Problem. Let \(K\) contain the complete graph on vertices \(\{0,\ldots,10\}\) and exactly \(45\) triangular faces. Among choices for which \(H_1(K;\mathbb Z)\) is finite, determine the largest possible order of \(H_1(K;\mathbb Z)\).
The current certified interval for the maximum torsion order is 51 through 54353589638.
2Remarks
Remark 1. There are C(11,3)=165 possible triangular faces.
Remark 2. The cycle rank of the complete one-skeleton is 45, so 45 face boundaries can kill all rational first homology and leave a finite torsion group.
3What counts as a solution
Give 45 faces attaining the maximum and a determinant or Smith-form certificate, together with an exhaustive or certified determinant bound for every other selection.
Saved packet status (The certified interval is 74 through 387,420,489). An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.[1]
1Packet records
4 records
Record
Kind
Assessment
An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Claim 1
An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.[1]
Relevance to this problem
For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-claim-certified-interval (“The certified interval is 74 through 387,420,489”) records a bound, answer, status fact, or structural consequence. The record states: An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
reported
Scope
all 45-face subcomplexes of the complete two-dimensional complex on eleven labeled vertices whose integral first homology is finite
Let \(M(K)\) be the maximum in the question. The exact replay in this record constructs a 45-face complex with
\[
H_1(K;\mathbb Z)\cong\mathbb Z/74\mathbb Z,
\]
so \(M(K)\ge74\).
Kalai's torsion theorem states that the torsion part of \(H_{d-1}\) for a \(d\)-complex on \(n\) vertices has order at most
\[
(\sqrt{d+1})^{\binom{n-2}{d}}.
\]
At \((n,d)=(11,2)\), this is
\[
(\sqrt3)^{\binom92}=3^{18}=387{,}420{,}489.
\]
Consequently
\[
\boxed{74\le M(K)\le387{,}420{,}489}.
\]
The construction improves the candidate record's lower bound of 51. The search used determinant-preserving basis exchanges followed by exact integer checks. It did not enumerate all \(\binom{165}{45}\) face sets, so the exact maximum remains open.
The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1
The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74.[1]
Relevance to this problem
For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-claim-order-74-construction (“A 45-face complex has H1 isomorphic to Z/74Z”) records a bound, answer, status fact, or structural consequence. The record states: The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
the displayed 45 triangular faces on vertices 0 through 10 and their complete one-skeleton
Delete the ten star edges \(0i\) from the edge chain group. The remaining edges \(ij\), with \(1\le i<j\le10\), give a 45-element basis for the cycle group after the usual spanning-tree reduction. For an oriented face \(a<b<c\), its reduced boundary column is
\[
[bc]-[ac]+[ab],
\]
with every term containing vertex 0 omitted. This gives the integer presentation matrix \(B\) used by the replay.
Fraction-free elimination gives \(\det B=74\). Deleting row 0, indexed by edge 12, and column 1, indexed by face 038, gives a 44 by 44 minor of determinant \(-1\). The gcd of all 44-row minors is therefore 1. The determinantal-divisor characterization of Smith normal form now gives
\[
\operatorname{SNF}(B)=\operatorname{diag}(1^{44},74).
\]
Thus the matrix has full rational rank, \(H_2(K;\mathbb Z)=0\), and
\[
H_1(K;\mathbb Z)\cong\operatorname{coker}B\cong\mathbb Z/74\mathbb Z.
\]
The face list and boundary matrix have SHA-256 digests `cdd0175fca03a927e3ae5b11da108dc2ca61d045f15a4c3442ecfa9bbc928f85` and `a3b4e556bd16ddd806f8199a4aad171acb521102d12e3a27e3f65beb8b991c15`.
Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
Trace
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
attempt · Route 1
Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.[2][3][1][4]
Relevance to this problem
For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-attempt-literature-and-search-audit (“The literature supplies the framework and universal bound, with no exact n=11 table located”) documents a concrete method, search boundary, or failed route. The record states: Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
completed
Scope
a focused status audit for maximum first-homology torsion among eleven-vertex rational 2-hypertrees
Kalai introduced rationally acyclic simplicial complexes and proved the torsion-weighted higher-dimensional version of Cayley's formula. In the present parameters, a full-rank selection of 45 triangle-boundary columns is a rational 2-hypertree. The same paper gives the universal torsion bound used above.
Linial and Peled restate the boundary-basis characterization and Kalai's torsion-weighted enumeration, then study unweighted enumeration and random constructions. Their work does not give a table of maximum torsion by vertex count. Newman studies the inverse problem of realizing prescribed torsion on few vertices. His concluding Question 3 asks for the largest torsion group realizable at fixed \(n\) and \(d\), and says the problem appears difficult even for small parameters.
