Minimum Distance in Spherical Fibonacci Lattices

Definitions and mathematical statement

Let $F_n$ denote the Fibonacci number intended in the supplied record, and write \[ Z_{F_n}=\{z_0,z_1,\ldots,z_{F_n-1}\} \] for its intended spherical Fibonacci point set. The record does not supply the construction, the sphere's radius, the indexing convention for Fibonacci numbers, or the admissible range of $n$. In particular, the labels $z_0,z_1$ cannot be identified from the supplied data alone.

For a finite set of distinct points in a Euclidean ambient space, define its least pairwise Euclidean distance by \[ \delta(Z_{F_n})=\min_{0\leq i<j<F_n}\|z_i-z_j\|_2, \qquad F_n\geq 2. \] Under the interpretation of the source's absolute-value notation as Euclidean chordal distance, the proposed identity is \[ \delta(Z_{F_n})=\|z_0-z_1\|_2=\frac{2}{\sqrt{F_n}}. \] Thus the question asks whether the indicated pair realizes the minimum and whether its distance has exactly the stated value, for the indices intended by the source. Equivalently, for those indices, it asks whether \[ \|z_0-z_1\|_2=\frac{2}{\sqrt{F_n}} \quad\text{and}\quad \|z_i-z_j\|_2\geq\frac{2}{\sqrt{F_n}} \quad\text{for every }0\leq i<j<F_n. \] The Euclidean interpretation, sphere normalization, coordinate construction, and index range require confirmation from the cited source before this becomes a fully specified conjecture. No geodesic-distance interpretation or particular sphere normalization is asserted here.

Short English statement

The supplied conjecture asks whether two designated points in a spherical Fibonacci configuration attain its least pairwise distance, with exact value $2/\sqrt{F_n}$. The configuration, distance convention, normalization, and index range are insufficiently specified in the supplied record. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30002163$, title Minimum Distance in Spherical Fibonacci Lattices, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2012 and identifies no individual proposer.
• Supplied report citation: Optimal and Near Optimal Configurations on Lattices and Manifolds (2013), Oberwolfach Reports, pp. 2429--2492, DOI 10.4171/owr/2012/40. https://doi.org/10.4171/owr/2012/40. This is the authoritative report reference supplied by the record; its contents have not been independently inspected here.
• Supplied literature source: OWR report 40. https://doi.org/10.4171/owr/2012/40.
• Supplied literature source: Spherical Fibonacci points. https://arxiv.org/abs/1109.3265.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem open and describes the least-distance formula as still conjectural, with later discrepancy and distribution results not proving this exact identity. This assessment and its claims are not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, accompanied by the assertion that the official report states this minimum-distance conjecture for the spherical Fibonacci construction. These are registry assertions, not mathematical or source verification performed for this draft.