# P2816: Integral torsion in scale-four hypercube Rips complexes

- ID: `P2816`
- Reference: `hypercube-rips-scale-four-torsion-free`
- Page: https://theoremdb.org/statements/P2816
- Record maturity: Reviewed problem with recorded work

## Problem

For \(n\ge1\), let \(Q_n=\{0,1\}^n\) with Hamming distance, and let \(\operatorname{VR}(Q_n;4)\) be the simplicial complex whose faces are the finite subsets of diameter at most four. Is \(H_j(\operatorname{VR}(Q_n;4);\mathbb Z)\) torsion-free for every \(n\) and every \(j\ge0\)?

### Definitions

- **Definition.** Hamming distance is the number of coordinates in which two binary strings differ.
- **Definition.** A Vietoris-Rips face at scale four is a set in which every pair has Hamming distance at most four.
- **Definition.** Integral homology is torsion-free when every homology group is a free abelian group.

### What counts as a solution

- Prove the stated torsion-freeness for all \(n,j\), or give specific \(n,j\) and an independently checkable integral boundary-matrix certificate for a nontrivial torsion invariant.

## Status

The complexes VR(Q_n;4) are torsion-free for n at most 5. For n=6, the rational reduced Betti numbers are 239 in degree 7 and 14 in degree 15, and every integral homology group has trivial 2-primary torsion. Odd-primary torsion at n=6 and the full torsion question for every n at least 7 remain unresolved. [3](#reference-3)

## Work

### Evidence for the current status

**Computation 1 (Torsion-free through n=5 and no 2-primary torsion at n=6).** The complexes VR(Q_n;4) are torsion-free for n at most 5. For n=6, the rational reduced Betti numbers are 239 in degree 7 and 14 in degree 15, and every integral homology group has trivial 2-primary torsion. Odd-primary torsion at n=6 and the full torsion question for every n at least 7 remain unresolved.

For \(1\leq n\leq4\), every pair of cube vertices has Hamming distance at most four. The Rips complex is a simplex, so its integral homology is \(H_0\cong\mathbb Z\) with all higher groups zero. For \(n=5\), the only forbidden pairs are the sixteen complementary pairs. The complex is the join of sixteen copies of \(S^0\), hence is \(S^{15}\). Thus \(H_0\cong H_{15}\cong\mathbb Z\), with all other groups zero.

At \(n=6\), Galetto, Montaño, and Wellner report
\[
\widetilde H_j(\operatorname{VR}(Q_6;4);\mathbb Q)\cong
\begin{cases}
\mathbb Q^{239},&j=7,\\
\mathbb Q^{14},&j=15,\\
0,&j\ne7,15,
\end{cases}
\]
and state that the Betti numbers over \(\mathbb F_2\) are the same. The universal coefficient theorem gives
\[
\dim_{\mathbb F_2}H_j(K;\mathbb F_2)
=b_j(K;\mathbb Q)+t_j+t_{j-1},
\]
where \(t_i\) is the number of cyclic 2-primary summands of \(H_i(K;\mathbb Z)\). Equality of the rational and mod-two Betti numbers in every degree forces every \(t_i\) to vanish.

The exact deletion-contraction replay supplies a separate Euler check. Its complete face count gives \(\widetilde\chi=-253\), agreeing with \(-239-14\). It also finds 2,932,100,733 faces including the empty face and maximum face cardinality 22.

This settles the prime 2 part at \(n=6\). The checked sources and computations supply no Smith certificate or odd-characteristic rank table that excludes odd-primary torsion. The canonical universal question therefore remains open.

### Background and intake notes

Existing work supplies homology generators and propagation maps at larger scales. The integral question asks whether those free classes exhaust the phenomenon or coexist with torsion invisible to rank-only calculations.

- Original intake status: UNKNOWN as of 2026-07-27. Scale two has a wedge-of-spheres description and recent work gives larger-scale rank information, while the checked papers do not classify integral torsion at scale four.
- 2026-07-27 prior-art search checked hypercube Rips complexes at scales two through four, homology propagation, group actions, and integral torsion. The scale-two homotopy theorem does not extend the requested conclusion to scale four.
- Betti-number lower bounds over a field do not distinguish free homology from torsion. Computations should compare several characteristics and retain integral Smith data.
- Coordinate deletion and cube-face inclusions may propagate a smallest torsion class to higher dimensions, so locating or excluding the first example is independently useful.

