Let \(R\) be the public file `c7/R367.txt`. It contains 367 distinct members of \(\mathbb Z_7^5\). For each unordered pair \(x,y\in R\), the exact checker finds a coordinate \(i\) with \[ \min\bigl((x_i-y_i)\bmod 7,(y_i-x_i)\bmod 7\bigr)\geq2. \] Thus \(R\) is independent in \(C_7^{\boxtimes5}\), and \[ \alpha(C_7^{\boxtimes5})\geq367. \] The check covers all \(\binom{367}{2}=67{,}161\) pairs and reproduces the construction in Polak and Schrijver.

The capacity comparison is exact before taking roots: \[ 367^2=134689<134753<135424=368^2. \] Consequently, the 2026 ten-dimensional construction improves the capacity bound obtained from 367 words, while a 368-word fifth-power construction would improve the ten-dimensional bound.

Polak and Schrijver give the explicit 367-word code and list \[ 367\leq\alpha(C_7^{\boxtimes5})\leq401. \] Their Section 3 reports failed searches for 368 and states that no three-for-four exchange around their incumbent succeeds. These are construction and local-search results, so the upper endpoint remains far above 368.

Itty, Rosin, Carstensen, and Reichman submitted arXiv:2607.21517v1 on 2026-07-23. Its Introduction and Section 3.1 still call 367 the largest known independent set in the fifth power. Their new bound uses 134,753 words in the tenth power. The paper's public repository includes `R367.txt` as the fifth-power base and supplies no 368-word file.

The checked primary sources and associated repository contain no 368-word witness and no global exclusion of one. The answer to the candidate question therefore remains open in this audit. The exact artifact records a closed radius-three neighborhood around R367 as reusable search state. It supplies no global bound.

The program downloads `R367.txt` at a fixed Git commit and checks its SHA-256 digest before parsing it. It verifies the alphabet, word length, distinctness, and all 67,161 codeword pairs.

For each of the 16,440 outside words \(v\), it records the set \(N_R(v)\) of codewords adjacent to \(v\). The conflict-count distribution for sizes 1 through 11 is \[ (8,254,1505,3039,3712,3897,2842,921,199,61,2). \] Every outside word has a conflict, so direct insertion cannot enlarge \(R\).

An improving \(k\)-exchange deletes \(D\subset R\), with \(|D|=k\), and inserts \(k+1\) mutually independent outside words. Every inserted word must satisfy \(N_R(v)\subseteq D\). The program enumerates every deletion set for \(k=1,2,3\), gathers every eligible outside word, and checks every possible insertion set. For \(k=3\), it examines all 8,171,255 deletion triples. Only 51 triples admit at least four eligible words, yielding 83 insertion quadruples to test. Each quadruple contains an adjacent pair. The smaller exchange searches also fail.

This independently reproduces the radius-three local-search statement in Polak and Schrijver. The certificate concerns one incumbent and this exchange radius. A 368-word code could lie outside that neighborhood.