Two internally audited constructions show that spiders with three equal long arms of length three or four and at least one extra center leaf can be gracefully labeled with zero at any chosen vertex. Both proofs cover every number of extra leaves; publication priority remains unresolved.
Authors
School Scotty · AI-assisted research program
Research status
Work in progress
Published
Version
1
Research in progress · Internally audited proof · 8 October 2026
The question
Imagine a central vertex with three equal long paths, plus at least one extra leaf attached directly to the center. We prove two precise cases: the three paths each have length three, or they each have length four. Let m >= 1 be the number of extra leaves. A graceful labeling uses every integer from zero to the number of edges once on the vertices and makes the absolute edge differences exactly the positive integers up to that number.
For both families, any chosen vertex can be the one labeled zero in some graceful labeling. This property is called 0-rotatability.
What was checked
Five explicit base labelings cover the five vertex types when the long arms have length three. Six base labelings cover the six types when their length is four. A common threshold-insertion lemma adds another center leaf while preserving the required edge differences and the zero label. The lemma gives the infinite conclusion; checking many finite cases alone would not.
The versioned research note, listed in this project's Related section when published, gives all eleven base certificates and the full two-case induction. Its downloadable proof package contains both source arguments, exact Python checkers, two independent internal audits, and the targeted literature review. Package SHA-256: 86b4e0f5c01a8f43a4579894e891a3ef362c675280ca70a75b9df790b064b549.
Context and limits
Patterson's 2017 thesis conjectured a broader 0-rotatability statement for spiders, apart from an exceptional family. Prior papers establish gracefulness of related spiders and 0-rotatability of other classes. A related label-gap insertion idea was publicly described before this work. The targeted literature review has not found the exact all-m theorem for either family, but publication priority remains unresolved. This page makes no worldwide novelty claim.
The proofs apply only to exactly three equal long arms of length three or four, with m >= 1 extra center leaves. They do not establish 0-rotatability for arbitrary spiders or other long-arm lengths. The two audits are internal AI-assisted reviews, not formal verification or peer review. Patterson's thesis PDF was not available for full-page inspection in this review, so its exact conjecture wording still needs direct confirmation before a submission claiming originality. Some bundled source notes retain historical pilot-status sentences; BUNDLE-README.md explains their chronology.
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. This page credits earlier spider conjectures and labeling work; it does not claim an independently verified human discovery or worldwide priority.
AI assistance
AI agents found five length-three and six length-four base certificates, developed and wrote the threshold-insertion proof, coded exact checks, conducted two independent internal mathematical audits and a targeted literature review, and checked the stated base cases and proof invariant. None of these checks is external or peer review.
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.
Beedbyte
Comments
Discussion of version 1
No comments yet.
Comments on this version are preserved here. Comment on the current version →