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.

The nonedges of the Rips graph on \(Q_6\) join pairs whose difference has Hamming weight 5 or 6. Write \(\mathbf1\) for the all-one vector and \(s_i=\mathbf1+e_i\) for \(1\leq i\leq6\). The six weight-five vectors \(s_i\) form a basis of \(\mathbb F_2^6\), and \[ \sum_{i=1}^6s_i=\mathbf1. \] In coefficient coordinates for this basis, the six weight-five generators become \(e_1,\ldots,e_6\), while the weight-six generator becomes \(\mathbf1\). Thus the forbidden-pair graph is isomorphic to the graph on \(\{0,1\}^6\) with the ordinary cube edges and the antipodal perfect matching. A Rips face is exactly an independent set in this graph.

Some authors use `folded cube` for this 64-vertex graph, while others reserve that name for an antipodal quotient. The description above fixes the convention used in this packet. The executable f-vector artifact checks the coordinate map against all 2,016 unordered vertex pairs.

The forbidden graph on \(Q_7\) joins words at Hamming distance 5, 6, or 7. It is regular of degree \[ \binom75+\binom76+\binom77=29 \] and has 1,856 edges. The exact independence polynomial in `hr4-artifact-exact-fvectors-six-seven` has degree 29 and coefficient sum 209,570,782,049. Its alternating evaluation gives ordinary Euler characteristic \(-3936\), hence reduced Euler characteristic \(-3937\).

The face count is coefficient-independent and supplies a checksum for any later boundary computation. Euler characteristic records only an alternating sum of free ranks. It neither detects torsion nor identifies individual Betti numbers. Combined with `hr4-claim-published-n7-rank-lower-bounds`, any complete rational table must have \(b_7\ge3107\), \(b_{15}\ge110\), and reduced Euler characteristic \(-3937\).

Under the graph reduction in this packet, \(\operatorname{VR}(Q_6;4)=I(G)\), where \(G\) is the 64-vertex graph \(Q_6\) with antipodal matching. The exact set-cover artifact proves \[ \gamma_t(G)=12. \] Theorem 2.3 quoted by Bendersky, Elia, and Grbić states that if \(\gamma_t(G)>2k\), then \(I(G)\) is \((k-1)\)-connected. Taking \(k=5\) gives 4-connectivity because \(12>10\).

This result proves integral homology vanishes through degree 4. It falls short of the degree-6 vanishing asserted elsewhere from a mod-two computation.

In the transformed coordinates of `hr4-claim-n6-forbidden-graph-reduction`, the twelve-vertex witness is `0, 15, 44, 28, 49, 50, 4, 40, 24, 11, 53, 54`. Its induced forbidden graph consists of exactly the six edges \[ \{0,4\},\ \{11,15\},\ \{24,28\},\ \{40,44\},\ \{49,53\},\ \{50,54\}. \] The full Rips subcomplex on these vertices is therefore the independence complex of six disjoint edges, which is the join of six copies of \(S^0\) and hence a copy of \(S^5\). An explicit integral fundamental cycle is the simplicial-join product \[ z=([0]-[4])*([11]-[15])*([24]-[28])*([40]-[44])*([49]-[53])*([50]-[54]), \] expanded into 64 oriented 5-simplices.

The published complete rational and mod-two tables both have \(H_5=0\), so the image of \([z]\) vanishes over \(\mathbb Q\) and \(\mathbb F_2\). The packet does not determine whether \([z]\) is zero integrally or has finite odd order. Testing an integral filling or annihilator for this small explicit cycle is a focused first case for the recommended Smith computation.

Adams and Virk report the same numerical bounds in Table 2 over the coefficient field used for the \(n=6\) computation. Their Theorems 4.1, 6.4, and 6.5 also apply to the later complete rational table.

Degree 7 first appears at cube dimension 6 because the complexes through dimension 5 have homology only in degrees 0 and 15. Theorem 6.4 therefore multiplies the rational rank 239 by \[ 1+2\binom65=13, \] giving \(b_7(\operatorname{VR}(Q_7;4);\mathbb Q)\ge3107\).

In degree 15, the twelve coordinate copies of \(Q_5\) in \(Q_6\) give twelve independent cross-polytopal classes. The rational rank 14 at \(n=6\) leaves a quotient of rank 2. At \(n=7\), Theorem 4.1 supplies 84 cross-polytopal classes and Theorem 6.5 propagates the two additional classes with factor 13. Thus \[ b_{15}(\operatorname{VR}(Q_7;4);\mathbb Q)\ge84+26=110. \] These are lower bounds. They do not give a complete rational homology table at \(n=7\).

