Realization Calculus

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1 \section{The quantum seed realization}\label{core:section:001}
2 
3 \paragraph{Conventions.}
4 Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying
5 \[
6 P_\alpha^*=P_\alpha,\qquad
7 P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad
8 \sum_\alpha P_\alpha=I.
9 \]
10 All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write $[A,B]=AB-BA$ and use units with $\hbar=1$. Parameter derivatives are real derivatives. A matrix $X$ is supported on a code projector $P$ when $PXP=X$.
11 
12 The common operator sequence of the series becomes
13 \[
14 \boxed{
15 \begin{gathered}
16 \text{quantum seed}
17 \longrightarrow
18 \text{gate or observable address}
19 \longrightarrow
20 \text{symmetry projection}\\
21 \longrightarrow
22 \text{packet transfer}
23 \longrightarrow
24 \text{constraint}
25 \longrightarrow
26 \text{refinement}
27 \longrightarrow
28 \text{reconstruction}
29 \end{gathered}
30 }
31 \]
32 
33 \begin{figure}[htbp]
34 \centering
35 \includegraphics[width=0.98\textwidth]{01_quantum_seed_architecture.pdf}
36 \caption{The diagram shows the quantum realization of the common sequence of operators in the seed calculus. The same finite packet language describes states, gates, Hamiltonians, channels, geometric projectors, and effective dynamics.}
37 \label{core:fig:architecture}
38 \end{figure}
39 
40 \begin{definition}[Finite quantum seed system]\label{core:definition:001}
41 A finite quantum seed system is a tuple
42 \[
43 \mathfrak Q=
44 \left(
45 \cH,
46 \psi_0,
47 \cO,
48 \cR,
49 \{P_{\ell,\alpha}\},
50 H,
51 \tau
52 \right),
53 \]
54 where $\cH$ is a finite-dimensional Hilbert space, $\psi_0\in\cH$ is a nonzero seed, $\cO$ is the declared operator grammar, $\cR$ is a collection of algebraic or physical constraints, $\{P_{\ell,\alpha}\}$ is an exact packet system at each refinement level, $H=H^*$ is a Hamiltonian when dynamics are present, and $\tau$ specifies the topology used in a later completion.
55 \end{definition}
56 
57 The generated state space is
58 \[
59 \cH(\psi_0;\cO)
60 =
61 \operatorname{span}
62 \left\{
63 O_r\cdots O_1\psi_0:
64 O_j\in\cO,
65 \ r\ge0
66 \right\}.
67 \]
68 The quantum seed is cyclic for the grammar when
69 \[
70 \cH(\psi_0;\cO)=\cH.
71 \]
72 This is a statement about the pair $(\psi_0,\cO)$, not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
73 
74 \section{Quantum seed rank and cyclicity}\label{core:section:002}
75 
76 \subsection{Finite observable algebras}
77 
78 Let a finite-dimensional unital $*$-subalgebra of operators act in its standard multiplicity form
79 \[
80 \cA
81 \cong
82 \bigoplus_{i=1}^r
83 \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
84 \]
85 on
86 \[
87 \cH
88 \cong
89 \bigoplus_{i=1}^r
90 \left(\bbC^{d_i}\otimes\bbC^{m_i}\right).
91 \]
92 Here every $d_i$ is a positive integer and every multiplicity $m_i$ is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector $\psi_i\in\bbC^{d_i}\otimes\bbC^{m_i}$ is identified with a $d_i\times m_i$ coefficient matrix $C_i$.
93 
94 \begin{theorem}[Orbit dimension of one quantum seed]\label{core:theorem:001}\label{core:thm:seed-orbit}
95 For integers $d\ge1$ and $m\ge0$, consider the action of $M_d(\bbC)\otimes I_m$ on $\bbC^d\otimes\bbC^m$. Let $C$ be the coefficient matrix of a vector $\psi$. Then
96 \[
97 \boxed{
98 \dim\operatorname{span}\{(A\otimes I_m)\psi:A\in M_d(\bbC)\}
99 =d\,\rank C.
100 }
101 \]
102 The seed is cyclic if and only if $\rank C=m$, which is possible only when $m\le d$.
103 \end{theorem}
104 
105 \begin{proof}\phantomsection\label{core:proof:001}
106 Under the identification by coefficient matrices, $(A\otimes I_m)\psi$ corresponds to $AC$. Every row of $AC$ lies in the row space of $C$. Conversely, any $d\times m$ matrix whose rows lie in the row space of $C$ equals $AC$ for a suitable $A$: choose each row of $A$ to produce the desired row combination. The orbit therefore consists of all matrices with $d$ independently chosen rows in the $\rank C$-dimensional row space of $C$, and has dimension $d\rank C$. It equals the full $dm$-dimensional space exactly when $\rank C=m$.
107 \phantomsection\label{core:proof:001:end}\end{proof}
108 
109 \begin{theorem}[Quantum seed-rank formula]\label{core:theorem:002}\label{core:thm:seed-rank}
110 For
111 \[
112 \cA=\bigoplus_i(M_{d_i}(\bbC)\otimes I_{m_i})
113 \quad\text{on}\quad
114 \cH=\bigoplus_i(\bbC^{d_i}\otimes\bbC^{m_i}),
115 \]
116 the minimum number of seeds required to generate $\cH$ is
117 \[
118 \boxed{
119 \operatorname{srank}_{\cA}(\cH)
120 =
121 \max_i\left\lceil\frac{m_i}{d_i}\right\rceil.
122 }
123 \]
124 \end{theorem}
125 
126 \begin{proof}\phantomsection\label{core:proof:002}
127 Each seed contributes a coefficient matrix of rank at most $d_i$ in the $i$th block. Thus $k$ seeds can cover at most a $kd_i$-dimensional subspace of the multiplicity factor $\bbC^{m_i}$, so $k\ge\lceil m_i/d_i\rceil$ for every $i$.
128 
129 For the converse, take
130 \[
131 k=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.
132 \]
133 In each block, partition a basis of $\bbC^{m_i}$ into at most $k$ groups of size at most $d_i$. Choose the coefficient matrix of seed $a$ so that its nonzero rows span the $a$th group. The combined row spaces of the $k$ seeds then span $\bbC^{m_i}$ in every block. The preceding theorem shows that their $\cA$-orbits generate the whole direct sum.
134 \phantomsection\label{core:proof:002:end}\end{proof}
135 
136 \begin{corollary}[Local orbit and Schmidt rank]\label{core:corollary:001}\label{core:thm:schmidt}
137 Let $\psi\in\cH_A\otimes\cH_B$, with $\dim\cH_A=d_A\ge1$, and let $r$ be its Schmidt rank. The reduced positive operator on $B$ is $\rho_B=\Tr_A(\ket\psi\bra\psi)$. Normalization of $\psi$ does not affect its support. Then
138 \[
139 \boxed{
140 \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in M_{d_A}(\bbC)\}
141 =d_A r.
142 }
143 \]
144 In particular, local operators on $A$ generate the complete bipartite Hilbert space exactly when the reduced state on $B$ has full support.
145 \end{corollary}
146 
147 \begin{proof}\phantomsection\label{core:proof:003}
148 A Schmidt decomposition of length $r$ gives a coefficient matrix of rank $r$. Applying Theorem~\ref{core:thm:seed-orbit} gives the orbit dimension $d_A r$. The range of $\rho_B$ is the span of the $r$ Schmidt vectors on $B$, so full support is equivalent to $r=\dim\cH_B$. Because $d_A>0$, this is equivalent to the orbit having the full bipartite dimension.
149 \phantomsection\label{core:proof:003:end}\end{proof}
150 
151 The conclusion concerns the algebraic span obtained from arbitrary local operators. A normalized unitary gate orbit need not contain every vector, and the span statement does not supply a deterministic protocol for preparing arbitrary remote states. Entanglement controls the multiplicity support appearing in that algebraic span.
