Number the cells modulo 24 and store cell \(i\) in bit \(i\) of an unsigned integer. In this convention the left neighbor of cell \(i\) is cell \(i+1\), the right neighbor is cell \(i-1\), and one synchronous update is \[ F(x)_i=x_{i+1}\mathbin{\mathtt{xor}}(x_i\mathbin{\mathtt{or}}x_{i-1}). \]

An exhaustive traversal of all \(2^{24}=16{,}777{,}216\) states gives maximum directed-cycle length \[ \boxed{185{,}040}. \] There is exactly one cycle of that length. Its least integer state is 449, which is hexadecimal `0001c1` and binary `000000000000000111000001` when bits 23 through 0 are printed left to right. Starting at 449 and applying \(F\) returns to 449 after exactly 185,040 updates.

The executable certificate uses two full-state methods. The first follows each unprocessed orbit and marks its current path. The second constructs every successor and indegree, removes all transient vertices with Kahn's algorithm, and scans the vertices that remain. Both methods produce the same sorted list of pairs `(least state on cycle, cycle length)`. Since every state belongs to the scan, a longer cycle cannot be omitted.

The value also appears in the published width table of Gage, Laub, and McGarry and in OEIS A334497.

A finite functional graph has one directed cycle in each weak component. The complete width-24 inventory contains 49 cycles, hence 49 weak components. Their lengths sum to 194,991, so that many of the 16,777,216 states are cyclic. Three cycles are fixed points. The maximum global indegree is three.

For a compact check, each cycle is represented by the least integer on that cycle and its length. Sort these 49 pairs by representative and then length. Serialize each unsigned 32-bit integer as four little-endian bytes, with the representative first, and apply 64-bit FNV-1a starting at `14695981039346656037`. The resulting digest is `8bb15c98ca3aeadf`. Both exhaustive algorithms produce this digest and agree on every pair before it is computed.

Dustin Gage, Elizabeth Laub, and Briana McGarry published `Cellular Automata: Is Rule 30 Random?` in the proceedings of the 2005 Midwest NKS Conference. Page 7, Table 2 records maximum cycle lengths for finite cyclic windows through width 36. Its width-24 entry is 185,040.

OEIS A334497 identifies its terms as the maximum eventual period over all starting configurations of cyclic Rule 30 at width \(n\). The entry cites the 2005 paper and lists \(a(24)=185040\). It says that Paolo Xausa added widths 23 through 36 in 2023 using page 7, Table 2 of that paper.

The focused search therefore resolves the candidate's novelty warning for the maximum value. The component count, cyclic-state count, uniqueness of the maximum cycle, representative 449, and compact inventory digest were obtained in the new exhaustive reproduction. They were absent from the sources inspected.

The path-marking scan and the indegree-peeling scan use separate state arrays and separate cycle-discovery logic. They evaluate the same explicit bitwise update, then compare all 49 sorted cycle records. The program also checks the maximum-cycle vector for widths 3 through 20 supplied with the candidate problem.

Its complete six-line output is 493 bytes including the final newline and has SHA-256 digest `be6fc56a241c6a445ea9d4bec0945b0ea466ecd9ed766ae4f670f66b774038cc`.