# References — Chapter ᚁ(B): The actual world *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-3, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch03.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 3.1 {#r-3.1} The seven SI defining constants — among them the Planck constant h, the speed of light c, the elementary charge e, the Boltzmann constant k_{B}, and the Avogadro constant N_{A} — and the exact derived relations have the quoted status and values. Since 20 May 2019 the ampere is defined by fixing the exact numerical value of the elementary charge e. ε0 was exact before 20 May 2019; since then ε0 = e²/(2αhc), and it carries the relative uncertainty of α. F = N_A e, the charge of one mole of electrons, named after Michael Faraday. R = N_A k_B. The Planck length is ℓP = √(ℏG/c³), so ℓP² = ℏG/c³. *Status: confirmed* - [BIPM, The International System of Units](https://www.bipm.org/en/measurement-units) — The SI defines its units through fixed constants from 20 May 2019. - [NIST, SI historical resolutions](https://www.nist.gov/pml/special-publication-330/sp-330-appendix-1) — The 1983 meter definition fixes the speed of light; the 2018 resolution establishes the revised SI effective in 2019. - [NIST 2022 CODATA complete listing](https://physics.nist.gov/cuu/Constants/Table/allascii.txt) — lists the exact SI constants and the quoted physical values. - [NIST Fundamental Physical Constants, front page](https://physics.nist.gov/cuu/Constants/) — the 2022 adjustment is the current one; the next regularly scheduled adjustment is 2026, for which data must be published or preprinted before 31 December 2026. Every CODATA-sourced value in this chapter is CODATA 2022; the cosmological, CMB, PDG, and electron-anomaly values are cited to their own sources at 3.2, 3.3, and 3.5. - [2019 revision of the SI, Wikipedia](https://en.wikipedia.org/wiki/2019_revision_of_the_SI) — “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (kB), and the Avogadro constant (NA), respectively.” - [Vacuum permittivity, Wikipedia](https://en.wikipedia.org/wiki/Vacuum_permittivity) — “The elementary charge was redefined exactly in terms of the coulomb as from 20 May 2019, with the effect that the vacuum electric permittivity and the magnetic vacuum permeability no longer have exactly determined values in SI units.”; displays ε0 = 1/(μ0c²) = e²/(2αhc), “with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the values of each of ε0 and μ0 are therefore the same as that for the fine-structure constant” - [Faraday constant, Wikipedia](https://en.wikipedia.org/wiki/Faraday_constant) — “It is named after the English scientist Michael Faraday. Since the 2019 revision of the SI, the Faraday constant has an exactly defined value, the product of the elementary charge (e, in coulombs) and the Avogadro constant (NA, in reciprocal moles)” and “One faraday is the amount of charge in one mole of electrons” - [Gas constant, Wikipedia](https://en.wikipedia.org/wiki/Gas_constant) — “The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB)”; “Since the 2019 revision of the SI, both N A and k are defined with exact numerical values when expressed in SI units. As a consequence, the SI value of the molar gas constant is exact.” - [Planck units, Wikipedia](https://en.wikipedia.org/wiki/Planck_units) — table of base units: “Planck length … length (L) … l_P = √(ħG/c³)” ### 3.2 {#r-3.2} The cosmological constant and density parameters have the displayed meanings and model dependence. In Planck units the measured cosmological constant is about 3 × 10⁻¹²². The Planck 2018 fit gives Λ ≈ 1.1 × 10^−52 m^−2; with the CODATA 2022 Planck length, ΛGℏ/c³ = Λ·ℓP² ≈ 1.1056 × 10^−52 × (1.616255 × 10^−35)² ≈ 2.9 × 10^−122 (arithmetic). Base ΛCDM assumes flat space (Ω_k = 0); curvature is fitted only as an extension, which shifts the other parameters. Planck’s total matter density includes the massive neutrinos. Ω_r, photons plus relativistic neutrinos, is about 9 × 10⁻⁵ today; radiation dominated until about fifty thousand years after the Big Bang. No accepted resolution of the cosmological constant problem is known; proposals include modified gravity and anthropic selection in the string theory landscape. *Status: confirmed* - [Cosmological constant, Wikipedia](https://en.wikipedia.org/wiki/Cosmological_constant) — confirms the 1917 term and vacuum-energy interpretation, and that “there is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.” - [Planck (spacecraft), Wikipedia](https://en.wikipedia.org/wiki/Planck_(spacecraft)) — confirms the base flat LambdaCDM fit and its parameters. - [Friedmann equations, Wikipedia](https://en.wikipedia.org/wiki/Friedmann_equations) — confirms the density balance and curvature term. - [Cosmological constant problem, Wikipedia](https://en.wikipedia.org/wiki/Cosmological_constant_problem) — gives the Planck-scale estimate as roughly 120 orders of magnitude too large, and attributes “probably the worst theoretical prediction in the history of physics” to Hobson, Efstathiou and Lasenby, General Relativity: An Introduction for Physicists (2006), p. 187. Under Proposed solutions: “Some proposals involve modifying gravity to diverge from general relativity.”; “Some propose an anthropic solution, and argue that we live in one region of a vast multiverse that has different regions with different vacuum energies.”; and “Anthropic arguments gradually gained credibility among many physicists after the discovery of dark energy and the development of the theoretical string theory landscape.” - [H. Padmanabhan and T. Padmanabhan, CosMIn: The Solution to the Cosmological Constant Problem, arXiv:1302.3226 (2013)](https://arxiv.org/abs/1302.3226) — “observations [1, 2] suggest that ΛL_P² ≈ 3.4 × 10⁻¹²²” - [Cosmological Constant Problem and H0 Tension in Void-dominated Cosmology, arXiv:2204.12180 (2022)](https://arxiv.org/abs/2204.12180) — quotes the Planck 2018 value (its ref. [39], Aghanim et al., A&A 641, A6): “The cosmological constant in the latest Planck data is reported as below [39], Λobs = 1.1056 × 10−52m−2.” - [Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209](https://arxiv.org/abs/1807.06209) — “We find good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper)”; §7.3: “The base-ΛCDM model assumes that the spatial hypersurfaces are flat”; Fig. 29: “Constraints on a non-flat universe as a minimal extension to the base-ΛCDM model. Points show samples from the Planck TT,TE,EE+lowE chains coloured by the value of the Hubble parameter” - [Lambda-CDM model, Wikipedia](https://en.wikipedia.org/wiki/Lambda-CDM_model) — parameter table lists Ω_m as “Total matter density today (including massive neutrinos)” - [K. Cahill, The eras of radiation, matter, and dark energy: new information from the Planck Collaboration, arXiv:1606.08865](https://arxiv.org/abs/1606.08865) — from Planck 2015 inputs: “The present ratio Ωr0 of the radiation density (2) to the critical density (6) is then” (9.0606 ± 0.1233) × 10⁻⁵; abstract: “The era of radiation ended, and the era of matter began, when the density of radiation dropped below that of matter. This happened 50,953 ± 2236 years after the time of infinite redshift” ### 3.3 {#r-3.3} The CMB and thermal constants have the quoted temperatures, laws, and identities, and Wien’s root is proven transcendental. Fixsen (2009) gives 2.72548 ± 0.00057 K as the determination from the measurements in the literature; the COBE FIRAS data alone, recalibrated with WMAP, give 2.7260 ± 0.0013 K. Wien’s constant b = 2.897771955… × 10⁻³ m K is exact; the 2.7255 K background peaks in the microwave, near 1 mm (b/T = 1.06 mm). The