152 
153 \subsection{A computational-basis seed}
154 
155 \begin{theorem}[One seed for the computational basis]\label{core:theorem:003}\label{core:thm:computational}
156 For an integer $n\ge0$, let
157 \[
158 \psi_0=\ket{0}^{\otimes n}
159 \]
160 and let $X_j$ denote the Pauli $X$ operator on qubit $j$. Then
161 \[
162 \left\{
163 X_1^{a_1}\cdots X_n^{a_n}\psi_0:
164 (a_1,\ldots,a_n)\in\{0,1\}^n
165 \right\}
166 \]
167 is the complete computational basis. Hence the seed is cyclic under the algebra generated by the $X_j$.
168 \end{theorem}
169 
170 \begin{proof}\phantomsection\label{core:proof:004}
171 For a binary address $a=(a_1,\ldots,a_n)$,
172 \[
173 X_1^{a_1}\cdots X_n^{a_n}\ket{0\cdots0}
174 =
175 \ket{a_1\cdots a_n}.
176 \]
177 Distinct binary addresses produce distinct orthonormal basis vectors. There are $2^n$ of them, equal to the dimension of the Hilbert space.
178 \phantomsection\label{core:proof:004:end}\end{proof}
179 
180 \begin{certificate}[Three-qubit seed orbit]\label{core:certificate:001}
181 The original numerical implementation generated all eight computational basis states from $\ket{000}$ using the three local $X$ address operators. It reported an orbit rank of $8/8$.
182 \end{certificate}
183 
184 \section{Cyclic symmetry packets}\label{core:section:003}
185 
186 \subsection{Character projectors}
187 
188 Let $N\ge1$ be an integer, and let $S$ be a unitary symmetry whose order divides $N$:
189 \[
190 S^N=I.
191 \]
192 Set $\omega=e^{2\pi i/N}$.
193 
194 \begin{theorem}[Cyclic quantum projectors]\label{core:theorem:004}\label{core:thm:cyclic}
195 The operators
196 \[
197 \boxed{
198 P_j
199 =
200 \frac1N\sum_{r=0}^{N-1}\omega^{-jr}S^r,
201 \qquad
202 0\le j<N,
203 }
204 \]
205 are self-adjoint, pairwise orthogonal projectors satisfying
206 \[
207 P_jP_k=\delta_{jk}P_j,
208 \qquad
209 \sum_{j=0}^{N-1}P_j=I,
210 \qquad
211 SP_j=\omega^jP_j.
212 \]
213 \end{theorem}
214 
215 \begin{proof}\phantomsection\label{core:proof:005}
216 Using $S^{r+s}=S^{r+s\bmod N}$ and the finite character orthogonality relation,
217 \[
218 \frac1N\sum_{r=0}^{N-1}\omega^{(k-j)r}=\delta_{jk},
219 \]
220 we obtain $P_jP_k=\delta_{jk}P_j$. Summing over $j$ cancels every term indexed by $r\ne0$ and retains the $r=0$ term, which proves completeness. Unitarity and the change of residue $r\mapsto-r$ give
221 \[
222 P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
223 \]
224 Finally, multiplication by $S$ shifts the summation index and contributes the factor $\omega^j$.
225 \phantomsection\label{core:proof:005:end}\end{proof}
226 
227 For parity,
228 \[
229 S=Z^{\otimes n},
230 \qquad
231 P_{\mathrm{even}}=\frac{I+S}{2},
232 \qquad
233 P_{\mathrm{odd}}=\frac{I-S}{2}.
234 \]
235 
236 \subsection{Tensor refinement}
237 
238 Fix $N\ge1$. Suppose every factor $a$ has a complete orthogonal family $P^{(a)}_0,\ldots,P^{(a)}_{N-1}$. The fine tensor packets are
239 \[
240 P_{\mathbf j}^{\mathrm{fine}}
241 =
242 P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
243 \]
244 
245 \begin{theorem}[Tensor/polyphase refinement]\label{core:theorem:005}\label{core:thm:tensor}
246 For each residue $r\in\bbZ/N\bbZ$, define
247 \[
248 \boxed{
249 P_r^{\mathrm{coarse}}
250 =
251 \sum_{j_1+\cdots+j_m\equiv r\pmod N}
252 P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
253 }
254 \]
255 Then the $P_r^{\mathrm{coarse}}$ are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
256 \end{theorem}
257 
258 \begin{proof}\phantomsection\label{core:proof:006}
259 The fine tensor projectors are mutually orthogonal and sum to the identity. Grouping the fine addresses by their total residue partitions the full address set. Sums over disjoint sets of orthogonal projectors remain orthogonal projectors, and the union of the residue classes is the full fine address set.
260 \phantomsection\label{core:proof:006:end}\end{proof}
261 
262 \begin{figure}[htbp]
263 \centering
264 \includegraphics[width=0.86\textwidth]{10_tensor_parity_refinement.pdf}
265 \caption{The eight computational packets of three qubits refine the two total-parity packets exactly.}
266 \label{core:fig:tensor-refinement}
267 \end{figure}
268 
269 \begin{certificate}[Exact parity refinement]\label{core:certificate:002}
270 For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with $(I\pm Z^{\otimes3})/2$ with residual $0$.
271 \end{certificate}
272 
273 \section{Quantum circuits as packet paths}\label{core:section:004}
274 
275 Let
276 \[
277 U=U_mU_{m-1}\cdots U_1
278 \]
279 be a circuit with $m\ge1$, and let $\{P_\alpha\}$ be a complete packet system. For the empty circuit $m=0$, set $U=I$. Its boundary block is $P_\beta P_\alpha=\delta_{\beta\alpha}P_\alpha$.
280 
281 \begin{theorem}[Exact circuit path expansion]\label{core:theorem:006}\label{core:thm:paths}
282 For source packet $\alpha_0$ and target packet $\alpha_m$,
283 \[
284 \boxed{
285 P_{\alpha_m}UP_{\alpha_0}
286 =
287 \sum_{\alpha_1,\ldots,\alpha_{m-1}}
288 P_{\alpha_m}U_mP_{\alpha_{m-1}}
289 \cdots
290 P_{\alpha_1}U_1P_{\alpha_0}.
291 }
292 \]
293 Summing over the source and target packets reconstructs the complete circuit operator.
294 \end{theorem}
295 
296 \begin{proof}\phantomsection\label{core:proof:007}
297 Insert the identity resolution $I=\sum_\alpha P_\alpha$ between every adjacent pair of gates. Distributing the finite sums gives the path expansion. Summing the left and right boundary projectors reconstructs $U$.
298 \phantomsection\label{core:proof:007:end}\end{proof}
299 
300 \begin{theorem}[Packetwise unitarity conservation]\label{core:theorem:007}\label{core:thm:unitarity}
301 If $U$ is unitary and $U_{\beta\alpha}=P_\beta UP_\alpha$, then
302 \[
303 \boxed{
304 \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma}
305 =
306 \delta_{\alpha\gamma}P_\alpha.
307 }
308 \]
309 Similarly,
310 \[
311 \sum_\alpha U_{\beta\alpha}U_{\delta\alpha}^*
312 =
313 \delta_{\beta\delta}P_\beta.
314 \]
315 \end{theorem}
316 
317 \begin{proof}\phantomsection\label{core:proof:008}
318 Using completeness and unitarity,
319 \[
320 \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma}
321 =
322 P_\alpha U^*\left(\sum_\beta P_\beta\right)UP_\gamma
323 =
324 P_\alpha P_\gamma.
325 \]
326 For the second identity, sum over the source packets and use unitarity in the reverse order.
327 \phantomsection\label{core:proof:008:end}\end{proof}
328 
329 \begin{example}[Bell seed circuit]\label{core:example:001}
330 In this example only, $H$ denotes the Hadamard gate, not a Hamiltonian. Starting from $\ket{00}$,
331 \[
332 (H\otimes I)\ket{00}
333 =
334 \frac{\ket{00}+\ket{10}}{\sqrt2},
335 \]
336 and then
337 \[
338 \operatorname{CNOT}(H\otimes I)\ket{00}
339 =
340 \frac{\ket{00}+\ket{11}}{\sqrt2}.
