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

AI summary

Overview

Research area: Physics of information and thermodynamics of computation, applied to astrobiology and the classification of technological civilizations (posted to arXiv under quantum physics, quant-ph).

Technical level: Intermediate to Advanced. The conceptual argument is accessible, but the paper assumes familiarity with the Landauer bound, constructor theory, the Margolus–Levitin theorem, and cosmological horizons.

Scope (one sentence): The paper proposes a single capability index, Ξ = K + B, that combines the Kardashev power scale with a new efficiency index measuring energy per "control operation," so that civilizations are ranked by how many specified binary manipulations of matter they can perform per second.

What This Paper Is About

The Kardashev scale ranks civilizations only by the power they command, and the Barrow scale ranks them only by how small a scale of matter they can manipulate; neither counts the operations a civilization actually performs. Gurses defines a "control operation" — setting one binary degree of freedom of matter to a value specified in advance, whatever its previous state — and builds a scale from the number of such operations per second, Ω = P/ε, where P is commanded power and ε is energy per operation. The Kardashev scale supplies the ceiling (power) and the Barrow scale the floor (least possible energy per operation at each level of matter), and the two are fused into the index Ξ = K + B.

Key Contributions

1. A unified operation-based scale. The capability index Ξ = K + B = (1/10) log₁₀(Ω/Ω₀) counts control operations per second, with Ω₀ = 10⁶ W / ε_{T₀} = 3.8 × 10²⁸ s⁻¹, where ε_{T₀} = k_B T₀ ln 2 = 2.61 × 10⁻²³ J at the cosmic microwave background temperature T₀ = 2.725 K. Kardashev's radio bit rate and earlier operation-rate estimates become special cases for transfer and processing.

2. A physical energy floor assigned to every Barrow level. Each level is given a length and a carrier by convention, and the least energy per operation is taken as the larger of a quantum confinement term E_loc(l,m) and a thermal term. The paper reports that the floor of every level is the thermal floor (26 yJ per bit at T₀) for operations that reset a thermally random degree of freedom, so descending the levels cannot lower it.

3. A classification of operations by constructor theory. Construction is the general class; processing (computation), storage and transfer are its special cases on information media, matching the computation, storage and communications capacities Hilbert & López measured. Operations that build the bit into the matter of a product remain in construction.

4. Placement of humanity and of an "ultimate technology." Humanity sits at K = 0.73 and B = −0.78 in primary energy; the ultimate technology is placed at Ξ = 7.58, dissipating the conjectured largest power c⁵/4G into an uncharged, non-rotating black hole near its largest mass in de Sitter space.

Main Findings

• Humanity's position: World total energy supply was 600.3 EJ in 2025, an average of 1.9 × 10¹³ W, giving K = 0.73. Humanity computes with CMOS logic gates at B = −0.78 in primary energy. Since 1950 the energy per logic operation has fallen 11.7 orders of magnitude, around 5 of them through miniaturization.

• Levels held by humanity: On the paper's criterion (setting a degree of freedom to a specified value) humanity has held III-minus since chemists first assembled molecules step by step, holds II-minus through recombinant DNA, holds IV-minus in the laboratory since Eigler & Schweizer positioned single atoms in 1990, and — contrary to Barrow's assessment — holds V-minus and VI-minus through laboratory control of single nuclear and electron spins. Bulk nuclear transmutation in reactors does not count toward V-minus because no state is specified for each nucleus.

• The floor is set by entropy, not length: Because a reset of a thermally random bit removes k_B ln 2 of entropy per bit regardless of carrier, the thermal floor is 26 yJ at T₀ and 2.9 zJ at 300 K. I-minus already has this floor, so descending the Barrow levels cannot lower it.

• Barrow levels and their model values (Table 1):

• Corresponding floors: The model values for the electron switch of atomic size, the nucleon and the created particle correspond to B_floor = −0.44, −1.11 and −1.51 respectively; the Planck energy lies 32 orders of magnitude above the floor at T₀.

• Heat rejection bounds Ξ ≤ K: For entropy leaving as heat radiated into the present sky, B ≤ B_floor ≤ 0 and Ξ ≤ K. An operation on the thermal floor at a rejection temperature T has B = (1/10) log₁₀(T₀/T), which is zero at T₀ and −0.20 at 300 K. Only a new entropy sink, or a sink colder than the microwave background such as a black hole or the de Sitter horizon, could raise Ξ above K.

• The maturity gap: Each civilization is described by (K, B_floor, μ) per class of operation, where μ = B − B_floor measures the distance of its operations from the floor, including a term (1/10) log₁₀(T₀/T) from its rejection temperature.

• Real hardware landmarks: ENIAC drew 150 kW for 5000 ten-digit additions per second, about 0.9 J per operation — 3 × 10²⁰ times the thermal floor at 300 K and 3 × 10²² times that at T₀. The IRDS switching event is CV² = 0.65 fJ in 2023 for a high-performance gate driving three others; published energies per switching event span 32 aJ (14 nm minimum-width transistor) to about 3 fJ (practical circuit with wiring); a 32-bit integer addition costs about 3 fJ at 45 nm and 0.9 fJ at 7 nm. El Capitan drew 29.7 MW at 1.81 × 10¹⁸ 64-bit floating-point operations per second in Linpack, or 2.6 × 10⁻¹³ J per operation, the energy of about 400 switching events.