CMB is the most perfect black-body spectrum ever measured. *Status: confirmed* - [ESA Planck CMB account](https://www.esa.int/Science_Exploration/Space_Science/Planck/Planck_reveals_first_stars_were_born_late) — confirms the CMB origin about 380,000 years after the beginning. - [Cosmic microwave background, Wikipedia](https://en.wikipedia.org/wiki/Cosmic_microwave_background) — (§ Discovery): supports 1964 — “On 20 May 1964 they made their first measurement clearly showing the presence of the microwave background”; and “the CMB is the most perfect black body spectrum ever measured.” - [Wikipedia, Discovery of cosmic microwave background radiation](https://en.wikipedia.org/wiki/Discovery_of_cosmic_microwave_background_radiation) — Penzias and Wilson reported the 3.5 K excess in their note “A Measurement of Excess Antenna Temperature at 4080 Mc/s,” Astrophysical Journal 142, 419–421, July 1965 (doi:10.1086/148307). - [Wien’s displacement law, Wikipedia](https://en.wikipedia.org/wiki/Wien%27s_displacement_law) — confirms the peak equation and root. - [Stefan-Boltzmann law, Wikipedia](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) — confirms the Stefan-Boltzmann expression. - [D. J. Fixsen, The Temperature of the Cosmic Microwave Background, Astrophysical Journal 707 (2009) 916–920](https://arxiv.org/abs/0911.1955) — the abstract: “The determination from the measurements from the literature is cosmic microwave background temperature of 2.72548 +/- 0.00057 K,” while “The FIRAS data are independently recalibrated using the WMAP data to obtain a CMB temperature of 2.7260 +/- 0.0013.” This is where the chapter’s 2.7255 kelvin comes from, and no later measurement supersedes it. - [Lindemann–Weierstrass theorem, Wikipedia](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — e^{a} is transcendental for every nonzero algebraic a. Wien’s root satisfies x = 5(1 − e^{−x}), so if it were algebraic then e^{−x} = 1 − x/5 would be algebraic too. It is therefore transcendental, which is the only proven-transcendental claim in this chapter. - [NIST 2022 CODATA complete listing](https://physics.nist.gov/cuu/Constants/Table/allascii.txt) — “Wien wavelength displacement law constant 2.897 771 955... e-3 (exact) m K” - [Cosmic Microwave Background, OpenSpace documentation](https://docs.openspaceproject.com/latest/content/universe/cosmic-microwave-background/index.html) — “Today, the peak CMB light has redshifted 1,000 times, so it’s now in the microwave with a wavelength of 1 millimeter (1 micron × 1,000 = 1 mm).” ### 3.4 {#r-3.4} The electromagnetic, quantum Hall, Josephson, and atomic identities use the displayed defining relations and CODATA values, and the Rydberg constant is among the most precisely measured constants rather than the single most precise. CODATA 2022 takes α chiefly from the electron’s magnetic-moment anomaly and from atom-interferometry measurements of h/m. R∞ = α²m_ec/(2h), inferred chiefly from hydrogen and deuterium spectroscopy. *Status: confirmed* - [NIST 2022 CODATA complete listing](https://physics.nist.gov/cuu/Constants/Table/allascii.txt) — lists the gas, Faraday, Josephson, flux, atomic, and electron constants. - [Quantum Hall effect, Wikipedia](https://en.wikipedia.org/wiki/Quantum_Hall_effect) — confirms the h/e^2 resistance relation. - [Josephson effect, Wikipedia](https://en.wikipedia.org/wiki/Josephson_effect) — confirms the 2eV/h relation. - [NIST CODATA 2022, Rydberg constant](https://physics.nist.gov/cgi-bin/cuu/Value?ryd) — 10 973 731.568 157(12) m^{−1}, relative standard uncertainty 1.1 × 10^{−12}. - [NIST CODATA 2022, electron g factor](https://physics.nist.gov/cgi-bin/cuu/Value?gem) — −2.002 319 304 360 92(36), relative standard uncertainty 1.8 × 10^{−13}, six times smaller than the Rydberg constant’s. So R_{∞} is among the most precisely measured constants and not the most precisely measured one. - [Mohr et al., CODATA recommended values of the fundamental physical constants: 2022, arXiv:2409.03787](https://arxiv.org/abs/2409.03787) — carries the inputs for α: the electron anomaly gives “one of the three most accurate determinations of the fine-structure constant α in the 2022 adjustment,” and “the two input data h/m(87Rb) and h/m(133Cs) in the 2022 adjustment obtained using atom interferometry are also of great importance because they provide the two other accurate values of α” - [Rydberg constant, Wikipedia](https://en.wikipedia.org/wiki/Rydberg_constant) — “The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c). ### 3.5 {#r-3.5} QED computes the electron anomaly; the weak mixing angle describes neutral electroweak field mixing; alpha_s is the running QCD coupling. The 2022 measurement of the electron magnetic moment was made by Gabrielse’s group at Northwestern University. Color confinement has no analytic proof in any non-abelian gauge theory. The CODATA 2022 value, 0.22305(23), is 1 − (m_W/m_Z)², which is the on-shell definition of sin²θ_W. Its value is not predicted by the Standard Model; the Weinberg angle is a free parameter. *Status: confirmed* - [Anomalous magnetic dipole moment, Wikipedia](https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment) — confirms QED agreement beyond ten significant figures. - [Weinberg angle, Wikipedia](https://en.wikipedia.org/wiki/Weinberg_angle) — confirms W3/B rotation and the on-shell value: “CODATA 2022 gives the value sin²θw = 1 − (mW/mZ)² = 0.22305(23)” (formula as typeset on the page); and “The specific value is not predicted by the Standard Model: The Weinberg angle θ_{w} is an open, free parameter.” - [X. Fan, T. G. Myers, B. A. D. Sukra and G. Gabrielse, Measurement of the Electron Magnetic Moment, Physical Review Letters 130 (2023) 071801](https://arxiv.org/abs/2209.13084) — g/2 = 1.001 159 652 180 59(13), the Northwestern measurement the a_{e} card quotes, taken in 2022 and published in 2023. - [Particle Data Group, Review of Particle Physics 2026, Quantum Chromodynamics review](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-qcd.pdf) — section 9, revised August 2025, gives α_{s}(m_{Z}^{2}) = 0.1180 ± 0.0009 and its running. This is the current edition; of its world average it says that it “changed only marginally as compared to the values reported in the last edition of this Review,” so the chapter’s number stands. - [Gabrielse group discovers new measurement of electron magnetic moment, Northwestern Physics and Astronomy](https://physics.northwestern.edu/about/news/2023/gabrielse-group-discovers-new-measurement-of-electron-magnetic-moment.html) — “A new measurement of the electron magnetic moment from Physics and Astronomy’s Board of Trustees Professor, Gerald Gabrielse and his research group at Northwestern University’s Center for Fundamental Physics has delivered the most precise measurement of a property of any elementary particle.” - [Color confinement, Wikipedia](https://en.wikipedia.org/wiki/Color_confinement) — “There is not yet an analytic proof of color confinement in any non-abelian gauge theory.” and “It is sometimes believed that the sole origin of confinement is the very large value of the strong coupling near the Landau pole.” … “It is however incorrect” - [Particle Data Group, Review of Particle Physics 2026, Electroweak Model and Constraints on New Physics review](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-standard-model.pdf) — “the OS scheme promotes the tree-level formula sin²θW = 1 − MW²/MZ² to a definition of the renormalized sin²θW to all orders in perturbation theory” - [Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, in The Millennium Prize Problems (Clay Mathematics