Blockchain Papers

Eric McLean

Solving Alpha — Version 5.2 The derivation of the fine structure constant, closed from two further directions. The fine structure constant α was derived from the self-reference axiom σ = 1/(1+σ) in the first paper of this series and reaffirmed across Versions 1 through 4. The Pentagon formula α⁻¹ = 360/φ² − 2/φ³ + 1/(3⁵φ⁵) + 1/(7⁷φ⁷) reproduces the Morel 2020 atomic recoil determination of α⁻¹ = 137.035999206(11) to within 0.05σ, with zero free parameters and no experimental input. That derivation stands as originally posted. Version 5.2 does not derive α again. It closes the proof from two further directions, each structurally independent of the original derivation and of each other. The first closure is internal uniqueness. Within a pre-specified coefficient pool drawn from the irreducible representations of the binary icosahedral group, the spectral structure of the 600-cell polytope, and the self-referential reciprocal-power family — defined before the formula is consulted and requiring no knowledge of α — the Pentagon formula is the unique 1σ match to Morel 2020. The nearest structurally distinct competitor sits 139× further from the measured value. The four prime exponents (2, 3, 5, 7) of the formula are independently attested by the seventh spectral moment of the 600-cell adjacency matrix, μ₇ = Tr(A⁷)/1440 = 50,400 = 2⁵ · 3² · 5² · 7. The Pentagon formula is not one of many φ-series that fit; it is the only structurally admissible one. The second closure is external overdetermination. The same number α⁻¹ = 137.036 that the Pentagon formula produces is independently recovered, with no electromagnetic input, from three disconnected non-electromagnetic sectors. The cosmological constant Λ from Planck CMB and BAO, the gravitational coupling G from CODATA torsion balance measurements, and the Hubble expansion rate H₀ from SH0ES distance ladders all sit on a single straight line whose slope is α⁻¹ and whose intercept is φ⁻². The horizontal coordinates of that line are forced by Dirichlet's 1837 class number theorem for the field ℚ(√5). Four disconnected experimental programmes, four independent determinations of α⁻¹, one common value. The original derivation gave the number. The first closure shows that no other formula in the structurally admissible space gives that number. The second closure shows that the same number is the unique slope on which four disconnected experimental sectors agree. The proof was complete in V1; it is now closed on three sides. The asymptotic series for α⁻¹ is presented in fully derived form, with coefficient C_k = 2^(k²) counting the directed coupling configurations among k self-referential modes at maximum entropy equilibrium. The series shares the asymptotic character of QED's own perturbation expansion, with optimal truncation near k = 6 settling within 1.65σ of the most precise measurement. A fifth term is pre-registered before any measurement at the required precision exists to test it. Confirmation of either the Parker 2018 caesium or Fan 2023 electron g−2 determinations as the correct value of α⁻¹ at high significance falsifies the formula at the current truncation order; the framework commits to Morel 2020 as the correct value. The fine structure constant is a theorem of self-referential geometry on the field ℚ(√5). The original derivation, the internal uniqueness closure, and the external overdetermination closure are now on the public record together. Ten revisions between V5 and V5.2 are documented inline; the bone-structure claims survive intact. Supplementary ablation scripts and machine-readable results are deposited alongside this record for full reproducibility. Keywords: fine structure constant, self-reference, 600-cell, binary icosahedral group, Dirichlet class number, asymptotic series, Pentagon Physics, derivation closure, falsifiable prediction, ℚ(√5)

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NFT (Non-Fungible Tokens) artworks are artistic works that are cryptographically recorded in the blockchain, with limited copy numbers. In the field of art, which has always been closely intertwined with socio-cultural, economic, and technological developments, blockchain technology is now being widely used through NFT artworks. This study aims to examine the NFT collection titled "Geometries" by American artist Frank Stella. The collection consists of 22 pieces and is available on Open Sea, one of the most popular NFT art markets. The evaluation of Frank Stella's NFT collection "Geometries" will focus on its form and content, as well as its relationship with the traditional art world.Frank Stella's art career offers one of the most original examples to evaluate the relationship that Modern and Postmodern art will establish with NFT technology. With this aspect, the artist has transferred the ongoing perspective from minimalism to his contemporary works. Although subjectivity is avoided in the collection, there is a unity of ideas and styles. Utopian-dystopian universes, urban phenomenon, biomorphic forms, space-space relationship come to the fore. In the “Geometries” series, form and form precede the subject in the studies.Studies on NFT artworks in the literature generally focus on the art market area. There is a lack of research that examines the content of NFT artworks or focuses on their relationship with art history. For this reason, it is thought that the study will contribute to the literature.