Section 6.4.4 of Adams and Virk reports a Ripser computation using about 180 GB of memory: \[ H_q(\operatorname{VR}(Q_6;4);\mathbb F_2)\cong \begin{cases} 0,&1\leq q\leq6,\\ \mathbb F_2^{239},&q=7,\\ 0,&8\leq q\leq14,\\ \mathbb F_2^{14},&q=15. \end{cases} \] Their displayed table ends at degree 15.

Appendix C.3 of Galetto, Montaño, and Wellner reports a Polymake computation over \(\mathbb Q\) taking about 18 hours and at least 800 GB of memory. It gives reduced rational homology of ranks 239 and 14 in degrees 7 and 15, respectively, and zero in every other degree. The authors then state that \(X^{6,4}\) has the same Betti numbers over \(\mathbb Z/2\) and \(\mathbb Q\) in every degree. That all-degree statement, rather than the earlier table truncated at degree 15, supports the universal-coefficient conclusion in the status record.

The associated repository was inspected at commit `15eebf49305d28a50cadc591aba6320f42e99bc3`. Its 454-byte `Polymake/n6r4_homology.poly` script constructs the full Rips complex and asks Polymake for rational homology. The repository supplies no stored output certificate and declares no repository-wide source license. This packet cites the paper's reported table and does not copy repository code.

Define \[ i_n:Q_n\longrightarrow Q_{n+1},\qquad i_n(x)=(x,0), \] and let \[ p_n:Q_{n+1}\longrightarrow Q_n \] delete the final coordinate. The first map preserves Hamming distance and the second is 1-Lipschitz. Both induce simplicial maps at scale four, and \(p_n\circ i_n\) is the identity on \(Q_n\). Hence \[ (p_n)_*\circ(i_n)_*=\operatorname{id} \] on integral homology. The map \((i_n)_*\) is split injective in every degree.

A torsion counterexample at a smallest dimension propagates to every larger dimension. Torsion-freeness at a fixed dimension supplies no converse implication.

The action was to compute \(\gamma_t\) exactly and feed it into the total-domination connectivity theorem. The search examined every candidate size allowed by the elementary lower bound until it found the optimum 12.

To obtain 6-connectivity from Theorem 2.3, set \(k=7\). Its strict hypothesis becomes \(\gamma_t(G)>14\). The exact value 12 violates that hypothesis. The same theorem certifies 4-connectivity and cannot certify 5-connectivity, whose hypothesis would be \(\gamma_t(G)>12\).

This failure boundary is exact. Improving the total-domination search or finding a different minimum witness cannot extend this theorem's result for the same graph. Another topological argument is required.

The source audit checked arXiv:2605.00705v2, revised 2026-07-17. Its introduction says that Ripser calculated the first 15 homology groups of \(\operatorname{VR}(Q_6;4)\) with \(\mathbb Z/2\) coefficients, found the first nonzero group in degree 7, and thereby showed 6-connectivity. Page 5 repeats exact 6-connectivity and states \(H_7(-;\mathbb Z)\ne0\).

Vanishing over \(\mathbb F_2\) in degrees 1 through 6 excludes free summands and 2-primary torsion in the corresponding universal-coefficient range. It permits odd-primary torsion. Homological vanishing over one field also supplies no direct vanishing of the homotopy groups needed for connectivity. The paper's rigorous total-domination bound gives only 3-connectivity, while the exact invariant in this packet improves that route to 4-connectivity.

The audit located no additional proof in the paper that closes the coefficient gap. This is a source-scope advisory about the connectivity sentence. It does not affect the canonical torsion question's definition.

The audit searched the exact target and nearby formulations using `VR(Q_6;4) homology`, `hypercube Vietoris-Rips torsion`, `scale four integral homology`, folded-cube independence-complex terminology, and facet terminology. It checked the complete current versions of five primary sources.