341 \]
342 The parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addresses.
343 \end{example}
344 
345 \begin{figure}[htbp]
346 \centering
347 \begin{minipage}[t]{0.48\textwidth}
348 \centering
349 \includegraphics[width=\textwidth]{02_bell_seed_probabilities.pdf}
350 \captionof{figure}{The Bell state generated from the seed $\ket{00}$.}
351 \label{core:fig:bell}
352 \end{minipage}\hfill
353 \begin{minipage}[t]{0.48\textwidth}
354 \centering
355 \includegraphics[width=\textwidth]{04_circuit_packet_paths.pdf}
356 \captionof{figure}{Circuit reconstruction as a sum over packet paths.}
357 \label{core:fig:circuit-paths}
358 \end{minipage}
359 \end{figure}
360 
361 \begin{certificate}[Bell and path reconstruction]\label{core:certificate:003}
362 The original numerical implementation reported residuals of $0$ for both the Bell state and its complete reconstruction from parity paths.
363 \end{certificate}
364 
365 \section{Symmetry-resolved Hamiltonians and leakage}\label{core:section:005}
366 
367 Let $H=H^*$ and let $P$ be a packet projector. If $[H,P]=0$, then $P\cH$ is invariant under $e^{-itH}$.
368 
369 \begin{theorem}[Exact symmetry packets]\label{core:theorem:008}\label{core:thm:symmetry}
370 If $[H,P_\alpha]=0$ for every packet, then
371 \[
372 H=\sum_\alpha P_\alpha HP_\alpha
373 \]
374 and
375 \[
376 e^{-itH}
377 =
378 \sum_\alpha P_\alpha e^{-itH}P_\alpha.
379 \]
380 No amplitude transfers between distinct packets.
381 \end{theorem}
382 
383 \begin{proof}\phantomsection\label{core:proof:009}
384 The commutation relation gives $P_\beta HP_\alpha=0$ for $\beta\ne\alpha$. Every polynomial in $H$ has the same block form, and the finite-dimensional exponential is the norm limit of its polynomial series.
385 \phantomsection\label{core:proof:009:end}\end{proof}
386 
387 For two qubits and real parameters $J,h$, consider
388 \[
389 H
390 =
391 JX\otimes X
392 +h(Z\otimes I+I\otimes Z).
393 \]
394 In the ordered even basis $\{\ket{00},\ket{11}\}$ and odd basis $\{\ket{01},\ket{10}\}$,
395 \[
396 H_{\mathrm{even}}
397 =
398 \begin{pmatrix}
399 2h&J\\
400 J&-2h
401 \end{pmatrix},
402 \qquad
403 H_{\mathrm{odd}}
404 =
405 \begin{pmatrix}
406 0&J\\
407 J&0
408 \end{pmatrix}.
409 \]
410 Therefore
411 \[
412 \Spec H_{\mathrm{even}}
413 =\{\pm\sqrt{4h^2+J^2}\},
414 \qquad
415 \Spec H_{\mathrm{odd}}=\{\pm J\}.
416 \]
417 
418 \begin{figure}[htbp]
419 \centering
420 \includegraphics[width=0.66\textwidth]{03_parity_block_hamiltonian.pdf}
421 \caption{The Hamiltonian becomes exactly block diagonal when the even and odd basis vectors are grouped.}
422 \label{core:fig:parity-hamiltonian}
423 \end{figure}
424 
425 \subsection{Leakage under broken symmetry}
426 
427 Set $Q=I-P$.
428 
429 \begin{theorem}[Commutator leakage bound]\label{core:theorem:009}\label{core:thm:leakage}
430 For every self-adjoint $H$, orthogonal projector $P$, and real time $t$,
431 \[
432 \boxed{
433 \norm{Qe^{-itH}P}
434 \le
435 |t|\,\norm{[H,P]}.
436 }
437 \]
438 In particular, exact commutation gives zero leakage for all time.
439 \end{theorem}
440 
441 \begin{proof}\phantomsection\label{core:proof:010}
442 Set $F(s)=e^{isH}Pe^{-isH}$. Differentiating the exponential gives
443 \[
444 F'(s)=i\,e^{isH}[H,P]e^{-isH},\qquad F(0)=P.
445 \]
446 Unitary conjugation preserves the operator norm, so $\|F'(s)\|=\|[H,P]\|$. The mean-value estimate on the interval between $0$ and $t$ yields
447 \[
448 \|F(t)-P\|\le |t|\,\|[H,P]\|.
449 \]
450 Finally, $e^{itH}Qe^{-itH}P=(P-F(t))P$ and $\|P\|\le1$. Left multiplication by $e^{itH}$ preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
451 \phantomsection\label{core:proof:010:end}\end{proof}
452 
453 \begin{figure}[htbp]
454 \centering
455 \includegraphics[width=0.78\textwidth]{05_leakage_bound.pdf}
456 \caption{A perturbation that breaks parity creates nonzero transfer between the packets. The measured leakage remains below the commutator bound.}
457 \label{core:fig:leakage}
458 \end{figure}
459 
460 \begin{certificate}[Symmetry and negative control]\label{core:certificate:004}
461 For the Hamiltonian that preserves parity, the norm of the block between distinct packets and the maximum measured leakage were both $0$. Adding an $X\otimes I$ perturbation broke parity and produced a maximum leakage of approximately $0.79196$. The maximum violation of the proved bound remained $0$.
462 \end{certificate}
463 
464 \section{Moving packets, connections, and curvature}\label{core:section:006}
465 
466 \subsection{Moving quantum coordinates}
467 
468 \begin{theorem}[Moving-frame Hamiltonian]\label{core:theorem:010}\label{core:thm:moving}
469 Let $U(t)$ and $\psi(t)$ be differentiable on an open real time interval, let $U(t)$ be unitary, and let $H(t)=H(t)^*$. Suppose
470 \[
471 i\dot\psi(t)=H(t)\psi(t).
472 \]
473 Writing
474 \[
475 \psi(t)=U(t)y(t)
476 \]
477 gives
478 \[
479 \boxed{
480 i\dot y(t)
481 =
482 \left(U(t)^*H(t)U(t)-iU(t)^*\dot U(t)\right)y(t).
483 }
484 \]
485 Both the connection term $-iU^*\dot U$ and the effective Hamiltonian $U^*HU-iU^*\dot U$ are Hermitian.
486 \end{theorem}
487 
488 \begin{proof}\phantomsection\label{core:proof:011}
489 Differentiate $\psi=Uy$:
490 \[
491 i\dot U y+iU\dot y=HUy.
492 \]
493 Multiplying by $U^*$ gives the formula. Differentiating $U^*U=I$ shows $U^*\dot U$ is skew-Hermitian, so $-iU^*\dot U$ is Hermitian.
494 \phantomsection\label{core:proof:011:end}\end{proof}
495 
496 \begin{figure}[htbp]
497 \centering
498 \includegraphics[width=0.70\textwidth]{08_moving_frame_connection.pdf}
499 \caption{The equation in the moving frame holds to numerical precision only when the connection term is retained.}
500 \label{core:fig:moving-frame}
501 \end{figure}
502 
503 \begin{certificate}[Moving-frame correction]\label{core:certificate:005}
504 The reported residual for the complete equation in the moving frame was $1.0645\times10^{-10}$. Omitting the connection increased the residual to $0.7000$.
