A capability scale from the Kardashev and Barrow scales

The Kardashev scale ranks civilizations by the power they command, and the Barrow scale ranks them by the smallest scale of matter they can manipulate. Neither scale says what a civilization can do with information. Here I want to combine the two scales into one number. It turns out that the combination is a ratio, the power commanded divided by the energy each elementary operation costs, and that the Barrow scale enters through that energy once each of its rungs is given a physical cost. Measured this way, almost all of humanity’s progress since 1950 has come from lowering the denominator, the reduction still available in the denominator is worth more than the rise to a Type I civilization, and below the scale of the atom the two scales stop being independent.

The two scales

Kardashev proposed his scale in 1964 (Kardashev, 1964). A Type I civilization commands the power available on its planet, a Type II the output of its star and a Type III the output of its galaxy, and Sagan interpolated between them with K = (log₁₀P − 6)/10, with P in watts, so that the three types sit at 10¹⁶, 10²⁶ and 10³⁶ W (Sagan, 1973). Humanity commanded an average of 1.9 × 10¹³ W in 2024 (Energy Institute, 2025), which is K = 0.73.

Barrow proposed his scale in 1998 as the inward counterpart (Barrow, 1998). A Type I-minus civilization manipulates objects at its own scale, a II-minus manipulates genes, a III-minus molecules, a IV-minus individual atoms, a V-minus the atomic nucleus, a VI-minus elementary particles and a Type Ω-minus the structure of space and time. We manipulate molecules at industrial scale and have positioned individual atoms since 1990 (Eigler & Schweizer, 1990), so I will treat the rungs as cumulative capabilities rather than a position. Neither scale has a unit for what the civilization does with its power or its devices.

The operation budget

A civilization that runs N devices, each performing f operations per second at an energy cost ε per operation, dissipates Nfε watts, and that cannot exceed the power it commands. Therefore the number of elementary operations per second it can perform is

Ω = P/ε,

with P in watts and ε in joules per operation. The Kardashev scale supplies the numerator, and the Barrow scale lowers the denominator, since the device is set by the scale of matter the civilization can control.

Two conventions are needed before Ω means anything. The first is what counts as an operation, and I will use the gate switching event, which costs about a femtojoule in a current process (Horowitz, 2014). The second is that Ω is a capacity rather than a measurement, since it assumes every watt goes into the best device the civilization can build. Data centers drew 47 GW in 2024 (International Energy Agency, 2025), 0.25% of the power we command, so the realized rate sits 2.6 orders of magnitude below the capacity.

Ω = P/ε is not new. Bradbury sized a Matrioshka brain with it (Bradbury, 1999; Bostrom, 2003), Sandberg computed Landauer-limited rates for megastructures (Sandberg, 1999), a 2026 preprint builds a cognitive Kardashev scale on it (Sharma, 2026), and Vidal drew the two scales as a plane (Vidal, 2011; Vidal, 2014). None of them gave the Barrow rungs an energy, and that energy turns the plane into a scale.

An energy for each rung

The energy an irreversible operation costs has two contributions, and the larger sets the cost at any rung. The first is thermal. Erasing a bit dissipates at least k_BT ln 2 (Landauer, 1961), and a switch that has to survive N operations without a thermal error needs a barrier of about k_BT ln N, which is 55 k_BT or 0.23 aJ for a large machine (Keyes & Landauer, 1970; Zhirnov et al., 2003). I will call the first the ultimate floor and the second the engineering floor, which differ by a factor of eighty. The temperature in both is the one at which the civilization finally rejects its heat, since refrigeration costs Carnot work rejected at the same place, as the post on information storage worked through, and it cannot be below the cosmic microwave background at 2.725 K today, where the ultimate floor is 26 yJ.

The second is the energy needed to control matter at the scale of the rung. Confining a carrier of mass m to a length l costs at least the localization energy ħ²/2ml², or ħc/l below the carrier’s Compton wavelength, and an irreversible operation at that scale commits at least that much energy, which is a model rather than a theorem. For an electron switch it equals the ultimate floor at l = 1.5 nm and the engineering floor at 0.16 nm (Zhirnov et al., 2003).

Together the two terms give each rung an energy once each rung is given a length, and the lengths below are my convention, evaluated on the engineering floor at room temperature. A I-minus device is a macroscopic switch whose cost is measured rather than bounded. A II-minus device works on a nucleotide at about 2 nm, a III-minus device on a molecule at about 1 nm and a IV-minus device on an atom at 0.1 nm, and all three sit on the thermal plateau at 0.23 aJ. A V-minus device works on a nucleon at 1 fm, where the localization energy is 20 MeV or 3.3 pJ, a VI-minus device works at 10⁻¹⁸ m, where it is 200 GeV or 32 nJ, and a Ω-minus device works at the Planck length, where it is the Planck energy, 2 × 10⁹ J. Therefore the energy per operation is flat from the meter to the atom and rises to seven, eleven and twenty-eight orders of magnitude above the plateau at the three rungs below it.

Biology offers a check on the plateau. An ATP hydrolysis delivers about 10⁻¹⁹ J, or 24 k_BT, and a growing E. coli cell spends about 10⁷ of them per second on 10⁻¹² W (Milo & Phillips, 2015), so a molecular machine runs within a factor of about two of the engineering floor per step.

