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| Format: | Recurso digital |
| Sprache: | Englisch |
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Zenodo
2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.19638426 |
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- <p><strong>Evidence Paper VII of the Existence Equation series.</strong></p> <p>The Gross-Pitaevskii equation governs Bose-Einstein condensates, superfluids, and superconductors — one of the most successful equations in condensed matter physics. This paper demonstrates that it is not an independent theory but the non-relativistic, low-energy limit of the Existence Equation:</p> <p><code>(1/c²)∂²Ψ/∂t² = ∇²Ψ + λΨ − α|Ψ|²Ψ</code></p> <p>When the relativistic term (1/c²)Ψ̈ becomes negligible, the ED equation reduces to the time-dependent Gross-Pitaevskii equation. The Josephson effect — the hallmark of macroscopic quantum coherence — then emerges as a direct structural consequence of a barrier potential, with no phenomenological input and no GP assumption.</p> <p><strong>Input: 4 items. Nothing else.</strong></p> <table> <tbody> <tr> <th>#</th> <th>Input</th> <th>Description</th> </tr> </tbody> <tbody> <tr> <td>1</td> <td>ED equation</td> <td>(1/c²)Ψ̈ = ∇²Ψ + λΨ − α|Ψ|²Ψ</td> </tr> <tr> <td>2</td> <td>1D lattice</td> <td>512 points, periodic boundary</td> </tr> <tr> <td>3</td> <td>Barrier potential</td> <td>Localized λ suppression (width=2.0, depth=85%)</td> </tr> <tr> <td>4</td> <td>Chemical potential Δμ</td> <td>Split-operator phase kick (standard technique)</td> </tr> </tbody> </table> <p><strong>No GP equation. No Josephson relations. No condensate order parameter. No phenomenological coupling.</strong></p> <p><strong>Output: Three tests, all passed.</strong></p> <table> <tbody> <tr> <th>Test</th> <th>Phenomenon</th> <th>Prediction</th> <th>Result</th> <th>Metric</th> </tr> </tbody> <tbody> <tr> <td><strong>A. DC Josephson</strong></td> <td>Supercurrent ∝ sin(Δφ)</td> <td>J = J_c sin(Δφ)</td> <td>Sin fit across 31 phase points</td> <td>R² > 0.99</td> </tr> <tr> <td><strong>B. AC Josephson</strong></td> <td>Phase rate ∝ Δμ</td> <td>dΔφ/dt = Δμ (NR limit)</td> <td>Linear regression, 6 values Δμ (0.03–0.30)</td> <td>R² > 0.99</td> </tr> <tr> <td><strong>C. f_Φ̇ channel</strong></td> <td>Temporal phase gradient force</td> <td>No GP counterpart (relativistic)</td> <td>Junction-localized force, ΔΦ̇/Δμ ≈ 1</td> <td>Ratio ≫ 1</td> </tr> </tbody> </table> <p><strong>Why this matters:</strong></p> <p>Tests A and B reproduce textbook Josephson physics from the ED equation alone. Test C goes beyond GP: the temporal phase gradient ∂_t(arg Ψ) generates an independent force channel that exists only because the ED equation is second-order in time — a relativistic term that the first-order GP equation cannot contain. This is a falsifiable prediction: any system where GP fails at high frequency should exhibit the f_Φ̇ channel.</p> <p><strong>When does GP break down? The ε-regime map:</strong></p> <p>The validity of GP is controlled by a single dimensionless parameter ε = k²/(4λ) = c²k²/(2μ²_ED), where μ_ED = c√(2λ) is the intrinsic Higgs gap of the ED equation. Where ε ≪ 1, GP is exact. Where ε ~ O(0.1–1), ED corrections become measurable — and where ε ~ O(1), GP is structurally inapplicable: the antiparticle channel and the amplitude inertial mode (Higgs sector) become dynamically active.</p> <table> <tbody> <tr> <th>Physical system</th> <th>μ_ED (estimate)</th> <th>ε (estimate)</th> <th>Regime</th> </tr> </tbody> <tbody> <tr> <td>SC Josephson, V ≪ Δ_gap</td> <td>Δ_gap (~meV)</td> <td>~10⁻⁶</td> <td>GP exact</td> </tr> <tr> <td>BEC Bragg, low k</td> <td>healing freq. ~kHz</td> <td>~10⁻³</td> <td>GP exact</td> </tr> <tr> <td>ULDM / fuzzy scalar dark matter</td> <td>m_φ ~ 10⁻²² eV</td> <td>~10⁻⁴⁰</td> <td>GP exact</td> </tr> <tr> <td><strong>BEC Bragg, high k</strong></td> <td><strong>healing freq.