Let \[ C=\{z^3:z\in\mathbf F_{127}^{\times}\} \] and join distinct \(x,y\) when \(x-y\in C\). The set \[ W=\{10,13,27,41,44,55,70,85,96,111,126\} \] is independent. Direct modular evaluation checks all \(\binom{11}{2}=55\) pairs and finds no difference in \(C\), so \(\alpha(G)\geq11\).

An independent set in \(G\) is a clique in \(\overline G\). The executable artifact `cgr127-artifact-exhaustive-size-12-search` performs a complete clique search in \(\overline G\). Its greedy coloring bound is valid at every branch and the run finds no clique of order 12. Any independent set with more than 12 vertices would contain an independent 12-subset, so this proves \(\alpha(G)\leq11\). Therefore \[ \boxed{\alpha(G)=11}. \]

This value was already known in computational Ramsey theory. Radziszowski's 2009 summary explicitly gives independence number 11 for the Hill-Irving graph \(G_{127}\). The independent artifact here supplies a short replay against the precise vertex and edge convention in the candidate.

The group \(\mathbf F_{127}^{\times}\) is cyclic of order 126. If \(g\) is a generator, then the cube subgroup is \[ C=\langle g^3\rangle, \qquad |C|=126/3=42. \] Also \[ -1=g^{63}=(g^{21})^3\in C. \] It follows that \(C=-C\). Thus \(x-y\in C\) if and only if \(y-x\in C\), which settles the direction convention. Since zero is excluded, the graph has no loops. Translation gives 42 neighbors at every vertex, hence 2,667 edges.

In the terminology of Lim and Praeger this is the generalized Paley graph whose connection subgroup has order 42. Their definition requires the connection subgroup to have even order in odd characteristic, exactly the condition that makes the relation symmetric.

Hill and Irving studied the cubic partition of \(\mathbf Z_{127}^{\times}\) in 1982 and obtained the graph now usually denoted \(G_{127}\). It is a \((4,12;127)\)-Ramsey graph: it contains no \(K_4\) and its complement contains no \(K_{12}\). Radziszowski's 2009 computational Ramsey notes identify the same Cayley graph and explicitly list 127 vertices, 2,667 edges, degree 42, and independence number 11.

Su, Li, Luo, and Li later developed an algorithm for cubic-residue graphs and their complements. Their 2002 paper reports computations for primes of the form \(6m+1\), and a 2017 account states that their clique-number computations extend through prime 1327. This places the parameter 127 well inside the previously computed range.

Generalized Paley clique bounds treat a single power-residue class. Here \(\alpha(G)=\omega(\overline G)\), while \(\overline G\) is the union of the other two cubic-character classes. The Ramsey-graph literature is therefore the direct exact source for this candidate.

The program constructs the 42-element cube set directly, then builds all 127 adjacency rows of \(G\) and \(\overline G\). Each row is a 127-bit integer. It checks primality of 127, the size and negation closure of the cube set, graph symmetry, every degree, and all 55 pairs of the displayed independent set before beginning the upper-bound search.

At each search node, `color_sort` greedily partitions the candidate-induced subgraph of \(\overline G\) into independent color classes. A clique uses at most one vertex of each class. The color number attached to each vertex therefore bounds the size of a clique in the corresponding candidate prefix. The search processes vertices in reverse color order. It explores the branch containing the current vertex, removes that vertex, then proceeds to the next branch. If the selected depth plus the prefix color bound is below 12, the whole prefix is safely discarded. These are the only cutoffs.

The full run finds no clique of order 12. It visits 117,239 nodes and makes 117,238 recursive branches. The node counts by selected depth are \[ [1,94,2306,18163,48068,38101,9387,1034,85,0,0,0,0]. \] Every terminal node ends at a coloring-bound cutoff. A SHA-256 trace mixes each node, branch, and cutoff with its depth and full 127-bit candidate mask, ending at `60d93eb7946529959e490bd7358c5826dd9b791496371ea4113efb5e5ae77452`. The stable eleven-line output has SHA-256 digest `d8211716032b2b4106352be3cc946ae5c6e85ae39dab2c25534b801e83e6242d`.