Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability
Computational Geometry
Summary
The authors study the problem of finding an unknown flat surface (an affine hyperplane) in spaces of various dimensions through a step-by-step search process. They analyze how the search strategy changes as the complexity of the space grows, revealing specific patterns in 1D and 2D, like a constant pattern and a spiral shape. In higher dimensions, they develop new geometric tools and formulas to understand the problem, though some results are proposed without full proof of optimality. They also prove that the best search method can be computed for any fixed dimension and provide a way to approximate it. Numerical tests up to dimension 10 support their theoretical work but are not considered rigorous proofs.
Affine hyperplaneOnline searchEqual-ripple principleLogarithmic spiralSupport functionHyperspherical parametrizationExponential orbitsTangency conditionComputabilityDelay system
Authors
Florentin Koch
Abstract
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.