TFPT scientific source excerpt
Original repository path: documents/articles--2026-08-30--mmst_charged_scaling_limit_en/sections/001-section-verdict-and-logical-form.tex
Original file SHA-256: e75e10cc33137187e6c494e8486ffdec91e43d6b636d434ce0735d97e56a9c7f
Inclusive original lines: 183-227
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.

000183 | \subsection{The finite charged field}
000184 | 
000185 | Let $\phi_N=(\phi_N^1,\ldots,\phi_N^8)$ be the bosonic current obtained
000186 | from the sixteen Majorana copies by edge-projected collar bosonization,
000187 | with zero mode fixed by the $\Z_4$ charge sector.  Each complex-fermion
000188 | pair gives one boson.  The longitudinal derivative uses the frozen
000189 | lattice symbol
000190 | \[
000191 |  \omega_N(m)=\frac{N}{\pi}\left|\sin\frac{\pi m}{N}\right|,
000192 |  \qquad 0<|m|\le\frac N2.
000193 | \tag{2.9}\label{eq:latticesymbol}
000194 | \]
000195 | The edge projection is the analytic top (or bottom) eigenbranch through
000196 | the zero-energy crossing; massive bulk-complement modes are not included
000197 | in $\phi_N$.  For a fixed normalized
000198 | two-cocycle $\varepsilon_\lambda$, define on the finite-energy vectors
000199 | \[
000200 |  \Psi_{\lambda,N}(f)
000201 |  =\sum_{m}\widehat f_m\,
000202 |    \varepsilon_\lambda
000203 |    :\!\exp(i\lambda\cdot\phi_N)\!:_m .
000204 |  \tag{2.10}\label{eq:chargedfield}
000205 | \]
000206 | Normal ordering is with respect to $C_N^{(0)}$.  Formula
000207 | \eqref{eq:chargedfield} is meaningful at every finite $N$.  The theorem
000208 | does not follow merely from that fact: the normal-ordering constant and
000209 | the graph norm of the exponential can grow with $N$.
000210 | 
000211 | \begin{definition}[Scaling convergence used below]
000212 | Let $V_N:\Hh_N\to\Hh_\infty$ be the canonical isometries of the chosen
000213 | operator-algebraic renormalization system.  A sequence of closable
000214 | operators $X_N$ converges on the finite-energy core
000215 | \[
000216 |  \Dfin=\bigcup_{E<\infty}\id_{[0,E]}(L_0)\Hh_\infty
000217 |  \tag{2.11}\label{eq:core}
000218 | \]
000219 | if for every $\xi\in\Dfin$, represented at sufficiently large scale,
000220 | \[
000221 |  V_NX_NV_N^*\xi\longrightarrow X\xi,\qquad
000222 |  V_NX_N^*V_N^*\xi\longrightarrow X^*\xi.
000223 |  \tag{2.12}\label{eq:graphconv}
000224 | \]
000225 | The adjoint clause is included because bare pointwise convergence of
000226 | unbounded fields is too weak to determine local von Neumann algebras.
000227 | \end{definition}