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| Главный автор: | |
|---|---|
| Формат: | Recurso digital |
| Язык: | английский |
| Опубликовано: |
Zenodo
2025
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| Предметы: | |
| Online-ссылка: | https://doi.org/10.5281/zenodo.16937428 |
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Оглавление:
- <p>We introduce and develop the <strong>φ–π Independence Conjecture</strong>, which asserts that the golden ratio<br><strong>φ = (1 + √5) / 2</strong><br>and the circular constant <strong>π</strong> admit no nontrivial exponential–algebraic relations beyond integer-phase identities. Formally, for any polynomial <strong>P(X,Y) ∈ Q̄[X,Y]</strong>,</p> <p><strong>exp(P(φ,π)) ∈ Q̄ ⇔ P(φ,π) ∈ 2πiQ.</strong></p> <p>The conjecture is motivated by two lines of evidence. First, in exponential–Diophantine equations, phases fall into a threefold pattern: rational multiples of <strong>π</strong> yield infinite oscillatory solution families, irrational phases collapse to triviality, and golden–circular phases exhibit exceptional rigidity. Second, structural analysis through the Resolver Normal Form (<strong>RNF</strong>) and Angle–Metric Concordance (<strong>AMC</strong>) frameworks we develop in companion papers shows that <strong>φ</strong> and <strong>π</strong> can align structurally only at icosahedral order.</p> <p>We prove unconditional theorems for large classes of polynomials — including linear exponentials, separable families, and degree-one cases — showing that all algebraic dependencies reduce to integer-phase relations. Under the <strong>Ax–Schanuel theorem</strong> for the exponential function, the conjecture holds in full generality. Integration with the <strong>Golden–Circular Resolver (GCR)</strong> framework reveals that <strong>φ–π independence</strong> is a manifestation of a universal harmonic law: golden scaling and circular curvature are irreducibly independent, except for one icosahedral resonance and a hexagonal echo.</p>