Helge Kragh

Ever since the famous Eddington-Dyson solar eclipse expedition in 1919, it has been known that massive bodies cause space (or rather space-time) to curve. This happens not only locally, in the vicinity of celestial bodies, but also on the largest possible global scale. Einstein's first cosmological model of 1917 represented the finite universe by the kind of 3D spherical space that had been familiar to mathematicians for more than half a century. According to Einstein, the constant curvature K and radius of curvature R were given by the average density ρ of matter in the universe by where G is Newton's gravitational constant. Although Einstein's model only survived to about 1930, curved space remained an element in most later cosmological models. The question to be decided by a combination of theory and observation was the size of the cosmic curvature, as expressed by the curvature constant k = R2K. In the Einstein universe, k = +1. The present consensus view, in part based on the inflationary scenario, is that we live in a flat or Euclidean space, corresponding to k = 0, which implies that the universe is infinite in extent. However, this is a view that can never be proved observationally, not even in principle. Whereas the reality of curved space belongs to the 20th century, as a mathematical hypothesis it was discussed many decades before Einstein. The first scientist who not only realized the possibility of a closed universe, but advocated it as a model of the real universe, is little known today. Few cosmologists have ever heard about the German astrophysicist Karl Friedrich Zöllner, who as early as 1872 argued that the universe is finite, in the sense that cosmic space is positively curved (Jaki 1969, Kragh 2012). Zöllner's remarkable cosmology based on non-Euclidean geometry deserves more than just a footnote in the annals of cosmological thought. Naturally, questions about the curvature of space could only be asked after the recognition, in the first half of the 19th century, that geometries other than Euclid's are possible. As early as about 1815, Karl Friedrich Gauss in Göttingen came to the conclusion that Euclidean geometry is not true by necessity but can be justified only empirically. According to an often repeated myth - but it is a myth - he attempted to test the validity of Euclidean geometry by measuring geodetically the sum of angles in a triangle extending between three mountain peaks in the state of Hanover (Breitenberger 1984). While Gauss anticipated non-Euclidean geometry, it was left to the Hungarian mathematician János Bolyai and, independently, his Russian colleague Nikolai Ivanovich Lobachevsky to establish geometrical systems different from the venerable one of Euclid. Of the two pioneers, Lobachevsky was the more empirically oriented. As he said in a paper of 1835, the truth of geometry “can only be verified, like all other laws of Nature, by experiment, such as astronomical observations” (Lobachevsky 1898). K F Zöllner, steel engraving from 1882. What Lobachevsky called “imaginary geometry” soon became known as hyperbolic geometry, characterized by a curvature constant k = −1 (and therefore an imaginary radius of curvature). Not only did he prove that in this kind of space the angle sum in a triangle always exceeds 180°, he also suggested that the geometry of physical space might be tested by considering stellar parallaxes. For example, while in Euclidean space the parallax of a star tends toward zero as its distance increases toward infinity, Lobachevsky showed that in hyperbolic space there is a minimum parallax for all stars irrespective of how far they are from the Earth. In his first paper on the new geometry, dating from 1829, he used a value of 1″.24 for the parallax of Sirius - three times as great as the real one - to conclude that space was flat to an approximation much closer than the error of measurement. Nonetheless, rather than concluding that space was Euclidean, he considered his calculations to be inconclusive. Perhaps, he speculated, a deviation from flat space would turn up in future measurements of much larger heavenly triangles. In a famous lecture of 1854, the young Göttingen mathematician Bernard Riemann completed and generalized the earlier ideas of Gauss, Lobachevsky and Bolyai. Emphasizing that curvature is an intrinsic property of space, he argued that although there is any number of possible geometries, there are only three that can represent physical space. These spaces of constant curvature correspond to the three values of the curvature constant, k = 0, ±1. Riemann paid particular attention to the case of a closed spherical space, pointing out that in such a space “we must distinguish between unboundedness and infinite extent.” A space of constant positive curvature “must