TFPT scientific source excerpt
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000192 | \begin{definition}[Primitive admissibility complex and primitive projector]
000193 | \label{def:admissibility-complex}
000194 | \label{def:primitive-admissibility-complex}
000195 | Let the primitive Hilbert space carry only the seam factor and the gap sector accessible from
000196 | $D_{\mathrm{rel}}$ alone:
000197 | \[
000198 | \mathcal H_{\mathrm{prim}}=\mathcal H_{\Seam}\otimes\mathcal H_t,
000199 | \qquad
000200 | \mathcal H_{\mathrm{prim}}^{\mathrm{ext}}
000201 | :=
000202 | \mathcal H_{\mathrm{prim}}\otimes\Lambda(\eta_\Sigma,\eta_g),
000203 | \]
000204 | with primitive seam selector
000205 | \[
000206 | P_{\Seam,+}:=\frac{\mathbf 1+\iC}{2}.
000207 | \]
000208 | Fix $\epsilon_{\mathrm{adm}}>0$ and let $D_{\mathrm{prim}}$ be the APS-realized restriction of
000209 | the primitive sector operator to $\mathrm{Ran}(P_{\Seam,+})$, controlled by $D_{\mathrm{rel}}$
000210 | relative to the geometric anchor. Define the primitive admissibility penalties
000211 | \[
000212 | K_\Sigma:=\sqrt{\lambda_\Sigma}\,\frac{\mathbf 1-\iC}{2},
000213 | \qquad
000214 | K_g:=\sqrt{\lambda_{\mathrm{gap}}}\,
000215 | \mathbf 1_{[0,\epsilon_{\mathrm{adm}}^2/4)}\!
000216 | \bigl(P_{\Seam,+}D_{\mathrm{prim}}^\dagger D_{\mathrm{prim}}P_{\Seam,+}\bigr),
000217 | \]
000218 | the primitive admissibility differential
000219 | \[
000220 | Q_{\mathrm{prim}}
000221 | :=
000222 | \eta_\Sigma^\dagger K_\Sigma
000223 | +\eta_g^\dagger K_g,
000224 | \]
000225 | and the primitive admissibility Laplacian
000226 | \[
000227 | \Delta_{\mathrm{prim}}:=\{Q_{\mathrm{prim}},Q_{\mathrm{prim}}^\dagger\}.
000228 | \]
000229 | The primitive admissibility projector is the Hodge projector
000230 | \[
000231 | \Pprim
000232 | :=
000233 | \Pi_{\ker\Delta_{\mathrm{prim}}}.
000234 | \]
000235 | \end{definition}
000236 | 
000237 | \begin{lemma}[Primitive nilpotency and primitive Hodge projector]
000238 | \label{lem:admissibility-nilpotency}
000239 | \label{lem:hodge-admissibility-projector}
000240 | \label{lem:primitive-admissibility-nilpotency}
000241 | On the primitive scaffold reconstructed from $\Tbound$,
000242 | \[
000243 | Q_{\mathrm{prim}}^2=0,
000244 | \qquad
000245 | \Delta_{\mathrm{prim}}
000246 | =
000247 | K_\Sigma^2+K_g^2,
000248 | \qquad
000249 | \Pprim
000250 | =
000251 | \operatorname*{s-lim}_{\beta\to\infty}e^{-\beta(K_\Sigma^2+K_g^2)}.
000252 | \]
000253 | In particular $\Pprim$ depends only on $\iC$, the APS realization, and $D_{\mathrm{rel}}$, and
000254 | is therefore reconstructed from the closed datum at the primitive level. Equivalently,
000255 | \[
000256 | \Pprim=P_{\Seam,+}P_{\mathrm{prim,gap}},
000257 | \qquad
000258 | P_{\mathrm{prim,gap}}
000259 | :=
000260 | \mathbf 1_{[\epsilon_{\mathrm{adm}}^2/4,\infty)}
000261 | \!\bigl(P_{\Seam,+}D_{\mathrm{prim}}^\dagger D_{\mathrm{prim}}P_{\Seam,+}\bigr).
000262 | \]
000263 | \end{lemma}
000264 | 
000265 | \begin{proof}[Proof sketch]
000266 | The auxiliary generators $\eta_\Sigma^\dagger$ and $\eta_g^\dagger$ anticommute, so the square
000267 | of $Q_{\mathrm{prim}}$ reduces to commutators of $K_\Sigma$ and $K_g$. Both penalties act on
000268 | commuting sectors of $\mathcal H_{\mathrm{prim}}$, hence the mixed terms vanish and
000269 | $\Delta_{\mathrm{prim}}=K_\Sigma^2+K_g^2$. The zero-temperature limit projects onto the
000270 | intersection of seam-even and gapped admissible vectors, which is the factorized form recorded
000271 | above.
000272 | \end{proof}
000273 | 
000274 | \begin{definition}[Carrier polarization on the finite block]
000275 | \label{def:carrier-polarization-involution}
000276 | Let $\mathcal H_{\mathrm{car}}$ denote the finite-dimensional carrier block selected after
000277 | primitive admissibility compression. Define
000278 | \[
000279 | \epscar:=\iC|_{\mathcal H_{\mathrm{car}}},
000280 | \qquad
000281 | E_-:=\ker(\epscar+1),
000282 | \qquad
000283 | E_+:=\ker(\epscar-1).
000284 | \]
000285 | Then the finite carrier decomposition is
000286 | \[
000287 | E=E_-\oplus E_+.
000288 | \]
000289 | \end{definition}