⚛️ general relativity

A technological capability scale for civilizations from the Kardashev and Barrow scales

This paper proposes a unified technological capability scale, defined as the sum of the Kardashev energy index and a Barrow efficiency index, which ranks civilizations by their control operations per second and extends from current human capabilities to a theoretical ultimate technology capable of dissipating the maximum possible power into a black hole in de Sitter space.

Original authors: Volkan Gurses

Published 2026-09-29

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Volkan Gurses

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand how far a civilization might travel, scientists have long looked at two different rulers. One measures how much energy a society can grab and use, from the power of a single star to the total output of an entire galaxy. The other measures how small a society can make its tools, from building houses to manipulating individual atoms. For decades, these two ways of measuring progress have been kept separate. One tells us about the size of a civilization's fuel tank, while the other tells us about the precision of its wrench. But a new study suggests that to truly understand the potential of intelligent life, we must combine them into a single, unified scale. This new approach asks a simple, fundamental question: how many times can a civilization change the state of a single piece of matter in one second?

The researchers, working from institutions including the Massachusetts Institute of Technology and the California Institute of Technology, propose a new way to rank civilizations based on their ability to perform "control operations." A control operation is defined as taking a tiny piece of matter—like an atom or a switch—and forcing it into a specific state, such as turning it on or off, regardless of what it was doing before. This is the basic act of writing information or building something. The study argues that the true measure of a civilization's power is not just how much energy it has, but how efficiently it can use that energy to perform these tiny acts of control. By counting how many of these operations a society can perform every second, the author creates a single number that captures both the scale of its energy and the precision of its technology.

The study builds on two famous ideas from the past. The first, known as the Kardashev scale, ranks civilizations by their total energy consumption. A Type I civilization uses all the energy that falls on its planet from its star. A Type II controls the energy of its entire star, and a Type III commands the energy of its whole galaxy. The second idea is the Barrow scale, which ranks civilizations by the smallest things they can manipulate. A civilization at the top of this scale might build cities, while one further down might manipulate genes, then molecules, then atoms, and finally the nuclei inside those atoms. The new paper unifies these by calculating how many times a civilization can reset a bit of information per second, given its total energy and the smallest scale it can reach.

The researchers found that for any civilization radiating heat into the cold emptiness of space, there is a hard physical limit to how efficiently it can work. This limit is set by the temperature of the universe itself, which is currently about 2.7 degrees above absolute zero. If a civilization tries to erase information or set a bit of matter, it must dump heat somewhere. If that heat goes into the current sky, the civilization cannot be more efficient than a certain baseline. This means that for any civilization currently radiating heat into space, its total capability is capped by its total energy. The more energy it has, the more operations it can perform, but it cannot break the efficiency ceiling set by the temperature of the universe.

When the author applied this scale to humanity, they found that we are currently at a level of 0.73 on the energy scale, meaning we use about half the energy of a full Type I civilization. However, our efficiency is still far from the theoretical limit. The study shows that most of our progress in computing power over the last century has not come from getting more energy, but from getting much better at using the energy we have. Since 1950, the energy required to perform a single logic operation has dropped by more than eleven orders of magnitude. This massive gain came largely from making devices smaller and more efficient, rather than from building bigger power plants. The author notes that while we have mastered the manipulation of atoms and even nuclei in the laboratory, our everyday machines still waste a tremendous amount of energy compared to the absolute minimum required by the laws of physics.

The paper also looks at what the ultimate future might hold. If a civilization could harness the maximum power allowed by the laws of physics and dump its waste heat into the coldest possible sink—a black hole near the edge of the universe—it could reach a capability index of 7.58. This is a theoretical limit, a corner of the map that no civilization has reached and perhaps never will. It represents a machine that performs operations at the speed of light with the maximum possible energy efficiency. The study suggests that while we are currently far from this limit, the path forward is clear: we must continue to shrink our devices and improve our efficiency, because simply adding more power will not be enough to reach the highest levels of capability.

One of the most striking findings is that the scale of the object being manipulated does not change the fundamental energy cost of erasing a bit of information. Whether you are flipping a switch on a giant machine or setting the spin of a single atomic nucleus, the minimum energy required is the same if the starting state is random. This means that moving to smaller scales does not automatically make a civilization more efficient; it only allows for denser storage and faster processing if the energy cost per operation is also reduced. The author shows that humanity has been getting better at this, but we are still far from the point where our machines operate at the absolute limit of efficiency.

The study also clarifies what is possible and what is not. It rules out the idea that a civilization can perform an infinite number of operations with a finite amount of energy. There is a hard stop, dictated by the second law of thermodynamics. Once a civilization reaches the point where it is operating at the minimum energy cost allowed by physics, the only way to do more work is to get more energy or to find a colder place to dump its heat. The author suggests that in the far future, if the universe continues to expand and cool, new opportunities might open up, but for now, the limits are set by the temperature of the cosmic background radiation.

In the end, this new scale offers a way to think about the future of technology not just as a race for more power, but as a journey toward greater precision. It suggests that the most advanced civilizations in the universe are not necessarily the ones with the biggest stars, but the ones that have learned to do the most with the least. By combining the measure of energy with the measure of efficiency, the researchers have provided a new map for understanding where we stand and where we might go. The journey from our current machines to the ultimate technology is a long one, but the path is now clearer, defined by the twin goals of gathering more energy and using it with perfect care.

