the Phase Coupling Equation}},
author = {Zhao Jun},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20674157},
url = {https://doi.org/10.5281/zenodo.20674157},
note = {CC BY},
}
RIS
TY - UNPB
TI - Dynamic Complex Geometry: Unifying Euclidean Space to Calabi-Yau Manifolds from the Phase Coupling Equation
AU - Zhao Jun
PY - 2026
DA - 2026/06/13
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20674157
UR - https://doi.org/10.5281/zenodo.20674157
AB - Classical differential geometry treats space as an a priori existing object, built upon the real number field—real numbers can only describe “how large” but not “how to turn”. This paper proposes Dynamic Complex Geometry, with the core proposition: All mathematical spaces are steady-state configurations of the phase field. A spiral curve is uniquely determined by its pitch function κ = dβ/dγ, with curvature and torsion as derived quantities; the topological charge I = (1/2π)∮ κ dγ reduces abstract invariants to measurable geometric displacements. Complex analysis, differential geometry, and topology are shown to be different-dimensional projections of the same spiral curve. Euclidean space, Riemann surfaces, and Calabi-Yau manifolds are steady-state solutions of the same equation under three different boundary conditions. On this basis, the paper reveals the generative origin of gravity: the field equations of General Relativity are not independent postulates but inevitable derivations from phase closure and topological conservation under emergent continuity; flat space is not the geometric ground state but a high-energy constrained state; gravity is not a force but the natural tendency of space to return to a low-energy curved state. Dynamic Complex Geometry further yields testable predictions in the strong-field limit—including vacuum dispersion, the emergent approximateness of the equivalence principle, the nature of black holes as dead states, and the emergent nature of the cosmological constant—these effects are necessary consequences rigorously derived from first principles, rather than phenomenological assumptions of traditional quantum gravity programs. This paper is the geometric chapter of the L1 layer of the Generativism series, with Dynamic Complex Analysis as its mathematical foundation and Dynamic Topology as its invariant support. Keywords: dynamic complex geometry; pitch function; steady-state configurations of the phase field; Calabi-Yau manifold; origin of gravity; testable predictions
ER -