Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability
Abstract
We study online search for an unknown affine hyperplane in βD, 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 π0 to πDβ1. 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 Ξ(Ο)=eΞΊΟΟ(Ο), 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 CD* is a computable real for every fixed finite D and that algebraic polygonal Ξ΅-optimal cells can in principle be synthesized. Numerical screening through D=10 is kept separate from the proved results.
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BibTeX
@misc{koch2026bellmanshoreline,
title = {{Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability}},
author = {Florentin Koch},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29060},
eprint = {2608.29060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29060},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability
AU - Florentin Koch
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29060
UR - https://arxiv.org/abs/2608.29060
AB - We study online search for an unknown affine hyperplane in β^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 π^0 to π^Dβ1. 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 Ξ(Ο)=e^ΞΊΟΟ(Ο), 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 Ξ΅-optimal cells can in principle be synthesized. Numerical screening through D=10 is kept separate from the proved results.
ER -