Institute)](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — the strong force is the gauge theory of SU(3) with quarks: “The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD)”; the prize problem is the pure theory: “as of yet there is no effective approach to the gauge theory without fermions.” ### 3.6 {#r-3.6} The six displayed laws express standard relativity, Maxwell, Noether, mass-energy, Bianchi, and charge-continuity identities. Luigi Bianchi published the identities named after him in 1902, rediscovering what Levi-Civita says Ricci had found in about 1889; the contracted form, which is the one that makes the divergence of the Einstein tensor vanish, had been published by Aurel Voss in 1880. Contracting the Bianchi identity with a pair of metric tensors gives the contracted form. No experiment has found evidence that magnetic monopoles exist. E = mc² has been called the world’s most famous equation, and it explains the source of the Sun’s energy. *Status: confirmed* - [Maxwell’s equations, Wikipedia](https://en.wikipedia.org/wiki/Maxwell%27s_equations) — confirms div B=0. - [Noether’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Noether%27s_theorem) — confirms symmetry-to-conservation within its hypotheses. - [Mass-energy equivalence, Wikipedia](https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence) — confirms E=mc^2 at rest. - [Einstein tensor, Wikipedia](https://en.wikipedia.org/wiki/Einstein_tensor) — confirms the contracted Einstein-tensor identity. - [Luigi Bianchi, Wikipedia](https://en.wikipedia.org/wiki/Luigi_Bianchi) — read as raw HTML: “In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor … According to Tullio Levi-Civita, these identities had first been discovered by Ricci in about 1889, but Ricci apparently forgot all about the matter, which led to Bianchi’s rediscovery. However, the contracted Bianchi identities, which are sufficient for the proof that the divergence of the Einstein tensor always vanishes, had been published by Aurel Voss in 1880.” The 1902 paper is given as Bianchi, Sui simboli a quattro indici e sulla curvatura di Riemann, Rend. Acc. Naz. Lincei 11 (1902). - [Continuity equation, Wikipedia](https://en.wikipedia.org/wiki/Continuity_equation) — confirms charge conservation. - [Contracted Bianchi identities, Wikipedia](https://en.wikipedia.org/wiki/Contracted_Bianchi_identities) — proof: “Contract both sides of the above equation with a pair of metric tensors,” arriving at ∇_m R^m_ℓ = ½ ∇_ℓ R - [Magnetic monopole, Wikipedia](https://en.wikipedia.org/wiki/Magnetic_monopole) — “Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.” - [Albert Einstein, Wikipedia](https://en.wikipedia.org/wiki/Albert_Einstein) — “His mass–energy equivalence formula E = mc², which arises from special relativity, has been called “the world’s most famous equation.”” - [Sun, Wikipedia](https://en.wikipedia.org/wiki/Sun) — “Albert Einstein provided the essential clue to the source of the Sun’s energy output with his mass–energy equivalence relation E = mc².” ### 3.7 {#r-3.7} The Planck units, Hubble values, atomic key values, and Fermi coupling have the displayed definitions and values. α_G = Gm_e²/(ℏc) = (m_e/m_P)² ≈ 1.7518 × 10⁻⁴⁵, computed from the CODATA 2022 values. The Hubble tension is the most statistically significant tension in the ΛCDM model. The proton-to-electron mass ratio is a fundamental dimensionless constant that no theory explains; it is measured. The Planck mass, about a flea’s egg, is the one Planck base unit on a human scale, and (m_e/m_P)² is the gravitational coupling constant α_G. The universe is usually given a temperature of about 10³² K at the Planck time. The Planck time is the time light takes to cross one Planck length, t_P = ℓ_P/c. The classical electron radius is e²/(4πε₀m_ec²), the radius of a charged sphere whose field energy is m_ec², also called the Thomson scattering length, and equals α²a₀; the real electron is treated as a point charge. At the Planck temperature the wavelength of thermal radiation reaches the Planck length. *Status: confirmed* - [NIST 2022 CODATA complete listing](https://physics.nist.gov/cuu/Constants/Table/allascii.txt) — lists the atomic, electron, magnetic, Fermi, and input constants. - [Planck units, Wikipedia](https://en.wikipedia.org/wiki/Planck_units) — confirms the Planck-unit definitions. These are the derived Planck length, time, and mass, built from ℏ, G, and c. The Planck constant h itself is a different thing — an SI defining constant, exact by definition — and is covered at 3.1. Of the Planck temperature: “At the Planck temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than T_{P}.” - [Planck (spacecraft), Wikipedia](https://en.wikipedia.org/wiki/Planck_(spacecraft)) — confirms the Planck Hubble and age values. - [A. G. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant…, Astrophysical Journal Letters 934 (2022) L7](https://arxiv.org/abs/2112.04510) — H_{0} = 73.04 ± 1.04 km s^{−1} Mpc^{−1}, and the paper states a five-sigma difference with the value Planck and ΛCDM predict. This is the late-universe half of the Hubble tension. - [Dimensionless physical constant, Wikipedia](https://en.wikipedia.org/wiki/Dimensionless_physical_constant) — “A gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted αG.” Displays αG = Gmₚ²/(ħc), mₚ the particle rest mass. - [E. Di Valentino et al., In the Realm of the Hubble tension – a Review of Solutions, arXiv 2103.01183 (2021)](https://arxiv.org/abs/2103.01183) — carries the ranking of the tension: “The most statistically significant tension is the 4−6σ disagreement between predictions of the Hubble constant H0 by early time probes with ΛCDM model, and a number of late time, model-independent determinations of H0 from local measurements” - [Planck units, OeisWiki](https://oeis.org/wiki/Planck_units) — carries the human scale and the flea’s egg: “Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. (It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.)” - [Ned Wright, Brief History of the Universe, UCLA](https://www.astro.ucla.edu/~wright/BBhistory.html) — carries the temperature at the Planck time (order of magnitude only): “The Planck time: 10⁻⁴³ seconds.” … “The temperature is T=10³² K.” - [Planck time, Simple English Wikipedia](https://simple.wikipedia.org/wiki/Planck_time) — “One Planck time is the time it would take a photon travelling at the speed of light to cross a distance equal to one Planck length.” - [Classical electron radius, Wikipedia](https://en.wikipedia.org/wiki/Classical_electron_radius) — carries the formula r_e = (1/4πε₀)e²/(m_ec²), the reading, the α-relations r_e = λ̄_e α = a₀α², and the point-like electron: “A classical charged sphere producing an electric field with energy equal to the electron’s rest mass energy would have a radius equal to the classical electron radius.”; “The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length.”; “Consequently, electrons are considered point charges in modern theories.” ### 3.8 {#r-3.8} The displayed product is approximately 1.3 times 10^-154 and finite positive. *Status: confirmed* - **Own computation** (the author's own work) — Substitute the displayed CODATA-scale values; exact-one brackets cancel, the thermal bracket is pi^2/60, and the remaining positive factors multiply to approximately 1.3e-154. ### 3.9 {#r-3.9} The Lambda wording is conventional shorthand, and the glossary supplies the dimensional distinction. *Status: judgement* - [Cosmological constant, Wikipedia](https://en.wikipedia.org/wiki/Cosmological_constant) — discusses Lambda and vacuum energy. - [Λ, the cosmological