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A Geometric Model of Infinity with Empirical Justification Through Analysis of the Deviation Between Experimental Data and Quantum Theory for Hardy's Paradox

Douglas Chesley Gill

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.33308.86401

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A Geometric Model of Infinity with Empirical Justification through Analysis of the Deviation between Experimental Data and Quantum Theory for Hardy’s Paradox

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The issue, examined across multiple disciplines, is that infinities cannot be represented as discrete collections. Incompleteness is unavoidable. The 1-D geometric model examines the paradoxical relationship that develops when attempting to define infinite states as contained structures. Two representations of the right triangle are constructed within the unit circle. The first follows standard geometric principles. The second assigns a counter-rational unitary value to each of the triangle’s vectors and incorporates emergence within its structure. Although the two formats of the unit circle have a null relationship, they are shown to share membership by calculating the cosine-square for the right triangle. The geometric structure serves as a model to understand the role of paradox in the representation of infinity. The model is applied to explain the mechanism of emergence in the collapse of the wavefunction, and to Hardy’s paradox, offering a rationale for the discrepancy between its formal calculations and experimental results.

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Bell's theorem states that the quantum mechanical description of physical quantities cannot be fully explained by local realistic theories, laying a solid basis for various quantum information applications. Hardy's paradox is celebrated as the simplest form of Bell's theorem concerning its "All versus Nothing" approach to test local realism. However, due to experimental imperfections, existing tests of Hardy's paradox require additional assumptions of the experimental systems, and these assumptions constitute potential loopholes for faithfully testing local realistic theories. Here, we experimentally demonstrate Hardy's nonlocality through a photonic entanglement source. By achieving a detection efficiency of 82.2%, a quantum state fidelity of 99.10%, and applying high-speed quantum random number generators for the measurement setting switching, the experiment is implemented in a loophole-free manner. During 6 h of running, a strong violation of P_{Hardy}=4.646×10^{-4} up to 5 standard deviations is observed with 4.32×10^{9} trials. A null hypothesis test shows that the results can be explained by local realistic theories with an upper bound probability of 10^{-16348}. These testing results provide affirmative evidence against local realism, and establish an advancing benchmark for quantum information applications based on Hardy's paradox.

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The issue, examined across multiple disciplines, is that infinities cannot be represented as discrete collections. Incompleteness is unavoidable. The 1-D geometric model examines the paradoxical relationship that develops when attempting to define infinite states as contained structures. Two representations of the right triangle are constructed within the unit circle. The first follows standard geometric principles. The second assigns a counter-rational unitary value to each of the triangle’s vectors and incorporates emergence within its structure. Although the two formats of the unit circle have a null relationship, they are shown to share membership by calculating the cosine-square for the right triangle. The geometric structure serves as a model to understand the role of paradox in the representation of infinity. The model is applied to explain the mechanism of emergence in the collapse of the wavefunction, and to Hardy’s paradox, offering a rationale for the discrepancy between its formal calculations and experimental results.
Bell's theorem states that the quantum mechanical description of physical quantities cannot be fully explained by local realistic theories, laying a solid basis for various quantum information applications. Hardy's paradox is celebrated as the simplest form of Bell's theorem concerning its "All versus Nothing" approach to test local realism. However, due to experimental imperfections, existing tests of Hardy's paradox require additional assumptions of the experimental systems, and these assumptions constitute potential loopholes for faithfully testing local realistic theories. Here, we experimentally demonstrate Hardy's nonlocality through a photonic entanglement source. By achieving a detection efficiency of 82.2%, a quantum state fidelity of 99.10%, and applying high-speed quantum random number generators for the measurement setting switching, the experiment is implemented in a loophole-free manner. During 6 h of running, a strong violation of P_{Hardy}=4.646×10^{-4} up to 5 standard deviations is observed with 4.32×10^{9} trials. A null hypothesis test shows that the results can be explained by local realistic theories with an upper bound probability of 10^{-16348}. These testing results provide affirmative evidence against local realism, and establish an advancing benchmark for quantum information applications based on Hardy's paradox.