505 \end{certificate}
506 
507 \subsection{A canonical projector connection}
508 
509 Let $P_1(t),\ldots,P_m(t)$ be differentiable on an open real time interval, with $P_a(t)^*=P_a(t)$ and
510 \[
511 P_aP_b=\delta_{ab}P_a,
512 \qquad
513 \sum_aP_a=I.
514 \]
515 
516 \begin{theorem}[Canonical packet connection]\label{core:theorem:011}\label{core:thm:canonical}
517 Define
518 \[
519 \boxed{
520 C(t)
521 =
522 \frac12\sum_a[\dot P_a(t),P_a(t)]
523 =
524 \sum_a\dot P_a(t)P_a(t).
525 }
526 \]
527 Then
528 \[
529 C(t)^*=-C(t)
530 \]
531 and
532 \[
533 \boxed{
534 [C(t),P_b(t)]=\dot P_b(t)
535 }
536 \]
537 for every packet $b$.
538 \end{theorem}
539 
540 \begin{proof}\phantomsection\label{core:proof:012}
541 Differentiating $P_a^2=P_a$ gives
542 \[
543 \dot P_aP_a+P_a\dot P_a=\dot P_a.
544 \]
545 Summing over $a$ and using $\sum_a\dot P_a=0$ yields
546 \[
547 \sum_aP_a\dot P_a=-\sum_a\dot P_aP_a,
548 \]
549 which proves both displayed formulas for $C$ and its skew-Hermiticity.
550 
551 For fixed $b$,
552 \[
553 CP_b=\dot P_bP_b.
554 \]
555 When $a\ne b$, differentiating $P_bP_a=0$ gives
556 \[
557 P_b\dot P_a=-\dot P_bP_a.
558 \]
559 Hence
560 \[
561 P_bC
562 =\sum_aP_b\dot P_aP_a
563 =-\dot P_b(I-P_b),
564 \]
565 and subtraction gives $[C,P_b]=\dot P_b$.
566 \phantomsection\label{core:proof:012:end}\end{proof}
567 
568 The connection generates infinitesimal transport of the moving packet frame. The packet family determines it directly, so its construction does not require a basis inside any packet.
569 
570 \subsection{Projector curvature}
571 
572 \begin{classical}[Projected connection]\label{core:classical:001}\label{core:thm:projected}
573 Let $M$ be a smooth finite-dimensional real manifold without boundary, let $V$ be a finite-dimensional real normed vector space, and let $P:M\to\operatorname{End}_{\bbR}(V)$ be smooth with $P(x)^2=P(x)$. In the quantum application, $V$ is the underlying real space of a finite complex Hilbert space and $P$ is an orthogonal complex-linear projector.
574 
575 The rank of $P$ is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section $s$, so that $Ps=s$, define
576 \[
577 \nabla_Xs=P\,ds(X).
578 \]
579 This is a connection on smooth range sections. For smooth vector fields $X,Y$, its curvature satisfies
580 \[
581 \boxed{
582 (\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})s
583 =P[dP(X),dP(Y)]P\,s.
584 }
585 \]
586 Equivalently, its endomorphism-valued curvature is
587 \[
588 \boxed{F=P(dP\wedge dP)P.}
589 \]
590 No global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalization concerns real smooth structures. It does not cover complex analytic manifolds or extensions to a boundary.
591 \end{classical}
592 
593 \begin{proof}\phantomsection\label{core:proof:013}
594 \textit{Range coordinates.} Fix $a\in M$ and put
595 \[
596 T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)).
597 \]
598 Then $T_a(a)=I$ and $P(x)T_a(x)=T_a(x)P(a)$. Invertibility is open, so on a neighborhood of $a$, $T_a(x)$ restricts to an isomorphism from $\Ran P(a)$ to $\Ran P(x)$. This proves local constancy of rank. The local coordinates and their transitions are
599 \[
600 (x,v)\longmapsto (x,T_a(x)^{-1}v),\qquad
601 u\longmapsto T_b(x)^{-1}T_a(x)u.
602 \]
603 Their smoothness follows from smoothness of $P$ and inversion. The transition maps satisfy the cocycle identity by cancellation. These charts construct the smooth range bundles on the open rank regions, including when different connected components have different ranks.
604 
605 \textit{Connection rules.} The smoothness of $P\,ds(X)$ and the identity $P^2=P$ show that $\nabla_Xs$ is a smooth range section. Differentiation and projection give
606 \[
607 \begin{aligned}
608 \nabla_{X+Y}s&=\nabla_Xs+\nabla_Ys,&
609 \nabla_{fX}s&=f\nabla_Xs,\\
610 \nabla_X(s+t)&=\nabla_Xs+\nabla_Xt,&
611 \nabla_X(fs)&=f\nabla_Xs+(Xf)s
612 \end{aligned}
613 \]
614 for smooth real $f$. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.
615 
616 \textit{Curvature.} Differentiating $P^2=P$ gives $P(dP(X))P=0$. Write $D_Xs=ds(X)$. For ambient smooth sections,
617 \[
618 D_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.
619 \]
620 Expansion of the projected derivatives therefore leaves
621 \[
622 R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs.
623 \]
624 Since $Ps=s$, differentiation gives $D_Ys=(dP(Y))s+PD_Ys$. The terms with a middle $P$ vanish. Thus
625 \[
626 R(X,Y)s=P\bigl(dP(X)dP(Y)-dP(Y)dP(X)\bigr)P\,s.
627 \]
628 This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation law for the Lie bracket make the local calculations agree on chart overlaps, which gives the global formula.
629 \phantomsection\label{core:proof:013:end}\end{proof}
630 
631 For an orthogonal rank-one complex projector, use the scalar connection convention $A=-i\langle u,du\rangle$ in a local unit frame. The associated scalar curvature density is
632 \[
633 \cF_{\mu\nu}
634 =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
635 \]
636 On the Bloch sphere,
637 \[
638 P(\theta,\phi)
639 =
640 \frac12\left(I+\mathbf n(\theta,\phi)\cdot\boldsymbol\sigma\right),
641 \]
642 where
643 \[
644 \mathbf n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).
645 \]
646 The Pauli multiplication law gives
647 \[
648 [\partial_\theta P,\partial_\phi P]
649 =\frac{i}{2}(\partial_\theta\mathbf n\mathbin{\times}\partial_\phi\mathbf n)\cdot\boldsymbol\sigma.
650 \]
651 Using $\Tr(P\,\mathbf a\cdot\boldsymbol\sigma)=\mathbf n\cdot\mathbf a$ therefore yields
652 \[
653 \boxed{\cF_{\theta\phi}
654 =\frac12\mathbf n\cdot(\partial_\theta\mathbf n\mathbin{\times}\partial_\phi\mathbf n)
655 =\frac12\sin\theta.}
656 \]
657 The scalar sign changes if the opposite convention for $A$ is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.
658 
659 \begin{figure}[htbp]
660 \centering
661 \includegraphics[width=0.76\textwidth]{09_bloch_curvature.pdf}
662 \caption{The curvature of the Bloch projector, evaluated by finite differences, is compared with the exact density $\frac12\sin\theta$.}
663 \label{core:fig:bloch}
664 \end{figure}
665 
666 \begin{certificate}[Geometric packet identities]\label{core:certificate:006}
667 The canonical connection reconstructed every packet derivative with residual $0$. The maximum residual for the Bloch curvature was $2.0803\times10^{-11}$.
668 \end{certificate}
669 
670 \section{Measurements and quantum channels}\label{core:section:007}
671 
672 \subsection{Packet measurements}
673 
674 Let $\rho$ be a density matrix and let $\{E_k\}$ be a POVM:
675 \[
676 E_k\ge0,
677 \qquad
678 \sum_kE_k=I.
679 \]
680 The outcome probabilities are
681 \[
682 p_k=\Tr(E_k\rho).
