Among the \(2^{36}=68{,}719{,}476{,}736\) labeled graphs on nine vertices, exactly \[ 50{,}766{,}340{,}113 \] have \(H_1(X(G);\mathbb F_2)\ne0\). The fraction is already reduced because its numerator is odd: \[ \Pr\bigl(H_1(X(G);\mathbb F_2)\ne0\bigr) =\frac{50{,}766{,}340{,}113}{68{,}719{,}476{,}736} \approx0.7387474777788157. \]

The full labeled distribution of \(\beta_1\) is \[ \begin{array}{c|rrrrrrrrrrrrr} \beta_1&0&1&2&3&4&5&6&7&8&9&10&11&12\\ \hline \#G&17{,}953{,}136{,}623&21{,}459{,}770{,}001&16{,}840{,}187{,}301&8{,}590{,}760{,}010&2{,}991{,}956{,}730&731{,}552{,}976&131{,}963{,}406&17{,}710{,}560&2{,}177{,}532&234{,}927&24{,}024&2{,}520&126. \end{array} \] The entries sum to \(2^{36}\). The top value has a simple check: each labeled \(K_{4,5}\) has no triangles and has cycle rank \(20-9+1=12\), and there are \(\binom94=126\) such graphs.

The computation generated all 274,668 isomorphism classes of nine-vertex simple graphs. For each representative \(G\), it computed \(\beta_1\), then added the orbit weight \(9!/|\operatorname{Aut}(G)|\) to the appropriate bin. The weights over all classes sum to \(9\)-vertex labeled graph count \(2^{36}\).

Let \(B\) be the edge-by-triangle matrix over \(\mathbb F_2\). A triangle column has a one in each of its three edge rows. The graph boundary map \(\partial_1:C_1\to C_0\) has rank \(n-c\), so \[ \dim\ker\partial_1=m-n+c. \] The image of \(\partial_2:C_2\to C_1\) is the column space of \(B\). Therefore \[ \beta_1(X(G);\mathbb F_2)=m-n+c-\operatorname{rank}_{\mathbb F_2}B. \] Cliques with four or more vertices add higher-dimensional simplices and do not change either map in this formula. The orbit computation stores every triangle boundary as a 36-bit word and performs exact Gaussian elimination with XOR.

Kahle defines \(X(n,p)\) as the clique complex of \(G(n,p)\), proves vanishing and nonvanishing regimes for higher homology, and gives estimates for expected Betti numbers. Kahle and Meckes prove Poisson and normal approximation theorems for random clique-complex Betti numbers. Kahle's later paper sharpens cohomology-vanishing thresholds. These results concern asymptotic regimes as \(n\) grows.

The audit checked the exact title phrase, `nine vertex random flag complex`, `exact Betti distribution random clique complex`, and combinations of \(n=9\), \(p=1/2\), \(\beta_1\), graph enumeration, and clique complexes. It also checked the three primary papers cited here. No exact finite numerator or beta histogram for these parameters was found. This leaves novelty of the table unverified, while the value itself is settled by the included exhaustive computation.

The checker uses tinygraph commit `97665350689943d64b90ca1501b3291db0538a0a`, whose generator is backed by nauty 2.6r6. `Graph::enumerate(9, ...)` visits one representative of every isomorphism class. `numLabeledGraphs()` supplies \(9!/|\operatorname{Aut}(G)|\).

For each representative, union-find gives the component count. Lexicographic edge indices identify each triangle boundary with a 36-bit word. Highest-pivot XOR elimination gives its exact rank over \(\mathbb F_2\). The resulting beta value receives the full labeled orbit weight.

The run found 196,389 positive classes and a labeled favorable weight of 50,766,340,113. All orbit weights sum to 68,719,476,736. The unlabeled beta histogram sums to 274,668, and its labeled counterpart sums to \(2^{36}\).

The generation-order serialization hashes, for every class, its graph6 record, a zero byte, eight little-endian bytes of orbit weight, then one byte each for component count, edge count, triangle count, boundary rank, and beta one. Its FNV-1a 64-bit checksum is `5771132747119421409`. The executed source has SHA-256 `d82620acd91bea3cf11f20f91aedb8c05cd7b7ec054fd3501f524b5a636d7289`. The four-line standard output has SHA-256 `f9f953771ea551bb9b2cdf126ded76fe31eaea090a32a7d41f38a8a3fb8e491b`.

This second implementation visits all 32,768 labeled six-vertex graphs directly. It constructs the graph components, creates one integer bit-vector for every triangle boundary, and computes rank by a Python dictionary of XOR pivots. It finds \[ (\#\beta_1=0,\ldots,\#\beta_1=4) =(20{,}326,10{,}317,2{,}010,105,10). \] The 12,442 positive cases agree with the candidate's independent six-vertex computation. The canonical report has SHA-256 `4d1011305015f6d7599ee8709932cce455bbd4458ab6b1daefa7fee098f42a63`.