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Original repository path: documents/articles--2026-08-30--det16_mirror_gap_theorem_en/sections/001-section-concrete-model-and-frozen-finite-input.tex
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000042 | \subsection{One DET16 cluster}
000043 | 
000044 | Let $K\cong\C^{16}$ carry the positive-chirality spin representation
000045 | $\rho_{16}$ of $\operatorname{Spin}(10)$ and let
000046 | $\F(K)=\bigoplus_{n=0}^{16}\bigwedge^nK$ be its fermionic Fock space.
000047 | For an ordered orthonormal basis $(e_1,\ldots,e_{16})$, write
000048 | \[
000049 |  \Omega^\dagger
000050 |  =a^\dagger(e_1)\cdots a^\dagger(e_{16}),\qquad
000051 |  |F\rangle=\Omega^\dagger|0\rangle ,
000052 | \]
000053 | and $\Omega=(\Omega^\dagger)^*$.  Put
000054 | \[
000055 | \begin{aligned}
000056 |  A_\phi
000057 |  &=P_0+P_{16}+e^{-i\phi}\Omega+e^{i\phi}\Omega^\dagger,\\
000058 |  |\Omega_\phi\rangle
000059 |  &=\frac{|0\rangle+e^{i\phi}|F\rangle}{\sqrt2},\\
000060 |  P_\phi&=\frac12A_\phi
000061 |  =|\Omega_\phi\rangle\langle\Omega_\phi|,\qquad
000062 |  q_\phi=\id-P_\phi.
000063 | \end{aligned}
000064 | \tag{1.1}\label{eq:cluster}
000065 | \]
000066 | The exact finite computation in
000067 | \texttt{verification/v1002\_spin10\_mirror\_projector.py} verifies
000068 | \[
000069 |  \Omega^\dagger\Omega=P_{16},\quad
000070 |  \Omega\Omega^\dagger=P_0,\quad
000071 |  P_\phi^2=P_\phi=P_\phi^*,\quad
000072 |  \rank P_\phi=1,
000073 | \tag{1.2}\label{eq:frozen}
000074 | \]
000075 | and all forty-five $\mathfrak{so}(10)$ commutators vanish.  These are
000076 | inputs to this memorandum; the algebraic consequences needed below
000077 | are also proved directly.