683 \]
684 
685 \begin{theorem}[Block-diagonal measurement invariance]\label{core:theorem:012}\label{core:thm:measurement}
686 Let $\{P_\alpha\}$ be a complete packet family, and suppose every measurement effect is packet diagonal:
687 \[
688 E_k=\sum_\alpha P_\alpha E_kP_\alpha.
689 \]
690 Define the packet-dephased state
691 \[
692 \cD_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha.
693 \]
694 Then
695 \[
696 \boxed{
697 \Tr(E_k\rho)
698 =
699 \Tr(E_k\cD_P(\rho))
700 }
701 \]
702 for every outcome $k$.
703 \end{theorem}
704 
705 \begin{proof}\phantomsection\label{core:proof:014}
706 Insert the packet decomposition of $E_k$ and use cyclicity of the trace:
707 \[
708 \Tr(E_k\rho)
709 =
710 \sum_\alpha\Tr(P_\alpha E_kP_\alpha\rho)
711 =
712 \sum_\alpha\Tr(E_kP_\alpha\rho P_\alpha).
713 \]
714 \phantomsection\label{core:proof:014:end}\end{proof}
715 
716 The theorem identifies which coherences a declared measurement can detect. Observables that are diagonal in the packet decomposition cannot detect coherence between distinct packets. Coherence within a packet remains available to such observables.
717 
718 \subsection{Kraus transfer}
719 
720 A quantum channel can transfer both populations and coherences between packet addresses. A Kraus representation describes this transfer by summing the contributions of its individual branches. The normalization of those branches ensures that the total trace is preserved.
721 
722 \begin{classical}[Finite Kraus channel]\label{core:classical:002}\label{core:thm:kraus}
723 Let $K_1,\ldots,K_r$ satisfy
724 \[
725 \sum_aK_a^*K_a=I.
726 \]
727 Then
728 \[
729 \boxed{
730 \Phi(\rho)=\sum_aK_a\rho K_a^*
731 }
732 \]
733 is completely positive and trace preserving.
734 \end{classical}
735 
736 \begin{proof}\phantomsection\label{core:proof:015}
737 Trace preservation follows from
738 \[
739 \Tr\Phi(\rho)
740 =\Tr\left(\rho\sum_aK_a^*K_a\right)
741 =\Tr\rho.
742 \]
743 For any auxiliary dimension and positive matrix $X$, each term
744 \[
745 (I\otimes K_a)X(I\otimes K_a)^*
746 \]
747 is positive, so their sum is positive. This is complete positivity.
748 \phantomsection\label{core:proof:015:end}\end{proof}
749 
750 Packet superoperator blocks are defined by
751 \[
752 \Phi_{\alpha\beta;\gamma\delta}(X)
753 =
754 P_\alpha
755 \Phi(P_\gamma XP_\delta)
756 P_\beta.
757 \]
758 Summing all four packet indices reconstructs $\Phi(X)$ exactly.
759 
760 For amplitude damping with a real parameter $0\le\gamma\le1$,
761 \[
762 K_0=
763 \begin{pmatrix}
764 1&0\\0&\sqrt{1-\gamma}
765 \end{pmatrix},
766 \qquad
767 K_1=
768 \begin{pmatrix}
769 0&\sqrt\gamma\\0&0
770 \end{pmatrix}.
771 \]
772 The excited state becomes
773 \[
774 \Phi(\ket1\bra1)
775 =
776 \gamma\ket0\bra0+(1-\gamma)\ket1\bra1.
777 \]
778 
779 \begin{figure}[htbp]
780 \centering
781 \includegraphics[width=0.56\textwidth]{16_amplitude_damping_channel.pdf}
782 \caption{Amplitude damping transfers population from the excited packet to the ground packet while preserving trace and positivity.}
783 \label{core:fig:damping}
784 \end{figure}
785 
786 \begin{certificate}[Channel reconstruction]\label{core:certificate:007}
787 For $\gamma=0.23$, the residuals for Kraus completeness and the output trace were $0$. The computed output from $\ket1\bra1$ was $\diag(0.23,0.77)$. In the Bell state test, packet dephasing left every measurement probability in the computational basis unchanged.
788 \end{certificate}
789 
790 \section{Quantum error correction as obstruction-packet annihilation}\label{core:section:008}
791 
792 Let $P=P^*=P^2$ project onto a code space $\cC\subset\cH$, and let $E_1,\ldots,E_r$ be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators
793 \[
794 P E_a^*E_bP
795 \]
796 are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of $P$, the environment can distinguish logical states and exact correction is impossible.
797 
798 We call a recovery complete when it is a Kraus map
799 \[
800 \cR(X)=\sum_jK_jXK_j^*,\qquad \sum_jK_j^*K_j=I.
801 \]
802 The family is \emph{exactly correctable} when one such recovery works for every coherent error combination $A_w=\sum_aw_aE_a$: for every $w$ there is a real $t_w\ge0$, independent of the logical input, such that
803 \[
804 PXP=X\quad\Longrightarrow\quad
805 \cR(A_wXA_w^*)=t_wX.
806 \]
807 The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix $A_w$.
808 
809 \begin{classical}[Finite Knill--Laflamme criterion]\label{core:classical:003}\label{core:thm:kl}
810 The finite error family is exactly correctable in the sense just defined if and only if there is a positive semidefinite complex matrix $C=(c_{ab})$ such that
811 \[
812 \boxed{PE_a^*E_bP=c_{ab}P\qquad\text{for every }a,b.}
813 \]
814 If $\sum_aE_a^*E_a=I$, the equivalent normalized-channel formulation is the existence of a complete recovery satisfying
815 \[
816 \cR\!\left(\sum_aE_aXE_a^*\right)=X
817 \quad\text{for every }X\text{ with }PXP=X.
818 \]
819 The zero code and an empty error family are included. An empty family cannot define a channel that preserves trace on a nonzero ambient space.
820 \end{classical}
821 
822 \begin{proof}\phantomsection\label{core:proof:016}
823 \textit{Necessity.} Fix an error $E_a$ and let $L_j=K_jE_aP$. Perfect recovery on rank-one code operators gives
824 \[
825 \sum_j\ket{L_j\psi}\bra{L_j\psi}
826 =t_a\ket\psi\bra\psi\qquad(\psi\in\cC).
827 \]
828 For every vector orthogonal to $\psi$, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each $L_j\psi$ therefore lies in the span of $\psi$. A linear operator that preserves every line in $\cC$ is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently
829 \[
830 K_jE_aP=d_{ja}P.
831 \]
832 Recovery completeness now gives
833 \[
834 \begin{aligned}
835 PE_a^*E_bP
836 &=\sum_j (K_jE_aP)^*(K_jE_bP)\\
837 &=\left(\sum_j\overline{d_{ja}}d_{jb}\right)P.
838 \end{aligned}
839 \]
840 The coefficients form a Gram matrix, hence are positive semidefinite. For a normalized channel, apply the same rank-one argument to all composite branches $K_jE_aP$ at once to obtain the necessary condition.
841 
842 \textit{Sufficiency.} Diagonalize $C=U\operatorname{diag}(\lambda_r)U^*$ with $U$ unitary and $\lambda_r\ge0$. Define compressed transformed errors
843 \[
844 F_r=\sum_aU_{ar}E_aP.
845 \]
846 Then $F_rP=F_r$ and $F_r^*F_s=\lambda_r\delta_{rs}P$. If $\lambda_r=0$, $F_r^*F_r=0$ forces $F_r=0$. For positive eigenvalues only, set $V_r=F_r/\sqrt{\lambda_r}$. These satisfy
847 \[
848 V_rP=V_r,\qquad V_r^*V_s=\delta_{rs}P.
849 \]
850 Let $S=\sum_{\lambda_r>0}V_rV_r^*$. Orthogonality makes $S$ an orthogonal projector. Use the recovery branches
851 \[
852 K_r=V_r^*,\qquad K_{\mathrm{unused}}=I-S.
853 \]
854 They are complete on the whole ambient space because
855 \[
856 \sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}}
857 =S+(I-S)^2=I.
