Proof · Python · Certificate
The Borough now has an explicit 15-generator layer-turn representation on $\Sigma=\{0,1,2,3,4\}^3$. The carrier and packing map deduplicate to existing Base-5 citizens; RUB5-C01…C08 add the new permutation, spectral, energy, low-band, and continuity identities.
The move action is geometric computation in its literal finite form:
$$ P:\mathbb R^{125}\to\mathbb R^{125},\qquad P^TP=I,\qquad P^4=I. $$
This page records an original mathematical re-typing of John Dee’s Sigillum Dei Aemeth as a finite spectral computation architecture.
Let the forty indexed positions be
$V=\{0,1,\dots,39\}$
with recurrence
$k_{j+1}=3k_j+s\pmod{40}.$
For fixed offset $s$, define the finite map
$F_s(j)=3j+s\pmod{40}$
and the permutation operator on the standard basis $\{e_j\}_{j=0}^{39}$ by
$P_se_j=e_{F_s(j)}.$
Because $\gcd(3,40)=1$, multiplication by $3$ is invertible modulo $40$, so $F_s$ is a bijection. Therefore $P_s$ is a permutation matrix and
$P_s^P_s=P_sP_s^=I.$
Associate to the wheel map its directed permutation graph, or its declared undirected comparison graph when a symmetric Laplacian is required. Let $A$ be the corresponding adjacency operator and $D$ the degree operator. Then