constant](ch03-cards.md#g-Lambda) — in this book — Λ has units of m^{−2}, and the vacuum-energy density that goes with it is built from Λ, c, and G. Editorial note: The chapter calls Λ the energy of empty space. That is the ordinary popular description rather than the literal one: Λ is a curvature, measured in m−2, and the vacuum-energy density that corresponds to it has to be built from Λ, c, and G, with the factor depending on whether mass or energy density is meant. The prose keeps the familiar phrase and the glossary card carries the distinction. ### 3.10 {#r-3.10} The phrase describing 1.3 times 10^-154 as close to nothing is rhetorical rather than a literal minimum claim. *Status: judgement* - **Own computation** (the author's own work) — The product evaluates to approximately 1.3e-154, but no fetched source can establish a global minimum among nonzero numbers. Editorial note: The product is finite, positive, and about 1.3 × 10−154. There is no smallest positive real number, so “as close to nothing as a number can get” cannot be literally true; it is a remark about scale, not a claim about a minimum, and it is left as the author wrote it. ### 3.11 {#r-3.11} Minkowski gave spacetime its invariant interval in 1908, and the interval between two events joined by a light ray is exactly zero. *Status: confirmed* - [Minkowski space, Wikipedia](https://en.wikipedia.org/wiki/Minkowski_space) — dates Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern to 1908 and the Cologne Raum und Zeit lecture to 1908, and defines a null or lightlike vector by c^2t^2 = r^2, which is the interval being exactly zero along a light ray; and: “Minkowski spacetime is constructed so that the speed of light will be the same constant regardless of the reference frame in which it is measured.” - [Hermann Minkowski, Wikipedia](https://en.wikipedia.org/wiki/Hermann_Minkowski) — read as raw wikitext: the address “called “Space and Time” delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908),” in which Minkowski set out the four-dimensional geometry of spacetime; this page does not name the city (the Minkowski space line, now the Minkowski spacetime article, gives Cologne). Editorial note: The chapter prose and the #dslight card both carry the year 1908 and both stood on 3.6, which sources div B = 0, Noether, E = mc², the Einstein tensor, and charge continuity but does not mention Minkowski at all. The date is correct; it simply had nothing behind it. Both the 1908 paper and the 1908 Cologne lecture are Minkowski’s; the lecture was published in 1909, which is why some sources give that year for Raum und Zeit. ### 3.12 {#r-3.12} The hierarchy problem is the unexplained disparity between the strength of gravity and that of the other forces, of which the ratio of the electromagnetic to the gravitational coupling between two electrons is one statement. The ratio e²/(4πε₀Gm_e²) ≈ 4.17 × 10⁴² is one of the large-number coincidences behind Dirac’s hypothesis. Between two electrons the ratio e²/(4πε₀Gmₑ²) = α/α_G is about 4 × 10⁴²; proposed answers include supersymmetry, extra dimensions, and composite Higgs models. *Status: confirmed* - [Hierarchy problem, Wikipedia](https://en.wikipedia.org/wiki/Hierarchy_problem) — defines the hierarchy problem as the large discrepancy between aspects of the weak force and gravity, and gives the weakness of gravity relative to the other forces as its primary numerical example, alongside the Higgs-mass fine-tuning form of the same problem: “the question is why the Higgs boson is so much lighter than the Planck mass”; “one would expect that the large quantum contributions to the square of the Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears unless there is an incredible fine-tuning cancellation between the quadratic radiative corrections and the bare mass.” - [Dirac large numbers hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis) — “LNH was Dirac’s personal response to a set of large number “coincidences” that had intrigued other theorists of his time.” The page displays e²/(4πε₀ G mₑ²) ≈ 4.1666763·10⁴², which is α/α_G. - [Composite Higgs models, Wikipedia](https://en.wikipedia.org/wiki/Composite_Higgs_models) — “Often referred to as “natural” composite Higgs models, CHMs are constructions that attempt to alleviate fine-tuning or “naturalness” problem of the Standard Model.” Editorial note: The chapter uses the broad sense: α/α_G is about 4 × 10⁴², so “about 10⁴²” is right, and it is the force-strength statement of the hierarchy problem rather than the Higgs-mass fine-tuning statement. The sentence stood on 3.7, which covers Planck units, the Hubble values, the atomic key values, and the Fermi coupling, and does not mention the hierarchy problem. ### 3.13 {#r-3.13} The chapter epigraph is Feynman’s, on the fine-structure constant, from QED: The Strange Theory of Light and Matter of 1985. *Status: confirmed* - [Fine-structure constant, Wikipedia](https://en.wikipedia.org/wiki/Fine-structure_constant) — read as raw HTML: it quotes Feynman at length — “It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics” — which is the epigraph and the source of the “greatest damn mysteries” line the #alpha card also uses. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. ### 3.14 {#r-3.14} The Standard Model is the quantum field theory whose defining internal symmetry is the local gauge group SU(3) × SU(2) × U(1), whose three factors give rise to the three non-gravitational interactions. Writing the most general Lagrangian with that symmetry leaves 19 parameters whose numerical values must be established by experiment; explaining neutrino mass is believed to require a further 7 or 8. *Status: confirmed* - [Standard Model, Wikipedia](https://en.wikipedia.org/wiki/Standard_Model) — read as raw HTML: “The local SU(3) × SU(2) × U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model… the three factors of the gauge symmetry give rise to the three fundamental interactions”; “the dynamics depends on 19 parameters, whose numerical values are established by experiment”; and “explaining neutrino mass will require an additional 7 or 8 constants.” The numbered parameter table runs to 19, ending at the Higgs mass. - [Review of Particle Physics, Electroweak model and constraints on new physics, Particle Data Group](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-standard-model.pdf) — the field’s own reference review of the model and of the measured inputs the chapter’s three couplings are quoted from. ### 3.15 {#r-3.15} ΛCDM is the standard cosmological model, with a cosmological constant Λ for dark energy and cold dark matter. The Planck collaboration’s version is fitted with six parameters — baryon density, dark matter density, the angular size of the sound horizon, the amplitude and the spectral index of the primordial fluctuations, and the reionization optical depth — six being the smallest number that gives an acceptable fit; the Hubble constant and the dark-energy density are then calculated from them rather than fitted, and the total density is fixed at 1 and the dark-energy equation of state at −1. *Status: confirmed* - [Lambda-CDM model, Wikipedia](https://en.wikipedia.org/wiki/Lambda-CDM_model) — read as raw HTML: “The Planck collaboration version of the ΛCDM model is based on six parameters… Six is the smallest number of parameters needed to give an acceptable fit to the observations; other possible parameters are fixed at ‘natural’ values, e.g. total density parameter = 1.00, dark energy equation of state = −1… From these six parameters, the other model values, such as the Hubble constant and the dark energy