For every integer m >= 1, the spiders S(3,3,3,1^m) and S(4,4,4,1^m) admit a graceful labeling with zero at any chosen vertex. Eleven base labelings and a threshold-insertion lemma prove both families.
Authors
School Scotty · AI-assisted research program
Research status
Work in progress
Published
Version
3
The result
For every integer m >= 1, both S(3,3,3,1^m) and S(4,4,4,1^m) are 0-rotatable: any chosen vertex can carry zero in some graceful labeling. Each tree has three equal long arms and m extra leaves at the center. If it has q edges, a graceful labeling uses the vertex labels 0,...,q and the edge differences 1,...,q, each exactly once.
How the proof works
Five explicit labelings cover the vertex types in the length-three family at m=1; six cover those in the length-four family. A threshold-insertion lemma then adds one center leaf at a time. It preserves both the required edge differences and the vertex labeled zero. Induction gives the result for every m, beyond the finite cases checked by the accompanying programs.
The research note contains the eleven labelings, the two-case insertion proof, and the relation to earlier work. The package contains the base data, Python checkers, and source notes. SHA-256: 86b4e0f5c01a8f43a4579894e891a3ef362c675280ca70a75b9df790b064b549.
The public repository provides the base certificates, checkers and a GitHub-readable copy of the existing research note.
Scope and earlier work
Patterson's 2017 thesis proposed a broader conjecture about 0-rotatable spiders. Other papers established gracefulness or 0-rotatability for related families, and a related label-gap insertion method was described before this work. Our targeted search did not locate the two exact all-m statements, but their publication priority remains unresolved.
The proof concerns exactly three long arms, all of length three or all of length four, with at least one extra center leaf.
Statements & evidence
SPIDER-333-LEAVES-0-ROTATABLEProof provided
For every integer m >= 1, S(3,3,3,1^m) is 0-rotatable: every prescribed vertex can receive label zero in a graceful labeling.
Formal verification
No formal verification
Review status
Internal check
Evidence
Five m=1 label certificates and the fixed-threshold insertion lemma; independent internal audit in graceful-spider-three-arms-proof.zip (SHA-256 86b4e0f5c01a8f43a4579894e891a3ef362c675280ca70a75b9df790b064b549), audit.md.
Scope
Exactly three arms of length three and at least one additional center leaf; all integer m >= 1. No result for arbitrary spiders or arbitrary numbers of length-three arms. Publication priority unresolved.
Assumptions
Graceful labels use each integer 0..m+9 once; edge differences use 1..m+9 once. The graph is S(3,3,3,1^m).
Toolchain
Elementary written induction; Python 3.12.1 checks the base certificates and finite invariant instances. No formal proof assistant.
Reviewer
Independent internal AI mathematical audit, 8 October 2026; no external or peer reviewer.
SPIDER-444-LEAVES-0-ROTATABLEProof provided
For every integer m >= 1, S(4,4,4,1^m) is 0-rotatable: every prescribed vertex can receive label zero in a graceful labeling.
Formal verification
No formal verification
Review status
Internal check
Evidence
Six m=1 label certificates and the fixed-threshold insertion lemma; independent internal audit in graceful-spider-three-arms-proof.zip (SHA-256 86b4e0f5c01a8f43a4579894e891a3ef362c675280ca70a75b9df790b064b549), audit-k4.md.
Scope
Exactly three arms of length four and at least one additional center leaf; all integer m >= 1. No result for arbitrary spiders or other arm lengths. Publication priority unresolved.
Assumptions
Graceful labels use each integer 0..m+12 once; edge differences use 1..m+12 once. The graph is S(4,4,4,1^m).
Toolchain
Elementary written induction; Python 3.12.1 checks the six base certificates and finite invariant instances. No formal proof assistant.
Reviewer
Independent internal AI mathematical audit, 8 October 2026; no external or peer reviewer.
Contributions & AI assistance
Contributions
School Scotty is an AI-assisted research program maintained by Jordi Gartner.
AI assistance
AI helped find the eleven base labelings, shape the insertion proof, build the checkers and translate selected passages of the page. Two separate AI audits checked the arguments; their reports are in the proof package. No external review has taken place.
A certificate-and-composition proof of 0-rotatability for S(k^n,1^m), k in {3,5,7,9,11}, n >= 2, m >= 0. Established ingredients and prior subfamilies are credited; publication priority remains unresolved.
For every even K≥20, attach two K-edge arms to the chosen zero-labeled root of any finite graph H with a supplied conventional graceful labeling. On either new arm, each depth 2≤d≤D8(K) separately admits a conventional graceful labeling with zero at that vertex. The guaranteed depth interval grows without bound.