</strong></td> <td><strong>0.05–0.2</strong></td> <td><strong>ED correction (5–20%)</strong></td> </tr> <tr> <td><strong>SC Josephson, V ~ Δ_gap</strong></td> <td><strong>Δ_gap</strong></td> <td><strong>0.1–1</strong></td> <td><strong>ED required</strong></td> </tr> <tr> <td><strong>Neutron-star superfluid core</strong></td> <td><strong>pairing gap</strong></td> <td><strong>O(0.1–1)</strong></td> <td><strong>ED required</strong></td> </tr> <tr> <td><strong>QCD chiral condensate</strong></td> <td><strong>f_π</strong></td> <td><strong>O(1)</strong></td> <td><strong>ED fundamental</strong></td> </tr> </tbody> </table> <p><em>Bold entries mark regimes where GP-ED deviations are experimentally or observationally accessible.</em> At ε ~ 0.1, systematic 5–20% deviations appear in dispersion observables. At ε ~ 1, the Ψ̈ term cannot be dropped — the antiparticle channel and Higgs sector become active, and GP is no longer a valid description. Four falsifiable predictions are identified: (1) high-momentum Bogoliubov dispersion shift in BEC Bragg spectroscopy, (2) asymmetric AC Josephson response at gap-scale voltages, (3) deep Shapiro delay correction, and (4) condensate fragmentation near ω ~ 2μ_ED.</p> <p><strong>Capstone of the Evidence Program:</strong></p> <p>The same equation that produces quantum scars (EP I), fractional quantum Hall states and anyon braiding (EP II), helical chirality (EP III), non-commutative algebra (EP IV), galaxy rotation curves from topological phase persistence (EP V), and the Tsirelson bound (EP VI) also contains macroscopic quantum coherence as its low-energy sector. Seven phenomena. One equation. No postulates beyond "existence = deviation; deviation smooths."</p> <p><strong>Evidence Program — Complete Summary:</strong></p> <table> <tbody> <tr> <th>EP</th> <th>ED mechanism</th> <th>What emerges</th> <th>Key result</th> </tr> </tbody> <tbody> <tr> <td>I</td> <td>α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion</td> <td>PXP Hamiltonian (not assumed, derived)</td> <td>‖T − H_PXP‖ = 0</td> </tr> <tr> <td>II</td> <td>Same exclusion on 2D torus; spatial phase closure → winding number sectors</td> <td>Topological degeneracy (without Coulomb interaction); anyon braiding</td> <td>q-fold at ν = p/q; 2π/q phase (0.0% error)</td> </tr> <tr> <td>III</td> <td>∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure</td> <td>Helical scars with spontaneous chirality</td> <td>Chirality 1.78 × 10⁻¹⁵</td> </tr> <tr> <td>IV</td> <td>α ≠ 0 → amplitude–phase feedback; commutative operators projected onto constrained subspace</td> <td>Non-commutative algebra with integer eigenvalues</td> <td>j_max = ⌊L/2⌋ (exact)</td> </tr> <tr> <td>V</td> <td>½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed</td> <td>Topological phase persistence as structural origin of dark matter</td> <td>⟨v_θ·r⟩ = 2.0000 ± 0.0003</td> </tr> <tr> <td>VI</td> <td>Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal</td> <td>Tsirelson bound S = 2√2 without Hilbert space</td> <td>Exact (analytic + numerical)</td> </tr> <tr> <td><strong>VII</strong></td> <td><strong>(1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary</strong></td> <td><strong>Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart)</strong></td> <td><strong>DC: R² > 0.99; AC: R² > 0.99; 4 falsifiable predictions at ε ~ 0.1</strong></td> </tr> </tbody> </table> <p>All code : <a href="https://github.com/Galileo-leo/existence-equation">https://github.com/Galileo-leo/existence-equation</a></p> <p>Part of the Existence Equation Evidence Paper series (EP I–VII).<br>Parent work: <a href="https://doi.org/10.5281/zenodo.18639317">The Existence Equation: The Grammar of Persistence</a> (doi:10.5281/zenodo.18639317)</p>