necessarily be finite provided this curvature has ever so small a positive value” (Riemann 1873). A physicist as well as a mathematician, he speculated that the metrical structure of space on a microscopic scale might be of importance for the physics of atoms and molecules. On the other hand, he did not take an interest in the space of the astronomers. Questions about the global properties of space he dismissed as “idle questions”. Non-Euclidean geometry circulated slowly in the mathematical community, and even more slowly among physicists and astronomers. Only in the 1870s, in large measure due to popular lectures by Hermann von Helmholtz and William Clifford, did Riemann's ideas become generally known and seen as a vision of a possible geometrization of physics. Johann Karl Friedrich Zöllner (1834–1882) is today recognized for his contributions to astrophysics and, in particular, his pioneering work in astrophotometry (Koerber 1899, Hermann 1982). A skilled experimentalist and designer of instruments, in 1858 Zöllner invented an astrophotometer to measure the feeble light from stars and planets. In 1862 he moved to Leipzig, where he was appointed professor and established an astrophysical research programme, the first of its kind. In addition to his experimental work, he also made important studies of theoretical problems in astronomy and physics. These included electrodynamics, solar theory, sunspots and the theory of comets. In his Natur der Cometen from 1872 (figure 2) he developed an electrical theory of comets that for a period was widely admired. Title page of Zöllner's 1872 book on the nature of comets, including his proposal of a closed-space universe. Zöllner was a tireless advocate of Heinrich Weber's theory of electrodynamics based on a fundamental force law acting between hypothetical charged particles. Not only did Zöllner accept Weber's force law and associated atomistic theory, he also argued that it was of universal significance and valid for all terrestrial and cosmic phenomena. He suggested that it could be translated into a law of gravitation superior to Newton's, in the sense that the latter was merely a special case of Weber's. In Zöllner's extended version of Weber's theory, the interaction between two charged particles of opposite sign differed slightly, by a factor of 1.7 × 10−40, from the interaction between two particles of the same sign. Thus, a very small residual force would remain between two bodies, and this residual electric force he identified with the gravitational attraction (Zöllner 1882). In effect, he recognized the later so famous (and still unexplained) ratio between the gravitational and the electromagnetic interaction, given by the pure number Fgrav/Fem ≅ 10−40. Among other things, he used his electro-gravitational theory in an attempt to explain the anomalous motion of Mercury's perihelion, one of the major problems in astronomy until it was finally solved by Einstein. Natur der Cometen (Zöllner 1872) was a remarkable work in more than one sense. The major part of the 600-page book was not about comets, but instead a strange mixture of philosophy of science and unconstrained, chauvinistic charges of plagiarism. Zöllner's main targets were British scientists, including luminaries such as William Thomson and Charles Darwin, but he also attacked Helmholtz, one of the most powerful men in German science. The book aroused a storm of controversy and had the effect that Zöllner became increasingly marginalized as a scientist. Although much of the last decade of Zöllner's troubled life was occupied with philosophical speculations, spiritualism and endless controversies, he continued doing scientific work. Thus, he developed a theory of the origin of the Earth's magnetism according to which the magnetism was due to electrical currents in the fluid core of the Earth. Natur der Cometen included a chapter on “The Finitude of Matter in Infinite Space” in which Zöllner offered an original solution to Olbers' paradox in terms of a universe of constant positive curvature (Jaki 1969). In his systematic discussion of the finite versus the infinite in the universe, he assumed, for the sake of discussion, that there is only a finite amount of matter in the world. He then argued that in an unbounded (and therefore infinite) Euclidean space any finite amount of matter would evaporate and dissolve to zero density in an infinity of time. Given the actual existence of matter of non-zero density, he concluded that either is space finite or the universe has only existed for a limited period of time. Unwilling to accept the latter hypothesis, he suggested that Riemann's geometry might provide the key that would unravel the secrets of the universe and dissolve the problems of a materially finite universe: “It seems to me that