任意维度下的Bellman-海岸线搜索：指数向量振荡器、主动记忆、进动与有效可计算性
Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability
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中文总结 AI 辅助
该研究探究任意维度下仿射超平面在线搜索的机制，推导不同维度下的相关恒等式，证明C_D^*为可计算实数，可合成代数多边形的ε-最优单元。
中文摘要 AI 辅助
我们研究在任意固定有限维的ℝ^D中对未知仿射超平面的在线搜索。在配套的自相似单元约简和支撑函数公式的基础上，我们探究当法空间从ℝ^0扩展到ℝ^{D-1}时，该机制会发生怎样的变化。在D=1时，交替性和生产率产生了等波纹原理以及精确的平稳常数9；在D=2时，类似的相对平衡是一条对数螺旋线，其瓶颈弦施加相切条件并选择螺距。对于指数轨道Γ(σ)=e^{κσ}ω(σ)，我们建立了对数方向几何、指数折现记忆、量规以及递归超球参数化。无需形状假设，瓶颈可由至多D个历史供应者支撑的证书来证明，在全局最差阶段，当前点位于活动面上。在正则腔室内，我们推导出精确的变分、相切、螺距、年龄，以及在D=3时的延迟系统恒等式。奇数维的障碍、对跖子类和谐波塔提供了约束和显式候选族，但未声称它们是全局最优的。最后，N-COMP定理表明，对于每个固定有限的D，C_D^*是一个可计算实数，原则上可合成代数多边形的ε-最优单元。通过D=10进行的数值筛选与已证明的结果分开处理。
英文摘要
We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $Γ(σ)=e^{κσ}ω(σ)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.
发表机构
- École Polytechnique（巴黎综合理工学院）
机构由 AI 辅助整理，请以论文原文为准。
