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NEW GREEN-DISTRICT EXHIBIT · Professor’s Cube — Shared-Carrier $S_3$ Symmetry Bridge. The Borough now exposes the exact hierarchy $125\supset47\supset5$: common $5^3$ carrier → E47 spectral sector → canonical rank-5 $S_3$-trivial spin-2 descendant. Enter the exhibit.
The correction remains binding: $\dim\ker L_{C_5^3}=1\neq47$. The bridge is shared carrier + factor-permutation symmetry, not equal kernels.
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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:
$$ P:\mathbb R^{125}\to\mathbb R^{125},\qquad P^TP=I,\qquad P^4=I. $$
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MACHINE RECONSTRUCTION ATTACHED · 2026-08-21
Drive validation source now supplies an explicit NumPy reconstruction of the forty-state affine map $F_s(j)=3j+s\pmod{40}$, its permutation matrix $P_s$, self-adjoint comparison Laplacian $L=I-(P_s+P_s^T)/2$, exact orbit projector $\Pi=(I+P+P^2+P^3)/4$, and contraction operator.
Independent rerun reproduces the displayed deterministic results exactly: for $s=1$, cycle lengths are ten 4-cycles, $\operatorname{rank}\Pi=10$, and $\sigma(L)=\{0^{(10)},1^{(20)},2^{(10)}\}$; for $s=2$, cycle lengths are $1,1,2,2,2,4,4,4,4,4,4,4,4$, $\operatorname{rank}\Pi=13$, and $\sigma(L)=\{0^{(13)},1^{(16)},2^{(11)}\}$. In both cases $\|P^4-I\|=\|P^TP-I\|=\|\Pi^2-\Pi\|=\|L\Pi\|=0$ in exact floating arithmetic for this construction.
With the displayed $\varepsilon=2/3$, the positive modes contract by factors $1/3$ and $-1/3$, so $(I-\frac23L)^n\to\Pi$ at spectral rate $1/3$.
Provenance boundary: this is a validated finite-operator realization of the existing Sigillum formalism, not a new historical attribution. Connected provenance search did not resolve an immutable GitHub implementation for this specific forty-state validator, so the Drive source is retained as the current scoped executable witness and no unrelated E47 commit is assigned.
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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
$P_s^P_s=P_sP_s^=I.$