Let \(N_{31}\) denote the requested norm. Exact rational evaluation of the integer vector in c31m-claim-exact-lower-witness gives \[ \sqrt{\frac{95224622960985697907125617859822805279079620426252}{53808054668649331334697257108564850111138302100375}} \leq N_{31}. \] The lower endpoint is \[ 1.3303042705991634737558623603323047716\ldots. \] The weighted Cauchy-Schwarz argument in c31m-claim-diagonal-upper-certificate proves \[ N_{31}\leq \sqrt{\frac{1916477}{720720}} =1.6306791519531046652193954473854436190\ldots. \] Thus the candidate's reported value is a certified lower bound to ten decimal places. This entry leaves the exact norm and the complete active-pattern exclusion open.

In cyclic coordinate order, take \[ \begin{aligned} x={}&(60240784093944,71756706270677,71756706270677,69429053137658,\\ &69429053137658,124946197568962,124946197568962,634350542282289,\\ &220384421138340,634350542282289,124946197568962,124946197568962,\\ &69429053137658,69429053137658,71756706270677,71756706270677,\\ &60240784093944,60240784093944,53164211021439,53164211021439,\\ &49020565351307,49020565351307,47095643667602,47095643667602,\\ &47095643667602,47095643667602,49020565351307,49020565351307,\\ &53164211021439,53164211021439,60240784093944). \end{aligned} \] Exact comparison of the sixteen averages at every coordinate selects the radii \[ (9,8,7,6,5,4,1,0,1,0,1,4,5,6,7,8,9,10,11,12,13,15,15,15,15,15,15,13,12,11,10). \] There are no ties. Direct rational summation gives \[ \frac{\|Mx\|_2^2}{\|x\|_2^2} =\frac{95224622960985697907125617859822805279079620426252}{53808054668649331334697257108564850111138302100375}. \] This quotient exceeds \((1.3303042705)^2\) by exact integer comparison.

Replace \(f\) by \(x=|f|\), which preserves the input norm and the maximal function. Set \[ d_0=1,\qquad d_k=d_{-k}=\frac1{4k}\quad(1\leq k\leq15). \] For a radius \(r\), the exact identity \[ \sum_{k=-r}^{r}\frac{1}{(2r+1)^2d_k} =\frac{1+8\sum_{k=1}^{r}k}{(2r+1)^2}=1 \] turns weighted Cauchy-Schwarz into \[ \left(\frac1{2r+1}\sum_{k=-r}^{r}x_{j+k}\right)^2 \leq \sum_{k=-r}^{r}d_kx_{j+k}^2 \leq \sum_{k=-15}^{15}d_kx_{j+k}^2. \] The right side is independent of the selected radius, so it also bounds \((Mx(j))^2\). Summing over all cyclic centers makes every input coordinate receive the coefficient \[ d_0+2\sum_{k=1}^{15}d_k =1+\frac12H_{15} =\frac{1916477}{720720}. \] Therefore \(\|Mx\|_2^2\leq(1916477/720720)\|x\|_2^2\) for every input. The artifact checks all sixteen coefficient identities over the rationals.

Gonzalez-Riquelme and Madrid define the centered maximal operator on a finite graph by taking averages over graph-metric balls. This specializes to the operator in the candidate record when the graph is \(C_{31}\). Their L2 results give exact formulas for complete graphs and star graphs. The paper supplies no corresponding cycle formula. A focused search by the cycle notation, graph name, and operator norm found nearby variation results, while the exact \(C_{31}\) constant remained unlocated. Novelty therefore remains unverified.

The replay evaluates each of the sixteen centered averages at each of the 31 coordinates. It verifies the active-radius vector, reduces the squared quotient to its exact fraction, and proves the stated decimal lower bound by cross multiplication. For the upper endpoint it checks \(\sum_{|k|\leq r}((2r+1)^2d_k)^{-1}=1\) at every radius. It then sums the radial weights and obtains \(1916477/720720\).

The witness together with its active radii has SHA-256 digest `d6e9301b3a7dc898333a7bb1c2d70a9516db17e08bc9650ea0830d128c9f5cab`. The six-line output has SHA-256 digest `5b55424e8e5e369a4b7034305fc525ae05286e9d9cd7c6f049841ecd565ca8b4`.