Network coefficient interval

The scalar successor page gives $I_{15}=[4196/12343,\,7039/20706)$. Let $H=32767$, $Z=25H=819175$, and $t_z=\lfloor Qz/(25S)\rfloor$. The exact interval preserving every canonical network output is

Its width is $25/491850297564$. The lower endpoint is active at $z=623548$ and the upper at $z=788793$. Both $q$ and $\lambda$ lie inside. The checker compares exact fractions, not rounded decimals.

For a fixed center $j$, twelve neighbors in $[0,H]$ realize every neighbor sum in $[0,12H]$. The local mixed integers cover every integer in $[0,25H]$. A condition on every local integer is therefore necessary and sufficient.

Two disagreements inside the cube

Set one vertex to zero, four neighbors to $32767,32767,32767,29512$, and every other vertex to zero. The mixed integer is 127813.

Five neighbors at 32767 and one at 1410 give $z=165245$. The dyadic candidate $22813643/2^{26}$ returns 2247; the canonical output is 2246. “First” means the least such local integer $z$, not an ordering of all 120-dimensional inputs.

$49136/144539$ and $846547/2490212$ agree with the canonical output on this domain.$169974885672389/(5\cdot 10^{14})$ is the same rational as $q$ globally. Quotient agreement does not imply equal remainders or equal record bytes. This note does not amend the scalar successor contract and does not adopt a new canonical factor.

The parameter checker and this archive’s run.