- Recorded example: For \(n\le4\), every pair of vertices has distance at most four, so the complex is a simplex. For \(n=5\), the only missing edges join complementary strings; the complex is the join of sixteen copies of \(S^0\), hence is homeomorphic to \(S^{15}\).

### Other known results

- **Computation 2** (reproduced): After an explicit linear change of coordinates over F_2, VR(Q_6;4) is the independence complex of the 64-vertex graph obtained by adding the antipodal perfect matching to the ordinary 6-cube.
- **Computation 3** (reproduced): Exact deletion-contraction gives every face count of VR(Q_7;4), maximum face cardinality 29, and reduced Euler characteristic -3937. These invariants constrain any future homology table and carry no torsion conclusion by themselves.
- **Computation 4** (reproduced): The exact value gamma_t=12 improves the published degree-only connectivity certificate: the cited Chudnovsky-Meshulam total-domination theorem proves that VR(Q_6;4) is 4-connected. [4](#reference-4)
- **Computation 5** (reproduced): The twelve-vertex total-domination witness induces six disjoint forbidden edges, so its full Rips subcomplex is a 5-sphere. The resulting 64-term integral cycle dies over Q and F_2; its integral class is a concrete odd-torsion test.
- **Proposition 1** (supported): The Adams-Virk propagation theorems, applied to the complete rational n=6 table, give b_7(VR(Q_7;4);Q) at least 3107 and b_15(VR(Q_7;4);Q) at least 110. [2](#reference-2)
- **Claim 1** (supported): Adams and Virk report mod-two homology through degree 15, with dimensions 239 in degree 7 and 14 in degree 15. Galetto, Montaño, and Wellner report the complete rational table with the same two dimensions and explicitly state that the rational and mod-two Betti numbers agree in every degree. [3](#reference-3)
- **Claim 2** (reported): For every n and j, appending a zero coordinate embeds H_j(VR(Q_n;4);Z) as a direct summand of H_j(VR(Q_{n+1};4);Z). Any torsion class at one cube dimension therefore persists at every larger dimension.

### Prior approaches

- **Route 1** (ruled out): The attractive total-domination route stops at 4-connectivity because the exact invariant is 12. The cited theorem would require gamma_t greater than 14 to certify 6-connectivity. [4](#reference-4)
- **Route 2** (supported): The 2026 cohomology paper infers 6-connectivity from mod-two homology vanishing through degree 6. That coefficient calculation leaves odd-primary integral torsion possible, so the inference needs another argument. [4](#reference-4)
- **Route 3** (supported): A 2026-07-28 audit found field-valued homology, connectivity bounds, propagation results, and representation decompositions in proved ranges. It found no integral Smith computation or theorem deciding odd-primary torsion at scale four. [2](#reference-2) [1](#reference-1) [5](#reference-5) [4](#reference-4) [3](#reference-3)

### Open directions

- **Route 4** (reported): The next computation should construct a certified Morse or symmetry reduction of the n=6 independence complex, then compute exact boundary ranks over F_3 and integral Smith data. A finite prime screen must retain its prime-by-prime scope.

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python replay computes every face count of VR(Q_6;4) and VR(Q_7;4), checks the n=6 graph isomorphism, and obtains reduced Euler characteristics -253 and -3937.
- **Artifact 2** (reproduced): A deterministic set-cover search proves that the total domination number of Q_6 with its antipodal matching is 12, with exhaustive failures at sizes 10 and 11 and an explicit size-12 witness.