Technical Summary: A Technological Capability Scale for Civilizations

Problem Statement
Existing frameworks for ranking technological civilizations, specifically the Kardashev scale (based on total power consumption) and the Barrow scale (based on the smallest scale of matter manipulated), operate independently. The Kardashev scale addresses the "ceiling" of available energy but ignores the efficiency of its use, while the Barrow scale addresses the "floor" of manipulation scale but lacks a unified metric for the rate of operations. Furthermore, neither scale explicitly counts the number of control operations a civilization performs per second, nor do they assign a physical minimum energy cost per operation for each level of matter. This paper introduces a unified scale that ranks civilizations by the total number of control operations they can perform per second, integrating power availability with thermodynamic efficiency.

Methodology
The author defines a control operation as an event that sets one binary degree of freedom of matter to a specified value, regardless of its previous state. Drawing on constructor theory, these operations are categorized into four classes: processing (overwriting bits), storage (holding bits), transfer (recreating bits elsewhere), and construction (building bits into matter).

The scale is constructed using two indices:

Kardashev Index ( $K$ ): Represents the power budget ( $P$ ) commanded by a civilization, defined as $K = \frac{1}{10} \log_{10} P - 6$ (with $P$ in watts).
Efficiency Index ( $B$ ): Represents the energy efficiency per operation ( $\varepsilon$ ), measured relative to the thermal floor at the cosmic microwave background temperature ( $T_0 = 2.725$ K). It is defined as $B = \frac{1}{10} \log_{10} \left( \frac{k_B T_0 \ln 2}{\varepsilon} \right)$ .

The Capability Index is the sum: $\Xi = K + B$ . The total operation budget is $\Omega = P/\varepsilon$ .

The paper establishes the thermal floor ( $\varepsilon_{floor} = k_B T \ln 2$ ) as the minimum energy required to reset a thermally random degree of freedom. It distinguishes between the thermal floor (theoretical minimum) and model values (energies required for specific carriers like nucleons or particles where quantum confinement energy exceeds the thermal term). The author analyzes humanity's historical trajectory from 3200 BCE to 2025, calculating $K$ based on total energy supply (including food/feed for pre-1950 data) and $\varepsilon$ based on the cheapest available technique for each operation class.

Key Contributions

Unified Metric: The paper proposes $\Xi$ as a single scalar to rank civilizations, where $\Xi$ is linear in $\log_{10} \Omega$ . It demonstrates that for operations radiating heat into the current sky, $\Xi \le K$ , as $B \le 0$ .
Thermodynamic Floors per Level: The author derives the minimum energy per operation for each Barrow level (from macroscopic objects down to elementary particles). They find that for operations resetting thermally random states, the floor is independent of the scale of matter and is determined solely by the temperature of heat rejection ( $k_B T \ln 2$ ).
Maturity Gap ( $\mu$ ): Defined as $\mu = B - B_{floor}$ , this metric quantifies the distance of a civilization's current technology from the theoretical thermodynamic limit.
Historical Decomposition: The paper decomposes humanity's growth in capability ( $\Xi$ ) into contributions from increased power ( $K$ ) and increased efficiency ( $B$ ).

Results

Humanity's Current State: In 2025, humanity has a Kardashev index of $K = 0.73$ . For processing (CMOS logic), the efficiency index is $B = -0.78$ , resulting in a capability index of $\Xi = -0.05$ .
Drivers of Progress: Since the first documented costs, humanity's capability growth has been driven almost entirely by efficiency gains ( $B$ $B$ ), not power increases ( $K$ $K$ ).
- For processing (since 1856), $B$ supplied 92% of the gain in $\Xi$ .
- For storage (since 793), $B$ supplied 82%.
- For transfer (since 1795), $B$ supplied 90%.
- For construction (since 1804), $B$ supplied 92% of the gain when measured per bit set by a ribosome, but only 63% when measured per transistor made by a fabrication plant.
Miniaturization: The paper estimates that miniaturization accounts for approximately 5 orders of magnitude of the 11.7 orders of magnitude drop in switching energy since 1950. The remaining reduction is attributed to physics changes (e.g., vacuum tubes to transistors) and generation efficiency.
The Ultimate Technology: The paper identifies a theoretical upper bound for a single system. A civilization utilizing the conjectured maximum power of one system ( $c^5/4G$ $c^{5} /4 G$ ) and dumping entropy into an uncharged, non-rotating black hole near the Nariai mass in de Sitter space (the coldest possible sink) would achieve:
- $K = 4.60$
- $B = 2.99$
- $\Xi = 7.58$
  This represents the "corner" of the $(K, B)$ plane.

Significance and Claims
The paper claims to provide the first scale that counts operations of every class on matter under a single energy budget and assigns a physical least energy per operation to each Barrow level. It asserts that:

Descending the Barrow levels does not lower the thermal floor: The minimum energy to reset a random bit remains $k_B T \ln 2$ regardless of whether the bit is a macroscopic object or a nuclear spin, provided the heat is rejected to the same temperature.
Efficiency is the primary limiter: For civilizations radiating heat into the current sky, the capability index $\Xi$ cannot exceed the power index $K$ . Gains in capability are currently limited by the "maturity gap" (inefficiency of devices and heat rejection temperatures) rather than power availability.
Future Limits: Once operations reach the thermal floor at their rejection temperature, further growth in $\Xi$ requires either increased power ( $K$ ), colder heat sinks, or new sinks for entropy (e.g., black holes).
Reversible Operations: The scale explicitly excludes reversible operations (like copying a design into prepared matter) from the "reset" cost, noting that such operations have no thermal floor, though they are bounded by other physical limits like the Margolus-Levitin bound.

The author maintains a modest scope, noting that the scale relies on current physical laws (thermodynamics, quantum mechanics, general relativity) and that discoveries of new physics or entropy sinks could alter the calculated floors and limits. The scale measures the capacity to perform operations, not the value or complexity of the information processed.

Technical Summary: A Technological Capability Scale for Civilizations

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