One Axiom : The Geometry of Quantum Search

2M: The Geometry of Quantum Search – $\sqrt{N}$ as Minimum Orbit Length and a Structural Unification of BBBV and Classical Simulability Abstract: Document 2M provides a rigorous geometric derivation of quantum search complexity lower bounds within the ONE AXIOM framework. It demonstrates that the $\Omega(\sqrt{N})$ quantum query lower bound is not merely a limitation of computational models, but the minimum orbit length required by $\sigma$-contractive dynamics in the coherence search space $(U_{\mathcal{M}},d_{\alpha})$ derived from Axiom M. The paper introduces and proves the Bridge Theorem (BT), which establishes a formal, bidirectional equivalence between the standard oracle query model and the $\sigma$-orbit model, preserving complexity across both tracks. By mapping queries to orbit steps, the $\Omega(\sqrt{N})$ lower bound is shown to follow from the geometric requirement of traversing a specific coherence angle to reach a target of measure $1/N$. Key Contributions: Geometric Lower Bound (T3): A first-principles derivation of the $\sqrt{N}$ bound as a traversal fact in a conservative Orbital Coherence Flow (OCF) field. Grover Uniqueness (T4): Formally proves that Grover’s algorithm is the unique $T_{min}$-optimal trajectory within the coherence geometry. BBBV–Schuster Unification (T8): 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 $M(x^*) = x^*$. The 13/64 Threshold (T8-III): Derives the algebraic boundary $f_F = 13/64$ (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 $f_F$ with the Schuster empirical threshold is an open calibration task. Context: Documents 0M: The M — One Axiom, ABC: Coherence, and 0A: Foundation