density, can be calculated.” - [Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 (A&A 641 (2020) A6)](https://arxiv.org/abs/1807.06209) — the same paper, free on arXiv: “good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology,” with H₀ = 67.4 ± 0.5 km/s/Mpc inferred from the fit rather than fitted. - [Planck 2018 results. VI, readable HTML (ar5iv)](https://ar5iv.labs.arxiv.org/html/1807.06209) — section 2.1 names the six fitted base-ΛCDM parameters: the baryon density ω_b, the cold dark matter density ω_c, θ_MC (“an approximation to the observed angular size of the sound horizon at recombination”), the reionization optical depth τ, the amplitude A_s and the spectral index n_s of the primordial perturbations — one amplitude. Editorial note: This is the model the chapter means by “taken consistently from the same cosmological model” in the Ω bracket. It is why the four density parameters may be added and why the individual values should not be mixed between fits. ### 3.16 {#r-3.16} The 26th General Conference on Weights and Measures voted unanimously on 16 November 2018 to redefine four SI base units, and the new definitions took effect on 20 May 2019, the 144th anniversary of the Metre Convention. The kilogram, ampere, kelvin, and mole are now defined by fixing exact numerical values for the Planck constant h, the elementary charge e, the Boltzmann constant k_B, and the Avogadro constant N_A; the second, meter, and candela had already been defined by physical constants. *Status: confirmed* - [2019 revision of the SI, Wikipedia](https://en.wikipedia.org/wiki/2019_revision_of_the_SI) — read as raw HTML: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (k_B), and the Avogadro constant (N_A), respectively. The second, metre, and candela had previously been redefined using physical constants,” and “The vote, which was held on 16 November 2018 at the 26th CGPM, was unanimous.” - [Measurement units, Bureau International des Poids et Mesures](https://www.bipm.org/en/measurement-units) — the defining body’s own page for the seven defining constants of the SI. ### 3.17 {#r-3.17} A dimensionless quantity, or quantity of dimension one, has no unit of measurement, so its numerical value is the same in every system of units; the fine-structure constant, about 1/137, and the proton-to-electron mass ratio, about 1836, are the standard examples, and CODATA lists both as pure numbers. *Status: confirmed* - [Dimensionless quantity, Wikipedia](https://en.wikipedia.org/wiki/Dimensionless_quantity) — read as raw wikitext: “Dimensionless quantities, or quantities of dimension one, are quantities with no associated units of measurement,” and of a ratio, “its value remains independent of the specific units of volume used”; its list of dimensionless quantities in physics and engineering includes the fine-structure constant, “α ≈ 1/137,” and “β (or μ) ≈ 1836, the proton-to-electron mass ratio.” - [NIST CODATA 2022, proton-electron mass ratio](https://physics.nist.gov/cgi-bin/cuu/Value?mpsme) — read as raw HTML: numerical value 1836.152 673 426(32), given with no unit. Editorial note: Added 18 September 2026. The #dimensionless card was the only one of 723 glossary cards that cited nothing at all; a sweep with citecheck.py found it. ### 3.18 {#r-3.18} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Bohr’s model of the hydrogen atom, whose innermost orbit is the Bohr radius, dates from 1913; Dirac’s equation of 1928 gives the electron a g-factor of exactly 2; Schwinger first computed the leading QED correction a_e = alpha/2pi in 1948; The Planck 2018 fit (final data release), combined with BAO, gives t0 = 13.787 +/- 0.020 billion years; Arnold Sommerfeld introduced the fine-structure constant in 1916; Ștefan Procopiu obtained the electron’s magnetic moment in 1913, and Bohr obtained it the same year from his atom model, and Pauli named the Bohr magneton in 1920; Pauli gave the Bohr magneton its name in 1920; Ole Rømer made the first estimate of the speed of light in 1676 from the moons of Jupiter; Arthur Compton explained the scattering of X-rays by electrons in 1923, with a shift proportional to h/(m_e c). The Bohr radius is about 0.53 Å, half an ångström. The age follows from H₀ and the density parameters through the Friedmann equation, and the oldest stars’ ages are compatible with the Planck 2018 value. The electron’s magnetic moment is 1.00115965218046(18) Bohr magnetons in magnitude, that is (1 + a_e) of one. Below the Compton wavelength an electron cannot be localized without enough energy to make a pair, and quantum field theory becomes essential. α is Sommerfeld’s ratio of the electron’s speed in the first Bohr orbit to c, and Feynman called its unexplained value one of the greatest damn mysteries of physics. The Bohr radius is approximately the most probable electron–nucleus distance in ground-state hydrogen. *Status: confirmed* - [Bohr radius, Wikipedia](https://en.wikipedia.org/wiki/Bohr_radius) — read raw: “In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction.”; “The Bohr radius (a_{0}) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state.” - [Dirac equation, Wikipedia](https://en.wikipedia.org/wiki/Dirac_equation) — read raw: “In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928..” - [Anomalous magnetic dipole moment, Wikipedia](https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment) — read raw: “where \alpha is the fine-structure constant. This result was first found by Julian Schwinger in 1948.” - [Planck (spacecraft), Wikipedia](https://en.wikipedia.org/wiki/Planck_(spacecraft)) — read raw: “Results from the final analysis of Planck data were released in July 2018..” - [Fine-structure constant, Wikipedia](https://en.wikipedia.org/wiki/Fine-structure_constant) — read raw: “The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom.”; it gives α as “The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom … to the speed of light in vacuum, c,” “Sommerfeld’s original physical interpretation”; and quotes Feynman: “Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans.” - [Bohr magneton, Wikipedia](https://en.wikipedia.org/wiki/Bohr_magneton) — read raw: “The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1913.”; “In 1920, Wolfgang Pauli gave the Bohr magneton its name in an article where he contrasted it with the magneton of the experimentalists which he called the Weiss magneton.”; “In the summer of 1913, the values for the natural units of atomic angular momentum and magnetic moment were obtained by the Danish physicist Niels Bohr as a consequence of his atom model.” - [Speed of light, Wikipedia](https://en.wikipedia.org/wiki/Speed_of_light) — read raw: “Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676..” - [Compton wavelength, Wikipedia](https://en.wikipedia.org/wiki/Compton_wavelength) — read raw: “It was introduced by Arthur Compton in 1923 in his explanation of the scattering of photons by electrons (a process known as Compton scattering)..” - [Angstrom, Wikipedia](https://en.wikipedia.org/wiki/Angstrom) — carries the size in ångströms: “the atomic unit of length, the bohr—about 0.53 Å” - [Age of the universe, Wikipedia](https://en.wikipedia.org/wiki/Age_of_the_universe) — carries the method and the stellar check: “With accurate measurements of these parameters, the age of the universe can be determined by using the Friedmann equation.”; “Both values are compatible with the Planck 2018 number.” (the Methuselah star, 14.27±0.80 billion years, and BD +17°3248, 13.8±4 billion years) - [Electron magnetic moment, Wikipedia](https://en.wikipedia.org/wiki/Electron_magnetic_moment) — carries