adjoint_dim

This module performs exact rational bookkeeping for the high-temperature Standard Model effective relativistic degrees of freedom

$$g_\star = g_b + \tfrac78 g_f = 28 + \tfrac78\cdot 90 = 106.75,$$

valid only for $T \gtrsim T_{\mathrm{EW}}$. Recognition Science supplies the gauge factor $SU(3)\times SU(2)\times U(1)$ (from $Q_3$ automorphisms) and the generation count 3 (from $D=3$); the matter representations and the $7/8$ thermal weight are imported textbook inputs.

In Lie theory the adjoint of $SU(n)$ is the real vector space of traceless Hermitian $n\times n$ matrices, whose real dimension is $n^2-1$. That integer multiplies the two transverse polarizations of each massless gauge boson above the electroweak scale, giving the bosonic contributions that enter $g_b$.

Pure definition: the body is the arithmetic expression $n^2-1$. No tactics, lemmas, or hypotheses. Downstream equalities such as su3_adjoint and su2_adjoint discharge the concrete cases $n=3$ and $n=2$ by native_decide.

Local glue for the bosonic half of $g_\star$. Downstream, gluon_dof is adjoint_dim 3 * 2 (8 gluons × 2 polarizations), weak_boson_dof_symmetric is adjoint_dim 2 * 2 + 1 * 2 (W triplet plus B, each with 2 polarizations), and the theorems su3_adjoint / su2_adjoint pin the integers 8 and 3. The parent theorem bosonic_traces_to_Q3 packages these into the claim that every bosonic DOF traces to $Q_3$ structure and that bosonic_dof = 28. In the RS forcing chain the gauge group itself is upstream (GaugeFromCube); this definition only converts that group data into the textbook adjoint dimensions used by the thermal count. Status remains bookkeeping over adopted SM content, not a novel RS prediction.