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Soryun-IV-Quantum-Memory-Noncommuting-Spatial-Fields

\chapter*{Notation and reading map}
\addcontentsline{toc}{chapter}{Notation and reading map}
\begingroup\small
\begin{longtable}{p{.23\linewidth}p{.70\linewidth}}
\toprule Symbol & Meaning\\\midrule\endhead
$N,n,b,\ell$ & Total qubits, a generic register count, maximum cell size, and a cell's occupancy.\\
$E,R,A(t)$ & Final erased set, final survivors, and currently active set; $R\subseteq A(t)$.\\
$B,F,L$ & Bath, untouched reference, and logical input system. A spatial field denoted $B(x,t)$ is a bath operator, not the bath Hilbert space.\\
$n_j,Z_j$ & Onsite occupation projector $\ket1\bra1_j$ and Pauli phase operator.\\
$P,D,\Omega$ & A chosen encoding-sector projector, its rank, and $P/D$.\\
$K,V$ & Logical dimension and its isometric embedding. $\log_2K$ counts logical qubits.\\
$\Delta$ & A proved upper bound on the deficit $N-\log_2D$, not a measured hardware cost.\\
$\cT_R$ & Final collective symmetry average, or conditional expectation onto an invariant observable algebra.\\
$\mu,p,T$ & An exogenous loss-pattern law, independent per-site erasure probability, and storage duration. A fixed endpoint $p$ is not a fixed loss hazard.\\
$\Lambda,\Gamma,\epsilon$ & Integrated exposure or sensitivity budgets and explicit approximation errors; the local definitions fix their units.\\
$m_{N,s}$ & Number of phase operators supported on at most $s$ sites in the longitudinal construction; the noncommuting chapters define their larger module dimension separately.\\
$\norm{\cdot}_1,\norm{\cdot},\norm{\cdot}_\diamond$ & Trace, operator, and diamond norms. A channel-distance budget uses one half of the diamond norm unless stated otherwise.\\
$S,\hbin$ & Von Neumann entropy and binary entropy in bits. Natural logarithms are denoted $\ln$.\\
\bottomrule
\end{longtable}

The main operational question and result statements are in Chapter~1. Exact bath extraction is the foundation for the finite coding theorem. Spectral windows supply a geometry-free finite-exposure theorem. The operator-envelope and noncommuting-field chapters contain the broadest capacity theorem and its module proof. The longitudinal continuous-field chapter provides a sharper specialized estimate. Geometry-specific constructions are retained because they give explicit moment labels and quantitative design alternatives. Exact charge and spin calculations provide solvable benchmarks. The last chapters explain extensions, implementation descriptions, counterexamples, and the distinction between proved rates and open computational questions.

All Hilbert spaces are finite dimensional for each fixed $N$. Their dimensions may increase with $N$. Bounded, integrable Hamiltonians are assumed whenever propagators are used. No conclusion about unbounded field operators follows without an additional domain or cutoff argument. Known spatial dictionaries and known cell membership are resources; the unknown quantum bath and its temporal behavior are not supplied to the decoder.
\endgroup

	\chapter*{Notation and reading map}
	\addcontentsline{toc}{chapter}{Notation and reading map}
	\begingroup\small
	\begin{longtable}{p{.23\linewidth}p{.70\linewidth}}
	\toprule Symbol & Meaning\\\midrule\endhead
	$N,n,b,\ell$ & Total qubits, a generic register count, maximum cell size, and a cell's occupancy.\\
	$E,R,A(t)$ & Final erased set, final survivors, and currently active set; $R\subseteq A(t)$.\\
	$B,F,L$ & Bath, untouched reference, and logical input system. A spatial field denoted $B(x,t)$ is a bath operator, not the bath Hilbert space.\\
	$n_j,Z_j$ & Onsite occupation projector $\ket1\bra1_j$ and Pauli phase operator.\\
	$P,D,\Omega$ & A chosen encoding-sector projector, its rank, and $P/D$.\\
	$K,V$ & Logical dimension and its isometric embedding. $\log_2K$ counts logical qubits.\\
	$\Delta$ & A proved upper bound on the deficit $N-\log_2D$, not a measured hardware cost.\\
	$\cT_R$ & Final collective symmetry average, or conditional expectation onto an invariant observable algebra.\\
	$\mu,p,T$ & An exogenous loss-pattern law, independent per-site erasure probability, and storage duration. A fixed endpoint $p$ is not a fixed loss hazard.\\
	$\Lambda,\Gamma,\epsilon$ & Integrated exposure or sensitivity budgets and explicit approximation errors; the local definitions fix their units.\\
	$m_{N,s}$ & Number of phase operators supported on at most $s$ sites in the longitudinal construction; the noncommuting chapters define their larger module dimension separately.\\
	$\norm{\cdot}_1,\norm{\cdot},\norm{\cdot}_\diamond$ & Trace, operator, and diamond norms. A channel-distance budget uses one half of the diamond norm unless stated otherwise.\\
	$S,\hbin$ & Von Neumann entropy and binary entropy in bits. Natural logarithms are denoted $\ln$.\\
	\bottomrule
	\end{longtable}

	The main operational question and result statements are in Chapter~1. Exact bath extraction is the foundation for the finite coding theorem. Spectral windows supply a geometry-free finite-exposure theorem. The operator-envelope and noncommuting-field chapters contain the broadest capacity theorem and its module proof. The longitudinal continuous-field chapter provides a sharper specialized estimate. Geometry-specific constructions are retained because they give explicit moment labels and quantitative design alternatives. Exact charge and spin calculations provide solvable benchmarks. The last chapters explain extensions, implementation descriptions, counterexamples, and the distinction between proved rates and open computational questions.

	All Hilbert spaces are finite dimensional for each fixed $N$. Their dimensions may increase with $N$. Bounded, integrable Hamiltonians are assumed whenever propagators are used. No conclusion about unbounded field operators follows without an additional domain or cutoff argument. Known spatial dictionaries and known cell membership are resources; the unknown quantum bath and its temporal behavior are not supplied to the decoder.
	\endgroup