the ratio: “In units of the Bohr magneton (μB), it is −1.00115965218046(18), which has a relative uncertainty of 1.8×10−13.” - [John Baez, Length Scales in Physics, UC Riverside](https://math.ucr.edu/home/baez/lengths.html) — carries both readings: “Trying to localize an electron to within less than its Compton wavelength makes its momentum so uncertain that it can have an energy large enough to make an extra electron-positron pair!” and “the length scale at which relativistic quantum field theory becomes crucial for its accurate description” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.19 {#r-3.19} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Maxwell’s equations of 1865 (the ‘Dynamical Theory’ paper), from which charge continuity follows; Coulomb measured/published the inverse-square law of electric force with a torsion balance in 1785; The four Maxwell equations were assembled in 1861–65; Millikan and Fletcher’s oil-drop experiment was begun in 1909 (final form published 1913); Einstein’s field equations date from November 1915; In a paper received in September 1905 and published that November, Einstein showed that a body’s inertia depends on its energy content (E = mc²); Faraday’s laws of electrolysis of 1834; Enrico Fermi’s 1934 theory of beta decay; In 1961 Deaver and Fairbank, and Doll and Näbauer, independently found flux quantization in steps of h/2e. Heaviside recast Maxwell’s equations in modern vector form in 1884. Faraday’s first law: the mass deposited at an electrode is proportional to the charge passed. Fermi’s theory was a four-fermion contact interaction; G_F is measured most precisely from the muon lifetime and is set by the mass of the heavy W boson. Deaver and Fairbank worked at Stanford; Doll and Näbauer in Herrsching. Mass–energy equivalence accounts for the energy released by nuclear fission in both nuclear weapons and nuclear power. *Status: confirmed* - [A Dynamical Theory of the Electromagnetic Field, Wikipedia](https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field) — read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..” - [Coulomb’s law, Wikipedia](https://en.wikipedia.org/wiki/Coulomb%27s_law) — read raw: “Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his application of a torsion balance to determine that the magnitude of the electric force between two point charges..” - [Maxwell’s equations, Wikipedia](https://en.wikipedia.org/wiki/Maxwell%27s_equations) — read raw: “Maxwell’s equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law..” - [Oil drop experiment, Wikipedia](https://en.wikipedia.org/wiki/Oil_drop_experiment) — read raw: “The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron)..” - [Einstein field equations, Wikipedia](https://en.wikipedia.org/wiki/Einstein_field_equations) — read raw: “The equations were published by Albert Einstein in 1915 in the form of a tensor equation.” - [Mass–energy equivalence, Wikipedia](https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence) — read raw: “Einstein did not write the exact formula E = mc² in his 1905 Annus Mirabilis paper “Does the Inertia of an object Depend Upon Its Energy Content?”; rather, the paper states that if a body gives off the energy L by emitting light, its mass diminishes by L/c².”; and “a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.” - [M. Faraday, Experimental researches in electricity — Seventh Series, Phil. Trans. R. Soc. 124 (1834) 77–122, Crossref record](https://api.crossref.org/works/10.1098/rstl.1834.0008) — read raw: “the publisher’s Crossref record gives the year of publication as 1834.” - [E. Fermi, Tentativo di una teoria dei raggi β, Il Nuovo Cimento 11 (1934) 1–19, Crossref record](https://api.crossref.org/works/10.1007/BF02959820) — read raw: “the publisher’s Crossref record gives the year of publication as 1934.” - [Magnetic flux quantum, Wikipedia](https://en.wikipedia.org/wiki/Magnetic_flux_quantum) — read raw: “The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank, and independently by R. Doll and M. Näbauer, in 1961..” - [Oliver Heaviside, Wikipedia](https://en.wikipedia.org/wiki/Oliver_Heaviside) — “In 1884 he recast Maxwell’s mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell’s equations.” - [Annus mirabilis papers, Wikipedia](https://en.wikipedia.org/wiki/Annus_mirabilis_papers) — “On 21 November Annalen der Physik published a fourth paper (received September 27): “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the Inertia of a Body Depend Upon Its Energy Content?”)” - [Faraday’s laws of electrolysis, Wikipedia](https://en.wikipedia.org/wiki/Faraday%27s_laws_of_electrolysis) — “Michael Faraday reported that the mass (m) of a substance deposited or liberated at an electrode during electrolysis is directly proportional to the charge (Q, for which the SI unit is the ampere-second or coulomb) passed through the electrolyte.” - [Fermi’s interaction, Wikipedia](https://en.wikipedia.org/wiki/Fermi%27s_interaction) — carries the contact interaction, the muon lifetime, and the heavy boson: “The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram).”; “In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents.”; “The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson).”; “MW is the mass of the W boson, which mediates the decay in question.” - [Wikipedia (German), Robert Doll](https://de.wikipedia.org/wiki/Robert_Doll) — places Doll at the Herrsching outstation of the low-temperature commission, where he and Martin Näbauer showed flux quantization in 1961: “Ab 1953 war er wissenschaftlicher Mitarbeiter an der Herrschinger Außenstelle der Kommission für Tieftemperaturforschung.” - [Wikipedia, William M. Fairbank](https://en.wikipedia.org/wiki/William_M._Fairbank) — the Stanford half of the 1961 experiments: “At Stanford, one of his most famous discoveries has been the flux quantization in superconducting tin.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.20 {#r-3.20} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Φ0 = h/(2e) exact since 2019; Newton’s Principia of 1687 stated the inverse-square law but never wrote G as a number; Cavendish’s torsion-balance experiment of 1798 weighed the Earth; CODATA 2018 gives G = 6.67430(15) x 10^-11 m^3 kg^-1 s^-2 (same as CODATA 2022); Clapeyron’s ideal-gas law (1834); Max Planck introduced h in 1900 to explain black-body radiation; Einstein took the quantum literally in 1905; Hubble’s 1929 law v = H0 d; Planck satellite’s 2018 fit gives H0 = 67.4 ± 0.5 km/s/Mpc. PV = nRT combines the laws of Boyle, Charles, and Avogadro. Planck regarded dividing energy into increments as a mathematical artifice. Since h is fixed, a Kibble balance measures the kilogram by fixing it. *Status: confirmed* - [Magnetic flux quantum, Wikipedia](https://en.wikipedia.org/wiki/Magnetic_flux_quantum) — read raw: “As a result of the 2019 revision of the SI, the Planck constant h has a fixed value.” - [Philosophiae Naturalis Principia Mathematica, Wikipedia](https://en.wikipedia.org/w/index.php?title=Philosophi%C3%A6_Naturalis_Principia_Mathematica) — read raw: “authorized (imprimatur) by Samuel Pepys, then-president of the Royal Society on 5 July 1686 and first published in 1687..” - [Gravitational constant, Wikipedia](https://en.wikipedia.org/wiki/Gravitational_constant) — read raw: “The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton’s death, by Henry Cavendish..” - [Fundamental Physical Constants — Complete Listing, 2018 CODATA adjustment, NIST](https://physics.nist.gov/cuu/Constants/ArchiveASCII/allascii_2018.txt) — read raw: “2018 CODATA adjustment ... Newtonian constant of