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Original repository path: documents/articles--2026-08-30--mmst_charged_scaling_limit_en/sections/002-section-l1-uniform-energy-bounds.tex
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000300 | $\max_{N,m}2|m|g_N(m)=1.029399027646<K_G$, verifies
000301 | $\delta_{\rm bulk}=1$ and the unit edge slope, and rejects both a
000302 | gapless transverse mutant (whose ratio grows from $64.829$ to $384.137$)
000303 | and the false symbol constant $\omega_N(m)\ge|m|$.
000304 | 
000305 | \section{L2: strong convergence and local algebras}
000306 | \label{sec:L2}
000307 | 
000308 | \subsection{Explicit dyadic mode and Fock isometries}
000309 | 
000310 | Let
000311 | \[
000312 |  \mathcal K_N
000313 |  =\ell^2\!\left(\{m\in\Z:0<|m|\le N/2\}\right)\otimes\C^8
000314 | \tag{5.1}\label{eq:oneparticleboson}
000315 | \]
000316 | with the single Nyquist representative convention of
000317 | $\mathbb M_N$.  Define
000318 | \[
000319 |  j_{N,2N}(e_m\otimes e_a)=e_m\otimes e_a,
000320 | \qquad m\in\mathbb M_N,\quad 1\le a\le8.
000321 | \tag{5.2}\label{eq:modeisometry}
000322 | \]
000323 | Thus $j_{N,2N}$ is the zero-extension/Shannon embedding of the old
000324 | Fourier modes into the refined lattice.  It is an isometry and
000325 | \[
000326 |  j_{2N,4N}j_{N,2N}=j_{N,4N}.
000327 | \tag{5.3}\label{eq:modecompat}
000328 | \]
000329 | On the charge-lattice sum of symmetric bosonic Fock spaces set
000330 | \[
000331 |  V_{N,2N}
000332 |  :=\id_{\ell^2(L_{\rm charge})}\otimes\Gamma_s(j_{N,2N}).
000333 | \tag{5.4}\label{eq:Fockisometry}
000334 | \]
000335 | Second quantization preserves isometries and composition, so
000336 | $V_{N,2N}^*V_{N,2N}=1$ and
000337 | $V_{2N,4N}V_{N,2N}=V_{N,4N}$.  The charge-lattice identity makes the
000338 | same statement valid in every one of the four $\lambda$ grades.
000339 |