### Computational notes

- A direct pair-distance check on \(Q_5\) found exactly sixteen forbidden complementary pairs and no others, verifying the cross-polytope-boundary description in the example.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hypercube-rips-scale-four-torsion-free`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>larger-scale open questions and submaximal cross-polytope description https://arxiv.org/abs/2103.01040
   - preprint; reference source; arXiv:2103.01040v3; checked 2026-07-28
   - Source use: citation_only
   - Proves the scale-two wedge-of-spheres theorem and identifies larger hypercube Rips scales as open.
2. <a id="reference-2"></a>Henry Adams and Žiga Virk, “Lower bounds on the homology of Vietoris-Rips complexes of hypercube graphs”. arXiv:2309.06222 (2023). Adamaszek-Adams arXiv:2103.01040v3; Adams-Virk arXiv:2309.06222v1; Briggs-Feng-Wells arXiv:2408.01288v2; Bendersky-Elia-Grbić arXiv:2605.00705v2; Galetto-Montaño-Wellner arXiv:2606.20784v1 https://arxiv.org/abs/2309.06222
   - Also cited at Section 6.4.4, Non-Example 7.4, and Section 8
   - Also cited at Adams and Virk, Theorems 4.1, 6.4, and 6.5 and Table 2; Galetto, Montaño, and Wellner arXiv:2606.20784v1, Appendix C.3
   - preprint; reference source; arXiv source revision v1; checked 2026-07-28
   - Source use: citation_only
   - Gives propagation theorems and rational rank bounds for hypercube Rips homology.
   - For Integral torsion in scale-four hypercube Rips complexes: The Adams-Virk propagation theorems, applied to the complete rational n=6 table, give b_7(VR(Q_7;4);Q) at least 3107 and b_15(VR(Q_7;4);Q) at least 110.
3. <a id="reference-3"></a>Federico Galetto, Jonathan Montaño, and Zoe Wellner, “Homology of Vietoris-Rips complexes of hypercube graphs via group actions”. arXiv:2606.20784 (2026). Appendix C.3 and https://github.com/galettof/VietorisRipsHypercube/tree/15eebf49305d28a50cadc591aba6320f42e99bc3/Polymake; Adams and Virk arXiv:2309.06222v1, Section 6.4.4 https://arxiv.org/abs/2606.20784
   - Also cited at abstract scope and Appendix C.3
   - Also cited at Galetto, Montaño, and Wellner, Appendix C.3; exact Euler replay in hr4-artifact-exact-fvectors-six-seven; universal-coefficient inference written here
   - preprint; reference source; arXiv source revision v1; checked 2026-07-28
   - Source use: citation_only
   - Provides field homology data that prove torsion-freeness through n=5 and exclude 2-primary torsion at n=6.
   - For Integral torsion in scale-four hypercube Rips complexes: The complexes VR(Q_n;4) are torsion-free for n at most 5. For n=6, the rational reduced Betti numbers are 239 in degree 7 and 14 in degree 15, and every integral homology group has trivial 2-primary torsion. Odd-primary torsion at n=6 and the full torsion question for every n at least 7 remain unresolved.
4. <a id="reference-4"></a>Martin Bendersky, Salvatore Elia, and Jelena Grbic, “Cohomological properties of the Vietoris--Rips Complex of a Hypercube Graph”. arXiv:2605.00705 (2026). Bendersky, Elia, and Grbić, Theorem 2.3; exact gamma_t computation in hr4-artifact-total-domination-twelve https://arxiv.org/abs/2605.00705
   - Also cited at Theorems 2.3 and 2.6 and the n=6 discussion
   - Also cited at Theorem 2.3 and exact total-domination artifact in this packet
   - Also cited at Introduction page 2, Theorem 2.3, Theorem 2.6, and discussion on page 5
   - preprint; reference source; arXiv:2605.00705v2; checked 2026-07-28
   - Source use: citation_only
   - Supplies connectivity bounds for hypercube Rips complexes while leaving the integral torsion question open.
5. <a id="reference-5"></a>Facets in the Vietoris-Rips complexes of hypercubes Introduction n=6 scale-four computation, Section 4.1, and Section 6 https://arxiv.org/abs/2408.01288v2
   - preprint; reference source; arXiv:2408.01288v2; checked 2026-07-28
   - Source use: citation_only
   - Studies facets and field-valued homology at larger scales without determining integral torsion.