Adamaszek and Adams prove the scale-two wedge-of-spheres theorem and list larger scales as open. Adams and Virk give homology lower bounds at every scale, publish the mod-two \(n=6,r=4\) table through degree 15, and retain the scale-four homotopy and collapse questions. Briggs, Feng, and Wells repeat that mod-two table, construct further nonzero field-valued classes, and leave the dimensions of nontrivial homology open. Bendersky, Elia, and Grbić give general connectivity and coconnectivity bounds, with the coefficient-scope concern recorded separately. Galetto, Montaño, and Wellner determine representation decompositions for \(r\leq3\) and \(r=n-1\); Appendix C.3 reports the rational \(n=6,r=4\) computation.

No checked source computes integral Smith form at \((n,r)=(6,4)\), discusses torsion in this complex, or proves torsion-freeness for every \(n\). Search silence is recorded as a dated audit result rather than evidence of nonexistence.

The unreduced complex has 2,932,100,733 faces including the empty face, so direct boundary assembly reproduces the published high-memory barrier. The graph description and exact f-vector provide fixed inputs and checksums for a smaller chain model. The explicit 64-term cycle in `hr4-claim-explicit-cross-polytopal-five-cycle` supplies a focused first integral filling test.

A useful next run has three stages. First, build an acyclic discrete Morse matching or an equivariant orbit-chain reduction for the independence complex of \(Q_6\) with antipodal matching, and emit the critical cells together with a checkable matching certificate. Second, map the explicit 5-cycle into the reduced complex, test whether it has an integral filling or finite annihilator, and compute ranks over \(\mathbb F_3\) in degrees 5 through 21. The 4-connectivity result handles lower degrees, while the exact f-vector shows that the complex has dimension 21. Third, compute Smith data or determinantal-divisor certificates for every reduced boundary matrix.

Use a first-pass budget of 48 wall hours, 256 GiB of memory, and 1 TiB of scratch storage. Stop and report an inconclusive reduction if the matching fails acyclicity validation, reaches a resource cap, or leaves a chain model too large to materialize every required boundary. Matching rational ranks over \(\mathbb F_3\) would exclude 3-primary torsion only. Repeating for selected odd primes remains a finite screen. A proof of complete torsion-freeness needs integral Smith form or a separate bound that limits possible torsion primes.

For each \(n\in\{6,7\}\), the program builds the graph whose edges join binary words at Hamming distance greater than four. Rips faces are independent sets of this graph. It computes the independence polynomial by \[ P_G(x)=P_{G-v}(x)+xP_{G-N[v]}(x), \] with exact integer coefficients, connected-component factorization, a maximum-induced-degree pivot, and memoization by the surviving vertex mask.

For \(n=6\), the 23 counts by face cardinality are `1, 64, 1792, 29120, 307440, 2239552, 11682944, 44769920, 128380880, 279211520, 464621248, 593908224, 582529360, 435648640, 245610720, 102886976, 31658620, 7189056, 1239840, 165760, 17584, 1408, 64`. They sum to 2,932,100,733 including the empty face. The maximum face cardinality is 22 and \(\widetilde\chi=-253\).

For \(n=7\), the 30 counts are `1, 128, 6272, 159488, 2409792, 23483264, 156322432, 742564352, 2605928992, 6949231744, 14412507648, 23711476992, 31506817664, 34424345984, 31535692288, 24757907456, 17050663168, 10523630208, 5903704576, 3018193920, 1393044352, 570251648, 202612992, 61051648, 15211392, 3040128, 467712, 51968, 3712, 128`. They sum to 209,570,782,049 including the empty face. The maximum face cardinality is 29 and \(\widetilde\chi=-3937\).

Two fresh-process replays produced byte-identical standard output.

A total dominating set is a collection of vertices whose open neighborhoods cover all 64 graph vertices. Every neighborhood has size 7, so the elementary covering bound starts at \(\lceil64/7\rceil=10\). Vertex transitivity lets a minimum set be translated to contain vertex 0.

For each candidate size, the program branches on an uncovered graph vertex and tries all seven vertices whose open neighborhoods could cover it. The state is the covered 64-bit mask and the remaining number of choices. Memoization prunes a repeated mask previously reached with at least as much remaining budget. The uncovered-cardinality lower bound prunes states that cannot be covered by the remaining choices.

The exact searches at sizes 10 and 11 exhaust 10,284 and 491,723 nodes without a cover. At size 12, the program returns `0, 15, 44, 28, 49, 50, 4, 40, 24, 11, 53, 54`, and directly checks that the union of their open neighborhoods is the full vertex set. Thus the total domination number is exactly 12. Two fresh-process replays produced byte-identical output.