gravitation 6.674 30 e-11 0.000 15 e-11 m^3 kg^-1 s^-2.” - [Émile Clapeyron, Wikipedia](https://en.wikipedia.org/w/index.php?title=%C3%89mile_Clapeyron) — read raw: “In 1834, he made his first contribution to the creation of modern thermodynamics by publishing a report entitled Mémoire sur la puissance motrice de la chaleur (Memoir on the Motive Power of Heat).” - [Planck constant, Wikipedia](https://en.wikipedia.org/wiki/Planck_constant) — read raw: “The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation.”; “Technologies of mass metrology such as the Kibble balance measure the kilogram by fixing the Planck constant.” - [Photoelectric effect, Wikipedia](https://en.wikipedia.org/w/index.php?title=Photoelectric_effect) — read raw: “In 1905, Albert Einstein proposed a theory of the photoelectric effect using a concept that light consists of tiny packets of energy known as photons or light quanta..” - [Hubble’s law, Wikipedia](https://en.wikipedia.org/wiki/Hubble%27s_law) — read raw: “The discovery of Hubble’s law is attributed to work published by Edwin Hubble in 1929,.” - [Planck (spacecraft), Wikipedia](https://en.wikipedia.org/wiki/Planck_(spacecraft)) — read raw: “Results from the final analysis of Planck data were released in July 2018. ... Planck measured the Hubble constant to be 67.4 ± 0.5 km/s/Mpc, increasing the tension with direct local measurements..” - [Ideal gas law, Wikipedia](https://en.wikipedia.org/wiki/Ideal_gas_law) — “It was first stated by Benoît Paul Émile Clapeyron and independently of him, Dmitry Mendeleev in 1834 as a combination of the empirical Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law.” and “The equation of state given here (PV = nRT) applies only to an ideal gas” - [Planck’s law, Wikipedia](https://en.wikipedia.org/wiki/Planck%27s_law) — “While Planck originally regarded the hypothesis of dividing energy into increments as a mathematical artifice, introduced merely to get the correct answer, other physicists including Albert Einstein built on his work” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.21 {#r-3.21} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Dirac wrote plain h for h/2π from 1926 until 1930, when he introduced the symbol ℏ in The Principles of Quantum Mechanics; Dirac’s 1930 Principles of Quantum Mechanics; K_J = 2e/h exact since 2019; Boltzmann’s 1877 relation S = k log W; Planck introduced k in the same 1900 paper as h; 1998 discovery of accelerated expansion; accelerated expansion recognized by the 2011 Nobel Prize; γ appeared in Lorentz’s transformations of 1904; Einstein’s special relativity of 1905; Karl Schwarzschild solved the field equations for a point mass in 1916, within months of their publication, and died on 11 May 1916 in Potsdam, having been invalided home from the Russian front that March. γ is 1 at rest and about 7000 for LHC protons. Brian Josephson predicted the effect in 1962. The Josephson voltage standard gives the standard representation of the volt. The Schwarzschild radius r_s = 2GM/c² is an event horizon, about 3 km for the Sun and 9 mm for the Earth, and equals twice the reduced Compton wavelength of one Planck mass. *Status: confirmed* - [Planck constant, Wikipedia](https://en.wikipedia.org/wiki/Planck_constant) — read raw: “Then, in 1926, in their seminal papers, Schrödinger and Dirac again introduced special symbols for it: K in the case of Schrödinger, and h in the case of Dirac. Dirac continued to use h in this way until 1930, when he introduced the symbol ℏ in his book The Principles of Quantum Mechanics..” - [The Principles of Quantum Mechanics, Wikipedia](https://en.wikipedia.org/w/index.php?title=The_Principles_of_Quantum_Mechanics) — read raw: “The Principles of Quantum Mechanics is an influential monograph written by Paul Dirac and first published by Oxford University Press in 1930..” - [2019 revision of the SI, Wikipedia](https://en.wikipedia.org/wiki/2019_revision_of_the_SI) — read raw: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e).” - [Boltzmann constant, Wikipedia](https://en.wikipedia.org/wiki/Boltzmann_constant) — read raw: “Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k”; “the relation was never expressed with a specific constant until Max Planck first introduced k ... in his derivation of the law of black-body radiation in 1900–1901..” - [Cosmological constant, Wikipedia](https://en.wikipedia.org/wiki/Cosmological_constant) — read raw: “In 1998, two teams of astrophysicists, the Supernova Cosmology Project and the High-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies’ recession in relation to the Milky Way increases over time.”; “For this work, Perlmutter, Schmidt, and Riess jointly received the Nobel Prize in Physics in 2011..” - [History of Lorentz transformations, Wikipedia](https://en.wikipedia.org/w/index.php?title=History_of_Lorentz_transformations) — read raw: “They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz..” - [Lorentz transformation, Wikipedia](https://en.wikipedia.org/wiki/Lorentz_transformation) — read raw: “All of the work up to 1905 assumed that the constant speed of light was a consequence of an aether and hypothesized the transformation. In that year Albert Einstein published what is now called special relativity.” - [Lorentz factor, Wikipedia](https://en.wikipedia.org/wiki/Lorentz_factor) — carries the value at rest: “Its initial value is 1 (when v = 0)” - [Large Hadron Collider, Wikipedia](https://en.wikipedia.org/wiki/Large_Hadron_Collider) — carries the LHC value at 6.5 TeV per proton: “At this energy, the protons have a Lorentz factor of about 6,930” - [Schwarzschild radius, Wikipedia](https://en.wikipedia.org/wiki/Schwarzschild_radius) — carries the solution, formula, horizon, and values: “In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body”; “The Schwarzschild radius (2GM/c²) of a given mass M equals twice its reduced Compton wavelength when M equals one Planck mass”; “The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body”; “the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), whereas Earth’s is approximately 9 mm (0.35 in)” - [Karl Schwarzschild, Wikipedia](https://en.wikipedia.org/wiki/Karl_Schwarzschild) — carries the timing: “an exact solution which he first set down in a letter to Einstein of 22 December 1915”; “In March 1916, Schwarzschild left military service because of his illness”; “Schwarzschild died of immune complications related to his illness on 11 May 1916, at the age of 42.” Its infobox gives the place of death as Potsdam; its “returned to Göttingen” is his leaving military service in March 1916. - [Karl Schwarzschild, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Schwarzschild/) — “Died 11 May 1916 Potsdam, Germany”; “after being invalided home in March 1916, he died two months later.” - [Josephson effect, Wikipedia](https://en.wikipedia.org/wiki/Josephson_effect) — read as raw wikitext: “The effect is named after the British physicist Brian Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.”; and “The Josephson voltage standard takes the caesium standard definition of frequency and gives the standard representation of a volt.