Growing prescribed-zero depths in equal-arm spiders

Version gd-2026-10-09-v1.1 · 9 October 2026.

Abstract

We give an explicit construction for an increasing set of prescribed zero positions in equal-arm spiders. Let s>=1 and r>=3s be integers with r!=3s+1, and set k=2r+1. The construction produces an alpha-labeling of the path P_(2k+1) with midpoint label k-1 and alpha threshold k-1, zero at depth 4s, and maximum 2k at the adjacent depth 4s+1 on the same arm. Its endpoint labels are (3k-1)/2 and (3k+1)/2. Established alpha-amalgamation and complementation then give, for every integer n>=2 and m>=0, a graceful labeling of S(k^n,1^m) with zero at any prescribed long-arm vertex of depth 4s or 4s+1. These are separate labelings. For every odd k>=11, at least 2 floor((k-5)/6) distinct depths are covered. This is partial prescribed-zero coverage; mathematical priority remains unresolved.

Scope

Here S(k^n,1^m) has n arms of length k and m arms of length one, with length measured in edges. Depth is distance from the designated center, including the path case n=2,m=0. For every permitted tuple and each specified vertex at one of the two depths, there exists a labeling assigning that vertex zero. The labeling may depend on the vertex; two vertices are not assigned zero in one labeling.

For a fixed r, the exact index set is I_r={s>=1:3s<=r and r!=3s+1}. The uniform bound uses all depths {4s,4s+1:1<=s<=floor((k-5)/6)}. When k=6t+1, the exact construction also admits s=t. The exclusion r=3s+1 restricts this formula and is not a nonexistence result. No coverage is asserted here for all depths, full zero-rotatability, all spiders, or an alpha-labeling at every zero position. Previous center and other boundary-depth constructions are separate established ingredients.

Evidence and review

A separate AI agent independently reconstructed the all-parameter proof, including the permutation, path extensions, and spider composition. It matched 156 saved paths and 960 saved spider labelings, checked 7,710 further path pairs and 7,040 spider labelings, and rejected 13 negative controls. The unbounded conclusion rests on the proof; finite checks support the constructions and file integrity. This was an internal audit, without human mathematical review. The focused literature comparison found no checked statement combining all the conditions, but access gaps in Cattell's full construction and older sources leave historical priority and possible subsumption unresolved.

Contributions and prior work

The classical Walecki order, earlier path and alpha-labeling constructions, center-zero labelings, and Huang–Kotzig–Rosa alpha-amalgamation are established ingredients. Patterson and Rofa document relevant constructions; the Huang–Kotzig–Rosa attribution currently rests on later sources. Earlier work already covers all prescribed zeros for n=2,m=0 and n=2,m=1.

An AI agent derived this formula and proof; separate AI agents reconstructed the mathematics, checked the certificates, and reviewed the literature. AI also assisted with English release wording and German and Chinese translations. School Scotty is the public research signature, and Jordi Gartner is responsible for editorial publication.

Proof and source files

Proof / Beweis / 证明 · Independent internal audit / Interne Gegenprüfung / 内部独立审查 · Source comparison / Quellenvergleich / 文献比较

Proof archive / Beweisarchiv / 证明归档 — SHA-256 3be4ef1da7645c2e8024ebf8b126434c1554e50a67dfbaa5051f7a830b86d450; 360302 bytes.