One Axiom : The Geometry of Quantum Search
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2026
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| _version_ | 1866902209982103552 |
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| author | Spychalski, Robert |
| author_facet | Spychalski, Robert |
| contents | <p>2M: The Geometry of Quantum Search – <span class="math-inline">$\sqrt{N}$</span> as Minimum Orbit Length and a Structural Unification of BBBV and Classical Simulability</p> <p><strong>Abstract:</strong></p> <p>Document 2M provides a rigorous geometric derivation of quantum search complexity lower bounds within the ONE AXIOM framework. It demonstrates that the <span class="math-inline">$\Omega(\sqrt{N})$</span> quantum query lower bound is not merely a limitation of computational models, but the minimum orbit length required by <span class="math-inline">$\sigma$</span>-contractive dynamics in the coherence search space <span class="math-inline">$(U_{\mathcal{M}},d_{\alpha})$</span> derived from Axiom M.</p> <p>The paper introduces and proves the <strong>Bridge Theorem (BT)</strong>, which establishes a formal, bidirectional equivalence between the standard oracle query model and the <span class="math-inline">$\sigma$</span>-orbit model, preserving complexity across both tracks. By mapping queries to orbit steps, the <span class="math-inline">$\Omega(\sqrt{N})$</span> lower bound is shown to follow from the geometric requirement of traversing a specific coherence angle to reach a target of measure <span class="math-inline">$1/N$</span>.</p> <p><strong>Key Contributions:</strong></p> <ul> <li> <p><strong>Geometric Lower Bound (T3):</strong> A first-principles derivation of the <span class="math-inline">$\sqrt{N}$</span> bound as a traversal fact in a conservative Orbital Coherence Flow (OCF) field.</p> </li> <li> <p><strong>Grover Uniqueness (T4):</strong> Formally proves that Grover’s algorithm is the unique <span class="math-inline">$T_{min}$</span>-optimal trajectory within the coherence geometry.</p> </li> <li> <p><strong>BBBV–Schuster Unification (T8):</strong> Bridges the 27-year gap between the BBBV theorem (1997) and recent results on the classical simulability of noisy circuits (Schuster et al., 2024). It proves they are two regimes of a single structural condition: the maintenance or destruction of the coherence fixed point <span class="math-inline">$M(x^*) = x^*$</span>.</p> </li> <li> <p><strong>The 13/64 Threshold (T8-III):</strong> Derives the algebraic boundary <span class="math-inline">$f_F = 13/64$</span> (from the ABC 51:13 partition) as a structural upper bound on the ALLOWED region. The ALLOWED/FORBIDDEN dichotomy is proven independently of this value (Proposition CS-impl); the exact numerical identification of <span class="math-inline">$f_F$</span> with the Schuster empirical threshold is an open calibration task.</p> </li> </ul> <p><strong>Context:</strong></p> <p><em>Documents <strong>0M: The M — One Axiom</strong>, <strong>ABC: Coherence</strong>, and <strong>0A: Foundation</strong></em></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19763825 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | One Axiom : The Geometry of Quantum Search Spychalski, Robert Quantum search Grover's algorithm BBBV theorem coherence geometry orbital metric oracle complexity classical simulability NISQ Axiom M orbit length 13/64 threshold OCF-Oracle Correspondence <p>2M: The Geometry of Quantum Search – <span class="math-inline">$\sqrt{N}$</span> as Minimum Orbit Length and a Structural Unification of BBBV and Classical Simulability</p> <p><strong>Abstract:</strong></p> <p>Document 2M provides a rigorous geometric derivation of quantum search complexity lower bounds within the ONE AXIOM framework. It demonstrates that the <span class="math-inline">$\Omega(\sqrt{N})$</span> quantum query lower bound is not merely a limitation of computational models, but the minimum orbit length required by <span class="math-inline">$\sigma$</span>-contractive dynamics in the coherence search space <span class="math-inline">$(U_{\mathcal{M}},d_{\alpha})$</span> derived from Axiom M.</p> <p>The paper introduces and proves the <strong>Bridge Theorem (BT)</strong>, which establishes a formal, bidirectional equivalence between the standard oracle query model and the <span class="math-inline">$\sigma$</span>-orbit model, preserving complexity across both tracks. By mapping queries to orbit steps, the <span class="math-inline">$\Omega(\sqrt{N})$</span> lower bound is shown to follow from the geometric requirement of traversing a specific coherence angle to reach a target of measure <span class="math-inline">$1/N$</span>.</p> <p><strong>Key Contributions:</strong></p> <ul> <li> <p><strong>Geometric Lower Bound (T3):</strong> A first-principles derivation of the <span class="math-inline">$\sqrt{N}$</span> bound as a traversal fact in a conservative Orbital Coherence Flow (OCF) field.</p> </li> <li> <p><strong>Grover Uniqueness (T4):</strong> Formally proves that Grover’s algorithm is the unique <span class="math-inline">$T_{min}$</span>-optimal trajectory within the coherence geometry.</p> </li> <li> <p><strong>BBBV–Schuster Unification (T8):</strong> Bridges the 27-year gap between the BBBV theorem (1997) and recent results on the classical simulability of noisy circuits (Schuster et al., 2024). It proves they are two regimes of a single structural condition: the maintenance or destruction of the coherence fixed point <span class="math-inline">$M(x^*) = x^*$</span>.</p> </li> <li> <p><strong>The 13/64 Threshold (T8-III):</strong> Derives the algebraic boundary <span class="math-inline">$f_F = 13/64$</span> (from the ABC 51:13 partition) as a structural upper bound on the ALLOWED region. The ALLOWED/FORBIDDEN dichotomy is proven independently of this value (Proposition CS-impl); the exact numerical identification of <span class="math-inline">$f_F$</span> with the Schuster empirical threshold is an open calibration task.</p> </li> </ul> <p><strong>Context:</strong></p> <p><em>Documents <strong>0M: The M — One Axiom</strong>, <strong>ABC: Coherence</strong>, and <strong>0A: Foundation</strong></em></p> |
| title | One Axiom : The Geometry of Quantum Search |
| topic | Quantum search Grover's algorithm BBBV theorem coherence geometry orbital metric oracle complexity classical simulability NISQ Axiom M orbit length 13/64 threshold OCF-Oracle Correspondence |
| url | https://doi.org/10.5281/zenodo.19763825 |