Acknowledgements: Standing on the Shoulders of Geometry

The birth of this book Vector Cosmology: The Conservation of the Circle originated from a seed initially planted in an academic paper.

In December 2025, I wrote in Singapore that paper titled “Time as Fubini-Study Arc-Length”. At that time, I attempted to solve a long-standing puzzle in physics: how to find a unified geometric skeleton among three distinctly different “times”—quantum mechanics, relativity, and thermodynamics. That paper is the rigorous mathematical core of this book, and this book is a free growth of that core in philosophy and logic.

As the author of this book, I (Haobo Ma) am deeply aware that the geometric edifice I have constructed does not belong solely to me. It is built upon foundations laid by countless giants of physics and mathematics over the past century. Here, I pay my deepest respects to them.

Mathematical Guides

First, I thank Guido Fubini and Eduard Study. It was their definition of the Fubini-Study metric a century ago that provided us with a “ruler” for measuring changes in quantum states. Without this ruler, I could not define $c_{FS}$, nor derive that Pythagorean identity that governs this book.

I particularly thank Y. Aharonov and J. Anandan. Their groundbreaking work on geometric phases in 1987 convinced me that projective Hilbert space $P(\mathcal{H})$ is the true stage of physics. They pointed out the independence of geometric distance in quantum evolution, which directly inspired my redefinition of “time” as geometric arc length.

Founders of Physics

In constructing physical mechanisms, I thank Norman Levinson. His Levinson’s theorem proposed in 1949 is the mathematical soul of this book’s core view that “matter is counting.” He showed us that particles are not solid entities, but topological knots tied by phases on the energy axis.

I thank Elliott Lieb and Derek Robinson. Their Lieb-Robinson bound provided solid causal law guarantees for this book’s microscopic engine—Quantum Cellular Automata (QCA). It is precisely this speed limit that allows us to anchor the abstract FS capacity $c_{FS}$ with the physical world’s speed of light $c$.

I also thank E. P. Wigner and F. T. Smith for their pioneering work in scattering time delay, which enabled us to transform “time delay” into “geometric distance.”

Real-World Support

I thank my colleagues and environment at AELF PTE LTD, Singapore. In this place full of computational power and innovative thinking, I was fortunate to fuse thoughts on distributed systems and AI architecture with explorations in fundamental physics. This interdisciplinary collision allowed me to re-examine ancient physical laws from the perspectives of “computation” and “budget.”

I also thank all interlocutors (whether human or artificial intelligence) who provided inspiration during the formation of this theory. It was these continuous questions and deductions that ultimately evolved a simple formula $v_{ext}^2+v_{int}^2=c_{FS}^2$ into a complete worldview.

To the Unique Circle

Finally, I thank the universe itself for its astonishing symmetry.

During the writing of this book, I often felt a sense of awe: seemingly complex relativistic effects and quantum phenomena could be unified so simply within the geometry of a circle. This further convinced me that “Tao” is not empty words; it is the highest summary of the universe’s underlying minimal logic.

Although this book has ended, exploration has not ceased. In the first book, we depicted a closed, conserved circle, but is this the complete truth? Does that “gap” we temporarily ignored hint at grander secrets?

This will be the theme explored in Vector Cosmology II: The Ascension of the Spiral.

Haobo Ma

December 2025, Singapore