any contradictions will disappear … if we ascribe to the constant curvature of space not the value zero but a positive value, however small … The assumption of a positive value of the spatial curvature measure involves us in no way in contradictions with the phenomena of the experienced world if only its value is taken to be sufficiently small.” In this way he made Olbers' paradox disappear without having to assume a limitation of either cosmic time or space. While he noted with satisfaction that energy conservation would apply to his finite material universe, he did not address the problem caused by the increase of entropy in a spatially finite but temporally infinite universe. Clearly inspired by Riemann, and happy to admit the inspiration, Zöllner further speculated that curved space was dynamically active, in the sense of determining the laws of Nature. Not even the divine force law of Weber was true a priori but somehow of cosmological origin, a speculation that bears some similarity to the later Mach's principle. And Zöllner went further than Riemann: whereas the Göttingen mathematician had declared that physics represented the “domain of another science”, the Leipzig astrophysicist maintained that the science of the physical world belonged entirely to the field of Riemann's investigations. Later in the century a few mathematicians attacked the problem of Mercury's anomalous precession by assuming space to be non-Euclidean. In 1885–1886 Wilhelm Killing and Carl Neumann derived orbits for Mercury moving in spherical space, and in 1902 Otto Liebmann did the same in the case of hyperbolic space. Zöllner's innovative cosmological speculations attracted some attention in German philosophical circles, but were ignored by most physicists and astronomers. Not only was cosmology considered a somewhat disreputable field that scarcely belonged to science, the idea of a closed space was also widely associated with the (even more disreputable) notion of a fourth space dimension. To understand the lack of scientific response to Zöllner's universe, one must take into account his controversial ideas of a fourth dimension as the site of spiritual phenomena (Zöllner 1880). In 1877, after meeting the chemist William Crookes in London, Zöllner turned wholeheartedly to spiritualism (Treitel 2004). Convinced of the reality behind spiritualist manifestations, he investigated them in great detail, attempting to integrate the spirits with both Weberian physics and his own highly unorthodox version of Christian theology. The first major result of his efforts in this area of unconventional research was an elaborate Transcendental Physics published in 1878 and translated into English two years later (Zöllner 1880). As Zöllner saw it, the project of a transcendental physics including both material and spiritual phenomena was but a natural extension of the astrophysical project of accommodating terrestrial and celestial phenomena within the same theoretical framework. It was a strictly scientific project. Not satisfied with simply accepting the spirits of deceased persons, as they appeared in séances, Zöllner argued that they were visitors from a hidden fourth dimension of space. During the last decades of the 19th century, beliefs of this kind were widespread; Zöllner only took them more seriously than most. It was sometimes contended that if our space is curved, it must be contained in a flat space of a higher dimension, in the same way that a 2D space is embedded in our 3D space. Although 4D “hyperspace” was often mixed up with ideas of non-Euclidean geometry, in reality there is no connection between them. William Clifford dismissed the connection as groundless, as did other mathematicians. A curved space does not need to be curved “in” another space. Zöllner's belief in a spiritual fourth dimension received inspiration from his knowledge of non-Euclidean geometry, which he sometimes used for purposes of illustration, but it did not depend on it. Nor did his claim of a fourth dimension rely exclusively on his belief in a spiritual world, for he held the claim even before his conversion to spiritualism. In a book of 1876 he argued that a fourth dimension was needed for epistemological reasons, in order to understand the symmetry between 3D objects, such as left- and right-handed gloves. The phenomenal objects in our 3D world must be “projections of objects in a space of four dimensions” (Zöllner 1876). He considered the insight to be of revolutionary importance to science as it heralded a change in the world view on a scale comparable to the one Copernicus had initiated. His colleagues in physics and astronomy were not immune to the fascination of the fourth dimension, but they rejected his interpretation of it. Zöllner was the only scientist in the 19th century who