Spectral Phenomena in Finite Approximations of the Syracuse Full Shift: Phase Transitions, Fake Cycles, and the Eigenvalue Ratio 1/√2
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| Format: | Recurso digital |
| Language: | English |
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2026
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| _version_ | 1866901309308796928 |
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| author | okabe, arata |
| author_facet | okabe, arata |
| contents | <p> The Syracuse map on the 2-adic integers Z_2 is shown to generate a full shift on the countable alphabet of step sizes {1,2,3,...}. For every s≥1, T(P_s) = <br> Z_2^{odd}, meaning all transitions between step sizes are admissible. Sarig's thermodynamic formalism applies directly, yielding a spectral gap, unique Gibbs <br> measure, and exponential mixing for the RPF operator with potential ψ_γ = -γ log|S|, for all γ ∈ (0,1).</p> <p> However, finite approximations mod 2^m exhibit rich spectral phenomena absent in the infinite system: fake Markov dependencies, fake cycles, spectral gap <br> collapse at γ=0, and a first-order phase transition with gap(γ,m) ≈ C(m)·γ. Computer-assisted certification using Bauer-Fike perturbation bounds with 50-digit <br> arithmetic confirms the spectral gap rigorously for m=8,10,12 with safety margins exceeding 10^5. The eigenvalue ratio λ₂/λ₁ converges to 1/√2 as m increases, <br> suggesting deep connections to 2-adic structure.</p> <p> This paper does NOT claim to prove the Collatz conjecture. The measure-zero barrier between Z_2 and the natural numbers N remains the fundamental open problem.</p> <p> Computational code for all results is included.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19607429 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Phenomena in Finite Approximations of the Syracuse Full Shift: Phase Transitions, Fake Cycles, and the Eigenvalue Ratio 1/√2 okabe, arata <p> The Syracuse map on the 2-adic integers Z_2 is shown to generate a full shift on the countable alphabet of step sizes {1,2,3,...}. For every s≥1, T(P_s) = <br> Z_2^{odd}, meaning all transitions between step sizes are admissible. Sarig's thermodynamic formalism applies directly, yielding a spectral gap, unique Gibbs <br> measure, and exponential mixing for the RPF operator with potential ψ_γ = -γ log|S|, for all γ ∈ (0,1).</p> <p> However, finite approximations mod 2^m exhibit rich spectral phenomena absent in the infinite system: fake Markov dependencies, fake cycles, spectral gap <br> collapse at γ=0, and a first-order phase transition with gap(γ,m) ≈ C(m)·γ. Computer-assisted certification using Bauer-Fike perturbation bounds with 50-digit <br> arithmetic confirms the spectral gap rigorously for m=8,10,12 with safety margins exceeding 10^5. The eigenvalue ratio λ₂/λ₁ converges to 1/√2 as m increases, <br> suggesting deep connections to 2-adic structure.</p> <p> This paper does NOT claim to prove the Collatz conjecture. The measure-zero barrier between Z_2 and the natural numbers N remains the fundamental open problem.</p> <p> Computational code for all results is included.</p> |
| title | Spectral Phenomena in Finite Approximations of the Syracuse Full Shift: Phase Transitions, Fake Cycles, and the Eigenvalue Ratio 1/√2 |
| url | https://doi.org/10.5281/zenodo.19607429 |