858 \]
859 The inverse unitary change of coordinates reconstructs each $E_aP$ from the $F_r$. Every recovery branch of a coherent combination is therefore scalar on the code, while the branch for the unused space vanishes. For all supported matrices, the resulting recovery is
860 \[
861 \cR(A_wXA_w^*)=(w^*Cw)X.
862 \]
863 This establishes the statement for arbitrary families without assuming normalization. Under $\sum_aE_a^*E_a=I$ and $P\ne0$, summing the diagonal KL equations gives $\Tr C=1$, so the normalized-channel output is $X$. When $P=0$, every supported $X$ is zero and the conclusion is automatic. The recovery is trace preserving by completeness and completely positive at every finite ancillary dimension by the Kraus calculation in Classical Input~\ref{core:thm:kraus}.
864 \phantomsection\label{core:proof:016:end}\end{proof}
865 
866 For the three-qubit bit-flip code,
867 \[
868 \ket{0_L}=\ket{000},
869 \qquad
870 \ket{1_L}=\ket{111},
871 \]
872 and the error family is
873 \[
874 I,
875 \quad X\otimes I\otimes I,
876 \quad I\otimes X\otimes I,
877 \quad I\otimes I\otimes X.
878 \]
879 The four error images of the code are pairwise orthogonal two-dimensional syndrome packets whose projectors sum to the identity on the eight-dimensional Hilbert space.
880 
881 \begin{figure}[htbp]
882 \centering
883 \includegraphics[width=0.88\textwidth]{19_error_correction_syndromes.pdf}
884 \caption{The four orthogonal syndrome packets of the three-qubit bit-flip code. Each packet identifies one correctable error without resolving the encoded logical state.}
885 \label{core:fig:error-correction}
886 \end{figure}
887 
888 \begin{certificate}[Exact bit-flip correction]\label{core:certificate:008}
889 The residuals for the Knill--Laflamme condition and completeness of the syndrome packets were both $0$. A mixed channel of single bit flip errors, applied to a nontrivial logical superposition, was reversed with a residual of $1.0385\times10^{-16}$ in the density matrix.
890 \end{certificate}
891 
892 \section{Exact effective dynamics}\label{core:section:009}
893 
894 \subsection{Schur--Feshbach reduction}
895 
896 Let $P$ be a retained packet projector and $Q=I-P$. In packet coordinates,
897 \[
898 H=
899 \begin{pmatrix}
900 H_{PP}&H_{PQ}\\
901 H_{QP}&H_{QQ}
902 \end{pmatrix}.
903 \]
904 
905 \begin{theorem}[Exact Feshbach resolvent]\label{core:theorem:013}\label{core:thm:feshbach}
906 Assume that both $zI_Q-H_{QQ}$ and $zI-H$ are invertible. Define
907 \[
908 \boxed{
909 H_{\mathrm{eff}}(z)
910 =
911 H_{PP}
912 +H_{PQ}(z-H_{QQ})^{-1}H_{QP}.
913 }
914 \]
915 Then
916 \[
917 \boxed{
918 P(z-H)^{-1}P
919 =
920 \left(z-H_{\mathrm{eff}}(z)\right)^{-1}
921 }
922 \]
923 where the left side is restricted to $P\cH$, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here $z-H_{QQ}$ and $z-H_{\mathrm{eff}}$ abbreviate the appropriately sized scalar matrices minus the indicated blocks.
924 \end{theorem}
925 
926 \begin{proof}\phantomsection\label{core:proof:017}
927 The block in the lower right corner of $z-H$ is $z-H_{QQ}$. Gaussian block elimination produces the Schur complement
928 \[
929 z-H_{PP}-H_{PQ}(z-H_{QQ})^{-1}H_{QP}.
930 \]
931 The assumed invertibility of this block and of the full matrix implies that the Schur complement is invertible. The block in the upper left corner of the full inverse is the inverse of that Schur complement, which is exactly the displayed operator.
932 \phantomsection\label{core:proof:017:end}\end{proof}
933 
934 The second premise cannot be dropped: at $z=0$ and $H=\operatorname{diag}(0,-1)$, the eliminated block is $1$ but $zI-H$ is singular. This counterexample is also included in the Lean module.
935 
936 The self-energy
937 \[
938 \Sigma_P(z)
939 =H_{PQ}(z-H_{QQ})^{-1}H_{QP}
940 \]
941 contains the complete influence of the eliminated packet. Bare truncation drops $\Sigma_P(z)$ and is therefore exact only when the coupling between packets vanishes or the self-energy cancels for another stated reason.
942 
943 \begin{figure}[htbp]
944 \centering
945 \begin{minipage}[t]{0.48\textwidth}
946 \centering
947 \includegraphics[width=\textwidth]{06_feshbach_vs_truncation.pdf}
948 \captionof{figure}{Exact Feshbach reduction versus bare truncation.}
949 \label{core:fig:feshbach}
950 \end{minipage}\hfill
951 \begin{minipage}[t]{0.48\textwidth}
952 \centering
953 \includegraphics[width=\textwidth]{07_packet_green_blocks.pdf}
954 \captionof{figure}{Norms of packet Green-function blocks.}
955 \label{core:fig:green-blocks}
956 \end{minipage}
957 \end{figure}
958 
959 \begin{certificate}[Effective resolvent]\label{core:certificate:009}
960 In the reported test with a random Hermitian matrix of dimension six, the exact Feshbach residual was $3.2291\times10^{-17}$. The corresponding residual for bare truncation was $1.3091\times10^{-2}$.
961 \end{certificate}
962 
963 \subsection{The hidden packet as memory}
964 
965 For a time-independent finite Hamiltonian $H$ and fixed projectors $P,Q$, write $\psi=x+y$, with $x=P\psi$ and $y=Q\psi$. Then
966 \[
967 \begin{aligned}
968 i\dot x&=H_{PP}x+H_{PQ}y,\\
969 i\dot y&=H_{QP}x+H_{QQ}y.
970 \end{aligned}
971 \]
972 Solving the second equation gives
973 \[
974 y(t)
975 =e^{-itH_{QQ}}y(0)
976 -i\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\,ds.
977 \]
978 Substitution yields the exact Volterra equation
979 \[
980 \boxed{
981 \begin{aligned}
982 i\dot x(t)
983 ={}&H_{PP}x(t)
984 +H_{PQ}e^{-itH_{QQ}}y(0)\\
985 &-i\int_0^t
986 K(t-s)x(s)\,ds,
987 \end{aligned}
988 }
989 \]
990 with memory kernel
991 \[
992 \boxed{
993 K(t)=H_{PQ}e^{-itH_{QQ}}H_{QP}.
994 }
995 \]
996 The influence of the eliminated degrees of freedom is therefore retained in a self-energy that depends on frequency, or equivalently in an exact memory term. For forward evolution the integral is causal. When the Hamiltonian depends on time, the corresponding propagator between two times replaces the stationary exponential used here. This Volterra derivation is not a separate entry in the finite Lean register.
997 
998 \section{Finite quantum geometry}\label{core:section:010}
999 
1000 \subsection{A projector-resolved geometry}
1001 
1002 At resolution $\ell$, define
1003 \[
1004 \boxed{
1005 \mathfrak Q_\ell
1006 =
1007 \left(
1008 \cA_\ell,
1009 \cH_\ell,
1010 D_\ell,
1011 \{P_{\ell,\alpha}\},
1012 E_\ell,
1013 C_\ell,
1014 \Pi_\ell^{\mathrm{phys}},
1015 T_\ell
1016 \right).
