Appendix C: Kardashev Index and Information Density Conversion Table

In Volume IV “Engineering” of Vector Cosmology II, we redefined the levels of civilization: not merely exponential growth in energy consumption, but geometric progression in $c_{FS}$ budget utilization. The traditional Kardashev index based on watts appears too crude to describe the essential characteristics of high-dimensional computational civilizations (such as Type III).

This appendix provides a revised civilization metric standard based on Information-Velocity Geometry. We convert energy consumption rates to Information Processing Density and provide a quantitative calculation formula to measure a civilization’s evolutionary rank in the spiral universe.

C.1 Beyond Energy: The Information Metric

The traditional Kardashev index $K$ is defined by:

$$K=\frac{{\log }_{10}P-6}{10}$$

where $P$ is the civilization’s total power (watts). Type I is $10^{16}$ W, Type II is $10^{26}$ W, Type III is $10^{36}$ W.

But in FS Geometry, power is merely a manifestation of $v_{ext}$ flow. A civilization that can perfectly utilize $v_{int}$ (such as a computational civilization miniaturized to Planck scale) may have very low macroscopic thermal radiation power, but its internal logic operation rate (ops/s) may reach astronomical numbers.

Therefore, we introduce the FS Civilization Index ($K_{FS}$), based on the ratio of Bit-Flip Rate ($B$) to the universe’s ultimate bandwidth.

According to Landauer’s principle and the uncertainty principle, the minimum energy flux required to process 1 bit of information per second has a lower bound. However, for super-civilizations capable of manipulating quantum entanglement and non-equilibrium thermodynamics, this limitation can be bypassed or optimized.

We define the Effective Computational Flux ${\Phi }_{comp}$:

$${\Phi }_{comp}=\eta \cdot \frac{P}{k_{B}T\ln 2}$$

where $\eta$ is the Geometric Efficiency, measuring the topological efficiency of converting physical energy into logic gates (0 to 1).

C.2 The Calculation Formula for $K_{FS}$

To be compatible with the traditional index, we define $K_{FS}$ as the logarithmic scale of effective computational flux. We set the universe’s underlying maximum single-unit computational capacity (e.g., the limit of a black hole computer) as the reference frame.

$$K_{FS}=\frac{{\log }_{10}({\Phi }_{comp}/{\Phi }_0)}{10}$$

where ${\Phi }_0$ is the baseline computational power, set to $10^{16}$ ops/s (approximately the computational power of the human brain or early supercomputers, corresponding to the threshold of Type 0 civilization).

• Correction of Geometric Efficiency Factor $\eta$:

  • Biological Civilization: $\eta \approx 10^{-9}$ (extremely low, much energy wasted on maintaining body temperature and other ineffective $v_{env}$).

  • Silicon-Based Civilization: $\eta \approx 10^{-3}$ (relatively high, limited by thermal resistance).

  • Quantum/Photon Civilization: $\eta \to 1$ (approaching the limit, directly operating $v_{int}$ in Hilbert space).

This means that a micro-civilization with extremely high $\eta$, even if its total power $P$ is only a small fraction of the Sun’s, may have a $K_{FS}$ level exceeding that of an energy-wasting interstellar empire.

C.3 Thresholds of Ascension

Based on the $K_{FS}$ formula, we can quantify the various evolutionary stages described in the second book:

C.4 The Theoretical Limit and Omega Point

Is there an upper limit to $K_{FS}$?

According to the Lieb-Robinson speed and Brillouin zone limitations we discuss in Appendix D, the maximum information processing rate in a finite volume $V$ of the universe is limited (Bremermann Limit):

$$B_{max}\approx \frac{m{c}^2}{h}$$

For the entire observable universe, this limit is approximately $10^{120}$ ops/s.

If we take ${\Phi }_{comp}=10^{120}$ in our formula, we obtain:

$$K_{FS}^{max}\approx \frac{120-16}{10}=10.4$$

This is the numerical embodiment of the Omega Point.

When a civilization’s $K_{FS}$ approaches 10, it has actually transformed every degree of freedom of every fundamental particle in the universe into computational units.

• At this point, Universe = Computer.

• At this point, $v_{int}=c_{FS}$ (full budget for internal computation), $v_{ext}=0$ (external time stops).

This is not only the limit of engineering, but also the geometric closure. A civilization reaching this point is no longer a passerby in the universe; it has become the total wave function of the universe itself. This is precisely the physical definition of “The Palm of the Buddha” in the final chapter of the book.