# Exact source question and indexing

Frozen UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e, record30002163, OWR-12012-001. Statement SHA256 f3e96bb3e8e5e88f36b3fbfb4fd7ebbfbd4f496cc2c18bf7ba650fb5b796e846.

Johann S. Brauchart, Spherical Nets, Sequences and Lattice rules -- construction of low-discrepancy sequences on spheres, OWR40/2012, pp2436--2439, conjecture on p2438 (PDF page10). Source DOI https://doi.org/10.4171/OWR/2012/40 . The report is for the2012 workshop; the frozen metadata labels its citation2013. Its Group Theory category is not an appropriate mathematical classification for this question.

F_0=0,F_1=F_2=1 and F_n=F_{n-1}+F_{n-2}. For n>=3, N=F_n and a=F_{n-1}, define z_k=(2sqrt((k/N)(1-k/N))cos(2piak/N),2sqrt((k/N)(1-k/N))sin(2piak/N),1-2k/N),0<=k<N. The question asks whether the least Euclidean chord distance is2/sqrt(N), attained by z_0,z_1. It does not ask whether this configuration is the optimal packing among all N-point sets, nor ask for discrepancy or integration bounds.

The report's displayed defining range is0<=k<F_n, but its immediately following set notation uses1,...,F_n. Section5.2 of Aistleitner--Brauchart--Dick, arXiv1109.3265, explicitly confirms0,...,F_n-1. Moreover the two endpoint conventions are isometric by(x,y,z)->(x,-y,-z), k->N-k. This editorial inconsistency does not change the separation value. N=1 has no distinct pair and is excluded, as required for the question to have meaning.

The accepted result proves the exact equality for every n>=3, with the unique unordered closest pair{z_0,z_1}, and a general sufficient Zaremba-index criterion. One exact source-question closure and one whole frozen-record closure are counted. There is no claim about the exact minimum for a pole-avoiding half-shifted irrational spiral.

# Detailed originating-system self-audit

1. The source uses Euclidean chords on the unit sphere, exact rational angle F_{n-1}/F_n and latitude1-2k/F_n. No replacement by the irrational golden angle is allowed.
2. The geometry is global: v=1-t_1t_2>=0, r>=0,h>=0 and v^2=r^2+h^2. Longitude may be arbitrary at a pole because r=0. There is no local tangent approximation.
3. For c<=0, v-cr>=v>=h. For c>=0, the two-dimensional Cauchy inequality with unit vector(c,sqrt(1-c^2)) yields the required bound. Squared chord distance is2(v-cr), with the factor2 retained.
4. Heights differ by2q/N. The reduced longitude d=||aq/N|| lies in[0,1/2]. Its cosine is nonpositive exactly when d>=1/4. Sine is bounded below by4d only for d<=1/4; the proof does not wrongly extend that bound to the other half-interval. The two branches agree at1/4.
5. Choose u=pN-aq with p nearest to aq/N. Coprimality and1<=q<N ensure u!=0, so the dual-lattice product is q|u|, not a max-convention edge case. Negative u and ties at half-integers are harmless.
6. Zaremba's exact index formula is a cited classical theorem, checked directly in two primary sources; it is not inferred from numerical samples. Only the weaker bound >=N/3 is needed. The inequality is non-strict at n4, but still strictly greater than N/4. n3,N2 is checked explicitly.
7. Obtuse-longitude pairs have distance squared>=4q/N>=4/N; acute-longitude pairs have distance squared>=16F_{n-2}/N^2>4/N. The north-pole pair attains4/N independently of its assigned longitude.
8. For uniqueness in the nonacute branch, equality forces q1 and v=h; the identity forces r0. Exactly one pole is in the0-based set, so the pair is{z0,z1}. No second south pole is mistakenly included.
9. The1-based endpoint convention is globally isometric, so its minimum is equal, with a different attaining pair. A genuine half-step latitude shift only preserves the lower bound, not the pole attainment.
10. The general rho>=N/4 condition is sufficient, not necessary. Dropping it is invalid: N12,a1,pair5,6 has squared distance2-sqrt(105)/6<1/3, checked exactly by105>100.
11. The symbolic identity and2,178,278 integer residue checks pass. Separately2,061,234 pair distances for n3...17 pass in float64. The latter are labelled diagnostics, not exact certificates or a general proof. Neither finite computation replaces the written theorem or the cited all-n arithmetic theorem.
12. All source cases n>=3 are covered. Complete proof accepted by the originating researcher; no external model/reviewer is required, and none is claimed. The bounded prior-art review found no identical resolution, but does not certify absolute priority.

# Bounded primary-source literature and novelty review

Cutoff:27 October2026. This is not an exhaustive search or absolute-priority certification.

1. Brauchart, OWR40/2012, p2438: exact unshifted rational spherical separation formula is posed as a conjecture. Its index convention is reconciled in canonical_statement.md. This establishes what question is answered; an old open statement alone is not proof of present-day novelty.
2. Aistleitner--Brauchart--Dick, Point sets on the sphere S^2 with small spherical cap discrepancy, DCG48(2012)990--1024, arXiv1109.3265: section5.2 supplies the construction and discrepancy results; planar minimum spacing is not the spherical minimum. No exact spherical separation theorem was found in the inspected sections or targeted full-text search.
3. Zaremba, A remarkable lattice generated by Fibonacci numbers, Fibonacci Quarterly8(1970)185--198, p186: the exact arithmetic index is PRIOR WORK. Dick--Pillichshammer, Integers5(3)(2005)A05, p7, gives a second primary account and explicitly discusses its ratio to N. These are credited inputs, not authored discoveries.
4. Hardin--Michaels--Saff, A comparison of popular point configurations on S^2, DRNA9(2016)16--49: section2 distinguishes rational Lambert lattices from shifted irrational lattices. Proposition2.2 and the near-pole numerical minimum about3.09207/sqrt(N) concern the latter, half-step shifted construction. They do not contradict or establish the exact2/sqrt(F_n) statement for the pole-including rational set here.
5. Marques--Bouville--Bouatouch--Blat, Extensible Spherical Fibonacci Grids: institutional manuscript, section4.4 explicitly shifts to the planar grid to derive a minimum-distance expression, and notes exceptional behavior at sphere poles. Its golden-angle shifted construction and extension method are different from the source's exact rational pole-including lattice. Its citation[5] is Brauchart--Dick's2012 Numerische Mathematik integration paper, not a later publication of the in-preparation Spherical Fibonacci points title.
6. Suryanarayana--Cools--Nuyens, Integration and Approximation with Fibonacci lattice points, journal-hosted PDF: Theorem2.2 states the classical exact Zaremba index with attribution to prior literature. It is used only to cross-check the arithmetic source, not as a spherical result.
7. Institution-hosted Brauchart2022/2023/2024 talk descriptions surfaced in search, concerning Weyl sums and cap discrepancy. These descriptions do not establish a separation resolution; no inference of their full contents is made.

Public searches covered the exact least-pairwise-distance phrase, spherical Fibonacci separation/minimum conjecture, Fibonacci Zaremba index, Lambert Zaremba separation, and the named closest comparison articles. Search outputs and the accessed primary-source URLs are preserved. No prior identical geometric resolution or general rho>=N/4 criterion was located in this bounded search. The geometry is elementary and the arithmetic is classical; novelty is attributed only to this presented combination and source resolution, with no claim that either classical tool is new. Unread unpublished work and unindexed literature remain limitations. No independent review or formal proof-assistant verification is claimed.