1017 }
1018 \]
1019 In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types $E_\ell:\cH_\ell\to\cH_{\ell+1}$ and $C_\ell:\cH_{\ell+1}\to\cH_\ell$. The left inverse relation $C_\ell E_\ell=I$ alone does not imply $C_\ell=E_\ell^*$ or make the embedding isometric.
1020 
1021 A coherent refinement should satisfy
1022 \[
1023 C_\ell E_\ell=I
1024 \]
1025 and
1026 \[
1027 \boxed{
1028 P_{\ell,\alpha}
1029 =
1030 C_\ell
1031 \left(
1032 \sum_{\beta\succ\alpha}P_{\ell+1,\beta}
1033 \right)
1034 E_\ell.
1035 }
1036 \]
1037 For an operator $D$, define the refinement defect
1038 \[
1039 \Delta_\ell(D)
1040 =C_\ell D_{\ell+1}E_\ell-D_\ell.
1041 \]
1042 Exact resolution independence is $\Delta_\ell(D)=0$. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
1043 
1044 \begin{figure}[htbp]
1045 \centering
1046 \includegraphics[width=0.84\textwidth]{17_quantum_spacetime_tower.pdf}
1047 \caption{The diagram organizes a conjectural nested packet tower for quantum geometry. It does not provide a numerical reconstruction of physical spacetime.}
1048 \label{core:fig:spacetime-tower}
1049 \end{figure}
1050 \FloatBarrier
1051 
1052 \subsection{Euclidean transfer}
1053 
1054 \begin{theorem}[Finite Euclidean transfer]\label{core:theorem:014}\label{core:thm:euclidean}
1055 Let $H$ be Hermitian on a nonzero finite-dimensional Hilbert space. For all real $\tau,\sigma\ge0$,
1056 \[
1057 T(\tau)=e^{-\tau(H-E_0I)},
1058 \qquad
1059 E_0=\min\Spec H,
1060 \]
1061 is positive definite and satisfies
1062 \[
1063 \boxed{
1064 T(\tau+\sigma)=T(\tau)T(\sigma).
1065 }
1066 \]
1067 Its spectrum is contained in $(0,1]$.
1068 \end{theorem}
1069 
1070 \begin{proof}\phantomsection\label{core:proof:018}
1071 Diagonalize $H=\sum_jE_jP_j$. Then
1072 \[
1073 T(\tau)=\sum_j e^{-\tau(E_j-E_0)}P_j.
1074 \]
1075 Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and $T(0)=I$. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.
1076 \phantomsection\label{core:proof:018:end}\end{proof}
1077 
1078 The paper tests a controlled volume-state model
1079 \[
1080 H_{nn}=2\kappa+V(n),
1081 \qquad
1082 H_{n,n+1}=H_{n+1,n}=-\kappa,
1083 \]
1084 with $\kappa>0$ and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
1085 
1086 \begin{figure}[htbp]
1087 \centering
1088 \begin{minipage}[t]{0.48\textwidth}
1089 \centering
1090 \includegraphics[width=\textwidth]{13_euclidean_transfer_spectrum.pdf}
1091 \captionof{figure}{Positive spectrum of the Euclidean transfer operator.}
1092 \label{core:fig:euclidean-spectrum}
1093 \end{minipage}\hfill
1094 \begin{minipage}[t]{0.48\textwidth}
1095 \centering
1096 \includegraphics[width=\textwidth]{14_causal_volume_ground_state.pdf}
1097 \captionof{figure}{Ground-state volume packet in the controlled toy model.}
1098 \label{core:fig:volume-ground}
1099 \end{minipage}
1100 \end{figure}
1101 
1102 \begin{figure}[htbp]
1103 \centering
1104 \includegraphics[width=0.68\textwidth]{15_causal_volume_transfer.pdf}
1105 \caption{The figure shows the finite Euclidean kernel for volume transfer. Positivity and semigroup composition hold exactly in the finite model. Its gravitational interpretation remains provisional.}
1106 \label{core:fig:volume-transfer}
1107 \end{figure}
1108 
1109 \begin{certificate}[Finite transfer model]\label{core:certificate:010}
1110 The minimum transfer eigenvalue was $0.0767054$, the semigroup residual was $0$, and reconstruction from four volume-residue packets had residual $0$.
1111 \end{certificate}
1112 
1113 \section{Spectral dimension and continuum diagnostics}\label{core:section:011}
1114 
1115 Let $L\ge0$ be a finite Laplacian on $N\ge1$ states, with eigenvalues $\lambda_j$ and at least one zero eigenvalue. For diffusion time $\tau>0$, define the normalized heat trace
1116 \[
1117 K(\tau)=\frac1N\sum_j e^{-\tau\lambda_j}
1118 \]
1119 and the scale-dependent spectral dimension
1120 \[
1121 \boxed{
1122 d_s(\tau)
1123 =-2\frac{d\log K(\tau)}{d\log\tau}.
1124 }
1125 \]
1126 For a finite system, $d_s(\tau)$ is a diagnostic, not a topological invariant. Indeed, finite differentiation gives
1127 \[
1128 d_s(\tau)=2\tau\,
1129 \frac{\sum_j\lambda_j e^{-\tau\lambda_j}}{\sum_j e^{-\tau\lambda_j}}.
1130 \]
1131 It tends to zero as $\tau\downarrow0$ and, under the stated assumption of a zero mode, as $\tau\to\infty$. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to $2\tau\lambda_{\min}$. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.
1132 
1133 \subsection{The circle dispersion bound}
1134 
1135 For a periodic grid with spacing $h>0$ and physical wave number $k\in\bbR$,
1136 \[
1137 \lambda_{k,h}
1138 =
1139 \frac4{h^2}\sin^2\left(\frac{kh}{2}\right).
1140 \]
1141 
1142 \begin{theorem}[Low-mode second-order bound]\label{core:theorem:015}\label{core:thm:lowmode}
1143 For every real $k$ and every nonzero real $h$, define $\lambda_{k,h}=4h^{-2}\sin^2(kh/2)$. Then
1144 \[
1145 \boxed{
1146 0\le k^2-\lambda_{k,h}
1147 \le
1148 \frac{k^4h^2}{12}.
1149 }
1150 \]
1151 \end{theorem}
1152 
1153 \begin{proof}\phantomsection\label{core:proof:019}
1154 Put $x=kh/2$. The inequalities $|\sin x|\le|x|$ and $|x-\sin x|\le |x|^3/6$, which hold for every real argument, give
1155 \[
1156 0\le x^2-\sin^2x
1157 =|x-\sin x|\,|x+\sin x|
1158 \le\frac{|x|^3}{6}\,2|x|
1159 =\frac{x^4}{3}.
1160 \]
1161 Multiplying by $4/h^2$ proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps $k$ fixed while $h\to0$, so it does not assert a uniformly small relative error across all grid modes.
1162 \phantomsection\label{core:proof:019:end}\end{proof}
1163 
1164 \begin{figure}[htbp]
1165 \centering
1166 \includegraphics[width=0.76\textwidth]{11_circle_low_mode_convergence.pdf}
1167 \caption{The first nonzero circle mode converges at second order and remains below the proved dispersion bound.}
1168 \label{core:fig:circle-convergence}
1169 \end{figure}
1170 
1171 \begin{certificate}[Continuum dispersion]\label{core:certificate:011}
1172 The reported convergence order was $1.99936$. For every tested mode, the error was below the bound, so none of those tests violated the theorem.
1173 \end{certificate}
1174 
1175 \subsection{One- and two-dimensional plateaus}
1176 
1177 At intermediate diffusion times, a finite cycle and a finite torus can exhibit plateaus near the dimensions of their underlying geometries. The following figure and numerical certificate show the reported behavior for these two examples. The plateaus occur between the small and large time regimes in which the spectral dimension returns to zero.
