E47 IN TWO PAGES · RESEARCH NOTE · 2026-09-22

The finite object, the recursive interpretation, and the boundary of the claim.

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PAGE 1 · What the object is

Why V₂ has dimension 5

The starting object is the spin-2 irreducible representation of SU(2), denoted $V_2$. A spin-$j$ irreducible representation has dimension

$$ \dim V_j = 2j+1. $$

For $j=2$,

$$ \dim V_2=5. $$

The carrier used here is three copies:

$$ V=V_2^{\otimes3}, \qquad \dim V=5^3=125. $$

So the numbers 5 and 125 are representation-theoretic dimensions, not fitted parameters.

The Casimir selector

Let

$$ C=J_{\mathrm{tot}}^2. $$

On this carrier its eigenvalues are

$$ \{0,2,6,12,20,30,42\}, $$

with state multiplicities

$$ \{1,9,25,28,27,22,13\}. $$