47-Degree Spectral Polynomial Framework — Cyclic Coordinates for a Rank-47 Space

47-Degree Spectral Polynomial — unmarked cyclic instrument

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VISUAL SYSTEM · MINI AI LABS

Near-black laboratory. Cyan live · bronze Λ-ghost · mint lock. No type on stills. Record: CITY-LABS-VISUAL-SYSTEM-20260915.

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47-Degree Spectral Polynomial Framework — concentric bronze Casimir rings. Mini AI Labs visual system.

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VISUAL SYSTEM · MINI AI LABS

Near-black laboratory. Cyan live · bronze Λ-ghost · mint lock. No type on stills.

Record: CITY-LABS-VISUAL-SYSTEM-20260915.

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Status: exact cyclic algebra combined with proposed computational and geometric interpretations.

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First-principles formalism

Define

$$ P_{47}(z)=z^{47}-1. $$

Its roots are

$$ z_k=e^{2\pi i k/47},\qquad k=0,\ldots,46, $$

which form the finite cyclic group

$$ \mu_{47}\cong\mathbb Z_{47}. $$

The natural basis is the discrete Fourier basis on $\mathbb Z_{47}$.

The 47th roots of unity provide a natural phase clock or coordinate system for a rank-47 space.
They may index basis vectors, iterative phases, or cyclic states.
They do not generate the canonical $E_{47}$ kernel, which arises from $V_2^{\otimes3}$.
A cube or torus visualization is a downstream representation choice.
A compiler or heterogeneous-architecture mapping is an engineering proposal.