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Zenodo
2026
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| Διαθέσιμο Online: | https://doi.org/10.5281/zenodo.19231149 |
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- <p>This paper demonstrates that <strong>π</strong> is not a fundamental constant of nature but a computable output of a deeper algebraic structure. We challenge the standard circular definition of π (the ratio of a circle's circumference to its diameter), which presupposes the very geometry it seeks to parameterize.</p> <p>The argument proceeds through a rigorous three-step derivation that requires no geometric input (no circles, no arc lengths, no Euclidean planes):</p> <ol> <li><strong>Algebraic Cosine:</strong> The cosine function is defined strictly by a combinatorial power series using only factorials and alternating signs.</li> <li><strong>Root Identification:</strong> π is identified as the minimal positive real root of the equation <code>cos(x) = -1</code>.</li> <li><strong>The Golden Bridge:</strong> Using the minimal self-referential polynomial <code>x² - x - 1 = 0</code>, we derive the identity <code>cos(π/5) = φ/2</code>, where <strong>φ</strong> is the golden ratio.</li> </ol> <p>The resulting closed-form expression, <strong>π = 5 · arccos(φ/2)</strong>, reveals that π is a derived quantity. We further argue that the ubiquity of π in physics—from the Einstein field equations (8πG) to the Bekenstein-Hawking entropy—is a <em>rendering artefact</em>.</p> <p>In the framework of <strong>Ontological Resolution Theory (ORT)</strong>, π represents the computational cost of projecting discrete icosahedral symmetry (FCC lattice) into a continuous manifold. By breaking the circular dependency of π, we provide a more parsimonious foundation for both mathematics and theoretical physics, where φ and discrete integers serve as the true irreducible seeds of reality.</p> <p><em>This work is part of the Ontological Resolution Theory (ORT) series, consolidating results on the discrete origin of fundamental physical constants.</em></p>