TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/002-subsection-internal-carrier-stabilizer-and-one-family-pa.tex
Original file SHA-256: ad8c8fced742cdbff008a9fb8b581d1aa75d92c3665ab9de21b4a821c74b42ef
Inclusive original lines: 188-301
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000188 | Hence the kernel is trivial. The quotient from $SU(3)\times SU(2)\times U(1)$ to
000189 | $S(U(3)\times U(2))$ already has kernel $\mathbb Z_6$ by \TFPTxref{thm:global-sm-theorem}, so this is
000190 | the exact physical global form.
000191 | \end{proof}
000192 | 
000193 | Throughout the manuscript, $S(U(3)\times U(2))$ is therefore read not only as the internal carrier
000194 | stabilizer but as the physical global gauge group selected by the faithful action on the admissible
000195 | matter-plus-Higgs package.
000196 | 
000197 | Matter is read from the positive half-spinor
000198 | \[
000199 | S^+=\Lambda^{\mathrm{even}}E,
000200 | \qquad
000201 | \dim S^+=16.
000202 | \]
000203 | 
000204 | \begin{theorem}[One carrier family from the exterior packet]
000205 | The positive half-spinor of the minimal carrier decomposes as
000206 | \[
000207 | S^+
000208 | \cong
000209 | (1,1)_0\oplus(3,2)_{1/6}\oplus(\bar 3,1)_{-2/3}\oplus(\bar 3,1)_{1/3}\oplus(1,2)_{-1/2}\oplus(1,1)_1,
000210 | \]
000211 | that is, exactly one chiral Standard Model family with $\nu^c$. Equivalently, in the familiar unified packaging language,
000212 | \[
000213 | S^+\cong \mathbf 1\oplus \mathbf{10}\oplus \overline{\mathbf 5}.
000214 | \]
000215 | \end{theorem}
000216 | 
000217 | \begin{proof}
000218 | One has
000219 | \[
000220 | S^+=\Lambda^0E\oplus\Lambda^2E\oplus\Lambda^4E.
000221 | \]
000222 | The $\Lambda^0E$ sector is immediately
000223 | \[
000224 | \Lambda^0E\cong (1,1)_0,
000225 | \]
000226 | which is identified with $\nu^c$ in left-chiral notation. For the minimal carrier,
000227 | \[
000228 | \Lambda^2E
000229 | \cong
000230 | \Lambda^2E_3\oplus(E_3\otimes E_2)\oplus\Lambda^2E_2.
000231 | \]
000232 | Here
000233 | \[
000234 | \Lambda^2E_3\cong \bar 3,
000235 | \qquad
000236 | E_3\otimes E_2\cong (3,2),
000237 | \qquad
000238 | \Lambda^2E_2\cong 1,
000239 | \]
000240 | and the hypercharges are obtained by adding the carrier charges from
000241 | \[
000242 | Y=\mathrm{diag}\!\left(-\frac13,-\frac13,-\frac13,\frac12,\frac12\right).
000243 | \]
000244 | Therefore
000245 | \[
000246 | \Lambda^2E
000247 | \cong
000248 | (\bar 3,1)_{-2/3}\oplus(3,2)_{1/6}\oplus(1,1)_1.
000249 | \]
000250 | Because the full determinant is trivial in $S(U(3)\times U(2))$, one has $\Lambda^5E\cong 1$ and hence
000251 | \[
000252 | \Lambda^4E\cong E^\ast.
000253 | \]
000254 | Equivalently,
000255 | \[
000256 | \Lambda^4E
000257 | \cong
000258 | (\Lambda^3E_3\otimes E_2)\oplus(\Lambda^2E_3\otimes\Lambda^2E_2).
000259 | \]
000260 | Using $\Lambda^3E_3\cong 1$, $\Lambda^2E_2\cong 1$, and $\Lambda^2E_3\cong E_3^\ast\cong\bar 3$, one gets
000261 | \[
000262 | \Lambda^4E\cong (1,2)_{-1/2}\oplus(\bar 3,1)_{1/3}.
000263 | \]
000264 | Collecting the three even-degree sectors yields
000265 | \[
000266 | S^+
000267 | \cong
000268 | (1,1)_0\oplus(\bar 3,1)_{-2/3}\oplus(3,2)_{1/6}\oplus(1,1)_1\oplus(\bar 3,1)_{1/3}\oplus(1,2)_{-1/2},
000269 | \]
000270 | with total multiplicity $1+3+6+1+3+2=16$. The $\Lambda^2E$ sector is the standard left-chiral $\mathbf{10}$ of $SU(5)$, the $\Lambda^4E$ sector is the standard left-chiral $\overline{\mathbf 5}$, and the $\Lambda^0E$ sector is the singlet $\mathbf 1$.
000271 | \end{proof}
000272 | 
000273 | In left-chiral notation the one-family decomposition takes the explicit form shown in \TFPTxref{tab:one-family}. This is the place where the manuscript must be explicit: the carrier theorem is meaningful only if the reader can see how the sixteen states are organized.
000274 | 
000275 | \begin{table}[t]
000276 | \centering
000277 | \footnotesize
000278 | \begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth}>{\raggedright\arraybackslash}p{0.19\linewidth}>{\raggedleft\arraybackslash}p{0.13\linewidth}>{\raggedright\arraybackslash}p{0.20\linewidth}>{\raggedleft\arraybackslash}p{0.08\linewidth}@{}}
000279 | \toprule
000280 | Exterior component & $\SM$ representation & Hypercharge & Field content & Mult. \\
000281 | \midrule
000282 | $\Lambda^0E$ & $(1,1)_0$ & $0$ & $\nu^c$ & $1$ \\
000283 | $\Lambda^2E_3$ & $(\bar 3,1)_{-2/3}$ & $-2/3$ & $u^c$ & $3$ \\
000284 | $E_3\otimes E_2$ & $(3,2)_{1/6}$ & $1/6$ & $Q_L$ & $6$ \\
000285 | $\Lambda^2E_2$ & $(1,1)_1$ & $1$ & $e^c$ & $1$ \\
000286 | \makecell[l]{$\Lambda^3E_3$\\$\otimes E_2$} & $(1,2)_{-1/2}$ & $-1/2$ & $L_L$ & $2$ \\
000287 | \makecell[l]{$\Lambda^2E_3$\\$\otimes\Lambda^2E_2$} & $(\bar 3,1)_{1/3}$ & $1/3$ & $d^c$ & $3$ \\
000288 | \midrule
000289 | \multicolumn{4}{r}{Total multiplicity} & $16$ \\
000290 | \bottomrule
000291 | \end{tabular}
000292 | \caption{One chiral family from $S^+=\Lambda^{\mathrm{even}}E$, written in left-chiral notation with charge-conjugate singlets.}
000293 | \label{tab:one-family}
000294 | \end{table}
000295 | 
000296 | \begin{proposition}[Exact parity-code realization of the family packet]
000297 | \label{prop:parity-code-family}
000298 | Identifying each wedge monomial in $\Lambda E$ with its occupation word
000299 | \[
000300 | x=(x_1,\dots,x_5)\in\{0,1\}^5,
000301 | \]