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PURPLE EXPRESS · QUANTUM INFORMATION / COORDINATION

This page begins only after the exact $E_{47}$ kernel has been selected and locked. The clock is an admissible dynamical structure acting on the certified 47-dimensional invariant space; it is not part of the theorem that creates $E_{47}$.

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Canonical route

$\boxed{\text{Invariant Grammar Grand Terminal}\rightarrow\text{Purple Express}\rightarrow\text{Quantum Information / Coordination}}$

$\text{representation}\rightarrow\text{spectral selection}\rightarrow E_{47}\rightarrow P_{47}\rightarrow\text{invariant lock}\rightarrow\text{clock action}$

Evidence status

$`boxed{begin{array}{ll}

E0: & E_{47}text{ kernel and projector exact} \

E0: & text{existence of an order-47 unitary exact} \

E1: & text{matrix construction and order tests verified} \

E2: & text{swarm and material behavior simulated} \

E3/E4: & text{physical clock realization unestablished}

end{array}}`$

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Evidence boundary

The dimension $47$ permits an exact order-47 clock. The kernel theorem does not uniquely select its basis, ordering, phase origin, physical tick duration, or observable readout.

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1. Exact invariant core

Let

$\Sigma=V_2^{\otimes3},\qquad \dim\Sigma=125.$

For the diagonal $SU(2)$ action,

$\Sigma=V_0\oplus3V_1\oplus5V_2\oplus4V_3\oplus3V_4\oplus2V_5\oplus V_6.$

Define

$C=(J_1+J_2+J_3)^2,\qquad K=(C-6I)(C-30I).$