found it probable, and not merely possible, that space is curved in accordance with Riemann's geometry. He was also the only one to use the hypothesis to solve a cosmological problem, namely Olbers' paradox of the dark night sky. From the late 1870s, non-Euclidean geometry attracted increasing interest among mathematicians and philosophers and a few astronomers followed suit. One of them was the Irishman Robert Stawell Ball, Royal Astronomer of Ireland and from 1892 professor of astronomy and geometry in Cambridge. Without committing himself, he suggested that parallax investigations might show space to be non-Euclidean. Characteristically, his guarded preference for a closed cosmic space turned up in his popular publications only. In The High Heavens of 1893, he expressed sympathy with the hypothesis, vaguely suggesting that a finite universe was more satisfactory than the consensus view of an infinite space filled with stars. Another astronomer of distinction, the American Simon Newcomb, also dealt with the possibility of a closed-space universe, if only cautiously and apparently without believing in it. In the first edition of his classical text Popular Astronomy, he discussed whether the heat radiated by the Sun and stars would be lost forever. Noting that this would not be the case in a spherical universe, he nonetheless denied taking a Riemannian cosmic space seriously. It was “too speculative to admit of discussion” he said (Newcomb 1878). He followed up on the subject in correspondence with the philosopher-scientist Charles S Peirce, who was much more sympathetic to curved space. Indeed, for a decade Peirce defended the idea enthusiastically, suggesting various astronomical methods by means of which the curvature might be measured. Newcomb advised him to calm down: “The task of getting the scientific world to accept any proof that space is not homoloidal [flat] is hopeless, and you could have no other satisfaction than that of doing a work for posterity” (Eisele 1957). The most elaborate pre-relativistic attempt to link astronomy with non-Euclidean geometry appeared in 1900, in a paper by the 26-year-old German astrophysicist Karl Schwarzschild (published in translation in 1998). I cannot go into the substance of this work, except noting its main results concerning the possible curvature of space. In the case of a hyperbolic space, Schwarzschild found R > 4×106 AU, and for the closed space he estimated a lower bound of R > 108 AU. Although he saw no way to go beyond this rather indefinite conclusion, from a philosophically point of view he preferred a closed universe, which he thought was more “satisfying to reason” (Schwarzschild 1998). So did Einstein, 17 years later. A knot experiment Zöllner made with the American medium Henry Slade. The ends of the cord were sealed together, yet Slade's “spirits” tied several knots in the cord. To Zöllner (1880), it proved the reality of a fourth space dimension. Following up on Schwarzschild's analysis, Paul Harzer at the University of Kiel argued that the universe might well consist of a finite stellar system located in a larger spherical space. He estimated the size of the entire universe by the time it would take a ray of light to circumnavigate it. For this journey round the world, Harzer (1908) gave the figure 8700 years. Neither Schwarzschild nor Harzer seems to have been aware of Zöllner's earlier work, at the time long forgotten. Ever since Lobachevsky, non-Euclidean geometry was associated with astronomy and yet it was a subject most astronomers were to were for one of them that space was not considered part of science. The motion of celestial bodies was the of not the space in which the motion took Newcomb for the of astronomers he among both and to of space as an in To interest in the astronomical community, of space would have to be or for problems of astronomical on both While astronomers realized that the curvature of space was they also realized that the kind of bound for the curvature that measurements was to distinguish curved from flat space. Given this no that they saw no to the Euclidean space that had them so well in the space be curved, the curvature radius would be so large that for all purposes it was infinite - that space could be considered So Among the few problems of cosmological that might have astronomers to curved space was the question of whether space is finite or infinite in extent. The question might be seen as merely as it often but it had such as Olbers' only in one Zöllner's discussion of was the problem by that the stellar universe might be closed in accordance with Riemann's His solution to the most of Olbers' in terms of and saw no between the dark night and an infinity of stars. The main for the to the of space non-Euclidean was just they had no need for the