The audit searched the exact parameter phrase, combinations of `11 vertices`, `2-hypertree`, `maximum torsion`, and determinant formulations. It also checked the cited hypertree and small-complex papers. No primary source settling this exact 45-face optimization was located. The value 74 should therefore be treated as a reproducible incumbent rather than a published optimum.
Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.
Relevance to this problem
For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-artifact-boundary-smith-replay (“Exact reduced boundary and Smith-form replay”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
all 2025 entries of the reduced boundary matrix, its determinant, and the specified 44 by 44 Smith minor
Entry point
join source_lines with newline and run with python3
Runtime
CPython 3, standard library only
Details
The program parses the face codes, orients each face increasingly, and constructs the reduced boundary matrix in lexicographic edge order. Its Bareiss routine uses integer division at every elimination step. It checks the full determinant 74 and the unit 44 by 44 minor.
These two determinants suffice for the claimed Smith form. The full determinant is the product of all invariant factors. The unit minor forces the 44th determinantal divisor to equal 1, so the first 44 invariant factors are 1. The final invariant factor is 74.
The program also binds the face sequence and all matrix entries to separate SHA-256 digests. Its canonical report has digest `bdce1e59cd40f418ee11dc99ca39eacbb63d0dc35029921ca098c2ad40ce620e`, and the standard output has digest `544c1f2666ee94e3df31aa6a10a652ffdb3cc83832b6c273cf295943fa60f013`.
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Original intake status. UNKNOWN as of 2026-07-25. An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18. The checked sources do not settle the full acceptance condition.
The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.
Recorded example 1. A 45-face complex with torsion order 51 is recorded in the computation field.
Computational notes
Ten thousand seeded determinant-exchange searches produced a full-rank boundary matrix of determinant -51. Its faces are 359,038,127,017,68A,57A,048,016,569,058,13A,578,129,67A,349,025,237,09A,45A,047,159,178,679,28A,568,146,89A,479,148,238,36A,26A,49A,789,036,245,246,15A,249,134,235,37A,02A,039,01A, where A denotes vertex 10 and each three-character word is a face. Fraction-free elimination independently replayed the determinant.
How the 4 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
ProblemLargest first-homology torsion from forty-five triangles on eleven vertices
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“Largest first-homology torsion from forty-five triangles on eleven vertices.” TheoremDB. P2684. Problem statement; statement text SHA-256 a20bce1a3e3a9b192d58608be4f980f7857a5258126e07e37a3f1733d57c52a8. https://theoremdb.org/statement/?ref=P2684
BibTeX
@misc{theoremdb-problem-a20bce1a3e3a9b192d58608be4f980f7857a5258126e07e37a3f1733d57c52a8,
title = {{Largest first-homology torsion from forty-five triangles on eleven vertices}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 a20bce1a3e3a9b192d58608be4f980f7857a5258126e07e37a3f1733d57c52a8},
url = {https://theoremdb.org/statement/?ref=P2684}
}
Once Lean accepts the draft, you can submit it while target review is pending. Your submission request is saved and continues after approval. Private checks stay private until you choose to submit.
1References
Packet source. Andrew Newman, “Small simplicial complexes with prescribed torsion in homology”. arXiv:1707.09271 (2017). Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record; Theorem 4 part 1 and concluding Question 3. ↗preprint · primary source · arXiv:1707.09271, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.The certified interval is 74 through 387,420,489. An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18. The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.Also cited at Theorem 4 part 1 and concluding Question 3.Also cited at Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record.Also cited at Exact integer certificate in cset11-artifact-boundary-smith-replay, produced 2026-07-25.For Largest first-homology torsion from forty-five triangles on eleven vertices: The certified interval is 74 through 387,420,489. An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18. The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.Source named by the research packet.
Nati Linial and Yuval Peled, “Enumeration and randomized constructions of hypertrees”. arXiv:1801.02423 (2018). Nati Linial and Yuval Peled, Enumeration and randomized constructions of hypertrees, Definition 1.1 and Theorem 1.2; Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 and Question 3; Definition 1.1, boundary-matrix characterization, and Theorem 1.2. ↗preprint · primary source · arXiv:1801.02423, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.Also cited at Definition 1.1, boundary-matrix characterization, and Theorem 1.2.For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
Sabine Grabner, “Seslerio-caricetum sempervirentis andCaricetum ferrugineae in the Northern Calcareous Alps”. Folia Geobotanica et Phytotaxonomica 32(3) (1997), 297-311. DOI 10.1007/BF02804009. Israel Journal of Mathematics 45 (1983), 337-351. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, “Simplicial matrix-tree theorems”. Trans. Amer. Math. Soc. 361 (2009), no. 11, 6073-6114. arXiv:0802.2576 (2008). Torsion-weighted cellular spanning-tree framework. ↗preprint · primary source · arXiv:0802.2576, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
Original CC0 determinant optimization interpreted as simplicial homology.
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