• Counting conventions: A choice among W equally likely alternatives counts log₂W operations; a polymerase counts log₂a (2 for an equimolar pool), a ribosome log₂20 = 4.32. A ten-digit addition is 33 operations, a w-bit addition or instruction w, a 64-bit floating-point operation 64, a SHA-256d hash 256.

• Reliability reference: A switch that must complete ν₀/H₀ = 4.6 × 10³⁰ operations over the Hubble time 1/H₀ = 4.6 × 10¹⁷ s needs a barrier of k_B T ln(ν₀/H₀) = 71 k_B T, which is 0.29 aJ at 300 K and lies 2.0 orders of magnitude above the thermal floor; the paper stresses this is a convention, not a bound.

• Where quantum terms take over: For an electron the quantum term equals the thermal floor at 15 nm at T₀ and 1.5 nm at 300 K, and equals the reliability reference at 1.5 nm at T₀ and 0.14 nm at 300 K. For a massless carrier ℏc/l crosses the thermal floor at 1.2 mm at T₀ and 11 μm at 300 K.

• Biological molecular machines: At 10⁻¹⁹ J (24 k_B T at 300 K) per ATP hydrolysis, a ribosome spends 9.3 × 10⁻²⁰ J (22 k_B T) per specified bit and a polymerase 10⁻¹⁹ J (24 k_B T). The 10⁻¹⁹ J figure is that implied by the 10⁻¹² W power of an E. coli cell and its 10⁷ ATP molecules hydrolysed per second.

• Data centres as a partial measure: Data centres used 1.75 EJ of electricity in 2025, an average of 55 GW, or about 130 GW of primary energy — 0.7% of humanity's total energy supply, giving K = 0.51.

• Not reported in the supplied excerpt: The details of Sections IV.4 (the "programmable quantum field"), Section V on storage and transfer, and Section VI's assumptions and scope are cut off in the provided text, as is the derivation behind the Ξ = 7.58 ultimate-technology figure beyond the abstract's description. The text also breaks off mid-sentence in the discussion of Bennett's estimate for biological costs.

Methodology in Plain English

The author sets up one accounting rule — you can never perform more operations per second than your power divided by the energy each operation costs — and then asks what the smallest possible cost is at each scale of matter. That minimum comes from two competing effects: the quantum energy needed to squeeze a particle into a small region, and the thermodynamic cost of erasing information, which is k_B T ln 2 per bit at temperature T. Because the second law requires that erased entropy be dumped somewhere, the temperature of the coldest available sink (here the 2.725 K cosmic microwave background) sets a universal floor for any operation that resets a randomly oriented bit. The two scales are then combined logarithmically, with each factor of ten in power or in efficiency adding one to the index. Humanity's current numbers are assembled from published energy statistics, the IRDS device roadmap, a TOP500 machine, and measured costs of ATP hydrolysis in cells, with explicit conversion factors from delivered electricity to primary energy (2.3 for 2025, 3.5 for 1950). Finally, the same bookkeeping is run to its physical limit to locate an "ultimate technology" on the (K, B) plane.

Why This Matters

Impact on research: The paper gives the Kardashev and Barrow scales a shared physical footing by tying both to thermodynamics of computation, and it supplies an explicit, falsifiable-style accounting for how far real devices sit from physical limits. Its argument that the Barrow floor does not improve as scale shrinks is a direct challenge to the common reading of Barrow's scale as a trend toward efficiency, and it corrects Barrow's own assessment of humanity's level.

Potential applications (as implications of the framework, not demonstrated uses):

• Benchmarking computing hardware by operations per joule against a floor, rather than by FLOP/s alone.
• Auditing data-centre energy use within a civilization-level energy budget (the paper's own example gives data centres 0.7% of world energy supply).
• Reinterpreting technosignature searches, where a single power estimate would be replaced by a power-plus-efficiency pair.
• Setting long-run research targets for reversible and low-dissipation computing, where the gap to the thermal floor is quantified.

Industry relevance: The conventions connect directly to semiconductor roadmap metrics (CV² switching energy, per-node integer

Authors’ abstract

We introduce a technological capability scale that ranks civilizations by the control operations they can perform per second. A control operation is defined as setting one binary degree of freedom of matter to a specified value, whatever its previous state. Building on constructor theory, we sort them into construction and its special cases on information media, namely processing, storage and transfer. The index $Ξ= K + B$ adds to the Kardashev index $K$ an efficiency index $B$ of energy per operation, measured from the thermal floor at the microwave background temperature. The Kardashev scale gives the ceiling for resets of thermally random bits with heat radiated into the present sky, and the Barrow scale gives the floor, which for such a reset is the thermal floor at the heat-rejection temperature at every level. Since 1950 the energy per logic operation has fallen 11.7 orders of magnitude, around 5 through miniaturization. Humanity, at $K = 0.73$, computes with CMOS logic gates at $B = -0.78$ in primary energy. We describe the ultimate technology, at $Ξ= 7.58$, which would dissipate the conjectured largest power $c^5/4G$ into an uncharged, non-rotating black hole near its largest mass in de Sitter space.