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.
Authors
Beedbyte
Research status
Work in progress
Published
Version
12
Source revision
bf1c1f21bcec75fe06edcc7140cd3dbdcf3a857b
Earlier three-arm results
For each k in {3,4,7} and integer m >= 1, S(k,k,k,1^m) is 0-rotatable: every prescribed vertex can receive zero in a graceful labeling. These are exactly three equal long arms and additional center leaves. No assertion is made for m=0, other lengths, other numbers of long arms, all spiders, or an alpha-labeling at every prescribed zero.
The existing note for lengths three and four and the separate K7 note give explicit base certificates and the leaf-insertion proof. The K7 proof uses five bases and four complements to cover nine zero orbits. The unbounded claim follows from the written interval argument; finite checker runs provide supplementary checks.
Earlier work
Brandon J. Patterson's 2017 master's thesis is now available in full. Theorem 3.3.8, printed pp. 13–14, already establishes alpha-gracefulness for all these families. Conjecture 5.4.4 is broader. Gracefulness and prescribed-zero rotatability are distinct. Patterson's other results also yield several K7 zero positions as deductions described in the K7 note.
Evidence and review
The public repository holds the existing certificates and historical V1 note copies. The K7 repository files supply the separate certificate proof and reproducibility records. The proof download for the existing note remains available separately. The K7 proof archive has SHA-256 f5bc44be564bdf7e2c3ad648e023c03ebd203c52873c35cb76c6e156cad0048e. These earlier files have internal project checks, without external peer review or proof-assistant verification. EN/DE/ZH language statuses remain draft until their corresponding human review is documented.
Equal-arm families
For every k in {3,5,7,9,11}, every integer n >= 2, every integer m >= 0, and every vertex v of S(k^n,1^m), there exists a graceful labeling f with f(v)=0. A separate family note gives the complete proof. The existing k=4 result remains restricted to three long arms and m >= 1. An additional partial lemma covers the center, depths 1,2,k-1,k and existing short leaves for every odd k >= 3; it does not prove every depth for all odd k. The even-length obstruction concerns the specified alpha-path method. Established center/leaf and composition ingredients, k=3,m=0, paths and other two-arm subfamilies are prior results. Source; separate proof archive, SHA-256 83e82afab5988ebb3cd3d19d9cc08d7945cf6bde5257b415f4ae662fd220be32.
Contiguous prescribed-zero prefix
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 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.
SPIDER-777-LEAVES-0-ROTATABLEProof provided
For every integer m >= 1 and every vertex v of S(7,7,7,1^m), there exists a graceful labeling f with f(v)=0.
Formal verification
No formal verification
Review status
Internal check
Evidence
Five bases, four complements and the written interval insertion proof cover nine zero orbits for all m >= 1. Proof archive SHA-256 f5bc44be564bdf7e2c3ad648e023c03ebd203c52873c35cb76c6e156cad0048e; download /downloads/8; repository commit 4aca7989c752870b353fcd40d43895ab54d073fe, k7/. Internal checkers and diagnostics in REPRO-REPORT.md.
Scope
Exactly three seven-edge long arms and m >= 1 center leaves. No m=0, other lengths, other arm counts, arbitrary spiders, alpha-labeling for every zero vertex, or novelty/priority assertion.
Assumptions
The graph is S(7,7,7,1^m), m >= 1, q=21+m; labels bijectively use 0..q, edge differences use 1..q once. A different labeling may be used for each chosen vertex.
Toolchain
Written elementary proof; exact Python checkers run with Python 3.12.1; diagnostics recorded in REPRO-REPORT.md. Python 3.10+ required. No proof assistant.
Reviewer
Separate internal AI mathematical review and checker, 9 October 2026; no external or peer reviewer.
For every k in {3,5,7,9,11}, every integer n >= 2, every integer m >= 0, and every vertex v of S(k^n,1^m), there exists a graceful labeling f with f(v)=0.
Formal verification
No formal verification
Review status
Internal check
Evidence
Two written certificate-and-composition proofs and ten exact path certificates: six for lengths 3,5,7,9 and four for length 11. Separate internal audits/checkers cover their stated scopes. Source families/ at commit 290c72cd3b79c955787c0b28e5f4fb2cd6556e4e; proof archive /downloads/9, SHA-256 83e82afab5988ebb3cd3d19d9cc08d7945cf6bde5257b415f4ae662fd220be32.
Scope
Exactly k in {3,5,7,9,11}, n >= 2, m >= 0. Includes paths n=2,m=0. Labeling may depend on the prescribed vertex. No all-odd-length, even-length, unequal-arm, alpha-at-every-zero, or arbitrary-spider theorem; no novelty or priority assertion.
Assumptions
S(k^n,1^m) has q=nk+m edges; vertex labels bijectively use 0..q and edge differences use 1..q once. n >= 2, m >= 0; short-leaf zero only when a short leaf exists.
Toolchain
Written unbounded constructions; Python standard-library certificate/labeling checkers. The constructor API supports only 3,5,7,9; separate length-11 checkers verify that extension. No proof assistant.
Reviewer
Separate internal AI mathematical audits and independently implemented verifiers for the first four lengths and the length-11 extension, 9 October 2026. No external peer review.
growing-prescribed-zero-depthsProof provided
Let s>=1 and r>=3s be integers with r!=3s+1, and k=2r+1. For every n>=2, m>=0 and every specified long-arm vertex of S(k^n,1^m) at depth 4s or 4s+1, there exists a graceful labeling assigning that vertex zero. The labeling may depend on the chosen vertex. Every odd k>=11 has at least 2 floor((k-5)/6) covered depths.
Formal verification
No formal verification
Review status
Internal check
Evidence
The supplied all-parameter proof passed independent reconstruction by a separate AI agent. The original 156 paths and 960 spiders matched; further path/spider cases and 13 negative controls passed. Finite checks support implementation; the unbounded result follows from the reconstructed proof. This is internal checking, without human peer review or proof-assistant verification. https://github.com/beedbyte/graceful-spiders/blob/8f2aac9c3cd9fd799b076779c4402ae19b88181a/growing-depth/v1/source/proof.md ; https://github.com/beedbyte/graceful-spiders/blob/8f2aac9c3cd9fd799b076779c4402ae19b88181a/growing-depth/v1/reviews/independent/audit-report.md
Scope
Partial prescribed-zero coverage. Depth means distance in edges from the designated center, including n=2,m=0. Exclusion r=3s+1 is a formula limitation. Earlier all-vertex theorem for k in {3,5,7,9,11} remains separate. Priority unresolved.
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.
Formal verification
Formally verified within the stated scope
Review status
Internal check
Evidence
Frozen v10 technical manuscript and D8 construction/separate internal reconstruction. Author report SHA-256 3150df1b1f6296c580eebf5c2bc4b7bba9b6ae97436b0ebfbb821f70c41658a6; separate report SHA-256 b9886f996b045aea344b4f62d3f6344801fa481024d943283b2fe5a7456f5c15. D8 formal author index a5aaf2d94b7eb81198f7a16e8e84b30cf21bf9e40259075ec94d195183d83619; separate formal replay index 3787f77fc7ef1a09282cb65d7b9e2765d12ddef7ce6b0cab3730d302dd492d2a. Source URLs must be bound to the released commit.
Scope
All integers k≥19,n≥2,m≥0; any selected long arm; every depth2≤d≤D8(k); separate labelings per request; both length parities. Sufficient prefix, not optimal or full zero-rotatability; novelty unresolved.
Assumptions
Exact designated-center graph S(k^n,1^m); graceful vertex and actual edge-difference bijections. D8 low branches11,27,25+16 floor((k−35)/18); high branch107+20a+2r, p=floor((k−3)/2)=61+11a+r,0≤r≤10.
Toolchain
Standard-library Python graph checks; Lean4.34.0 exact actual-spider proof and separate source replay, using propext, Classical.choice and Quot.sound. Formal ratio scope is an integer precision criterion.
Reviewer
Separate internal mathematical reconstruction; no external reviewer or peer review recorded.
Contributions & AI assistance
Contributions
Jordi Gartner publishes this work through Beedbyte.
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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