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.22 {#r-3.22} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: J. J. Thomson identified the electron in 1897 and measured its charge-to-mass ratio; the mole was adopted as an SI base unit in 1971, then defined by the atoms in 0.012 kilograms of carbon-12; Maxwell’s 1865 result c^2 = 1/(μ0 ε0); from 1948 to 2019 the ampere’s definition fixed μ0 at exactly 4π × 10^-7 H/m; since 2019 μ0 = 2αh/(e^2 c), with the uncertainty of α; Avogadro’s 1811 hypothesis (equal volumes, equal numbers of molecules); Jean Perrin coined the name Avogadro’s number in 1909; Emmy Noether’s work on the invariance theorem began in 1915, when she was helping Felix Klein and David Hilbert with general relativity; Emmy Noether’s Invariante Variationsprobleme was presented for her by Felix Klein and published in 1918. Klein presented it on 26 July 1918; Noether presumably did not because she was not a member of the society. An electron and a positron annihilating with negligible momentum give photons whose energies total 1.022 MeV, twice m_ec². *Status: confirmed* - [Electron, Wikipedia](https://en.wikipedia.org/wiki/Electron) — read raw: “In 1897, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson, performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.”; and of annihilation: “If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons whose energies total 1.022 MeV.” - [Mole (unit), Wikipedia](https://en.wikipedia.org/wiki/Mole_(unit)) — read raw: “The mole was made the seventh SI base unit in 1971 by the 14th CGPM.”; and of the definition that replaced oxygen-16 “during the 1960s”: “The International Bureau of Weights and Measures defined the mole as “the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12.”” - [A Dynamical Theory of the Electromagnetic Field, Wikipedia](https://en.wikipedia.org/w/index.php?title=A_Dynamical_Theory_of_the_Electromagnetic_Field) — read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..” - [Impedance of free space, Wikipedia](https://en.wikipedia.org/wiki/Impedance_of_free_space) — read raw: “Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of μ0 to be exactly 4π × 10^-7 H/m..” - [Vacuum permeability, Wikipedia](https://en.wikipedia.org/wiki/Vacuum_permeability) — read raw: “Since the revision of the SI in 2019 (when the values of e and h were fixed as defined quantities), μ0 is an experimentally determined constant with its value proportional to the dimensionless fine-structure constant.” - [Avogadro constant, Wikipedia](https://en.wikipedia.org/wiki/Avogadro_constant) — read raw: “The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas.”; “The name Avogadro’s number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 32 grams of oxygen gas.”; “Perrin himself determined the Avogadro number, which he called “Avogadro’s constant” (constante d’Avogadro), by several different experimental methods.” - [Noether’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Noether%27s_theorem) — read raw: “Emmy Noether’s work on the invariance theorem began in 1915 when she was helping Felix Klein and David Hilbert with their work related to Albert Einstein’s theory of general relativity”; “This is the first of two theorems (see Noether’s second theorem) published by the mathematician Emmy Noether in 1918..” - [Emmy Noether, Wikipedia](https://en.wikipedia.org/wiki/Emmy_Noether) — “The paper, Invariante Variationsprobleme, was presented by Felix Klein on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.23 {#r-3.23} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Max Planck proposed his natural units in 1899, using h where the modern definitions use ℏ; Klaus von Klitzing discovered the quantized Hall resistance in 1980; the quantum Hall effect won the 1985 Nobel Prize; R_K = h/e^2 exact since 2019 because h and e are; Johannes Rydberg’s 1888 formula for spectral lines; Bohr’s 1913 atom explained the constant; Jožef Stefan found the fourth-power law in 1879, and his student Boltzmann derived it in 1884; Planck’s 1900 law gave σ its value; the Weinberg angle of the 1967 electroweak theory. Bohr’s model gives R∞ = m_e e⁴/(8ε₀²h³c). *Status: confirmed* - [Planck units, Wikipedia](https://en.wikipedia.org/wiki/Planck_units) — read raw: “In 1899, one year before the advent of quantum theory, Max Planck introduced what later became known as the Planck constant. At the end of the paper, he proposed the base units that were later named in his honor.”; and “At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate.” - [Quantum Hall effect, Wikipedia](https://en.wikipedia.org/wiki/Quantum_Hall_effect) — read raw: “In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance was exactly quantized.”; “For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics.”; “Later, the 2019 revision of the SI fixed exact values of h and e, resulting in an exact R_K = h/e^2.” - [Rydberg formula, Wikipedia](https://en.wikipedia.org/wiki/Rydberg_formula) — read raw: “It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg, then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics..” - [Josef Stefan, MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Stefan_Josef/) — read raw: “Stefan showed empirically, in 1879, that total radiation from a blackbody is proportional to the fourth power of its absolute temperature.”; “Boltzmann, who was one of Stefan’s students, showed in 1884 that this Stefan-Boltzmann law could be demonstrated mathematically.” - [Stefan–Boltzmann law, Wikipedia](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) — read raw: “The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck’s law as formulated in 1900..” - [Steven Weinberg, Wikipedia](https://en.wikipedia.org/w/index.php?title=Steven_Weinberg) — read raw: “In 1967 he was a visiting professor at MIT. It was in that year at MIT that Weinberg proposed his model of unification of electromagnetism and nuclear weak forces.” - [Rydberg constant, Wikipedia](https://en.wikipedia.org/wiki/Rydberg_constant) — “The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c). Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.24 {#r-3.24} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Wilhelm Wien found the displacement law in 1893; Z0 was exact before 2019 and is now derived from alpha. Since 2019 μ0, and so Z0 = μ0c, is fixed by the measured fine-structure constant. *Status: confirmed* - [Wien’s displacement law, Wikipedia](https://en.wikipedia.org/wiki/Wien%27s_displacement_law) — read raw: “The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument..” - [Impedance of free space, Wikipedia](https://en.wikipedia.org/wiki/Impedance_of_free_space) — read raw: “This chain of dependencies changed when the ampere was redefined on 20 May 2019..” - [Vacuum permeability, Wikipedia](https://en.wikipedia.org/wiki/Vacuum_permeability) — “In the 2019 revision of the SI, the ampere is defined exactly in terms of the elementary charge and the second, and the value of μ0 is now determined experimentally (based on the measured value of the fine-structure constant)” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 3.26 {#r-3.26} Quantum electrodynamics is the quantum field theory of the electromagnetic interaction, developed by Feynman, Schwinger, and Tomonaga in the late 1940s, and its prediction for the electron’s anomalous magnetic moment agrees with measurement to more than ten significant figures. Schwinger first computed the anomalous moment in 1948, and Feynman, Schwinger, and Tomonaga shared the 1965 Nobel Prize. *Status: confirmed* - [Quantum electrodynamics, Wikipedia](https://en.wikipedia.org/wiki/Quantum_electrodynamics) — describes the theory’s development and its precision agreement for the electron’s magnetic moment. - [Anomalous magnetic dipole moment, Wikipedia](https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment) — carries Schwinger’s 1948 one-loop result and the agreement: “found by Julian Schwinger in 1948” and theory and experiment “agree to more than 10 significant figures,” “making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics”