Realization Calculus

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1 \section{Introduction}\label{strong:section:001}\label{strong:sec:introduction}
2 
3 Quantum mechanics repeatedly uses the same operation. A Hilbert space is split into sectors, an operator is resolved between those sectors, and the pieces are propagated or recombined. Symmetry sectors, eigenspaces, syndrome spaces, adiabatic subspaces, and retained model spaces are familiar examples. This article treats them as instances of one finite packet calculus.
4 
5 The elementary coordinate object is a complete family of orthogonal projectors
6 \[
7 P_aP_b=\delta_{ab}P_a,
8 \qquad
9 \sum_{a\in I}P_a=I.
10 \]
11 Every vector has packet coordinates \(P_a\psi\). Every operator has transfer blocks
12 \[
13 A_{ba}=P_bAP_a.
14 \]
15 These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.
16 
17 Quantum information and operator theory in finite dimensions provide the standard background \cite{NielsenChuang2010,Watrous2018,Bhatia1997}. Finite group projectors come from character theory \cite{Serre1977,FultonHarris1991}. Moving spectral subspaces and geometric phase originate in the work of Kato, Berry, Wilczek, and Zee \cite{Kato1950,Kato1995,Berry1984,WilczekZee1984}. Completely positive maps and quantum dynamical semigroups are treated in \cite{Stinespring1955,Choi1975,Kraus1983,Lindblad1976}. Exact error correction is governed by the Knill--Laflamme condition \cite{KnillLaflamme1997,SchumacherNielsen1996,BenyKempfKribs2007}. Projection methods and hidden state memory are linked by the Feshbach, Nakajima, and Zwanzig constructions \cite{Feshbach1958,Nakajima1958,Zwanzig1960,BravyiDiVincenzoLoss2011}.
18 
19 The first part begins with quantum seeds. A state is cyclic only relative to a declared observable algebra. For an algebra in multiplicity form, the local orbit dimension is controlled by the rank of a coefficient matrix. This recovers the relation between Schmidt rank and the dimension reached by local operations. The finite group section then constructs central isotypic projectors. Cyclic symmetry gives a sharper charge selection rule. An operator with charge \(q\) can move a state only from packet \(j\) to packet \(j+q\).
20 
21 A circuit is resolved by inserting the identity between successive gates. This produces a sum over packet paths. The target probability contains diagonal path terms and interference between distinct paths. Symmetry can remove entire path families before any matrix multiplication is performed. Unitarity gives exact row and column relations among the transfer blocks.
22 
23 The moving packet section treats time dependent projectors. The canonical generator
24 \[
25 K(t)=\sum_a\dot P_a(t)P_a(t)
26 \]
27 is skew adjoint and transports every packet. In the moving frame, the physical Hamiltonian acquires the geometric term \(-\ii W(t)^*\dot W(t)\). The leakage estimate must account for the motion of the packet itself. The physical coupling \(QHP\) is therefore replaced by
28 \[
29 Q(t)\bigl(H(t)-\ii K(t)\bigr)P(t).
30 \]
31 This is the exact finite distinction between following a moving subspace and remaining in a fixed one.
32 
33 Density matrices require separate packet labels for their output and input coordinates. The superoperators
34 \[
35 \Pcal_{ab}(X)=P_aXP_b
36 \]
37 form a complete orthogonal packet system on Hilbert--Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Choi matrix carries the equivalent symmetry under \(U\otimes\overline U\).
38 
39 Error correction is expressed through compression to a code. Let \(P\) be the code projector. The class of \(PAP\) modulo scalar multiples of \(P\) measures the obstruction to making \(A\) undetectable within the code. The Knill--Laflamme criterion says that this obstruction vanishes for every product \(E_a^*E_b\) from the error family. Diagonalizing the resulting positive coefficient matrix produces orthogonal syndrome packets and an explicit recovery on the error span.
40 
41 The effective dynamics section returns to ordinary Hilbert space packets. Schur complementation gives the exact retained resolvent and a frequency dependent self energy. The same hidden block gives a causal memory kernel in time. For a self adjoint hidden Hamiltonian, the self energy has a fixed sign in the upper half plane, and the time kernel is positive definite. Uniform spectral separation gives direct bounds that can be carried through a refinement tower.
42 
43 The geometric part develops two finite constructions. A strictly positive contraction gives a reflection positive Hankel kernel, and its logarithm reconstructs a nonnegative Hamiltonian. A finite Laplacian gives a heat trace from which the spectral dimension is defined at each diffusion scale. The article proves convergence from periodic lattices to the continuum torus at every fixed positive diffusion time. Poisson summation then determines the continuum dimension as diffusion time tends to zero.
44 
45 Constraint closure needs a different obstruction. If \(C_a\) are self adjoint constraints and \(P_{\phys}\) projects onto their common kernel, then
46 \[
47 P_{\phys}[C_a,C_b]P_{\phys}=0
48 \]
49 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of \([C_a,C_b]\) in the quotient of \(\End(\Hcal)\) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.
50 
51 The last part gives two theorems for towers of bounded Hamiltonians. Exactly compatible finite Hamiltonians with a uniform norm bound define a bounded self adjoint operator on the Hilbert direct limit. When successive bounded self adjoint Hamiltonians on a common Hilbert space have norm summable differences, they converge to a bounded self adjoint limit. Their unitary evolutions, and their Euclidean evolutions under the stated positivity assumption, converge uniformly on bounded time intervals. Reflection positivity and closure with fixed structure coefficients pass to the limit under the stated convergence assumptions.
52 
53 These results do not construct an interacting continuum quantum field theory. They do not prove anomaly freedom on an unbounded common domain, Osterwalder--Schrader reconstruction for a continuum measure, an Einstein infrared action, or a positive massless spin two sector. Each of those conclusions needs hypotheses absent from the finite calculations.
54 
55 \begin{figure}[t]
56 \centering
57 \includegraphics[width=\textwidth]{01_quantum_architecture.pdf}
58 \caption{The finite mathematical layers developed in the article and the separate limit criteria needed at the continuum boundary.}
59 \label{strong:fig:architecture}
60 \end{figure}
61 
62 \begin{table}[t]
63 \centering
64 \small
65 \begin{tabular}{>{\raggedright\arraybackslash}p{.20\textwidth}>{\raggedright\arraybackslash}p{.31\textwidth}>{\raggedright\arraybackslash}p{.39\textwidth}}
66 \toprule
67 Object & Packet coordinate & Exact structural condition \\
68 \midrule
69 State & \(P_a\psi\) & completeness of the projector family \\
70 Operator & \(P_bAP_a\) & block multiplication \\
71 Circuit & intermediate packet path & insertion of complete resolutions \\
72 Density matrix & \(P_a\rho P_b\) & two sided operator packets \\
73 Channel & superoperator block & complete positivity and covariance when declared \\
74 Code & syndrome packet & scalar compression of error products \\
75 Effective model & Feshbach map & invertibility of the hidden resolvent \\
76 Euclidean model & positive transfer matrix & positivity and self adjointness \\
77 Constraint system & quotient obstruction & commutator closure modulo the constraint span \\
78 Continuum tower & direct or norm limit & compatibility plus explicit convergence \\
79 \bottomrule
80 \end{tabular}
81 \caption{The packet coordinates used in the article.}
82 \label{strong:tab:dictionary}
83 \end{table}
84 
85 \section{Quantum seeds and observable algebras}\label{strong:section:002}\label{strong:sec:seeds}
86 
87 \begin{definition}[Finite quantum seed system]\label{strong:definition:001}\label{strong:def:quantum_seed}
88 A finite quantum seed system is a triple
89 \[
90 (\Hcal,\Acal,S),
91 \]
92 where \(\Hcal\) is a finite dimensional Hilbert space, \(\Acal\subseteq\End(\Hcal)\) is a unital star subalgebra, and \(S\subseteq\Hcal\) is a finite set. Its generated hull is
93 \[
94 \Acal S
95 =
96 \operatorname{span}\{A\psi:A\in\Acal,\ \psi\in S\}.
97 \]
98 The seed set is cyclic when \(\Acal S=\Hcal\). The quantum seed rank is the least cardinality of a cyclic seed set.
99 \end{definition}
100 
101 The algebra is part of the data. A vector may be cyclic for \(\End(\Hcal)\) and fail to be cyclic for a restricted observable or gate algebra.
102 
103 Let
104 \[
105 \Acal
106 \cong
107 \bigoplus_{r=1}^{s}
108 \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
109 \]
110 act on
111 \[
112 \Hcal
113 \cong
114 \bigoplus_{r=1}^{s}
115 \bigl(\CC^{d_r}\otimes\CC^{m_r}\bigr).
116 \]
117 Assume \(d_r,m_r\ge1\) for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a \(d_r\) by \(m_r\) coefficient matrix.
118 
119 \begin{theorem}[Orbit dimension of one seed]\label{strong:theorem:001}\label{strong:thm:orbit_dimension}
120 For integers \(d\ge1\), \(m\ge0\), and the action of \(M_d(\CC)\otimes I_m\) on \(\CC^d\otimes\CC^m\), let \(C_\psi\) be the coefficient matrix of \(\psi\). Then
121 \[
122 \dim\operatorname{span}\{(A\otimes I_m)\psi:A\in M_d(\CC)\}
123 =
124  d\rank C_\psi.
125 \]
126 The vector is cyclic exactly when \(\rank C_\psi=m\).
127 \end{theorem}
128 
129 \begin{proof}\phantomsection\label{strong:proof:001}
130 Under the coefficient matrix identification, \((A\otimes I_m)\psi\) corresponds to \(AC_\psi\). Every row of \(AC_\psi\) lies in the row space of \(C_\psi\). Conversely, a matrix whose rows lie in that row space equals \(AC_\psi\) for a suitable \(A\). There are \(d\) independent row choices in a space of dimension \(\rank C_\psi\). The orbit dimension is therefore \(d\rank C_\psi\). It equals \(dm\) exactly when \(C_\psi\) has full column rank.
131 \phantomsection\label{strong:proof:001:end}\end{proof}
132 
133 \begin{theorem}[Quantum seed rank]\label{strong:theorem:002}\label{strong:thm:quantum_seed_rank}
134 For the displayed multiplicity decomposition,
135 \[
136 \operatorname{srank}_{\Acal}(\Hcal)
137 =
138 \max_{1\le r\le s}
139 \left\lceil\frac{m_r}{d_r}\right\rceil.
140 \]
141 \end{theorem}
142 
143 \begin{proof}\phantomsection\label{strong:proof:002}
144 A single seed contributes a row space of dimension at most \(d_r\) to the \(r\)th multiplicity factor. Hence \(k\) seeds cover at most \(kd_r\) multiplicity directions, which gives the lower bound.
145 
146 Set
147 \[
148 k=\max_r\left\lceil\frac{m_r}{d_r}\right\rceil.
149 \]
150 For each \(r\), partition a basis of \(\CC^{m_r}\) into at most \(k\) groups of size at most \(d_r\). Choose the coefficient matrix of the \(j\)th seed so that its row space spans the \(j\)th group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the \(k\) seeds generate every summand.
151 \phantomsection\label{strong:proof:002:end}\end{proof}
152 
153 \begin{corollary}[Local orbit and Schmidt rank]\label{strong:corollary:001}\label{strong:cor:schmidt_orbit}
154 Let \(\psi\in\Hcal_A\otimes\Hcal_B\), let \(d_A=\dim\Hcal_A\ge1\), and let \(r\) be the Schmidt rank of \(\psi\). Then
155 \[
156 \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in\End(\Hcal_A)\}
157 =
158  d_A r.
159 \]
160 Local operators on \(A\) generate the whole bipartite Hilbert space exactly when the reduced state on \(B\) has full support.
161 \end{corollary}
162 
163 \begin{proof}\phantomsection\label{strong:proof:003}
164 The Schmidt rank is the rank of the coefficient matrix in any product basis. Substituting that rank into \cref{strong:thm:orbit_dimension} gives the stated orbit dimension.
165 \phantomsection\label{strong:proof:003:end}\end{proof}
166 
167 \begin{figure}[t]
168 \centering
169 \includegraphics[width=.68\textwidth]{02_seed_rank.pdf}
170 \caption{The dimension reached by all local operators is linear in Schmidt rank.}
171 \label{strong:fig:seed_rank}
172 \end{figure}
173 
174 \begin{proposition}[Computational address orbit]\label{strong:proposition:001}\label{strong:prop:computational_orbit}
175 For \(n\) qubits, let
176 \[
177 \psi_0=\ket{0}^{\otimes n}
178 \]
179 and let \(X_j\) be the Pauli \(X\) operator on the \(j\)th qubit. Then
180 \[
181 \left\{
182 X_1^{a_1}\cdots X_n^{a_n}\psi_0:
183 (a_1,\ldots,a_n)\in\{0,1\}^n
184 \right\}
185 \]
186 is the computational basis.
187 \end{proposition}
188 
189 \begin{proof}\phantomsection\label{strong:proof:004}
190 The addressed vector is \(\ket{a_1\cdots a_n}\). Distinct binary addresses produce the \(2^n\) distinct basis vectors.
191 \phantomsection\label{strong:proof:004:end}\end{proof}
192 
193 \begin{remark}\label{strong:remark:001}\label{strong:rem:seed_scope}
194 The seed rank records generation by a declared algebra. It does not measure circuit depth, locality cost, control complexity, or physical reachability under a restricted time dependent Hamiltonian.
195 \end{remark}
196 
197 \section{Symmetry packets and selection rules}\label{strong:section:003}\label{strong:sec:symmetry}
198 
199 Let \(G\) be a finite group and let
200 \[
201 U:G\longrightarrow\Ucal(\Hcal)
202 \]
203 be a unitary representation. Write \(\widehat G\) for the set of irreducible unitary representations. For \(\lambda\in\widehat G\), let \(d_\lambda\) and \(\chi_\lambda\) denote its dimension and character.
204 
205 \begin{theorem}[Finite group isotypic projectors]\label{strong:theorem:003}\label{strong:thm:isotypic_projectors}
206 The operators
207 \[
208 P_\lambda
209 =
210 \frac{d_\lambda}{\abs{G}}
211 \sum_{g\in G}
212 \overline{\chi_\lambda(g)}U(g)
213 \]
214 are orthogonal projectors. They satisfy
215 \[
216 P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda,
217 \qquad
218 \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
219 \]
220 The range of \(P_\lambda\) is the \(\lambda\) isotypic component of \(\Hcal\).
221 \end{theorem}
222 
223 \begin{proof}\phantomsection\label{strong:proof:005}
224 Complete reducibility gives a unitary decomposition
225 \[
226 \Hcal
227 \cong
228 \bigoplus_{\mu\in\widehat G}
229 V_\mu\otimes M_\mu,
230 \qquad
231 U(g)
232 \cong
233 \bigoplus_\mu
234 \pi_\mu(g)\otimes I_{M_\mu}.
235 \]
236 On the \(\mu\) summand, the displayed average is central in the irreducible representation \(\pi_\mu\), so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when \(\mu=\lambda\) and zero otherwise. The asserted identities follow on the direct sum.
237 \phantomsection\label{strong:proof:005:end}\end{proof}
238 
239 The central projectors separate irreducible types but do not choose coordinates inside the representation factor. Matrix units provide that finer resolution.
240 
241 \begin{theorem}[Representation matrix units]\label{strong:theorem:004}\label{strong:thm:representation_matrix_units}
242 Fix an irreducible unitary representation \(\pi_\lambda:G\to U(d_\lambda)\) and an orthonormal basis of its representation space. Define
243 \[
244 E_{ij}^{\lambda}
245 =
246 \frac{d_\lambda}{\abs{G}}
247 \sum_{g\in G}
248 \bigl[\pi_\lambda(g^{-1})\bigr]_{ji}U(g).
249 \]
250 Then
251 \[
252 E_{ij}^{\lambda}E_{k\ell}^{\mu}
253 =
254 \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda},
255 \qquad
256 \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
257 \]
258 and
259 \[
260 P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}.
261 \]
262 \end{theorem}
263 
264 \begin{proof}\phantomsection\label{strong:proof:006}
265 Apply the matrix coefficient form of Schur orthogonality to the two group averages. It gives the matrix unit multiplication law. Unitarity gives the adjoint relation after replacing \(g\) by \(g^{-1}\). Summing the diagonal coefficients gives the character average in \cref{strong:thm:isotypic_projectors}.
266 \phantomsection\label{strong:proof:006:end}\end{proof}
267 
268 \subsection{Cyclic charge packets}\label{strong:subsec:cyclic_charge}
269 
270 Let \(N\ge1\) be an integer, \(S\) unitary with \(S^N=I\), and \(\omega=\e^{2\pi\ii/N}\).
271 
272 \begin{corollary}[Cyclic charge projectors]\label{strong:corollary:002}\label{strong:cor:cyclic_projectors}
273 The operators
274 \[
275 P_j
276 =
277 \frac1N
278 \sum_{r=0}^{N-1}
279 \omega^{-jr}S^r,
280 \qquad
281 j\in\ZZ/N\ZZ,
282 \]
283 form a complete orthogonal packet system and satisfy
284 \[
285 SP_j=\omega^jP_j.
286 \]
287 \end{corollary}
288 
289 \begin{proof}\phantomsection\label{strong:proof:007}
290 Apply \cref{strong:thm:isotypic_projectors} to the cyclic group. Its irreducible representations are the one dimensional characters \(r\mapsto\omega^{jr}\).
291 \phantomsection\label{strong:proof:007:end}\end{proof}
292 
293 \begin{definition}[Operator charge]\label{strong:definition:002}\label{strong:def:operator_charge}
294 An operator \(A\in\End(\Hcal)\) has cyclic charge \(q\in\ZZ/N\ZZ\) when
295 \[
296 SAS^*=\omega^qA.
297 \]
298 \end{definition}
299 
300 \begin{theorem}[Charge selection rule]\label{strong:theorem:005}\label{strong:thm:charge_selection}
301 If \(A\) has charge \(q\), then
302 \[
303 P_kAP_j=0
304 \]
305 unless \(k=j+q\) in \(\ZZ/N\ZZ\). In fact,
306 \[
307 AP_j=P_{j+q}AP_j.
308 \]
309 \end{theorem}
310 
311 \begin{proof}\phantomsection\label{strong:proof:008}
312 For \(x\in P_j\Hcal\),
313 \[
314 S(Ax)
315 =(SAS^*)Sx
316 =\omega^{q+j}Ax.
317 \]
318 Thus \(Ax\in P_{j+q}\Hcal\). The block statement follows.
319 \phantomsection\label{strong:proof:008:end}\end{proof}
320 
321 \begin{corollary}[Charge addition]\label{strong:corollary:003}\label{strong:cor:charge_addition}
322 If \(A\) and \(B\) have charges \(q\) and \(r\), then \(AB\) has charge \(q+r\). The adjoint \(A^*\) has charge \(-q\).
323 \end{corollary}
324 
325 \begin{proof}\phantomsection\label{strong:proof:009}
326 Conjugation by \(S\) is a star automorphism. Applying it to \(AB\) adds the two charges, while applying it to \(A^*\) changes the sign of the charge.
327 \phantomsection\label{strong:proof:009:end}\end{proof}
328 
329 \begin{figure}[t]
330 \centering
331 \includegraphics[width=.82\textwidth]{03_symmetry_selection.pdf}
332 \caption{An operator of charge one moves each cyclic packet to the next packet and has no other transfer blocks.}
333 \label{strong:fig:symmetry_selection}
334 \end{figure}
335 
336 \subsection{Tensor refinement}\label{strong:subsec:tensor_refinement}
337 
338 Suppose the \(a\)th tensor factor carries a cyclic packet family
339 \[
340 \bigl(P_j^{(a)}\bigr)_{j\in\ZZ/N\ZZ}.
341 \]
342 The fine tensor packets are
343 \[
344 P_{\bm j}^{\rm fine}
345 =
346 P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
347 \]
348 
349 \begin{theorem}[Tensor charge refinement]\label{strong:theorem:006}\label{strong:thm:tensor_refinement}
350 For \(r\in\ZZ/N\ZZ\), define
351 \[
352 P_r^{\rm total}
353 =
354 \sum_{j_1+\cdots+j_m=r}
355 P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
356 \]
357 Then the \(P_r^{\rm total}\) form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.
358 \end{theorem}
359 
360 \begin{proof}\phantomsection\label{strong:proof:010}
361 The fine tensor projectors are mutually orthogonal and sum to the identity. The total residue classes partition the fine address set. Summing the fine projectors over each class gives the result.
362 \phantomsection\label{strong:proof:010:end}\end{proof}
363 
364 \begin{corollary}[Tensor selection rule]\label{strong:corollary:004}\label{strong:cor:tensor_selection}
365 If \(A_a\) has local charge \(q_a\), then
366 \[
367 A_1\otimes\cdots\otimes A_m
368 \]
369 has total charge \(q_1+\cdots+q_m\).
370 \end{corollary}
371 
372 \begin{proof}\phantomsection\label{strong:proof:011}
373 Conjugate the tensor product by the total cyclic symmetry and multiply the local phase factors.
374 \phantomsection\label{strong:proof:011:end}\end{proof}
375 
376 \begin{remark}\label{strong:remark:002}\label{strong:rem:nonabelian_fusion}
377 For a nonabelian group, tensor products are refined by irreducible fusion multiplicities rather than residue addition. The central isotypic projectors remain exact, but a complete tensor refinement also requires a choice of Clebsch--Gordan intertwiners.
378 \end{remark}
379 
380 \section{Quantum circuits as packet paths}\label{strong:section:004}\label{strong:sec:circuits}
381 
382 Let
383 \[
384 U=U_mU_{m-1}\cdots U_1
385 \]
386 be a finite circuit with \(m\ge1\), and let \((P_a)_{a\in I}\) be a complete orthogonal packet family. For the empty circuit, \(U=I\) and its transfer block is \(P_bP_a=\delta_{ba}P_a\), whose endpoint sum reconstructs \(I\).
387 
388 \begin{theorem}[Exact circuit path expansion]\label{strong:theorem:007}\label{strong:thm:circuit_path}
389 For source packet \(a_0\) and target packet \(a_m\),
390 \[
391 P_{a_m}UP_{a_0}
392 =
393 \sum_{a_1,\ldots,a_{m-1}}
394 P_{a_m}U_mP_{a_{m-1}}
395 \cdots
396 P_{a_1}U_1P_{a_0}.
397 \]
398 Summing over \(a_0\) and \(a_m\) reconstructs \(U\).
399 \end{theorem}
400 
401 \begin{proof}\phantomsection\label{strong:proof:012}
402 Insert \(I=\sum_aP_a\) between every adjacent pair of gates and distribute the finite sums. Completeness at the two endpoints gives the full operator.
403 \phantomsection\label{strong:proof:012:end}\end{proof}
404 
405 For a packet path
406 \[
407 \gamma=(a_0,a_1,\ldots,a_m),
408 \]
409 write
410 \[
411 U_\gamma
412 =
413 P_{a_m}U_mP_{a_{m-1}}
414 \cdots
415 P_{a_1}U_1P_{a_0}.
416 \]
417 
418 \begin{theorem}[Packet unitarity identities]\label{strong:theorem:008}\label{strong:thm:packet_unitarity}
419 If \(U\) is unitary and \(U_{ba}=P_bUP_a\), then
420 \[
421 \sum_bU_{ba}^*U_{bc}=\delta_{ac}P_a
422 \]
423 and
424 \[
425 \sum_aU_{ba}U_{da}^*=\delta_{bd}P_b.
426 \]
427 \end{theorem}
428 
429 \begin{proof}\phantomsection\label{strong:proof:013}
430 Use completeness and unitarity:
431 \[
432 \sum_bU_{ba}^*U_{bc}
433 =
434 P_aU^*\left(\sum_bP_b\right)UP_c
435 =P_aP_c.
436 \]
437 Summing instead over the source packets and using unitarity in the reverse order gives the second identity.
438 \phantomsection\label{strong:proof:013:end}\end{proof}
439 
440 \begin{corollary}[Probability conservation from one packet]\label{strong:corollary:005}\label{strong:cor:packet_probability}
441 If \(\psi\in P_a\Hcal\), then
442 \[
443 \sum_b\norm{P_bU\psi}^2=\norm{\psi}^2.
444 \]
445 \end{corollary}
446 
447 \begin{proof}\phantomsection\label{strong:proof:014}
448 Take the inner product of the first identity in \cref{strong:thm:packet_unitarity} with \(\psi\) and set \(c=a\).
449 \phantomsection\label{strong:proof:014:end}\end{proof}
450 
451 \begin{theorem}[Path interference formula]\label{strong:theorem:009}\label{strong:thm:path_interference}
452 Fix a source vector \(\psi\in P_{a_0}\Hcal\) and a target packet \(b\). Let \(\Gamma_b\) be the set of packet paths from \(a_0\) to \(b\), and set
453 \[
454 \eta_\gamma=U_\gamma\psi.
455 \]
456 Choose any total order on this finite path set. Then
457 \[
458 \norm{P_bU\psi}^2
459 =
460 \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2
461 +
462 2\operatorname{Re}
463 \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}}
464 \ip{\eta_\gamma}{\eta_{\gamma'}}.
465 \]
466 Consequently,
467 \[
468 \norm{P_bU\psi}
469 \leq
470 \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}.
471 \]
472 If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
473 \end{theorem}
474 
475 \begin{proof}\phantomsection\label{strong:proof:015}
476 The path expansion gives
477 \[
478 P_bU\psi=\sum_{\gamma\in\Gamma_b}\eta_\gamma.
479 \]
480 Expanding the squared norm and grouping conjugate cross terms gives the interference formula. The triangle inequality gives the norm bound. When the path outputs are orthogonal, every cross term vanishes and only the individual path probabilities remain.
481 \phantomsection\label{strong:proof:015:end}\end{proof}
482 
483 \begin{proposition}[Charge pruning of circuit paths]\label{strong:proposition:002}\label{strong:prop:charge_pruning}
484 Suppose every gate has a finite charge decomposition
485 \[
486 U_r=\sum_qU_r^{(q)},
487 \qquad
488 SU_r^{(q)}S^*=\omega^qU_r^{(q)}.
489 \]
490 A term with gate charges \(q_1,\ldots,q_m\) can contribute from source charge \(a_0\) only to target charge
491 \[
492 a_m=a_0+q_1+\cdots+q_m.
493 \]
494 Every packet path violating one of the intermediate charge updates vanishes.
495 \end{proposition}
496 
497 \begin{proof}\phantomsection\label{strong:proof:016}
498 Apply \cref{strong:thm:charge_selection} after each gate component. The packet address is forced at every step.
499 \phantomsection\label{strong:proof:016:end}\end{proof}
500 
501 \begin{figure}[t]
502 \centering
503 \includegraphics[width=.88\textwidth]{04_circuit_paths.pdf}
504 \caption{A circuit block is the coherent sum of its allowed packet paths. Selection rules remove forbidden paths before amplitudes are evaluated.}
505 \label{strong:fig:circuit_paths}
506 \end{figure}
507 
508 \begin{example}[Bell circuit]\label{strong:example:001}\label{strong:ex:bell_circuit}
509 Starting from \(\ket{00}\), apply a Hadamard gate to the first qubit and then a controlled NOT gate. The final state is
510 \[
511 \frac{\ket{00}+\ket{11}}{\sqrt2}.
512 \]
513 It lies in the even parity packet. The intermediate state has one even and one odd computational contribution, so the final even amplitude is reconstructed by more than one packet path.
514 \end{example}
515 
516 \section{Symmetry resolved dynamics and leakage}\label{strong:section:005}\label{strong:sec:dynamics}
517 
518 Let \(H=H^*\) and let \((P_a)_{a\in I}\) be a complete orthogonal packet system.
519 
520 \begin{theorem}[Exact invariant packets]\label{strong:theorem:010}\label{strong:thm:invariant_packets}
521 The following conditions are equivalent:
522 \begin{enumerate}
523 \item The Hamiltonian commutes with each projector: \([H,P_a]=0\) for every \(a\).
524 \item Transfer blocks between distinct packets vanish: \(P_bHP_a=0\) whenever \(a\neq b\).
525 \item Every packet \(P_a\Hcal\) is invariant under \(\e^{-\ii tH}\) for every real \(t\).
526 \end{enumerate}
527 When they hold,
528 \[
529 H=\sum_aP_aHP_a
530 \]
531 and
532 \[
533 \e^{-\ii tH}=\sum_aP_a\e^{-\ii tH}P_a.
534 \]
535 \end{theorem}
536 
537 \begin{proof}\phantomsection\label{strong:proof:017}
538 Commutation is equivalent to vanishing of every off diagonal block. The same block decomposition passes to polynomials in \(H\), then to the exponential by norm convergence. Invariance of every packet implies invariance under both \(\e^{-\ii tH}\) and its adjoint. Differentiation at \(t=0\) gives commutation with \(H\).
539 \phantomsection\label{strong:proof:017:end}\end{proof}
540 
541 Fix one packet projector \(P\) and set \(Q=I-P\).
542 
543 \begin{lemma}[Off diagonal norm]\label{strong:lemma:001}\label{strong:lem:off_diagonal_norm}
544 For self adjoint \(H\),
545 \[
546 \norm{[H,P]}=\norm{QHP}=\norm{PHQ}.
547 \]
548 \end{lemma}
549 
550 \begin{proof}\phantomsection\label{strong:proof:018}
551 Relative to \(P\Hcal\oplus Q\Hcal\),
552 \[
553 [H,P]
554 =
555 \begin{pmatrix}
556 0&-PHQ\\
557 QHP&0
558 \end{pmatrix}.
559 \]
560 Its square has diagonal blocks \(-PHQHP\) and \(-QHPHQ\). The norm is therefore the largest singular value of \(QHP\).
561 \phantomsection\label{strong:proof:018:end}\end{proof}
562 
563 \begin{theorem}[Fixed packet leakage]\label{strong:theorem:011}\label{strong:thm:fixed_leakage}
564 For every real \(t\),
565 \[
566 \norm{Q\e^{-\ii tH}P}
567 \leq
568 \abs{t}\norm{QHP}
569 =
570 \abs{t}\norm{[H,P]}.
571 \]
572 \end{theorem}
573 
574 \begin{proof}\phantomsection\label{strong:proof:019}
575 Let
576 \[
577 H_{\rm d}=PHP+QHQ,
578 \qquad
579 H_{\rm o}=PHQ+QHP.
580 \]
581 The propagator generated by \(H_{\rm d}\) preserves \(P\Hcal\) and \(Q\Hcal\). Duhamel's formula relative to \(H_{\rm d}\) gives
582 \[
583 Q\e^{-\ii tH}P
584 =
585 -\ii
586 \int_0^t
587 Q\e^{-\ii(t-s)H}H_{\rm o}\e^{-\ii sH_{\rm d}}P
588 \dd s.
589 \]
590 Since \(H_{\rm o}P=QHP\) and both propagators are unitary, the norm is at most \(\abs{t}\norm{QHP}\). Use \cref{strong:lem:off_diagonal_norm} for the last equality.
591 \phantomsection\label{strong:proof:019:end}\end{proof}
592 
593 \begin{figure}[t]
594 \centering
595 \includegraphics[width=.72\textwidth]{05_leakage_bound.pdf}
596 \caption{Exact transition amplitude in a two packet model and the linear commutator bound.}
597 \label{strong:fig:leakage}
598 \end{figure}
599 
600 \begin{corollary}[Leakage probability]\label{strong:corollary:006}\label{strong:cor:leakage_probability}
601 If \(\psi\in P\Hcal\) is a unit vector, then
602 \[
603 \norm{Q\e^{-\ii tH}\psi}^2
604 \leq
605  t^2\norm{QHP}^2.
606 \]
607 \end{corollary}
608 
609 \begin{proof}\phantomsection\label{strong:proof:020}
610 Apply \cref{strong:thm:fixed_leakage} to the unit vector, then square the resulting nonnegative bound.
611 \phantomsection\label{strong:proof:020:end}\end{proof}
612 
613 \section{Moving packets and geometric transport}\label{strong:section:006}\label{strong:sec:moving_packets}
614 
615 Let \((P_a(t))_{a\in I}\) be a twice continuously differentiable family of complete orthogonal packet systems on a finite dimensional Hilbert space.
616 
617 \begin{theorem}[Canonical packet connection]\label{strong:theorem:012}\label{strong:thm:kato_connection}
618 Define
619 \[
620 K(t)
621 =
622 \sum_a\dot P_a(t)P_a(t).
623 \]
624 Then
625 \[
626 K(t)^*=-K(t)
627 \]
628 and
629 \[
630 [K(t),P_a(t)]=\dot P_a(t)
631 \]
632 for every \(a\).
633 \end{theorem}
634 
635 \begin{proof}\phantomsection\label{strong:proof:021}
636 Differentiating \(P_a^2=P_a\) gives
637 \[
638 \dot P_aP_a+P_a\dot P_a=\dot P_a.
639 \]
640 Summing over \(a\) and using \(\sum_a\dot P_a=0\) yields
641 \[
642 \sum_aP_a\dot P_a=-\sum_a\dot P_aP_a.
643 \]
644 This proves skew adjointness. For fixed \(b\),
645 \[
646 KP_b=\dot P_bP_b.
647 \]
648 Differentiating \(P_bP_a=0\) for \(a\neq b\) gives
649 \[
650 P_b\dot P_a=-\dot P_bP_a.
651 \]
652 Hence
653 \[
654 P_bK=-\dot P_b(I-P_b).
655 \]
656 Subtracting gives \([K,P_b]=\dot P_b\).
657 \phantomsection\label{strong:proof:021:end}\end{proof}
658 
659 \begin{theorem}[Exact Kato transport]\label{strong:theorem:013}\label{strong:thm:kato_transport}
660 Let \(W(t)\) solve
661 \[
662 \dot W(t)=K(t)W(t),
663 \qquad
664 W(0)=I.
665 \]
666 Then \(W(t)\) is unitary and
667 \[
668 P_a(t)W(t)=W(t)P_a(0)
669 \]
670 for every packet \(a\).
671 \end{theorem}
672 
673 \begin{proof}\phantomsection\label{strong:proof:022}
674 Skew adjointness gives
675 \[
676 \frac{\dd}{\dd t}W(t)^*W(t)=0.
677 \]
678 Thus \(W(t)\) is unitary. Next,
679 \[
680 \frac{\dd}{\dd t}
681 \left(W(t)^*P_a(t)W(t)\right)
682 =
683 W(t)^*
684 \left(
685 \dot P_a-[K,P_a]
686 \right)
687 W(t)=0.
688 \]
689 The expression equals \(P_a(0)\) at \(t=0\), which proves the intertwining identity.
690 \phantomsection\label{strong:proof:022:end}\end{proof}
691 
692 \begin{theorem}[Moving frame Hamiltonian]\label{strong:theorem:014}\label{strong:thm:moving_frame}
693 Let \(U(t)\) solve
694 \[
695 \ii\dot U(t)=H(t)U(t),
696 \qquad
697 U(0)=I,
698 \]
699 where \(H(t)=H(t)^*\) is continuous on the compact time interval under consideration. Set
700 \[
701 V(t)=W(t)^*U(t).
702 \]
703 Then
704 \[
705 \ii\dot V(t)=H_{\rm mov}(t)V(t),
706 \]
707 where
708 \[
709 H_{\rm mov}(t)
710 =
711 W(t)^*H(t)W(t)-\ii W(t)^*\dot W(t)
712 =
713 W(t)^*\bigl(H(t)-\ii K(t)\bigr)W(t).
714 \]
715 The moving frame Hamiltonian is self adjoint.
716 \end{theorem}
717 
718 \begin{proof}\phantomsection\label{strong:proof:023}
719 Differentiate \(V=W^*U\). Since \(\dot W=KW\) and \(K^*=-K\),
720 \[
721 \dot W^*=-W^*K.
722 \]
723 Substitution gives the equation. The first term in \(H_{\rm mov}\) is self adjoint. The operator \(-\ii W^*\dot W\) is self adjoint because \(W^*\dot W\) is skew adjoint.
724 \phantomsection\label{strong:proof:023:end}\end{proof}
725 
726 \begin{theorem}[Moving packet leakage]\label{strong:theorem:015}\label{strong:thm:moving_leakage}
727 Fix a packet \(P(t)=P_a(t)\) and put \(Q(t)=I-P(t)\). For \(0\leq t\leq T\),
728 \[
729 \norm{Q(t)U(t)P(0)}
730 \leq
731 \int_0^t
732 \norm{Q(s)\bigl(H(s)-\ii K(s)\bigr)P(s)}
733 \dd s.
734 \]
735 In particular, the physical evolution follows the moving packet exactly on \([0,T]\) when the displayed off packet block vanishes for every time in that interval.
736 \end{theorem}
737 
738 \begin{proof}\phantomsection\label{strong:proof:024}
739 In the Kato frame, \(P(0)\) is fixed and
740 \[
741 Q(t)U(t)P(0)
742 =
743 W(t)Q(0)V(t)P(0).
744 \]
745 Apply the time dependent version of the Duhamel argument in \cref{strong:thm:fixed_leakage} to \(H_{\rm mov}(t)\). The off diagonal block is
746 \[
747 Q(0)H_{\rm mov}(s)P(0)
748 =
749 W(s)^*Q(s)\bigl(H(s)-\ii K(s)\bigr)P(s)W(s).
750 \]
751 Unitary conjugation preserves the norm.
752 \phantomsection\label{strong:proof:024:end}\end{proof}
753 
754 The exact coupling test therefore includes the geometric term. Even if the Hamiltonian is diagonal in the packet decomposition at every fixed time, its evolution can fail to follow a packet whose range rotates.
755 
756 \subsection{Discrete packet transport}\label{strong:subsec:discrete_transport}
757 
758 Let \((P_a)\) and \((Q_a)\) be two complete packet systems with the same index set. Define
759 \[
760 S(\Qcal,\Pcal)=\sum_aQ_aP_a.
761 \]
762 
763 \begin{theorem}[Direct packet rotation]\label{strong:theorem:016}\label{strong:thm:direct_rotation}
764 Assume
765 \[
766 \norm{S(\Qcal,\Pcal)-I}<1.
767 \]
768 Then \(S(\Qcal,\Pcal)\) is invertible. Its unitary polar factor
769 \[
770 \Ucal(\Qcal,\Pcal)
771 =
772 S(S^*S)^{-1/2}
773 \]
774 satisfies
775 \[
776 Q_a\Ucal(\Qcal,\Pcal)
777 =
778 \Ucal(\Qcal,\Pcal)P_a
779 \]
780 for every \(a\).
781 \end{theorem}
782 
783 \begin{proof}\phantomsection\label{strong:proof:025}
784 The norm condition gives invertibility by the Neumann series. The identity
785 \[
786 Q_aS=SP_a=Q_aP_a
787 \]
788 shows that \(S^*S\) commutes with every \(P_a\). Its inverse square root also commutes with every \(P_a\). Hence
789 \[
790 Q_a\Ucal
791 =Q_aS(S^*S)^{-1/2}
792 =SP_a(S^*S)^{-1/2}
793 =\Ucal P_a.
794 \]
795 The polar factor is unitary because \(S\) is invertible.
796 \phantomsection\label{strong:proof:025:end}\end{proof}
797 
798 \begin{theorem}[Convergence of short packet rotations]\label{strong:theorem:017}\label{strong:thm:short_rotation_convergence}
799 Let \((P_a(t))\) be twice continuously differentiable on \([0,T]\). For a partition
800 \[
801 0=t_0<t_1<\cdots<t_n=T,
802 \]
803 let
804 \[
805 W_\pi
806 =
807 \Ucal\bigl(\Pcal(t_n),\Pcal(t_{n-1})\bigr)
808 \cdots
809 \Ucal\bigl(\Pcal(t_1),\Pcal(t_0)\bigr).
810 \]
811 For sufficiently fine partitions the factors are defined, and
812 \[
813 \norm{W_\pi-W(T)}
814 \leq
815 C_T\abs{\pi}
816 \]
817 for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
818 \end{theorem}
819 
820 \begin{proof}\phantomsection\label{strong:proof:026}
821 Taylor expansion gives
822 \[
823 S\bigl(\Pcal(t+h),\Pcal(t)\bigr)
824 =
825 I+hK(t)+O(h^2)
826 \]
827 uniformly on \([0,T]\). Since \(K(t)^*=-K(t)\),
828 \[
829 S^*S=I+O(h^2),
830 \]
831 so the polar factor satisfies
832 \[
833 \Ucal\bigl(\Pcal(t+h),\Pcal(t)\bigr)
834 =
835 I+hK(t)+O(h^2).
836 \]
837 This gives a first order approximation to \(\dot W=KW\) over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times \(\sum_j(t_{j+1}-t_j)^2\), which is at most \(T\abs{\pi}\).
838 \phantomsection\label{strong:proof:026:end}\end{proof}
839 
840 \begin{figure}[t]
841 \centering
842 \includegraphics[width=.58\textwidth]{06_kato_transport.pdf}
843 \caption{A moving rank one packet on the Bloch sphere. The Kato evolution transports the range without choosing a phase convention for its eigenvector.}
844 \label{strong:fig:kato_transport}
845 \end{figure}
846 \FloatBarrier
847 
848 \subsection{Local frames and curvature}\label{strong:subsec:curvature}
849 
850 Let \(P(x)\) be a smooth rank \(r\) orthogonal projector over a parameter manifold. Choose a local isometry
851 \[
852 V(x):\CC^r\longrightarrow\Hcal,
853 \qquad
854 V(x)V(x)^*=P(x).
855 \]
856 
857 \begin{theorem}[Nonabelian packet connection]\label{strong:theorem:018}\label{strong:thm:nonabelian_connection}
858 The one form with matrix values
859 \[
860 \Acal=V^*\dd V
861 \]
862 is skew adjoint. Under a frame change \(V'=VG\), where \(G(x)\in U(r)\),
863 \[
864 \Acal'
865 =
866 G^*\Acal G+G^*\dd G.
867 \]
868 Its curvature
869 \[
870 \Fcal=\dd\Acal+\Acal\wedge\Acal
871 \]
872 satisfies
873 \[
874 \Fcal=V^*(\dd P\wedge\dd P)V.
875 \]
876 \end{theorem}
877 
878 \begin{proof}\phantomsection\label{strong:proof:027}
879 Differentiate \(V^*V=I\) to obtain \(\Acal^*=-\Acal\). The gauge transformation law follows by differentiating \(VG\). Since \(P=VV^*\) and \((I-P)\dd V=\dd P\,V\), direct expansion gives
880 \[
881 \dd\Acal+\Acal\wedge\Acal
882 =
883 \dd V^*(I-P)\wedge\dd V
884 =
885 V^*(\dd P\wedge\dd P)V.
886 \]
887 \phantomsection\label{strong:proof:027:end}\end{proof}
888 
889 \begin{corollary}[Bloch sphere curvature]\label{strong:corollary:007}\label{strong:cor:bloch_curvature}
890 Let
891 \[
892 P(\theta,\phi)
893 =
894 \frac12\left(I+\bm n(\theta,\phi)\cdot\bm\sigma\right),
895 \]
896 where
897 \[
898 \bm n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).
899 \]
900 With the convention
901 \[
902 \Fcal_{\theta\phi}
903 =-\ii\Tr\bigl(P[\partial_\theta P,\partial_\phi P]\bigr),
904 \]
905 one has
906 \[
907 \Fcal_{\theta\phi}=\frac12\sin\theta
908 \]
909 and
910 \[
911 \frac1{2\pi}
912 \int_0^{2\pi}\int_0^\pi
913 \Fcal_{\theta\phi}\dd\theta\dd\phi
914 =1.
915 \]
916 \end{corollary}
917 
918 \begin{proof}\phantomsection\label{strong:proof:028}
919 The Pauli identity
920 \[
921 (\bm a\cdot\bm\sigma)(\bm b\cdot\bm\sigma)
922 =(\bm a\cdot\bm b)I
923 +\ii(\bm a\times\bm b)\cdot\bm\sigma
924 \]
925 gives
926 \[
927 -\ii\Tr\bigl(P[\partial_\theta P,\partial_\phi P]\bigr)
928 =\frac12
929 \bm n\cdot
930 (\partial_\theta\bm n\times\partial_\phi\bm n)
931 =\frac12\sin\theta.
932 \]
933 Integrating the displayed curvature over the sphere and applying the stated normalization gives one.
934 \phantomsection\label{strong:proof:028:end}\end{proof}
935 
936 \section{Operator packets, measurements, and quantum channels}\label{strong:section:007}\label{strong:sec:channels}
937 
938 Density matrices and observables carry both an input packet and an output packet. Equip \(\End(\Hcal)\) with the Hilbert--Schmidt inner product
939 \[
940 \ip{X}{Y}_{\HS}=\Tr(X^*Y).
941 \]
942 
943 \begin{definition}[Two sided operator packet]\label{strong:definition:003}\label{strong:def:operator_packet}
944 For a complete orthogonal family \((P_a)\), define
945 \[
946 \Pcal_{ab}(X)=P_aXP_b.
947 \]
948 The range
949 \[
950 \Bcal_{ab}=P_a\End(\Hcal)P_b
951 \]
952 is the operator packet with output address \(a\) and input address \(b\).
953 \end{definition}
954 
955 \begin{theorem}[Complete operator packet system]\label{strong:theorem:019}\label{strong:thm:operator_packets}
956 The superoperators \((\Pcal_{ab})_{a,b\in I}\) are mutually orthogonal projections on Hilbert--Schmidt operator space. They satisfy
957 \[
958 X=\sum_{a,b}P_aXP_b
959 \]
960 and
961 \[
962 \norm{X}_{\HS}^2
963 =
964 \sum_{a,b}\norm{P_aXP_b}_{\HS}^2.
965 \]
966 \end{theorem}
967 
968 \begin{proof}\phantomsection\label{strong:proof:029}
969 Idempotence and completeness follow from the projector identities. For Hilbert--Schmidt orthogonality,
970 \[
971 \begin{aligned}
972 \ip{P_aXP_b}{P_cYP_d}_{\HS}
973 &=
974 \Tr(P_bX^*P_aP_cYP_d)\\
975 &=
976 \delta_{ac}\delta_{bd}
977 \Tr(P_bX^*P_aYP_b).
978 \end{aligned}
979 \]
980 Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.
981 \phantomsection\label{strong:proof:029:end}\end{proof}
982 
983 \begin{figure}[t]
984 \centering
985 \includegraphics[width=.58\textwidth]{07_operator_packets.pdf}
986 \caption{A density matrix or observable carries two packet addresses. Diagonal packets contain populations and internal observables. Off diagonal packets contain coherences.}
987 \label{strong:fig:operator_packets}
988 \end{figure}
989 
990 \begin{definition}[Packet dephasing]\label{strong:definition:004}\label{strong:def:dephasing}
991 The packet dephasing map is
992 \[
993 \Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
994 \]
995 \end{definition}
996 
997 \begin{theorem}[Dephasing as conditional expectation]\label{strong:theorem:020}\label{strong:thm:dephasing_expectation}
998 The map \(\Dcal_{\Pcal}\) is completely positive, trace preserving, unital, and idempotent. It is the Hilbert--Schmidt orthogonal projection onto the block diagonal algebra
999 \[
1000 \Acal_{\Pcal}
1001 =
1002 \{X:[X,P_a]=0\text{ for every }a\}.
1003 \]
1004 In particular,
1005 \[
1006 \norm{\Dcal_{\Pcal}(X)}_{\HS}
1007 \leq
1008 \norm{X}_{\HS}.
1009 \]
1010 \end{theorem}
1011 
1012 \begin{proof}\phantomsection\label{strong:proof:030}
1013 The projectors \(P_a\) form a Kraus family because \(\sum_aP_a^*P_a=I\). Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from \cref{strong:thm:operator_packets}. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.
1014 \phantomsection\label{strong:proof:030:end}\end{proof}
1015 
1016 \begin{corollary}[Block diagonal measurement invariance]\label{strong:corollary:008}\label{strong:cor:measurement_invariance}
1017 Let \((E_k)\) be a positive operator valued measurement, and suppose
1018 \[
1019 E_k=\Dcal_{\Pcal}(E_k)
1020 \]
1021 for every \(k\). Then every density matrix \(\rho\) satisfies
1022 \[
1023 \Tr(E_k\rho)
1024 =
1025 \Tr\bigl(E_k\Dcal_{\Pcal}(\rho)\bigr).
1026 \]
1027 \end{corollary}
1028 
1029 \begin{proof}\phantomsection\label{strong:proof:031}
1030 Since \(\Dcal_{\Pcal}\) is self adjoint for the Hilbert--Schmidt inner product,
1031 \[
1032 \Tr(E_k\rho)
1033 =
1034 \Tr\bigl(\Dcal_{\Pcal}(E_k)\rho\bigr)
1035 =
1036 \Tr\bigl(E_k\Dcal_{\Pcal}(\rho)\bigr).
1037 \]
1038 \phantomsection\label{strong:proof:031:end}\end{proof}
1039 
1040 \subsection{Channel superblocks}\label{strong:subsec:channel_blocks}
1041 
1042 Let \(\Phi:\End(\Hcal_X)\to\End(\Hcal_Y)\) be linear. Choose packet systems \((P_a)\) on \(\Hcal_X\) and \((Q_c)\) on \(\Hcal_Y\). Define
1043 \[
1044 \Phi_{cd,ab}
1045 =
1046 \Qcal_{cd}\Phi\Pcal_{ab}.
1047 \]
1048 
1049 \begin{theorem}[Exact channel block calculus]\label{strong:theorem:021}\label{strong:thm:channel_blocks}
1050 Every linear map has the exact reconstruction
1051 \[
1052 \Phi=\sum_{a,b,c,d}\Phi_{cd,ab}.
1053 \]
1054 If \(\Psi:\End(\Hcal_Y)\to\End(\Hcal_Z)\), then
1055 \[
1056 (\Psi\Phi)_{ef,ab}
1057 =
1058 \sum_{c,d}
1059 \Psi_{ef,cd}\Phi_{cd,ab}.
1060 \]
1061 \end{theorem}
1062 
1063 \begin{proof}\phantomsection\label{strong:proof:032}
1064 Insert the complete operator packet resolutions before and after \(\Phi\). For composition, insert the middle resolution between \(\Psi\) and \(\Phi\) and collect the output and input packets.
1065 \phantomsection\label{strong:proof:032:end}\end{proof}
1066 
1067 \begin{standardfact}[Finite Kraus and Choi criteria]\label{strong:standardfact:001}\label{strong:fact:kraus_choi}
1068 A linear map \(\Phi:M_n(\CC)\to M_m(\CC)\) is completely positive exactly when it has a Kraus representation
1069 \[
1070 \Phi(X)=\sum_rK_rXK_r^*.
1071 \]
1072 It is trace preserving exactly when
1073 \[
1074 \sum_rK_r^*K_r=I.
1075 \]
1076 Equivalently, its Choi matrix
1077 \[
1078 J(\Phi)
1079 =
1080 \sum_{i,j}\Phi(\ket{i}\bra{j})\otimes\ket{i}\bra{j}
1081 \]
1082 is positive, and trace preservation is
1083 \[
1084 \Tr_{\rm out}J(\Phi)=I_{\rm in}.
1085 \]
1086 \end{standardfact}
1087 
1088 These finite criteria are classical \cite{Stinespring1955,Choi1975,Kraus1983,Watrous2018}.
1089 
1090 \subsection{Covariance and coherence order}\label{strong:subsec:channel_covariance}
1091 
1092 Let \(S\) be unitary and \(N\ge1\) an integer with \(S^N=I\), and define the conjugation action
1093 \[
1094 \Ad_S(X)=SXS^*.
1095 \]
1096 
1097 \begin{theorem}[Coherence order projectors]\label{strong:theorem:022}\label{strong:thm:coherence_projectors}
1098 For \(q\in\ZZ/N\ZZ\), define
1099 \[
1100 \Lcal_q(X)
1101 =
1102 \frac1N
1103 \sum_{r=0}^{N-1}
1104 \omega^{-qr}S^rXS^{-r}.
1105 \]
1106 Then the \(\Lcal_q\) form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying
1107 \[
1108 SXS^*=\omega^qX.
1109 \]
1110 If \(P_j\) are the state charge projectors, then
1111 \[
1112 \Ran\Lcal_q
1113 =
1114 \bigoplus_jP_{j+q}\End(\Hcal)P_j.
1115 \]
1116 \end{theorem}
1117 
1118 \begin{proof}\phantomsection\label{strong:proof:033}
1119 Apply the cyclic character projector formula to the unitary representation \(\Ad_S\) on operator space. The last identity follows from \cref{strong:thm:charge_selection} applied to each operator packet.
1120 \phantomsection\label{strong:proof:033:end}\end{proof}
1121 
1122 \begin{theorem}[Covariant channels preserve coherence order]\label{strong:theorem:023}\label{strong:thm:covariant_channel}
1123 Suppose
1124 \[
1125 \Phi(SXS^*)=S\Phi(X)S^*
1126 \]
1127 for every \(X\). Then
1128 \[
1129 \Phi\Lcal_q=\Lcal_q\Phi
1130 \]
1131 for every \(q\). Thus \(\Phi\) cannot mix distinct coherence orders.
1132 \end{theorem}
1133 
1134 \begin{proof}\phantomsection\label{strong:proof:034}
1135 The covariance identity says that \(\Phi\) commutes with every power of \(\Ad_S\). It therefore commutes with each finite character average \(\Lcal_q\).
1136 \phantomsection\label{strong:proof:034:end}\end{proof}
1137 
1138 \begin{theorem}[Choi symmetry criterion]\label{strong:theorem:024}\label{strong:thm:choi_covariance}
1139 Let \(U:G\to\Ucal(\Hcal)\) be a finite group representation. A channel \(\Phi\) is covariant,
1140 \[
1141 \Phi\bigl(U(g)XU(g)^*\bigr)
1142 =
1143 U(g)\Phi(X)U(g)^*,
1144 \]
1145 if and only if
1146 \[
1147 \bigl(U(g)\otimes\overline{U(g)}\bigr)
1148 J(\Phi)
1149 \bigl(U(g)^*\otimes U(g)^{\mathsf T}\bigr)
1150 =
1151 J(\Phi)
1152 \]
1153 for every \(g\in G\).
1154 \end{theorem}
1155 
1156 \begin{proof}\phantomsection\label{strong:proof:035}
1157 Let
1158 \[
1159 \Omega=\sum_i\ket{i}\otimes\ket{i}.
1160 \]
1161 Then
1162 \[
1163 J(\Phi)=(\Phi\otimes\id)(\ket{\Omega}\bra{\Omega}).
1164 \]
1165 The vector \(\Omega\) is invariant under \(U\otimes\overline U\). Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of
1166 \[
1167 \Ad_{U(g)}\circ\Phi\circ\Ad_{U(g)^*}.
1168 \]
1169 The Choi correspondence is injective, so invariance implies that this map equals \(\Phi\), which is covariance.
1170 \phantomsection\label{strong:proof:035:end}\end{proof}
1171 
1172 \begin{corollary}[Choi packet decomposition]\label{strong:corollary:009}\label{strong:cor:choi_packets}
1173 The Choi matrix of a covariant channel is block diagonal with respect to the isotypic decomposition of \(U\otimes\overline U\).
1174 \end{corollary}
1175 
1176 \begin{proof}\phantomsection\label{strong:proof:036}
1177 Apply \cref{strong:thm:isotypic_projectors} to the representation \(U\otimes\overline U\) and use \cref{strong:thm:choi_covariance}.
1178 \phantomsection\label{strong:proof:036:end}\end{proof}
1179 
1180 \begin{example}[Amplitude damping]\label{strong:example:002}\label{strong:ex:amplitude_damping}
1181 For \(0\leq\gamma\leq1\), let
1182 \[
1183 K_0=
1184 \begin{pmatrix}
1185 1&0\\
1186 0&\sqrt{1-\gamma}
1187 \end{pmatrix},
1188 \qquad
1189 K_1=
1190 \begin{pmatrix}
1191 0&\sqrt\gamma\\
1192 0&0
1193 \end{pmatrix}.
1194 \]
1195 The channel
1196 \[
1197 \Phi_\gamma(\rho)=K_0\rho K_0^*+K_1\rho K_1^*
1198 \]
1199 is trace preserving. It sends the off diagonal packet \(\ket0\bra1\) to \(\sqrt{1-\gamma}\ket0\bra1\), while the excited population transfers partly into the ground population.
1200 \end{example}
1201 
1202 \begin{figure}[t]
1203 \centering
1204 \includegraphics[width=.82\textwidth]{08_channel_coherence.pdf}
1205 \caption{Amplitude damping transfers population and contracts the coherence packet.}
1206 \label{strong:fig:channel_coherence}
1207 \end{figure}
1208 
1209 
1210 \subsection{Quantum dynamical semigroups}\label{strong:subsec:lindblad}
1211 
1212 \begin{standardfact}[Finite Lindblad generator]\label{strong:standardfact:002}\label{strong:fact:lindblad}
1213 A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr\"odinger picture, has a generator of the form
1214 \[
1215 \Gcal(X)
1216 =
1217 -\ii[H,X]
1218 +
1219 \sum_r
1220 \left(
1221 L_rXL_r^*
1222 -
1223 \frac12L_r^*L_rX
1224 -
1225 \frac12XL_r^*L_r
1226 \right),
1227 \]
1228 where \(H=H^*\). Conversely, every operator of this form generates a completely positive trace preserving semigroup \cite{Lindblad1976}.
1229 \end{standardfact}
1230 
1231 \begin{theorem}[Charge preservation by a Lindblad generator]\label{strong:theorem:025}\label{strong:thm:lindblad_charge}
1232 Let \(S\) be unitary and \(N\ge1\) an integer with \(S^N=I\). Assume
1233 \[
1234 SHS^*=H
1235 \]
1236 and, for every jump operator, there is a charge \(q_r\) such that
1237 \[
1238 SL_rS^*=\omega^{q_r}L_r.
1239 \]
1240 Then the Lindblad generator \(\Gcal\) preserves every coherence order:
1241 \[
1242 \Gcal\Lcal_p=\Lcal_p\Gcal.
1243 \]
1244 The quantum dynamical semigroup \(\e^{t\Gcal}\) has the same property.
1245 \end{theorem}
1246 
1247 \begin{proof}\phantomsection\label{strong:proof:037}
1248 Let \(X\) have coherence order \(p\). The Hamiltonian commutator has order \(p\) because \(H\) has charge zero. The jump term \(L_rXL_r^*\) has charge
1249 \[
1250 q_r+p-q_r=p.
1251 \]
1252 Also \(L_r^*L_r\) has charge zero, so both anticommutator terms have charge \(p\). Thus \(\Gcal\) preserves \(\Ran\Lcal_p\). The exponential series then shows that \(\e^{t\Gcal}\) preserves the same packets.
1253 \phantomsection\label{strong:proof:037:end}\end{proof}
1254 
1255 \begin{corollary}[Independent coherence evolution]\label{strong:corollary:010}\label{strong:cor:no_coherence_mixing}
1256 Under the hypotheses of \cref{strong:thm:lindblad_charge}, each coherence component of an initial operator evolves within its own order. In particular, a density matrix in the zero order sector remains in that sector.
1257 \end{corollary}
1258 
1259 \begin{proof}\phantomsection\label{strong:proof:038}
1260 Decompose the initial operator by \cref{strong:thm:coherence_projectors} and apply the invariant packet statement to each component.
1261 \phantomsection\label{strong:proof:038:end}\end{proof}
1262 
1263 Preservation of coherence order does not by itself give a closed classical Markov chain for the packet populations. Such scalar population equations require additional isotropy within each packet.
1264 
1265 \section{Quantum error correction as a compression obstruction}\label{strong:section:008}\label{strong:sec:qec}
1266 
1267 Let \(P\) be the orthogonal projector onto a code space
1268 \[
1269 \Ccal=P\Hcal,
1270 \qquad
1271 k=\dim\Ccal\ge1.
1272 \]
1273 The compression of \(A\in\End(\Hcal)\) to the code is \(PAP\in\End(\Ccal)\). The zero code has only the zero supported operator and no normalized code state. Part III treats that branch in the mathematical completion.
1274 
1275 \begin{definition}[Code compression obstruction]\label{strong:definition:005}\label{strong:def:code_obstruction}
1276 Let
1277 \[
1278 q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
1279 \]
1280 be the quotient map. Define
1281 \[
1282 \Ob_P(A)=q_P(PAP).
1283 \]
1284 Equivalently, one may use the traceless representative
1285 \[
1286 \ob_P(A)
1287 =
1288 PAP-
1289 \frac{\Tr(PAP)}{k}P.
1290 \]
1291 \end{definition}
1292 
1293 The class vanishes exactly when \(A\) acts as a scalar after compression to the code. Such an operator cannot distinguish normalized code vectors by expectation value.
1294 
1295 Exact correction of an error span means that one completely positive trace preserving recovery $\Rcal$ satisfies $\Rcal(F\rho F^*)=q_F\rho$ for every $F$ in that span and every code-supported matrix $\rho$, with a nonnegative scalar $q_F$ depending on $F$ but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.
1296 
1297 \begin{theorem}[Knill--Laflamme as obstruction vanishing]\label{strong:theorem:026}\label{strong:thm:kl_obstruction}
1298 Let \(E_1,\ldots,E_r\) be an error family. The code \(\Ccal\) corrects their linear span exactly if and only if
1299 \[
1300 \Ob_P(E_a^*E_b)=0
1301 \]
1302 for all \(a,b\). Equivalently, there is a matrix \(c=(c_{ab})\) such that
1303 \[
1304 PE_a^*E_bP=c_{ab}P
1305 \]
1306 for all \(a,b\).
1307 \end{theorem}
1308 
1309 \begin{proof}\phantomsection\label{strong:proof:039}
1310 The equivalence between the quotient statement and the scalar compression identity is immediate from the definition. The latter identity is the finite Knill--Laflamme criterion \cite{KnillLaflamme1997,SchumacherNielsen1996}.
1311 \phantomsection\label{strong:proof:039:end}\end{proof}
1312 
1313 The next result turns the scalar compression identities into a family of orthogonal syndrome packets. Those packets provide the coordinates used by the recovery map.
1314 
1315 \begin{theorem}[Orthogonal syndrome construction]\label{strong:theorem:027}\label{strong:thm:syndrome_construction}
1316 Assume the Knill--Laflamme identities. Then the coefficient matrix \(c\) is positive semidefinite. Choose a unitary matrix \(W\) such that
1317 \[
1318 WcW^*=\diag(\lambda_1,\ldots,\lambda_r),
1319 \qquad
1320 \lambda_j\geq0,
1321 \]
1322 and define
1323 \[
1324 F_j=\sum_a\overline{W_{ja}}E_a.
1325 \]
1326 For every \(j\) with \(\lambda_j>0\), the operator
1327 \[
1328 V_j=\lambda_j^{-1/2}F_jP
1329 \]
1330 is an isometry from the code into \(\Hcal\). Its range projector
1331 \[
1332 Q_j=V_jV_j^*
1333 \]
1334 is a syndrome packet, and the nonzero \(Q_j\) are pairwise orthogonal.
1335 \end{theorem}
1336 
1337 \begin{proof}\phantomsection\label{strong:proof:040}
1338 For \(z\in\CC^r\), let \(F_z=\sum_az_aE_a\). Then
1339 \[
1340 PF_z^*F_zP=(z^*cz)P\geq0,
1341 \]
1342 so \(c\geq0\). The unitary change of error coordinates gives
1343 \[
1344 PF_j^*F_\ell P
1345 =
1346 \lambda_j\delta_{j\ell}P.
1347 \]
1348 Thus
1349 \[
1350 V_j^*V_\ell=\delta_{j\ell}P.
1351 \]
1352 Each \(V_j\) is an isometry on \(P\Hcal\), and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
1353 \phantomsection\label{strong:proof:040:end}\end{proof}
1354 
1355 \begin{theorem}[Exact recovery on the error span]\label{strong:theorem:028}\label{strong:thm:exact_recovery}
1356 Under the hypotheses of \cref{strong:thm:syndrome_construction}, define on the syndrome support
1357 \[
1358 Q=\sum_{\lambda_j>0}Q_j
1359 \]
1360 the completely positive map
1361 \[
1362 \Rcal_0(X)
1363 =
1364 \sum_{\lambda_j>0}
1365 V_j^*Q_jXQ_jV_j.
1366 \]
1367 It extends to a quantum channel \(\Rcal\) on \(\End(\Hcal)\). If
1368 \[
1369 \Phi(X)=\sum_aE_aXE_a^*
1370 \]
1371 is trace preserving on code states, then
1372 \[
1373 \Rcal\Phi(\rho)=\rho
1374 \]
1375 for every density matrix \(\rho=P\rho P\).
1376 \end{theorem}
1377 
1378 \begin{proof}\phantomsection\label{strong:proof:041}
1379 The map \(\Rcal_0\) is completely positive. It is trace preserving on operators supported in \(Q\Hcal\) because
1380 \[
1381 \Tr\bigl(V_j^*Q_jXQ_jV_j\bigr)
1382 =
1383 \Tr(Q_jXQ_j).
1384 \]
1385 Fix a code state \(\rho_0\) and write \(Q^\perp=I-Q\). Then
1386 \[
1387 \Rcal(X)
1388 =
1389 \Rcal_0(QXQ)
1390 +
1391 \Tr(Q^\perp X)\rho_0
1392 \]
1393 is completely positive and trace preserving. It extends \(\Rcal_0\) on operators supported in \(Q\Hcal\).
1394 
1395 Unitary mixing of a Kraus family does not change the channel, so
1396 \[
1397 \Phi(\rho)
1398 =
1399 \sum_jF_j\rho F_j^*
1400 =
1401 \sum_{\lambda_j>0}
1402 \lambda_jV_j\rho V_j^*.
1403 \]
1404 Recovery gives
1405 \[
1406 \Rcal\Phi(\rho)
1407 =
1408 \left(\sum_j\lambda_j\right)\rho.
1409 \]
1410 Trace preservation on a code state implies \(\sum_j\lambda_j=1\).
1411 \phantomsection\label{strong:proof:041:end}\end{proof}
1412 
1413 \begin{corollary}[Syndrome measurement preserves logical information]\label{strong:corollary:011}\label{strong:cor:syndrome_preserves_logic}
1414 For a channel trace preserving on code states as in the preceding theorem, the completed projective measurement consisting of \((Q_j)_{\lambda_j>0}\) and \(I-\sum_{\lambda_j>0}Q_j\) identifies the orthogonal error syndrome without resolving the encoded state inside the code. The complement outcome has probability zero on the corrupted code state.
1415 \end{corollary}
1416 
1417 \begin{proof}\phantomsection\label{strong:proof:042}
1418 On syndrome \(j\), the encoded state is carried by the isometry \(V_j\). The outcome probability is \(\lambda_j\), independent of the normalized code state, while the postmeasurement state is \(V_j\rho V_j^*\).
1419 \phantomsection\label{strong:proof:042:end}\end{proof}
1420 
1421 \begin{definition}[Approximate Knill--Laflamme defect]\label{strong:definition:006}\label{strong:def:approximate_kl}
1422 For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero.
1423 \[
1424 \varepsilon_{\rm KL}
1425 =
1426 \max_{a,b}
1427 \inf_{c\in\CC}
1428 \norm{PE_a^*E_bP-cP}.
1429 \]
1430 \end{definition}
1431 
1432 \begin{proposition}[Meaning of the approximate defect]\label{strong:proposition:003}\label{strong:prop:approximate_kl_meaning}
1433 For each pair \((a,b)\), choose a minimizing scalar \(c_{ab}\). For unit code vectors \(\psi,\phi\),
1434 \[
1435 \abs{
1436 \ip{E_a\psi}{E_b\phi}
1437 -c_{ab}\ip{\psi}{\phi}
1438 }
1439 \leq
1440 \varepsilon_{\rm KL}.
1441 \]
1442 Thus \(\varepsilon_{\rm KL}\) controls the deviation of error overlaps from their scalar code form.
1443 \end{proposition}
1444 
1445 \begin{proof}\phantomsection\label{strong:proof:043}
1446 The left side is
1447 \[
1448 \abs{
1449 \ip{\psi}{(PE_a^*E_bP-c_{ab}P)\phi}
1450 },
1451 \]
1452 which is bounded by the operator norm.
1453 \phantomsection\label{strong:proof:043:end}\end{proof}
1454 
1455 No recovery fidelity estimate is inferred from this norm alone. Approximate error correction requires a separate continuity theorem and a declared channel metric.
1456 
1457 \begin{example}[Three qubit repetition code]\label{strong:example:003}\label{strong:ex:bit_flip_code}
1458 Let
1459 \[
1460 \ket{0_L}=\ket{000},
1461 \qquad
1462 \ket{1_L}=\ket{111},
1463 \]
1464 and take the errors
1465 \[
1466 I,
1467 \quad
1468 X\otimes I\otimes I,
1469 \quad
1470 I\otimes X\otimes I,
1471 \quad
1472 I\otimes I\otimes X.
1473 \]
1474 Their images of the code are four pairwise orthogonal two dimensional syndrome packets whose sum is the full eight dimensional Hilbert space.
1475 \end{example}
1476 
1477 \begin{figure}[t]
1478 \centering
1479 \includegraphics[width=.68\textwidth]{09_qec_syndromes.pdf}
1480 \caption{The syndrome packets of the three qubit repetition code. Each packet has dimension two and carries the full logical state.}
1481 \label{strong:fig:qec_syndromes}
1482 \end{figure}
1483 
1484 \begin{standardfact}[Correctable observable algebras]\label{strong:standardfact:003}\label{strong:fact:operator_qec}
1485 Let \(\Phi\) be a finite completely positive trace preserving channel with Kraus operators \((E_a)\). A finite dimensional unital star subalgebra \(\Mcal\subseteq P\End(\Hcal)P\) is correctable when a completely positive trace preserving recovery \(\Rcal\) satisfies \(P(\Phi^*\circ\Rcal^*)(X)P=X\) for every \(X\in\Mcal\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \(PE_a^*E_bP\) belongs to the commutant \(\Mcal'\) on the code. The scalar code criterion is the special case \(\Mcal=P\End(\Hcal)P\) \cite{BenyKempfKribs2007}.
1486 \end{standardfact}
1487 
1488 \section{Feshbach effective dynamics and hidden packet memory}\label{strong:section:009}\label{strong:sec:feshbach}
1489 
1490 Let \(P\) be an orthogonal projector, put \(Q=I-P\), and write a self adjoint Hamiltonian as
1491 \[
1492 H=
1493 \begin{pmatrix}
1494 A&B\\
1495 B^*&D
1496 \end{pmatrix}
1497 \]
1498 relative to \(P\Hcal\oplus Q\Hcal\).
1499 
1500 \begin{definition}[Feshbach map and self energy]\label{strong:definition:007}\label{strong:def:feshbach_map}
1501 For \(z\notin\Spec D\), define
1502 \[
1503 \Sigma(z)=B(z-D)^{-1}B^*
1504 \]
1505 and
1506 \[
1507 F_P(z)=z-A-\Sigma(z).
1508 \]
1509 \end{definition}
1510 
1511 \begin{theorem}[Exact projected resolvent]\label{strong:theorem:029}\label{strong:thm:feshbach_resolvent}
1512 If \(z\notin\Spec D\) and \(F_P(z)\) is invertible, then
1513 \[
1514 P(z-H)^{-1}P=F_P(z)^{-1}
1515 \]
1516 as an operator on \(P\Hcal\).
1517 \end{theorem}
1518 
1519 \begin{proof}\phantomsection\label{strong:proof:044}
1520 The lower right block of \(z-H\) is \(z-D\). Block Gaussian elimination gives
1521 \[
1522 \begin{pmatrix}
1523 I&B(z-D)^{-1}\\
1524 0&I
1525 \end{pmatrix}
1526 (z-H)
1527 \begin{pmatrix}
1528 I&0\\
1529 (z-D)^{-1}B^*&I
1530 \end{pmatrix}
1531 =
1532 \begin{pmatrix}
1533 F_P(z)&0\\
1534 0&z-D
1535 \end{pmatrix}.
1536 \]
1537 Invert the factors. The retained resolvent is the block in the upper left corner of the resulting inverse.
1538 \phantomsection\label{strong:proof:044:end}\end{proof}
1539 
1540 \begin{theorem}[Spectral equivalence]\label{strong:theorem:030}\label{strong:thm:feshbach_spectrum}
1541 Let \(z\notin\Spec D\). Then
1542 \[
1543 z\in\Spec H
1544 \quad\Longleftrightarrow\quad
1545 0\in\Spec F_P(z).
1546 \]
1547 The map
1548 \[
1549 x\longmapsto
1550 \begin{pmatrix}
1551 x\\
1552 (z-D)^{-1}B^*x
1553 \end{pmatrix}
1554 \]
1555 is an isomorphism from \(\Ker F_P(z)\) onto \(\Ker(z-H)\).
1556 \end{theorem}
1557 
1558 \begin{proof}\phantomsection\label{strong:proof:045}
1559 The equation \((z-H)(x,y)^{\mathsf T}=0\) is
1560 \[
1561 (z-A)x-By=0,
1562 \qquad
1563 -B^*x+(z-D)y=0.
1564 \]
1565 The second equation gives \(y=(z-D)^{-1}B^*x\). Substitution into the first gives \(F_P(z)x=0\). The construction is reversible.
1566 \phantomsection\label{strong:proof:045:end}\end{proof}
1567 
1568 \begin{figure}[t]
1569 \centering
1570 \includegraphics[width=.78\textwidth]{10_feshbach_spectrum.pdf}
1571 \caption{The zeros of the Feshbach determinant recover the eigenvalues of the full finite Hamiltonian away from the hidden spectrum.}
1572 \label{strong:fig:feshbach_spectrum}
1573 \end{figure}
1574 
1575 \begin{theorem}[Upper half plane sign]\label{strong:theorem:031}\label{strong:thm:self_energy_sign}
1576 If \(\operatorname{Im}z>0\), then
1577 \[
1578 \operatorname{Im}\Sigma(z)
1579 =
1580 -(\operatorname{Im}z)
1581 B(z-D)^{-1}(\overline z-D)^{-1}B^*
1582 \leq0.
1583 \]
1584 Thus \(-\Sigma\) is an operator valued Herglotz function.
1585 \end{theorem}
1586 
1587 \begin{proof}\phantomsection\label{strong:proof:046}
1588 For \(R(z)=(z-D)^{-1}\), the resolvent identity gives
1589 \[
1590 R(z)-R(z)^*
1591 =(\overline z-z)R(z)R(z)^*.
1592 \]
1593 Divide by \(2\ii\), then multiply the result on the left by \(B\) and on the right by \(B^*\).
1594 \phantomsection\label{strong:proof:046:end}\end{proof}
1595 
1596 \begin{proposition}[Uniform self energy bounds]\label{strong:proposition:004}\label{strong:prop:self_energy_bounds}
1597 Let
1598 \[
1599 \delta=\dist(z,\Spec D)>0,
1600 \qquad
1601 g=\norm B.
1602 \]
1603 Then
1604 \[
1605 \norm{\Sigma(z)}\leq\frac{g^2}{\delta}
1606 \]
1607 and
1608 \[
1609 \norm{\Sigma'(z)}\leq\frac{g^2}{\delta^2}.
1610 \]
1611 For real \(E,E'\) in an interval at distance at least \(\delta\) from \(\Spec D\),
1612 \[
1613 \norm{\Sigma(E)-\Sigma(E')}
1614 \leq
1615 \frac{g^2}{\delta^2}\abs{E-E'}.
1616 \]
1617 \end{proposition}
1618 
1619 \begin{proof}\phantomsection\label{strong:proof:047}
1620 The self adjoint resolvent satisfies
1621 \[
1622 \norm{(z-D)^{-1}}=\frac1{\dist(z,\Spec D)}.
1623 \]
1624 Differentiate the resolvent to obtain
1625 \[
1626 \Sigma'(z)=-B(z-D)^{-2}B^*.
1627 \]
1628 Taking operator norms gives the first two estimates. Applying the derivative bound along the real interval gives the final mean value estimate.
1629 \phantomsection\label{strong:proof:047:end}\end{proof}
1630 
1631 \begin{proposition}[Low energy expansion]\label{strong:proposition:005}\label{strong:prop:low_energy_expansion}
1632 Assume \(D\) is invertible and \(\abs z\norm{D^{-1}}<1\). Then
1633 \[
1634 \Sigma(z)
1635 =-
1636 \sum_{n=0}^{\infty}
1637 z^nBD^{-n-1}B^*
1638 \]
1639 with norm convergence. In particular,
1640 \[
1641 F_P(z)
1642 =
1643  z-A+BD^{-1}B^*+O\bigl(\abs z\norm B^2\norm{D^{-1}}^2\bigr).
1644 \]
1645 \end{proposition}
1646 
1647 \begin{proof}\phantomsection\label{strong:proof:048}
1648 Write
1649 \[
1650 (z-D)^{-1}
1651 =-D^{-1}(I-zD^{-1})^{-1}
1652 =-\sum_{n=0}^{\infty}z^nD^{-n-1}
1653 \]
1654 and multiply on the left by \(B\) and on the right by \(B^*\).
1655 \phantomsection\label{strong:proof:048:end}\end{proof}
1656 
1657 \begin{proposition}[Bare truncation error identity]\label{strong:proposition:006}\label{strong:prop:bare_truncation}
1658 If both \(F_P(z)\) and \(z-A\) are invertible, then
1659 \[
1660 P(z-H)^{-1}P-(z-A)^{-1}
1661 =
1662 F_P(z)^{-1}\Sigma(z)(z-A)^{-1}.
1663 \]
1664 \end{proposition}
1665 
1666 \begin{proof}\phantomsection\label{strong:proof:049}
1667 Use the inverse difference identity
1668 \[
1669 X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}
1670 \]
1671 with \(X=F_P(z)\) and \(Y=z-A\).
1672 \phantomsection\label{strong:proof:049:end}\end{proof}
1673 
1674 \subsection{Time memory}\label{strong:subsec:time_memory}
1675 
1676 Write a Schrödinger state as \(\psi=x+y\), where \(x=P\psi\) and \(y=Q\psi\).
1677 
1678 \begin{theorem}[Exact retained memory equation]\label{strong:theorem:032}\label{strong:thm:memory_equation}
1679 The retained coordinate satisfies
1680 \[
1681 \begin{aligned}
1682 \ii\dot x(t)
1683 ={}&Ax(t)+B\e^{-\ii tD}y(0)\\
1684 &-\ii\int_0^tK(t-s)x(s)\dd s,
1685 \end{aligned}
1686 \]
1687 where
1688 \[
1689 K(t)=B\e^{-\ii tD}B^*.
1690 \]
1691 For \(\operatorname{Re}s>0\),
1692 \[
1693 \int_0^\infty\e^{-st}K(t)\dd t
1694 =
1695 B(s+\ii D)^{-1}B^*.
1696 \]
1697 \end{theorem}
1698 
1699 \begin{proof}\phantomsection\label{strong:proof:050}
1700 The hidden equation is
1701 \[
1702 \ii\dot y=B^*x+Dy.
1703 \]
1704 Variation of constants gives
1705 \[
1706 y(t)=\e^{-\ii tD}y(0)
1707 -\ii\int_0^t\e^{-\ii(t-s)D}B^*x(s)\dd s.
1708 \]
1709 Substitute into \(\ii\dot x=Ax+By\). The Laplace formula follows by diagonalizing \(D\) or integrating the norm convergent finite dimensional exponential.
1710 \phantomsection\label{strong:proof:050:end}\end{proof}
1711 
1712 \begin{theorem}[Positive definite quantum memory]\label{strong:theorem:033}\label{strong:thm:positive_memory}
1713 The kernel \(K(t-s)\) is positive definite. For times \(t_1,\ldots,t_m\) and retained vectors \(v_1,\ldots,v_m\),
1714 \[
1715 \sum_{j,k=1}^{m}
1716 \ip{v_j}{K(t_j-t_k)v_k}
1717 \geq0.
1718 \]
1719 \end{theorem}
1720 
1721 \begin{proof}\phantomsection\label{strong:proof:051}
1722 Set
1723 \[
1724 w=\sum_{k=1}^{m}\e^{\ii t_kD}B^*v_k.
1725 \]
1726 Then
1727 \[
1728 \norm w^2
1729 =
1730 \sum_{j,k}
1731 \ip{B^*v_j}{\e^{-\ii(t_j-t_k)D}B^*v_k}
1732 =
1733 \sum_{j,k}
1734 \ip{v_j}{K(t_j-t_k)v_k}.
1735 \]
1736 \phantomsection\label{strong:proof:051:end}\end{proof}
1737 
1738 \begin{corollary}[Finite oscillatory memory expansion]\label{strong:corollary:012}\label{strong:cor:oscillatory_memory}
1739 If
1740 \[
1741 D=\sum_{r}E_rQ_r
1742 \]
1743 is the spectral decomposition of the hidden Hamiltonian, then
1744 \[
1745 K(t)=\sum_r\e^{-\ii tE_r}BQ_rB^*.
1746 \]
1747 Every coefficient \(BQ_rB^*\) is positive semidefinite.
1748 \end{corollary}
1749 
1750 \begin{proof}\phantomsection\label{strong:proof:052}
1751 Inserting the spectral decomposition into the exponential gives the stated sum. The spectral projectors are positive semidefinite, so multiplication by the coupling and its adjoint preserves positivity of each coefficient.
1752 \phantomsection\label{strong:proof:052:end}\end{proof}
1753 
1754 \begin{figure}[t]
1755 \centering
1756 \includegraphics[width=.76\textwidth]{11_memory_kernel.pdf}
1757 \caption{A finite hidden spectrum produces an oscillatory memory kernel. The retained dynamics therefore depend on the history of the retained state.}
1758 \label{strong:fig:memory_kernel}
1759 \end{figure}
1760 
1761 \begin{theorem}[Exact local closure criterion]\label{strong:theorem:034}\label{strong:thm:local_closure}
1762 Assume \(y(0)=0\). The retained equation is local and equal to
1763 \[
1764 \ii\dot x=Ax
1765 \]
1766 for every retained initial state exactly when \(B=0\). Equivalently, exact local closure holds exactly when the retained and hidden packets reduce \(H\).
1767 \end{theorem}
1768 
1769 \begin{proof}\phantomsection\label{strong:proof:053}
1770 If \(B=0\), the retained and hidden equations decouple. Conversely, suppose that every solution with \(y(0)=0\) satisfies \(x(t)=\e^{-\ii tA}x(0)\). The full block equation gives
1771 \[
1772 \ddot x(0)
1773 =
1774 -\bigl(A^2+BB^*\bigr)x(0),
1775 \]
1776 while the declared local equation gives
1777 \[
1778 \ddot x(0)=-A^2x(0).
1779 \]
1780 Since this holds for every retained initial vector, \(BB^*=0\). Hence \(B=0\).
1781 \phantomsection\label{strong:proof:053:end}\end{proof}
1782 
1783 \subsection{Stability across models}\label{strong:subsec:feshbach_stability}
1784 
1785 \begin{theorem}[Stability of the self energy]\label{strong:theorem:035}\label{strong:thm:self_energy_stability}
1786 Let \(D=D^*\) and \(\widetilde D=\widetilde D^*\) act on one fixed hidden Hilbert space, and let \(B,\widetilde B\) map that hidden space to one fixed retained Hilbert space. Define
1787 \[
1788 \Sigma(z)=B(z-D)^{-1}B^*,
1789 \qquad
1790 \widetilde\Sigma(z)=\widetilde B(z-\widetilde D)^{-1}\widetilde B^*.
1791 \]
1792 Assume
1793 \[
1794 \norm B,\norm{\widetilde B}\leq g
1795 \]
1796 and
1797 \[
1798 \dist(z,\Spec D),
1799 \dist(z,\Spec\widetilde D)
1800 \geq\delta>0.
1801 \]
1802 Then
1803 \[
1804 \norm{\Sigma(z)-\widetilde\Sigma(z)}
1805 \leq
1806 \frac{2g}{\delta}\norm{B-\widetilde B}
1807 +
1808 \frac{g^2}{\delta^2}\norm{D-\widetilde D}.
1809 \]
1810 \end{theorem}
1811 
1812 \begin{proof}\phantomsection\label{strong:proof:054}
1813 Add and subtract \(\widetilde B(z-D)^{-1}B^*\) and \(\widetilde B(z-D)^{-1}\widetilde B^*\). The first two differences give the coupling term. For the last, use
1814 \[
1815 (z-D)^{-1}-(z-\widetilde D)^{-1}
1816 =(z-D)^{-1}(D-\widetilde D)(z-\widetilde D)^{-1}.
1817 \]
1818 Apply the resolvent bounds.
1819 \phantomsection\label{strong:proof:054:end}\end{proof}
1820 
1821 \begin{corollary}[Uniform Feshbach control]\label{strong:corollary:013}\label{strong:cor:uniform_feshbach}
1822 For a family represented on fixed retained and hidden Hilbert spaces, with self adjoint hidden Hamiltonians, a common hidden spectral separation \(\delta\), and a common coupling bound \(g\), the self energies are uniformly bounded and uniformly Lipschitz on the declared spectral region. If the couplings and hidden Hamiltonians converge in norm, then the self energies converge uniformly there.
1823 \end{corollary}
1824 
1825 \begin{proof}\phantomsection\label{strong:proof:055}
1826 The common gap and coupling bounds allow \cref{strong:prop:self_energy_bounds,strong:thm:self_energy_stability} to be applied uniformly across the family.
1827 \phantomsection\label{strong:proof:055:end}\end{proof}
1828 
1829 This result identifies hypotheses that make Feshbach control uniform in the cutoff. Whether a chosen geometric refinement tower satisfies those hypotheses remains a separate question.
1830 
1831 \section{Euclidean transfer and finite reflection positivity}\label{strong:section:010}\label{strong:sec:euclidean}
1832 
1833 Let \(H=H^*\) on a nonzero finite dimensional Hilbert space and let
1834 \[
1835 E_0=\min\Spec H.
1836 \]
1837 For a lattice time step \(a>0\), define
1838 \[
1839 T=\e^{-a(H-E_0I)}.
1840 \]
1841 
1842 \begin{theorem}[Finite Euclidean transfer]\label{strong:theorem:036}\label{strong:thm:euclidean_transfer}
1843 The operator \(T\) is positive definite, self adjoint, and contractive. Its spectrum lies in \((0,1]\), and
1844 \[
1845 T^n=\e^{-na(H-E_0I)}
1846 \]
1847 for every \(n\in\NN\).
1848 \end{theorem}
1849 
1850 \begin{proof}\phantomsection\label{strong:proof:056}
1851 Diagonalize \(H\). Each eigenvalue of \(T\) is
1852 \[
1853 \e^{-a(E_j-E_0)}
1854 \]
1855 and therefore lies in \((0,1]\). The power law follows from the functional calculus.
1856 \phantomsection\label{strong:proof:056:end}\end{proof}
1857 
1858 \begin{theorem}[Hamiltonian reconstruction from transfer]\label{strong:theorem:037}\label{strong:thm:transfer_logarithm}
1859 Let \(T\) be positive definite, self adjoint, and contractive. Then
1860 \[
1861 H_T=-a^{-1}\log T
1862 \]
1863 is a nonnegative self adjoint operator and
1864 \[
1865 T^n=\e^{-naH_T}.
1866 \]
1867 It is the unique nonnegative self adjoint operator with \(T=\e^{-aH_T}\).
1868 \end{theorem}
1869 
1870 \begin{proof}\phantomsection\label{strong:proof:057}
1871 Apply the real logarithm to the positive spectrum of \(T\). The spectral calculus gives existence and the power identity. Injectivity of the scalar exponential on \([0,\infty)\) gives uniqueness.
1872 \phantomsection\label{strong:proof:057:end}\end{proof}
1873 
1874 \begin{theorem}[Finite reflection positivity]\label{strong:theorem:038}\label{strong:thm:finite_reflection_positivity}
1875 For vectors \(v_1,\ldots,v_m\in\Hcal\) and nonnegative integers \(n_1,\ldots,n_m\), the matrix
1876 \[
1877 G_{jk}=\ip{v_j}{T^{n_j+n_k}v_k}
1878 \]
1879 is positive semidefinite. Equivalently,
1880 \[
1881 \sum_{j,k}\overline{c_j}c_k
1882 \ip{v_j}{T^{n_j+n_k}v_k}
1883 \geq0
1884 \]
1885 for every \(c\in\CC^m\).
1886 \end{theorem}
1887 
1888 \begin{proof}\phantomsection\label{strong:proof:058}
1889 Since \(T\) is positive and self adjoint,
1890 \[
1891 \ip{v_j}{T^{n_j+n_k}v_k}
1892 =
1893 \ip{T^{n_j}v_j}{T^{n_k}v_k}.
1894 \]
1895 The quadratic form is the squared norm of
1896 \[
1897 \sum_kc_kT^{n_k}v_k.
1898 \]
1899 \phantomsection\label{strong:proof:058:end}\end{proof}
1900 
1901 \begin{corollary}[Reflection positive correlators]\label{strong:corollary:014}\label{strong:cor:reflection_correlators}
1902 Fix a vector \(\Omega\) and operators \(A_1,\ldots,A_m\). The matrix
1903 \[
1904 C_{jk}
1905 =
1906 \ip{A_j\Omega}{T^{n_j+n_k}A_k\Omega}
1907 \]
1908 is positive semidefinite.
1909 \end{corollary}
1910 
1911 \begin{proof}\phantomsection\label{strong:proof:059}
1912 Apply \cref{strong:thm:finite_reflection_positivity} with \(v_j=A_j\Omega\).
1913 \phantomsection\label{strong:proof:059:end}\end{proof}
1914 
1915 \begin{figure}[t]
1916 \centering
1917 \includegraphics[width=.68\textwidth]{12_reflection_positivity.pdf}
1918 \caption{Eigenvalues of a finite reflection positive Gram matrix. Small numerical negative values would signal either roundoff or failure of the transfer assumptions.}
1919 \label{strong:fig:reflection_positivity}
1920 \end{figure}
1921 
1922 \begin{proposition}[Ground packet convergence]\label{strong:proposition:007}\label{strong:prop:ground_packet}
1923 Let \(P_0\) be the spectral projector of \(H\) at \(E_0\), and let
1924 \[
1925 \Delta=\min\{E_j-E_0:E_j>E_0\}.
1926 \]
1927 If an excited eigenvalue exists, then
1928 \[
1929 \norm{T^n-P_0}=\e^{-na\Delta}.
1930 \]
1931 \end{proposition}
1932 
1933 \begin{proof}\phantomsection\label{strong:proof:060}
1934 In the spectral decomposition, \(T^n-P_0\) has eigenvalues zero on the ground space and \(\e^{-na(E_j-E_0)}\) on the excited eigenspaces. The largest is \(\e^{-na\Delta}\).
1935 \phantomsection\label{strong:proof:060:end}\end{proof}
1936 
1937 \begin{proposition}[Symmetry resolved transfer]\label{strong:proposition:008}\label{strong:prop:symmetry_transfer}
1938 If \([H,P_a]=0\) for a complete packet family, then
1939 \[
1940 T=\sum_aP_aTP_a
1941 \]
1942 and each reflection positive Gram matrix decomposes as the sum of its packet Gram matrices.
1943 \end{proposition}
1944 
1945 \begin{proof}\phantomsection\label{strong:proof:061}
1946 The functional calculus preserves commutation. Decompose every vector into packet components and use orthogonality.
1947 \phantomsection\label{strong:proof:061:end}\end{proof}
1948 
1949 Finite reflection positivity is a necessary structural property of transfer data. It is not by itself the continuum Osterwalder--Schrader theorem, which also requires a suitable algebra of Euclidean fields, covariance, regularity, clustering, and a controlled continuum limit \cite{OsterwalderSchrader1973,OsterwalderSchrader1975,GlimmJaffe1987}.
1950 
1951 \section{Finite spectral geometry and diffusion dimension}\label{strong:section:011}\label{strong:sec:spectral_geometry}
1952 
1953 A finite spectral datum consists of a finite dimensional Hilbert space, a nonnegative self adjoint operator \(L\), and any declared packet family. One may take \(L=D^2\) for a finite Dirac type operator or a graph Laplacian. Related geometric viewpoints appear in \cite{Connes1994,Vassilevich2003}.
1954 
1955 \begin{definition}[Heat trace and spectral dimension]\label{strong:definition:008}\label{strong:def:spectral_dimension}
1956 Let the eigenvalues of \(L\) be \(\lambda_1,\ldots,\lambda_N\), with at least one zero eigenvalue. Define
1957 \[
1958 Z_L(t)=\Tr(\e^{-tL})
1959 \]
1960 and
1961 \[
1962 d_L(t)
1963 =-2\frac{\dd\log Z_L(t)}{\dd\log t}
1964 =
1965 2t
1966 \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
1967 \]
1968 \end{definition}
1969 
1970 \begin{proposition}[Finite scale limits]\label{strong:proposition:009}\label{strong:prop:finite_spectral_limits}
1971 For every finite nonnegative \(L\) with nontrivial kernel,
1972 \[
1973 \lim_{t\downarrow0}d_L(t)=0
1974 \]
1975 and
1976 \[
1977 \lim_{t\to\infty}d_L(t)=0.
1978 \]
1979 For every \(t>0\),
1980 \[
1981 0\leq d_L(t)\leq2t\norm L.
1982 \]
1983 \end{proposition}
1984 
1985 \begin{proof}\phantomsection\label{strong:proof:062}
1986 The formula in \cref{strong:def:spectral_dimension} is twice \(t\) times a weighted average of the eigenvalues. This gives the bound and the limit at zero. At large time, the denominator converges to \(\dim\Ker L\), while every positive eigenvalue contribution in the numerator decays exponentially.
1987 \phantomsection\label{strong:proof:062:end}\end{proof}
1988 
1989 Thus a finite system cannot have a nonzero spectral dimension at the extreme ultraviolet or infrared ends. A geometric dimension can appear only in a refinement window.
1990 
1991 \begin{proposition}[Heat packet decomposition]\label{strong:proposition:010}\label{strong:prop:heat_packets}
1992 If \([L,P_a]=0\) for every packet, then
1993 \[
1994 Z_L(t)=\sum_aZ_{L,a}(t),
1995 \qquad
1996 Z_{L,a}(t)=\Tr(P_a\e^{-tL}).
1997 \]
1998 The global spectral dimension is the average of the packet dimensions weighted by their heat traces:
1999 \[
2000 d_L(t)
2001 =
2002 \sum_{a:Z_{L,a}(t)>0}
2003 \frac{Z_{L,a}(t)}{Z_L(t)}
2004  d_{L,a}(t),
2005 \]
2006 where
2007 \[
2008 d_{L,a}(t)
2009 =2t
2010 \frac{\Tr(P_aL\e^{-tL})}{Z_{L,a}(t)}
2011 \]
2012 for nonempty packet heat traces.
2013 \end{proposition}
2014 
2015 \begin{proof}\phantomsection\label{strong:proof:063}
2016 Commutation gives a block diagonal heat operator. Add the traces and the corresponding first spectral moments, then divide by the total heat trace.
2017 \phantomsection\label{strong:proof:063:end}\end{proof}
2018 
2019 \subsection{Periodic lattices}\label{strong:subsec:periodic_lattices}
2020 
2021 Let \(N\geq2\) and let \(\Lambda_N=(\ZZ/N\ZZ)^d\) be the unit flat torus grid with spacing \(1/N\). The standard periodic difference Laplacian has eigenvalues
2022 \[
2023 \lambda_N(k)
2024 =
2025 4N^2
2026 \sum_{j=1}^{d}
2027 \sin^2\left(\frac{\pi k_j}{N}\right),
2028 \]
2029 where \(k\) runs through centered residue representatives. This is the standard periodic finite difference Laplacian \cite{Strikwerda2004}.
2030 
2031 \begin{theorem}[Low mode dispersion in dimension \(d\)]\label{strong:theorem:039}\label{strong:thm:dispersion_bound}
2032 For every centered mode \(k\),
2033 \[
2034 0
2035 \leq
2036 4\pi^2\abs{k}^2-\lambda_N(k)
2037 \leq
2038 \frac{4\pi^4}{3N^2}
2039 \sum_{j=1}^{d}k_j^4.
2040 \]
2041 \end{theorem}
2042 
2043 \begin{proof}\phantomsection\label{strong:proof:064}
2044 For \(x\in[0,\pi/2]\),
2045 \[
2046 \sin x\leq x
2047 \]
2048 and
2049 \[
2050 \sin x\geq x-\frac{x^3}{6}.
2051 \]
2052 Thus
2053 \[
2054 0\leq x^2-\sin^2x\leq\frac{x^4}{3}.
2055 \]
2056 Apply this with \(x=\pi\abs{k_j}/N\), multiply by \(4N^2\), and sum over \(j\).
2057 \phantomsection\label{strong:proof:064:end}\end{proof}
2058 
2059 \begin{theorem}[Fixed time heat trace convergence]\label{strong:theorem:040}\label{strong:thm:heat_trace_convergence}
2060 For every \(t>0\),
2061 \[
2062 \sum_{k\in\Lambda_N^*}
2063 \e^{-t\lambda_N(k)}
2064 \longrightarrow
2065 \sum_{k\in\ZZ^d}
2066 \e^{-4\pi^2t\abs{k}^2}
2067 \]
2068 as \(N\to\infty\). The first spectral moments also converge:
2069 \[
2070 \sum_{k\in\Lambda_N^*}
2071 \lambda_N(k)\e^{-t\lambda_N(k)}
2072 \longrightarrow
2073 \sum_{k\in\ZZ^d}
2074 4\pi^2\abs{k}^2\e^{-4\pi^2t\abs{k}^2}.
2075 \]
2076 Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
2077 \end{theorem}
2078 
2079 \begin{proof}\phantomsection\label{strong:proof:065}
2080 For a fixed integer mode, \cref{strong:thm:dispersion_bound} gives
2081 \[
2082 \lambda_N(k)\longrightarrow4\pi^2\abs{k}^2.
2083 \]
2084 For centered modes, the inequality
2085 \[
2086 \sin x\geq\frac{2x}{\pi},
2087 \qquad
2088 0\leq x\leq\frac\pi2,
2089 \]
2090 gives
2091 \[
2092 \lambda_N(k)\geq16\abs{k}^2.
2093 \]
2094 Also \(\lambda_N(k)\leq4\pi^2\abs{k}^2\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by
2095 \[
2096 \e^{-16t\abs{k}^2},
2097 \]
2098 and the moment terms by
2099 \[
2100 4\pi^2\abs{k}^2\e^{-16t\abs{k}^2}.
2101 \]
2102 Both dominating sequences are summable on \(\ZZ^d\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.
2103 \phantomsection\label{strong:proof:065:end}\end{proof}
2104 
2105 \begin{standardfact}[Theta transformation]\label{strong:standardfact:004}\label{strong:fact:theta_transform}
2106 The continuum flat torus heat trace satisfies
2107 \[
2108 Z_d(t)
2109 =
2110 \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2}
2111 =
2112 (4\pi t)^{-d/2}
2113 \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
2114 \]
2115 This is the Poisson summation formula for a Gaussian.
2116 \end{standardfact}
2117 
2118 \begin{theorem}[Continuum torus small time dimension]\label{strong:theorem:041}\label{strong:thm:continuum_dimension}
2119 Let
2120 \[
2121 d_{\rm torus}(t)
2122 =-2\frac{\dd\log Z_d(t)}{\dd\log t}.
2123 \]
2124 Then
2125 \[
2126 d_{\rm torus}(t)
2127 =
2128  d+O\left(t^{-1}\e^{-1/(4t)}\right)
2129 \]
2130 as \(t\downarrow0\).
2131 \end{theorem}
2132 
2133 \begin{proof}\phantomsection\label{strong:proof:066}
2134 By \cref{strong:fact:theta_transform},
2135 \[
2136 Z_d(t)
2137 =(4\pi t)^{-d/2}(1+R(t)),
2138 \]
2139 where
2140 \[
2141 R(t)=\sum_{n\neq0}\e^{-\abs{n}^2/(4t)}
2142 =O\bigl(\e^{-1/(4t)}\bigr).
2143 \]
2144 Differentiation gives
2145 \[
2146 tR'(t)=O\left(t^{-1}\e^{-1/(4t)}\right).
2147 \]
2148 Substituting this estimate into the logarithmic derivative gives the stated small time expansion.
2149 \phantomsection\label{strong:proof:066:end}\end{proof}
2150 
2151 \begin{corollary}[Rigorous intermediate dimension window]\label{strong:corollary:015}\label{strong:cor:dimension_window}
2152 For every \(\varepsilon>0\), there is a diffusion time \(t_\varepsilon>0\) and an integer \(N_\varepsilon\) such that
2153 \[
2154 \abs{d_{L_N}(t_\varepsilon)-d}<\varepsilon
2155 \]
2156 for every \(N\geq N_\varepsilon\).
2157 \end{corollary}
2158 
2159 \begin{proof}\phantomsection\label{strong:proof:067}
2160 Choose \(t_\varepsilon\) so that \cref{strong:thm:continuum_dimension} is within \(\varepsilon/2\) of \(d\). Then apply \cref{strong:thm:heat_trace_convergence} to choose \(N_\varepsilon\) so that the discrete spectral dimension is within \(\varepsilon/2\) of the continuum value.
2161 \phantomsection\label{strong:proof:067:end}\end{proof}
2162 
2163 \begin{figure}[t]
2164 \centering
2165 \includegraphics[width=.78\textwidth]{13_spectral_dimension.pdf}
2166 \caption{Finite periodic lattices show an intermediate dimension window. The ultraviolet and infrared limits return to zero because the state space is finite.}
2167 \label{strong:fig:spectral_dimension}
2168 \end{figure}
2169 
2170 The order of limits matters. Taking \(t\downarrow0\) at fixed \(N\) gives zero. Taking \(N\to\infty\) at fixed positive \(t\), then taking \(t\downarrow0\), gives the continuum dimension.
2171 
2172 \section{Constraint closure and anomaly packets}\label{strong:section:012}\label{strong:sec:constraints}
2173 
2174 Let \(C_1,\ldots,C_m\) be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory \cite{Dirac1964,HenneauxTeitelboim1992}. Write
2175 \[
2176 \mathfrak c=\operatorname{span}_{\CC}\{C_1,\ldots,C_m\}
2177 \subseteq\End(\Hcal).
2178 \]
2179 
2180 \begin{definition}[Universal closure obstruction]\label{strong:definition:009}\label{strong:def:closure_obstruction}
2181 Let
2182 \[
2183 q_{\mathfrak c}:\End(\Hcal)\longrightarrow\End(\Hcal)/\mathfrak c
2184 \]
2185 be the quotient map. Define
2186 \[
2187 \Ob_{ab}^{\rm cl}
2188 =
2189 q_{\mathfrak c}([C_a,C_b]).
2190 \]
2191 \end{definition}
2192 
2193 \begin{theorem}[Finite constraint closure criterion]\label{strong:theorem:042}\label{strong:thm:constraint_closure}
2194 The constraint span \(\mathfrak c\) is closed under commutators exactly when
2195 \[
2196 \Ob_{ab}^{\rm cl}=0
2197 \]
2198 for all \(a,b\).
2199 \end{theorem}
2200 
2201 \begin{proof}\phantomsection\label{strong:proof:068}
2202 The quotient class vanishes exactly when \([C_a,C_b]\in\mathfrak c\). This holds for every pair exactly when \(\mathfrak c\) is a Lie algebra under the commutator.
2203 \phantomsection\label{strong:proof:068:end}\end{proof}
2204 
2205 If structure coefficients \(f_{ab}^{\phantom{ab}c}\) are prescribed, define the fixed structure anomaly
2206 \[
2207 \mathfrak A_{ab}
2208 =
2209 [C_a,C_b]
2210 -\ii\sum_cf_{ab}^{\phantom{ab}c}C_c.
2211 \]
2212 Exact realization of the prescribed algebra is the stronger condition \(\mathfrak A_{ab}=0\).
2213 
2214 \begin{proposition}[Physical compression is not a closure test]\label{strong:proposition:011}\label{strong:prop:physical_compression_trivial}
2215 Let
2216 \[
2217 \Hcal_{\phys}=\bigcap_a\Ker C_a
2218 \]
2219 and let \(P_{\phys}\) be its orthogonal projector. Then
2220 \[
2221 P_{\phys}[C_a,C_b]P_{\phys}=0
2222 \]
2223 for every pair \(a,b\), whether or not the constraint span closes.
2224 \end{proposition}
2225 
2226 \begin{proof}\phantomsection\label{strong:proof:069}
2227 Self adjointness and \(C_aP_{\phys}=0\) give \(P_{\phys}C_a=0\). Both terms in the compressed commutator therefore vanish.
2228 \phantomsection\label{strong:proof:069:end}\end{proof}
2229 
2230 The quotient in \cref{strong:def:closure_obstruction} is needed because closure concerns the operator algebra. Compression to physical states tests only the action on the common kernel, where the constraints already vanish.
2231 
2232 \begin{figure}[t]
2233 \centering
2234 \includegraphics[width=.88\textwidth]{15_constraint_obstruction.pdf}
2235 \caption{The closure obstruction lives in a quotient by the constraint span. Physical compression is automatic for self adjoint constraints and cannot replace this quotient.}
2236 \label{strong:fig:constraint_obstruction}
2237 \end{figure}
2238 
2239 \begin{theorem}[Invariance of the physical space]\label{strong:theorem:043}\label{strong:thm:physical_invariance}
2240 Let \(H=H^*\). Suppose
2241 \[
2242 [H,C_a]
2243 =
2244 \sum_cb_a^{\phantom a c}C_c
2245 \]
2246 for every \(a\). Then \(H\) preserves \(\Hcal_{\phys}\), and so does \(\e^{-\ii tH}\) for every real \(t\).
2247 \end{theorem}
2248 
2249 \begin{proof}\phantomsection\label{strong:proof:070}
2250 For \(\psi\in\Hcal_{\phys}\),
2251 \[
2252 C_aH\psi
2253 =HC_a\psi-[H,C_a]\psi
2254 =0.
2255 \]
2256 Thus \(H\psi\in\Hcal_{\phys}\). The exponential preserves every invariant finite dimensional subspace.
2257 \phantomsection\label{strong:proof:070:end}\end{proof}
2258 
2259 \subsection{Constraint refinement}\label{strong:subsec:constraint_refinement}
2260 
2261 Let \((\Hcal_\ell,C_{\ell,a})\) and \((\Hcal_{\ell+1},C_{\ell+1,a})\) be two levels, and let
2262 \[
2263 R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1}
2264 \]
2265 be an isometry.
2266 
2267 \begin{theorem}[Exact constraint transport]\label{strong:theorem:044}\label{strong:thm:constraint_transport}
2268 Assume
2269 \[
2270 C_{\ell+1,a}R_\ell=R_\ell C_{\ell,a}
2271 \]
2272 for every \(a\). Then
2273 \[
2274 [C_{\ell+1,a},C_{\ell+1,b}]R_\ell
2275 =
2276 R_\ell[C_{\ell,a},C_{\ell,b}].
2277 \]
2278 If the same structure coefficients are used at both levels, then
2279 \[
2280 \mathfrak A_{\ell+1,ab}R_\ell
2281 =
2282 R_\ell\mathfrak A_{\ell,ab}.
2283 \]
2284 The isometry maps coarse physical states into fine physical states.
2285 \end{theorem}
2286 
2287 \begin{proof}\phantomsection\label{strong:proof:071}
2288 Apply the intertwining identity twice to each product in the commutator. The anomaly identity follows by subtracting the same linear combination of constraints. If \(C_{\ell,a}\psi=0\), then
2289 \[
2290 C_{\ell+1,a}R_\ell\psi=R_\ell C_{\ell,a}\psi=0.
2291 \]
2292 \phantomsection\label{strong:proof:071:end}\end{proof}
2293 
2294 \begin{proposition}[Approximate constraint transport]\label{strong:proposition:012}\label{strong:prop:approx_constraint_transport}
2295 Assume
2296 \[
2297 \norm{C_{\ell+1,a}R_\ell-R_\ell C_{\ell,a}}
2298 \leq\varepsilon_a
2299 \]
2300 and
2301 \[
2302 \norm{C_{\ell,a}},\norm{C_{\ell+1,a}}\leq M.
2303 \]
2304 Then a coarse physical unit vector \(\psi\) satisfies
2305 \[
2306 \norm{C_{\ell+1,a}R_\ell\psi}\leq\varepsilon_a.
2307 \]
2308 Also,
2309 \[
2310 \begin{aligned}
2311 &\norm{
2312 [C_{\ell+1,a},C_{\ell+1,b}]R_\ell
2313 -R_\ell[C_{\ell,a},C_{\ell,b}]
2314 }\\
2315 &\hspace{4em}\leq
2316 2M(\varepsilon_a+\varepsilon_b).
2317 \end{aligned}
2318 \]
2319 \end{proposition}
2320 
2321 \begin{proof}\phantomsection\label{strong:proof:072}
2322 The first claim follows by applying the intertwining defect to \(\psi\). Write
2323 \[
2324 E_a=C_{\ell+1,a}R_\ell-R_\ell C_{\ell,a}.
2325 \]
2326 Expanding the two commutators gives four error terms:
2327 \[
2328 E_aC_{\ell,b},
2329 \quad
2330 C_{\ell+1,a}E_b,
2331 \quad
2332 -E_bC_{\ell,a},
2333 \quad
2334 -C_{\ell+1,b}E_a.
2335 \]
2336 The triangle inequality and the stated bounds on the constraints and defects give the claimed estimate.
2337 \phantomsection\label{strong:proof:072:end}\end{proof}
2338 
2339 Approximate transport controls finite violations. It does not turn a nonzero anomaly into an exact continuum algebra without a limit theorem and a common operator domain.
2340 
2341 \section{Quantum refinement towers}\label{strong:section:013}\label{strong:sec:towers}
2342 
2343 \begin{definition}[Finite quantum refinement tower]\label{strong:definition:010}\label{strong:def:quantum_tower}
2344 A finite quantum refinement tower consists of Hilbert spaces \(\Hcal_\ell\), isometries
2345 \[
2346 R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
2347 \]
2348 self adjoint Hamiltonians \(H_\ell\), packet projectors \(P_{\ell,a}\), constraints \(C_{\ell,r}\), and positive transfer operators \(T_\ell\). Exact compatibility means that every declared object intertwines with \(R_\ell\). Approximate compatibility is recorded by the corresponding operator norm defects.
2349 \end{definition}
2350 
2351 \begin{figure}[t]
2352 \centering
2353 \includegraphics[width=.82\textwidth]{14_continuum_tower.pdf}
2354 \caption{A refinement tower carries Hamiltonians, packets, constraints, and transfer data. Each family has its own compatibility defect.}
2355 \label{strong:fig:continuum_tower}
2356 \end{figure}
2357 
2358 \subsection{Exact bounded direct limits}\label{strong:subsec:exact_direct_limit}
2359 
2360 Let \(\Hcal_\infty\) be the Hilbert direct limit of the isometric system, and let
2361 \[
2362 \iota_\ell:\Hcal_\ell\longrightarrow\Hcal_\infty
2363 \]
2364 be the canonical isometries.
2365 
2366 \begin{theorem}[Bounded Hamiltonian direct limit]\label{strong:theorem:045}\label{strong:thm:bounded_direct_limit}
2367 Assume
2368 \[
2369 H_{\ell+1}R_\ell=R_\ell H_\ell
2370 \]
2371 and
2372 \[
2373 \sup_\ell\norm{H_\ell}<\infty.
2374 \]
2375 Then there is a unique bounded self adjoint operator \(H_\infty\) on \(\Hcal_\infty\) such that
2376 \[
2377 H_\infty\iota_\ell=\iota_\ell H_\ell
2378 \]
2379 for every \(\ell\). It satisfies
2380 \[
2381 \e^{-\ii tH_\infty}\iota_\ell
2382 =
2383 \iota_\ell\e^{-\ii tH_\ell}
2384 \]
2385 and
2386 \[
2387 \e^{-tH_\infty}\iota_\ell
2388 =
2389 \iota_\ell\e^{-tH_\ell}
2390 \]
2391 when the Hamiltonians are nonnegative.
2392 \end{theorem}
2393 
2394 \begin{proof}\phantomsection\label{strong:proof:073}
2395 Define \(H_\infty\) on the algebraic union of the embedded levels by
2396 \[
2397 H_\infty\iota_\ell x=\iota_\ell H_\ell x.
2398 \]
2399 Compatibility makes the definition independent of the chosen level. The uniform norm bound makes the operator bounded on the dense union, so it extends uniquely to the completion. It is symmetric because every \(H_\ell\) is self adjoint. A bounded symmetric operator defined on the whole Hilbert space is self adjoint. The functional calculus intertwining follows first for polynomials and then for exponentials by uniform convergence.
2400 \phantomsection\label{strong:proof:073:end}\end{proof}
2401 
2402 \begin{theorem}[Exact packet direct limit]\label{strong:theorem:046}\label{strong:thm:packet_direct_limit}
2403 Assume the packet families have one fixed finite address set at every level and each packet address \(a\) is compatible in the sense
2404 \[
2405 P_{\ell+1,a}R_\ell=R_\ell P_{\ell,a}.
2406 \]
2407 Then there is an orthogonal projector \(P_{\infty,a}\) on \(\Hcal_\infty\) satisfying
2408 \[
2409 P_{\infty,a}\iota_\ell=\iota_\ell P_{\ell,a}.
2410 \]
2411 Finite completeness and orthogonality pass to the direct limit.
2412 \end{theorem}
2413 
2414 \begin{proof}\phantomsection\label{strong:proof:074}
2415 Use the same construction as in \cref{strong:thm:bounded_direct_limit}. Every projector has norm at most one, so no additional boundedness assumption is needed. The projector identities hold on the dense algebraic union and hence on the completion.
2416 \phantomsection\label{strong:proof:074:end}\end{proof}
2417 
2418 \subsection{Norm summable approximation}\label{strong:subsec:norm_summable}
2419 
2420 The next theorem places all approximating Hamiltonians on a common Hilbert space. It applies to bounded effective Hamiltonians. An unbounded field Hamiltonian requires a different convergence argument.
2421 
2422 \begin{theorem}[Norm summable Hamiltonian tower]\label{strong:theorem:047}\label{strong:thm:norm_summable_tower}
2423 Let \((H_n)\) be bounded self adjoint operators on one Hilbert space and assume
2424 \[
2425 \sum_{n=0}^{\infty}\norm{H_{n+1}-H_n}<\infty.
2426 \]
2427 Then \(H_n\) converges in norm to a bounded self adjoint operator \(H\). For every \(T>0\),
2428 \[
2429 \sup_{\abs t\leq T}
2430 \norm{\e^{-\ii tH_n}-\e^{-\ii tH}}
2431 \leq
2432 T\norm{H_n-H}.
2433 \]
2434 If every \(H_n\geq0\), then
2435 \[
2436 \sup_{0\leq t\leq T}
2437 \norm{\e^{-tH_n}-\e^{-tH}}
2438 \leq
2439 T\norm{H_n-H}.
2440 \]
2441 \end{theorem}
2442 
2443 \begin{proof}\phantomsection\label{strong:proof:075}
2444 The summability condition makes \((H_n)\) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives
2445 \[
2446 \e^{-\ii tH_n}-\e^{-\ii tH}
2447 =-\ii\int_0^t
2448 \e^{-\ii(t-s)H_n}(H_n-H)\e^{-\ii sH}\dd s.
2449 \]
2450 Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.
2451 \phantomsection\label{strong:proof:075:end}\end{proof}
2452 
2453 \begin{corollary}[Limit of packet leakage]\label{strong:corollary:016}\label{strong:cor:limit_leakage}
2454 Let \(P_n\) be orthogonal projectors and \(H_n\) bounded self adjoint operators on one Hilbert space, with \(P_n\to P\) and \(H_n\to H\) in norm. Then for every bounded time interval,
2455 \[
2456 (I-P_n)\e^{-\ii tH_n}P_n
2457 \longrightarrow
2458 (I-P)\e^{-\ii tH}P
2459 \]
2460 uniformly in time.
2461 \end{corollary}
2462 
2463 \begin{proof}\phantomsection\label{strong:proof:076}
2464 Add and subtract terms so that one factor changes at a time. The projector terms converge uniformly because \(P_n\to P\) in norm. Duhamel's identity gives
2465 \[
2466 \sup_{\abs t\leq T}
2467 \norm{\e^{-\ii tH_n}-\e^{-\ii tH}}
2468 \leq
2469 T\norm{H_n-H},
2470 \]
2471 which proves uniform convergence of the middle factor.
2472 \phantomsection\label{strong:proof:076:end}\end{proof}
2473 
2474 \begin{theorem}[Reflection positivity is closed under limits]\label{strong:theorem:048}\label{strong:thm:reflection_limit}
2475 Suppose \(C_n(j,k)\) are reflection positive kernels on \(\NN\), meaning every finite matrix
2476 \[
2477 \bigl(C_n(j_r,j_s)\bigr)_{r,s}
2478 \]
2479 is positive semidefinite. If
2480 \[
2481 C_n(j,k)\longrightarrow C(j,k)
2482 \]
2483 for every pair, then \(C\) is reflection positive.
2484 \end{theorem}
2485 
2486 \begin{proof}\phantomsection\label{strong:proof:077}
2487 For a fixed finite set of times and coefficients, the corresponding quadratic forms are nonnegative and converge to the quadratic form of \(C\). The limit is nonnegative.
2488 \phantomsection\label{strong:proof:077:end}\end{proof}
2489 
2490 \begin{theorem}[Constraint closure under norm convergence]\label{strong:theorem:049}\label{strong:thm:constraint_limit}
2491 Let \(C_{n,a}\to C_a\) in norm for every \(a\), and fix structure coefficients \(f_{ab}^{\phantom{ab}c}\). If
2492 \[
2493 \norm{
2494 [C_{n,a},C_{n,b}]
2495 -\ii\sum_cf_{ab}^{\phantom{ab}c}C_{n,c}
2496 }
2497 \longrightarrow0,
2498 \]
2499 then
2500 \[
2501 [C_a,C_b]
2502 =\ii\sum_cf_{ab}^{\phantom{ab}c}C_c.
2503 \]
2504 \end{theorem}
2505 
2506 \begin{proof}\phantomsection\label{strong:proof:078}
2507 Operator multiplication and the commutator are continuous in operator norm. Pass to the limit in the anomaly identity.
2508 \phantomsection\label{strong:proof:078:end}\end{proof}
2509 
2510 \begin{theorem}[Heat trace convergence from trace norm]\label{strong:theorem:050}\label{strong:thm:trace_norm_heat}
2511 Let \(L_n,L\) be nonnegative self adjoint operators on one Hilbert space. For a fixed \(t>0\), assume their heat operators are trace class and that
2512 \[
2513 \e^{-tL_n}\longrightarrow\e^{-tL}
2514 \]
2515 in trace norm. Then
2516 \[
2517 \Tr(\e^{-tL_n})\longrightarrow\Tr(\e^{-tL}).
2518 \]
2519 If also the heat moment operators are trace class and
2520 \[
2521 L_n\e^{-tL_n}\longrightarrow L\e^{-tL}
2522 \]
2523 in trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.
2524 \end{theorem}
2525 
2526 \begin{proof}\phantomsection\label{strong:proof:079}
2527 The trace is continuous in trace norm. Apply this to the heat operators and to the first heat moments, then take the quotient in \cref{strong:def:spectral_dimension}.
2528 \phantomsection\label{strong:proof:079:end}\end{proof}
2529 
2530 \begin{remark}\label{strong:remark:003}\label{strong:rem:unbounded_limit}
2531 The theorems above produce bounded continuum Hamiltonians. A physically useful continuum Hamiltonian is often unbounded. Its construction generally requires strong resolvent convergence, convergence of closed quadratic forms, or another method that controls operator domains \cite{Kato1995,ReedSimon1972}. None of those conclusions follows from finite matrix convergence without additional hypotheses.
2532 \end{remark}
2533 
2534 \section{Continuum milestones and exact boundary}\label{strong:section:014}\label{strong:sec:boundary}
2535 
2536 The finite theorems establish exact identities and criteria for bounded limits. Passing from a finite transfer model to a continuum theory still requires several separate arguments. This section states those remaining mathematical tasks.
2537 
2538 \begin{definition}[Anomaly free continuum constraint system]\label{strong:definition:011}\label{strong:def:anomaly_free_continuum}
2539 A continuum constraint system consists of a Hilbert space, a common invariant dense domain \(\Dcal\), and operators \(C_a\) on \(\Dcal\). It is anomaly free for prescribed structure coefficients when
2540 \[
2541 [C_a,C_b]\psi
2542 =
2543 \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
2544 \]
2545 for every \(\psi\in\Dcal\), and when the associated gauge evolution is defined on the physical quotient or physical subspace.
2546 \end{definition}
2547 
2548 Norm closure of bounded finite constraints, as in \cref{strong:thm:constraint_limit}, is one special case. It does not treat unbounded constraints, changing domains, structure functions, or renormalized operator products.
2549 
2550 \begin{definition}[Reflection positive Euclidean limit]\label{strong:definition:012}\label{strong:def:reflection_positive_limit}
2551 A reflection positive Euclidean limit requires a limiting algebra of observables, Euclidean time translations, a reflection operation, and correlation functions whose reflected quadratic forms are nonnegative. A field theoretic reconstruction also requires the regularity and covariance assumptions needed to construct a Hilbert space, a vacuum, and a self adjoint time translation generator.
2552 \end{definition}
2553 
2554 \Cref{strong:thm:reflection_limit} shows that positivity of each fixed finite Gram matrix survives pointwise convergence. It does not prove existence, locality, clustering, or the remaining Osterwalder--Schrader axioms.
2555 
2556 \begin{definition}[Einstein infrared criterion]\label{strong:definition:013}\label{strong:def:einstein_criterion}
2557 A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form
2558 \[
2559 \Gamma_{\rm eff}[g]
2560 =
2561 \int
2562 \sqrt{\abs g}
2563 \left(
2564 \Lambda
2565 +
2566 \frac{1}{16\pi G}R
2567 +
2568 \text{controlled higher order terms}
2569 \right)
2570 \dd^dx.
2571 \]
2572 Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an estimate for the omitted terms on the declared infrared regime.
2573 \end{definition}
2574 
2575 A spectral dimension close to four does not imply this criterion. A heat trace records eigenvalue growth. It does not determine the tensor structure, gauge identities, sign, or interaction coefficients of an effective action.
2576 
2577 \begin{definition}[Massless spin two criterion]\label{strong:definition:014}\label{strong:def:spin_two_criterion}
2578 A positive massless spin two sector requires a physical two point function or resolvent with a massless pole in the transverse traceless sector, positive residue on physical states, absence of negative norm modes, and compatibility with the limiting gauge constraints. Interactions must preserve the constraint structure.
2579 \end{definition}
2580 
2581 A low energy eigenvalue, a spectral dimension plateau, or a finite rank two tensor packet does not establish a massless spin two particle.
2582 
2583 \begin{figure}[t]
2584 \centering
2585 \includegraphics[width=.96\textwidth]{16_continuum_boundary.pdf}
2586 \caption{The finite identities occupy the first step. Every later step needs a separate convergence or reconstruction theorem.}
2587 \label{strong:fig:continuum_boundary}
2588 \end{figure}
2589 
2590 \begin{table}[t]
2591 \centering
2592 \small
2593 \begin{tabular}{>{\raggedright\arraybackslash}p{.25\textwidth}>{\raggedright\arraybackslash}p{.31\textwidth}>{\raggedright\arraybackslash}p{.34\textwidth}}
2594 \toprule
2595 Milestone & Finite result available here & Missing continuum input \\
2596 \midrule
2597 Symmetry and constraints & exact packets and quotient obstruction & common unbounded domain and renormalized closure \\
2598 Effective dynamics & exact Feshbach map and uniform gap bounds & cutoff independent spectral separation and locality \\
2599 Euclidean transfer & positive finite transfer and finite reflection positivity & convergent Euclidean fields and reconstruction axioms \\
2600 Spectral dimension & lattice heat trace convergence & relation to a physical metric and interacting observables \\
2601 Lorentzian theory & bounded unitary direct limits & unbounded generator, causality, locality, and field algebra \\
2602 Einstein regime & a definition of the required expansion & derivation and error control for the effective action \\
2603 Spin two sector & a precise spectral criterion & positive physical pole and interaction consistency \\
2604 \bottomrule
2605 \end{tabular}
2606 \caption{What the finite results establish and what remains outside the article.}
2607 \label{strong:tab:continuum_milestones}
2608 \end{table}
2609 
2610 \section{Conclusion}\label{strong:section:015}\label{strong:sec:conclusion}
2611 
2612 A complete packet family gives one coordinate language for finite quantum structure. Group characters produce symmetry packets. Charged operators obey exact selection rules. Circuits become coherent sums over packet paths. The canonical moving connection separates geometric transport from physical mixing. Density matrices and channels use two sided packets, while channel covariance becomes block preservation on operator space and symmetry of the Choi matrix.
2613 
2614 Compression to a code gives a natural error correction obstruction. Its vanishing is the Knill--Laflamme condition. Diagonalizing the resulting coefficient matrix produces orthogonal syndrome packets and an exact recovery on the error span. Schur--Feshbach elimination gives an equally exact description of hidden Hamiltonian packets. Their frequency response is the self energy, and their time response is a positive definite memory kernel.
2615 
2616 Positive Euclidean transfer gives finite reflection positivity. Periodic lattice heat traces converge to the continuum torus at every fixed positive diffusion time, and the continuum small time spectral dimension is recovered after the refinement limit. Constraint closure is measured by a quotient of operator space, not by compression to the physical kernel.
2617 
2618 The theorems for bounded towers show how exact compatibility and summable differences in operator norm produce limiting quantum evolutions. Their hypotheses also mark the scope of the argument. Unbounded field Hamiltonians, continuum gauge constraints, reflection positive field measures, the Einstein infrared action, and a massless spin two sector each require additional theorems. The finite packet calculus gives a common framework in which those questions can be stated and studied, while leaving their physical resolution open.
2619 
2620 

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