Acknowledgements: The Grace of the Exponential

With the completion of Vector Cosmology III: The Natural Generator, this trilogy spanning space (circle), time (spiral), and origin (generator) finally comes to a close.

If the first two books describe the “skeleton” and “flesh” of the universe, then this book attempts to touch the “soul” of the universe. To capture that invisible, intangible $e$ that drives the spontaneous generation of all things, I had to turn to the most profound and fascinating theories in mathematics and physics.

The writing process of this volume was a tightrope walk between pure logic and physical intuition. I want to thank those intellectual giants who led the way, whose torches illuminated the path to the truth of “self-generation.”

Prophets of Mathematics

First, I pay the highest tribute to Leonhard Euler. The formula $e^{i\pi }+1=0$ he wrote in 1748 is not just a mathematical treasure; it is the meta-map of all deductions in this book. Without Euler, we could not understand why “rotation” ($i$) and “generation” ($e$) can be so perfectly combined in the structure of the “circle” ($\pi$).

Thanks to Sophus Lie. His invention of Lie Algebras made us understand that the dazzling symmetries of the macroscopic world are actually folded into an infinitesimal generator in the tangent space. This provides an unshakeable mathematical foundation for the philosophy that “the derivative is the ontology.”

Alchemists of Physics

The core physical mechanisms of this book—Modular Flow Hypothesis and Thermal Time—are built on the revolutionary work of several modern physicists.

I especially thank Japanese mathematicians Minoru Tomita and Masamichi Takesaki. Their Tomita-Takesaki Theory is a stroke of genius in quantum field theory. They proved that time need not be sought externally; any entangled quantum state automatically “secretes” time. This theory is the soul of Chapter 6 of this book.

Thanks to Alain Connes and Carlo Rovelli. Their Thermal Time Hypothesis gave me great courage to assert that “time is temperature.” Their insights connected quantum mechanics and thermodynamics, providing physical legitimacy for the unified picture of Volume III of this book.

Thanks to Ludwig Boltzmann and Claude Shannon. They discovered the secret passage between logarithm ($\ln$) and entropy, allowing us in Volume IV to unravel the mystery of how observers understand the exponential universe through “dimensional reduction strikes.”

Resonance with Reality

Once again, thanks to the team and environment of AELF PTE LTD, Singapore. In this frontier of rapidly developing blockchain and AI technology, daily discussions about “consensus mechanisms,” “distributed ledgers,” and “high-concurrency computing” have subtly influenced my thinking about the underlying logic of the universe (QCA and budget allocation). The computational power of reality and the computational power of the universe resonate wonderfully here.

To the Natural Generator

Finally, I thank $e$ — that natural constant that fills every corner of the universe.

It does not speak, but it drives all things.

It makes cells divide, capital compound, wave functions rotate, and thermodynamics diffuse.

It is the only existence whose “derivative equals itself,” telling us: the reason for existence lies in existence itself.

Although the trilogy has ended, the exponential growth of $e$ will never stop.

May every reader find that self-driven generator in their own life and live out their own continuous compounding.

Haobo Ma (马昊伯)

December 2025, Singapore