For every integer k≥19, all n≥2,m≥0 and any selected long arm of S(k^n,1^m), each depth 2≤d≤D8(k) can receive zero in a graceful labeling. Separate requests may use separate labelings. The explicit sufficient prefix includes both length parities and has limiting proportion 10/11; no optimality or historical priority is claimed.
For any finite graph H with a conventional graceful labeling and a chosen root labeled 0, attach two named arms of K=23+24t edges, where t≥1. On each new arm, a separate conventional graceful labeling can place zero at depth 20+22t or 21+22t.
For k=23+24t, n≥2 and m≥0, every selected named long-arm vertex at depth 17+20t through 20+22t can be zero in its own graceful labeling for every integer t≥0. For t≥1, the same holds at the additional depth 21+22t. For t≥2, it also holds at depth 22+22t. Consequently, for t≥5 the q24/D8 prefix and q48 point combine to give all depths 1≤d≤22+22t.
Given a finite graph with a supplied graceful labeling that assigns zero to a chosen root, attach two new 119-edge arms there. Every interior depth 1–118 on either arm can receive zero in its own labeling. The source also proves eight specified near-tip depths for K=30·4^r−1, r≥1.
Given any finite graph with a conventional graceful labeling that assigns zero to a chosen root, attach two 95-edge arms at that root. Each interior vertex on either new arm can receive zero in its own graceful labeling.
For every integer m≥1 and every prescribed vertex of S(7,7,7,1^m), there exists a graceful labeling that assigns zero to that vertex. Five base certificates, four complements, and a leaf-insertion lemma prove the result; publication priority remains unresolved.
For every n ≥ 2 and m ≥ 0, each actual vertex of S(23^n,1^m) can be assigned zero in a graceful labeling, with a separate labeling for each selected vertex.
For every n ≥ 2 and m ≥ 0, every actual vertex of S(25^n,1^m) can receive zero in a graceful labeling, with a separate labeling for each selected vertex.
For every n ≥ 2 and m ≥ 0, each actual vertex of S(47^n,1^m) can be the unique zero in a graceful labeling; a separate labeling may be used for each target.
For every n ≥ 2 and m ≥ 0, each actual vertex of S(71^n,1^m) can be the unique zero in a graceful labeling; a separate labeling may be used for each target.
For every n≥2 and m≥0, each vertex of S(95^n,1^m)—the hub, any of its m original leaves, or any position on any named 95-edge arm—can be assigned zero in its own graceful labeling.
For every even k ≥ 2, n ≥ 2, and m ≥ 0, each named vertex of S(k^n,1^m) can receive zero in a graceful labeling, with a separate labeling for each chosen vertex.
For every integer m >= 1, both S(3,3,3,1^m) and S(4,4,4,1^m) are 0-rotatable. Eleven explicit base labelings and one threshold-insertion lemma give a written proof for all m.
Added a contiguous prescribed-zero prefix for every integer arm length k≥19 with limiting proportion 10/11, completed additional residue classes and supplied explicit depth bounds and reproducible proof sources.
Adds a construction for prescribed zeros at long-arm depths 4s and 4s + 1 in S(k^n,1^m), where k = 2r + 1, s ≥ 1, r ≥ 3s, r ≠ 3s + 1, n ≥ 2, and m ≥ 0. The number of guaranteed depths grows with k; this is not a full zero-rotatability result. The two zero positions use separate spider labelings.
Added a proof note for the five odd arm lengths k = 3, 5, 7, 9, and 11, for n ≥ 2 and m ≥ 0. Archived wording
Adds a separate two-parameter family note for k in {3,5,7,9,11}, n >= 2, m >= 0. The previous k=3,4 and K7 notes retain their own versions and archives. Publication priority unresolved; DE/ZH remain translation drafts.
Added the separate K7 certificate proof and extended the project scope to k in {3,4,7}, m >= 1; corrected Patterson full-text availability and prior alpha-gracefulness. DE/ZH remain translation drafts.
Added German and Chinese translations of the existing claim evidence, scope, assumptions, toolchain and reviewer information. The claims, proofs, statuses and English source text are unchanged; the review status stayed the same. Archived wording
Added German and Chinese translations for existing claim evidence, scope, assumptions, toolchain and reviewer metadata only; no claim, proof, status, source text or review-status promotion.
Revised the English, German and Chinese project overview and linked the existing research note and public repository; the theorem claims, evidence and references are unchanged.
Initial project page for two internally audited 0-rotatability theorems, for three long arms of length three or four with m>=1 center leaves; novelty and priority remain unresolved.
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