Three numbers for a civilization

The natural index for Ω is the same logarithm as Sagan’s K, normalized so that a civilization whose devices sit on the ultimate floor at room temperature scores its Kardashev number. I define

B = (1/10) log₁₀(k_BT₀ ln 2 / ε), with T₀ = 300 K, and Ξ = K + B = (1/10)(log₁₀Ω − 26.54), with Ω in operations per second.

B reads as an efficiency deficit or surplus in Kardashev units, zero on the ultimate floor at room temperature, −0.19 on the engineering floor, −0.55 for a femtojoule gate and +0.20 at the microwave background. B splits into the floor the rung allows, which is zero from I-minus to IV-minus and negative below, and the maturity gap between the device a civilization has built and that floor. For humanity in 2024 the three numbers are K = 0.73, a rung floor of 0 and a maturity gap of −0.55 at the gate level, for Ξ = 0.17 and Ω = 1.9 × 10²⁸ gate switchings per second.

The cold rows carry a cost the table does not show, since rejecting 10²⁶ W at 300 K needs a radiator of radius 0.88 AU, a Dyson sphere, and rejecting it at 2.725 K needs one of radius 0.17 light years. Lloyd’s ultimate computer is not on the table because it is not a power budget, since its 5.4 × 10⁵⁰ orthogonalizing steps per second (Lloyd, 2000; Margolus & Levitin, 1998) would take 1.6 × 10³⁰ W, four thousand suns, as irreversible operations at the room-temperature floor.

The trajectory since 1950

In 1950 the world commanded 3.2 × 10¹² W, K = 0.65, and ENIAC spent about 30 J per addition, against 1.9 × 10¹³ W and 1.3 × 10⁻¹² J per DGX operation in 2024 (NVIDIA, 2024). ε fell 13.4 orders of magnitude while P rose 0.8, so Ξ rose by 1.41 and K contributed 0.08 of it. Ninety-five percent of the gain in capability since 1950 came from the denominator, whose doubling time was 1.57 years from 1946 to 2009 (Koomey et al., 2011) and has lengthened to about 2.6 years since 2000 (Naffziger & Koomey, 2016). The whole of that fall in ε took place on the plateau, since the rungs humanity reached after 1950 all lie on it, so the gain was engineering maturity above a floor that never moved.

What remains

A femtojoule gate is 3.6 orders of magnitude above the engineering floor and 5.5 above the ultimate floor, while our power is 2.7 orders of magnitude below a Type I civilization’s. Therefore closing the maturity gap at the power we already command is worth more than reaching Type I with the devices we already have, and reaching the ultimate floor is worth twice as much. Applying the post-2000 doubling time to the gate energy, we reach the engineering floor in about thirty years and the ultimate floor in about fifty. After that, Ξ can grow only as fast as K, which has grown at 0.001 per year for seventy years. Reversible logic goes below the floor by trading energy for area and latency (Bennett, 1973; Athas et al., 1994), and Ω does not apply to it. Whether a civilization gains by waiting for the background radiation to cool is disputed (Sandberg et al., 2016; Bennett et al., 2019), but neither side changes the room-temperature floor that a civilization living on a planet faces.

Below the atom

The rungs below IV-minus do not lower ε, so on this scale a civilization gains nothing in Ω by reaching them. A Type II civilization that spent its power on nucleon-scale operations at 20 MeV each would perform 3 × 10³⁷ per second, seven orders of magnitude fewer than on the atomic plateau. Those rungs act on the numerator instead. Manipulating nuclei means fusion and transmutation, which are sources of P (Ćirković, 2015), manipulating elementary particles gives neutronium as a storage material, with some fifteen orders of magnitude more bits per volume than a bit per atom (Sandberg, 1999), and manipulating spacetime gives a black hole engine (Vidal, 2011). Therefore the two scales are independent above the atom and coupled below it.

What the scale leaves out

Ω counts irreversible operations only, in aggregate, and says nothing about serial depth, algorithms or heat removal. Storage and communication need scales of their own, because they draw on different budgets. Storage draws on mass, so the number of bits a civilization can hold is the mass it commits divided by the mass per bit, which is set by the rung. Humanity holds about 10²³ bits (Hilbert & López, 2011), the Earth at one bit per atom would hold 1.4 × 10⁵⁰, and the Bekenstein bound for the Earth’s mass is reached only by a black hole, at 1.4 × 10⁶⁶ bits (Bekenstein, 1981). Communication draws on power, like processing, but a single channel cannot use a large power budget, since its capacity grows only as the square root of the power (Pendry, 1983), so a Type II civilization that put 10²⁶ W into one channel would send 1.4 × 10³⁰ bits per second, sixteen orders of magnitude below its Landauer-limited 3.5 × 10⁴⁶.

What is new

Ćirković named the interrelation between the two scales as an open question (Ćirković, 2015), Smart argued for efficiency over power as the measure of development (Smart, 2012), and five things here are new. Each Barrow rung now has an energy. That energy is flat from the meter to the atom and rises steeply below it. A civilization’s position decomposes into its Kardashev power, the floor its rung allows and the maturity of its engineering above that floor. Humanity’s gain since 1950 was maturity above a floor that never moved, and the remaining maturity gap counts for more than the rise to Type I. Below the atom the two scales couple, because the lower rungs raise the power and the storage density instead of lowering the cost of an operation.