TFPT scientific source excerpt
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000128 | \begin{theorem}[Joint discrete admissible sector from the boundary primitive kernel]
000129 | \label{thm:joint-discrete-admissible-sector}
000130 | \label{thm:discrete-sector-from-primitive-kernel}
000131 | For every admissible boundary primitive kernel $\Tker^\partial$, the compatible discrete datum
000132 | \[
000133 | \mathfrak D_{\mathrm{disc}}(\Tker^\partial)
000134 | \]
000135 | is a singleton. Its unique element is
000136 | \[
000137 | d^\star_{\mathrm{disc}}
000138 | :=
000139 | \Bigl(
000140 | E_3\oplus E_2,\,
000141 | Y,\,
000142 | S^+,\,
000143 | X_f^\circ,\,
000144 | [\nabla_F^\star],\,
000145 | c_1(L_2),\,
000146 | c_1(L_3),\,
000147 | N_{\mathrm{fam}},\,
000148 | \theta_{\mathrm{eff}}
000149 | \Bigr)
000150 | \]
000151 | with
000152 | \[
000153 | X_f^\circ\cong\mathbb P^1\setminus\mu_4,
000154 | \qquad
000155 | N_{\mathrm{fam}}=3,
000156 | \qquad
000157 | (g,b,s)=(5,3,2),
000158 | \qquad
000159 | \dim S^+=16,
000160 | \]
000161 | \[
000162 | \begin{aligned}
000163 | S^+&=\Lambda^{\mathrm{even}}(E_3\oplus E_2),\\
000164 | Y&=\mathrm{diag}\!\left(-\tfrac13,-\tfrac13,-\tfrac13,\tfrac12,\tfrac12\right),\\
000165 | (c_1(L_2),c_1(L_3))&=(1,0),\qquad
000166 | \theta_{\mathrm{eff}}=0.
000167 | \end{aligned}
000168 | \]
000169 | Equivalently, carrier data, family data, determinant data, and the effective strong angle are fixed
000170 | jointly from the same boundary primitive kernel.
000171 | \end{theorem}
000172 | 
000173 | \begin{proof}
000174 | We prove the discrete blocks in the only order that uses no downstream uniqueness claim.
000175 | 
000176 | \textbf{Step 1: primitive winding.}
000177 | By \TFPTxref{thm:primitive-seam-generator}, the primitive seam class is the positive generator
000178 | \[
000179 | [u_\Sigma]=1\in K^1(S^1)\cong\mathbb Z,
000180 | \]
000181 | and its associated spectral flow is
000182 | \[
000183 | \SF(U_\Sigma)=1.
000184 | \]
000185 | Hence the primitive branch carries exactly one elementary seam winding. Write this winding number as
000186 | \[
000187 | n:=\SF(U_\Sigma)=1.
000188 | \]
000189 | 
000190 | \textbf{Step 2: family geometry and family multiplicity.}
000191 | The lifted spin action on the oriented normal two-plane closes only after $4\pi$, so one primitive
000192 | winding is realized by four quarter-turn corners. Therefore the compact family quotient carries
000193 | exactly four $\mathbb Z_4$ orbifold points, and removing them produces
000194 | \[
000195 | X_f^\circ\cong\mathbb{CP}^1\setminus\{q_1,q_2,q_3,q_4\}.
000196 | \]
000197 | The lifted seam symmetry acts by the symmetries of a square, so the marked configuration admits a
000198 | faithful $D_4$ action. By the classification of finite subgroups of $PGL_2(\mathbb C)$, every
000199 | faithful $D_4$ action on four marked points of $\mathbb{CP}^1$ is M\"obius conjugate to the square
000200 | configuration
000201 | \[
000202 | \{1,-1,i,-i\}.
000203 | \]
000204 | Hence
000205 | \[
000206 | X_f^\circ\cong\mathbb P^1\setminus\mu_4.
000207 | \]
000208 | The corner deficit count is then
000209 | \[
000210 | N_{\mathrm{fam}}
000211 | :=
000212 | \sum_{j=1}^{4}\left(1-\frac14\right)
000213 | =
000214 | 4\cdot\frac34
000215 | =
000216 | 3.
000217 | \]
000218 | Because $\chi(X_f^\circ)=2-4=-2$, one also has
000219 | \[
000220 | H_1(X_f^\circ,\mathbb Z)\cong\mathbb Z^3,
000221 | \qquad
000222 | H^1(X_f^\circ,\mathbb C)\cong\mathbb C^3.
000223 | \]
000224 | Thus the family multiplicity and the family geometry are fixed already at the discrete level.