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Status: exact core with independent arithmetic and geometric observations shown separately.

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First-principles formalism

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

$$ C=(J_1+J_2+J_3)^2, $$

$$ K=(C-6I)(C-30I). $$

Hence

$$ \ker K=E_6\oplus E_{30}=5V_2\oplus2V_5, $$

and

$$ \dim E_{47}=25+22=47. $$

Logical deductions

• The selected Casimir eigenvalues are $6$ and $30$.
• They correspond to total spin $J=2$ and $J=5$.
• The eigenspaces include multiplicity, so $E_6\oplus E_{30}$ is more precise than treating them as single irreducibles.
• The remaining Casimir sectors form a $78$-dimensional complement.

Interpretation

The term “unit” is a visual metaphor for the invariant decomposition and not an additional theorem.

Open obligation

Golden-ratio, Babylonian, arithmetic, combinatorial, and symbolic correspondences displayed around the core remain independent observations unless derived from the representation theory.

Plate audit III provenance update