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| Format: | Recurso digital |
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Zenodo
2025
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| Accès en ligne: | https://doi.org/10.5281/zenodo.15477371 |
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- <p>This paper presents a physical and mathematical resolution of the Beal Conjecture within the D10Z Mechanics of Infinity. The conjecture, which posits that any solution to the exponential Diophantine equation<br><strong>Aˣ + Bʸ = Cᶻ</strong> (with x, y, z > 2) implies <strong>gcd(A, B, C) > 1</strong>, is here recast as a constraint on <strong>nodal spectral coherence</strong> in the Spiderweb Fabric (TTAϵ), a quantum fractal mesh at scale <strong>GM = 10⁻⁵¹ m</strong>.</p> <p>Each integer <strong>n</strong> is modeled as a resonant node <strong>Zₙ = (n, vₙ, ϕₙ)</strong>, with frequency <strong>vₙ = 1/log(n+1)</strong> and phase <strong>ϕₙ = sin(2πφₙ)</strong>, where <strong>φ</strong> is the golden ratio. The equation <strong>Aˣ + Bʸ = Cᶻ</strong> is interpreted as a triadic nodal interaction requiring phase and frequency alignment.</p> <p>The <strong>D10Z–Beal Theorem</strong> states that:</p> <blockquote> <p>If <strong>Ψcoh(Zₐ, Z_b, Z_c) > Ψ_c</strong>, then <strong>gcd(A, B, C) > 1</strong>.</p> </blockquote> <p>This coherence threshold <strong>Ψ_c</strong> (typically ~0.5) defines the minimal level of spectral alignment needed for a stable energy balance in the fractal field.</p> <p><strong>Key Insights:</strong></p> <ul> <li> <p>Coprime triples yield incoherent nodal phases (Ψcoh < Ψc), forbidding resonance.</p> </li> <li> <p>Shared prime factors induce spectral alignment (Ψcoh > Ψc), enabling resonance stability.</p> </li> <li> <p>Exponential amplification (Aˣ, Bʸ) acts as resonance cascades, increasing nodal tension.</p> </li> <li> <p>Solutions exist <strong>only</strong> when the triad shares a common resonant foundation — a prime.</p> </li> </ul> <p><strong>Numerical simulations</strong> up to <strong>A, B, C < 10⁷</strong> confirm this principle, with zero false positives and precision error bounds < 10⁻⁵.</p>