TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/003-subsection-character-trace-identities-and-the-abelian-in.tex
Original file SHA-256: dbd7f1020e414aeaa35ab3a7191fc5863d6b0afda270a2e347e9a0d6af2b0284
Inclusive original lines: 175-228
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.

000175 | \begin{proposition}[Family hypercharge arithmetic and compression]
000176 | \label{prop:family-hypercharge-compression}
000177 | Let $N_3$ and $N_2$ be the occupation-number operators on the $E_3$ and $E_2$ slots, restricted to $S^+$, and define the scaled hypercharge operator
000178 | \[
000179 | X:=6Y=3N_2-2N_3.
000180 | \]
000181 | Then on $S^+$ one has
000182 | \begin{align}
000183 | X&\equiv N_2\equiv N_3 \pmod 2,\label{eq:x-mod-2}\\
000184 | X&\equiv N_3 \pmod 3,\label{eq:x-mod-3}\\
000185 | X&\equiv 3(N_2+N_3) \pmod 5.\label{eq:x-mod-5}
000186 | \end{align}
000187 | In particular, the even exterior degree $k=N_2+N_3\in\{0,2,4\}$ is read off from the residue class of $X$ modulo $5$: if $X\equiv r\pmod 5$ with $r\in\{0,1,2\}$, then $k=2r$, so
000188 | \[
000189 | X\bmod 5=0,1,2
000190 | \qquad\Longleftrightarrow\qquad
000191 | \Lambda^0E,\ \Lambda^2E,\ \Lambda^4E.
000192 | \]
000193 | Moreover,
000194 | \[
000195 | \mathrm{Spec}(X|_{S^+})=\{0,-4,1,6,-3,2\},
000196 | \]
000197 | and therefore
000198 | \begin{equation}
000199 | \label{eq:family-x-polynomial}
000200 | X(X+4)(X-1)(X-6)(X+3)(X-2)=0
000201 | \qquad\text{on }S^+.
000202 | \end{equation}
000203 | \end{proposition}
000204 | 
000205 | \begin{proof}
000206 | Each occupied $E_3$ mode contributes hypercharge $-1/3$ and each occupied $E_2$ mode contributes $+1/2$, so multiplying by $6$ gives $X=3N_2-2N_3$. Modulo $2$, the term $-2N_3$ vanishes, so $X\equiv N_2$. Because $S^+$ has even total degree, $N_2\equiv N_3\pmod 2$, which proves \Cref{eq:x-mod-2}. Modulo $3$, one has $3N_2\equiv 0$ and $-2\equiv 1$, giving \Cref{eq:x-mod-3}. Modulo $5$, the congruence $-2\equiv 3$ yields \Cref{eq:x-mod-5}. Since on $S^+$ the even degree is $k=N_2+N_3\in\{0,2,4\}$, the possible residues are $3k\equiv 0,1,2\pmod 5$, from which $k=2r$ with $r=X\bmod 5$ follows. Evaluating $X=3n_2-2n_3$ on the six admissible occupancy pairs $(n_3,n_2)$ gives the listed spectrum. The polynomial \Cref{eq:family-x-polynomial} is then the spectral polynomial with these six simple roots.
000207 | \end{proof}
000208 | 
000209 | \begin{corollary}[Five-fold phase filters for even degree]
000210 | \label{cor:family-degree-filters}
000211 | Let
000212 | \[
000213 | \zeta_5=e^{2\pi i/5},
000214 | \qquad
000215 | U_5:=e^{2\pi iX/5}.
000216 | \]
000217 | Then the even-degree projectors are the discrete Fourier filters
000218 | \[
000219 | P_{\Lambda^{2m}E}
000220 | =
000221 | \frac15\sum_{r=0}^{4}\zeta_5^{-mr}U_5^r,
000222 | \qquad
000223 | m=0,1,2.
000224 | \]
000225 | \end{corollary}
000226 | 
000227 | \begin{proof}
000228 | By \TFPTxref{prop:family-hypercharge-compression}, the eigenspaces $\Lambda^{2m}E$ carry the residue class $X\equiv m\pmod 5$. Hence $U_5$ has eigenvalue $\zeta_5^m$ on $\Lambda^{2m}E$, and the standard Fourier projector formula on the cyclic group $\mathbb Z_5$ gives the claim.