1178 
1179 \begin{figure}[htbp]
1180 \centering
1181 \includegraphics[width=0.80\textwidth]{12_spectral_dimension.pdf}
1182 \caption{The curves show intermediate windows of spectral dimension for a finite cycle and a finite two-dimensional torus. Departures in the ultraviolet and infrared reflect the finite size of the models rather than additional dimensions.}
1183 \label{core:fig:spectral-dimension}
1184 \end{figure}
1185 
1186 \begin{certificate}[Spectral dimension]\label{core:certificate:012}
1187 The median intermediate plateaus were
1188 \[
1189 d_s^{(1)}=1.00062,
1190 \qquad
1191 d_s^{(2)}=2.03781.
1192 \]
1193 These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
1194 \end{certificate}
1195 
1196 \section{Constraint packets and the quantum-spacetime boundary}\label{core:section:012}
1197 
1198 At one finite level, let $\widehat C_A$ be a finite family of matrix constraints, with fixed scalar structure coefficients, and let $\Pi^{\mathrm{phys}}$ be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients $f_{AB}^{\ \ C}$, define the anomaly operator
1199 \[
1200 \boxed{
1201 \mathfrak A_{AB}
1202 =
1203 [\widehat C_A,\widehat C_B]
1204 -i\sum_C f_{AB}^{\ \ C}\widehat C_C.
1205 }
1206 \]
1207 Its physical packet is
1208 \[
1209 \Pi^{\mathrm{phys}}\mathfrak A_{AB}\Pi^{\mathrm{phys}}.
1210 \]
1211 In this exact finite setting, $\widehat C_C\Pi^{\mathrm{phys}}=0$ for every $C$. Associativity gives
1212 \[
1213 \mathfrak A_{AB}\Pi^{\mathrm{phys}}=0,
1214 \qquad
1215 \Pi^{\mathrm{phys}}\mathfrak A_{AB}\Pi^{\mathrm{phys}}=0
1216 \]
1217 automatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, $C_1=\operatorname{diag}(1,0,0)$ and $C_2=E_{12}+E_{21}$ annihilate the same third coordinate, while their commutator $E_{12}-E_{21}$ is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression vanishes.
1218 
1219 The closure target must be stated at the operator level, not inferred from that automatic compression. Unbounded constraints additionally require a specified common invariant domain. Neither such a domain theory nor gravitational structure functions are supplied by the finite check.
1220 
1221 \begin{conjecture}[Anomaly-free refinement]\label{core:conjecture:001}
1222 For a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure
1223 \[
1224 \mathfrak A^{(\ell)}_{AB}=0
1225 \]
1226 at every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure.
1227 \end{conjecture}
1228 The substantive research target is the operator condition. The earlier condition stated only in terms of compression was too weak. A choice of model, dynamics, and refinement maps is still needed before this program becomes a fully specified physical existence problem.
1229 
1230 \begin{conjecture}[Uniform Feshbach control]\label{core:conjecture:002}
1231 For the same specified tower, the self-energies from exact elimination of fine packets admit bounds uniform in cutoff and volume on a common spectral domain. Together with consistent identifications of the Hilbert spaces and a proved resolvent convergence criterion, they yield a nontrivial continuum limit in the strong resolvent sense. Uniform boundedness alone is not asserted to imply convergence.
1232 \end{conjecture}
1233 
1234 \begin{conjecture}[Lorentzian reconstruction]\label{core:conjecture:003}
1235 The Euclidean packet theory satisfies a positivity or reconstruction principle strong enough to produce a Lorentzian Hilbert space with causal propagation and positive physical norm.
1236 \end{conjecture}
1237 
1238 \begin{conjecture}[Einstein infrared limit]\label{core:conjecture:004}
1239 There exist semiclassical packet states $\Psi_{g,\ell}$ such that the effective action converges to
1240 \[
1241 \frac1{16\pi G}
1242 \int\sqrt{-g}\,(R-2\Lambda)\,d^4x
1243 \]
1244 with controlled higher-order corrections.
1245 \end{conjecture}
1246 
1247 \begin{conjecture}[Massless spin-two sector]\label{core:conjecture:005}
1248 The continuum two-point function possesses a positive transverse-traceless massless spin-two pole, while all unphysical packet sectors are removed by the constraints.
1249 \end{conjecture}
1250 
1251 The exact finite identities establish none of these physical conclusions, and no formal theorem of logical independence is claimed. Exact packet reconstruction provides a kinematic and computational advantage. It does not determine the physical action, prove existence of the continuum theory, or guarantee the limit of general relativity.
1252 
1253 \begin{figure}[htbp]
1254 \centering
1255 \includegraphics[width=0.86\textwidth]{18_proof_boundary.pdf}
1256 \caption{The proof boundary. Finite packet, circuit, leakage, Feshbach, connection, and spectral identities are established here. Constraint closure and the physical continuum claims remain conjectural.}
1257 \label{core:fig:proof-boundary}
1258 \end{figure}
1259 
1260 \section{Numerical certificates}\label{core:section:013}
1261 
1262 The numerical study reports CPU experiments and a suite of 22 tests. The values below come from the accompanying numerical record. They are separate from the 323 supporting declarations in the formal companion. A displayed floating-point residual of zero describes the reported computation and does not independently prove equality in exact arithmetic.
1263 
1264 \begin{table}[htbp]
1265 \centering
1266 \caption{Reported finite numerical checks, retained from the original manuscript and not rerun for this edition.}
1267 \label{core:tab:certificates}
1268 \begin{tabular}{L{0.64\textwidth}r}
1269 \toprule
1270 Certificate & Value\\
1271 \midrule
1272 Bell-state residual & $0$\\
1273 Three-qubit seed-orbit rank & $8/8$\\
1274 Parity off-block norm & $0$\\
1275 Circuit path reconstruction residual & $0$\\
1276 Exact-symmetry maximum leakage & $0$\\
1277 Maximum leakage-bound violation & $0$\\
1278 Feshbach projected-resolvent residual & $3.2291\times10^{-17}$\\
1279 Bare-truncation projected-resolvent residual & $1.3091\times10^{-2}$\\
1280 Moving-frame residual with connection & $1.0645\times10^{-10}$\\
1281 Moving-frame residual without connection & $7.0000\times10^{-1}$\\
1282 Canonical packet-connection residual & $0$\\
1283 Bloch-curvature residual & $2.0803\times10^{-11}$\\
1284 Kraus completeness residual & $0$\\
1285 Output trace residual & $0$\\
1286 Knill--Laflamme residual & $0$\\
1287 Bit-flip recovery residual & $1.0385\times10^{-16}$\\
1288 Tensor parity-refinement residual & $0$\\
1289 Circle low-mode convergence order & $1.99936$\\
1290 One-dimensional spectral plateau & $1.00062$\\
1291 Two-dimensional spectral plateau & $2.03781$\\
1292 Minimum Euclidean-transfer eigenvalue & $0.0767054$\\
1293 Transfer semigroup residual & $0$\\
1294 Volume-packet reconstruction residual & $0$\\
1295 \bottomrule
1296 \end{tabular}
1297 \end{table}
1298 
1299 The original reported count is
1300 \[
1301 \boxed{22/22\text{ historical numerical tests, not rerun here}.}
1302 \]
1303 The 19 original figures are retained from the recovered numerical companion. The separate visual guide supplies a reading route through these constructions and the later strengthened results.
1304 
1305 \setcounter{section}{14}
1306 \section{Conclusion}\label{core:section:014}
1307 
1308 The quantum realization of the seed calculus begins with a primitive state and a declared operator grammar. Symmetry projectors resolve the state and every operator into exact packets. In these coordinates, circuit products become sums over packet paths, and Hamiltonian symmetry appears as a diagonal block structure. When that symmetry is broken, the resulting transfer is measured by the leakage estimates.
1309 
1310 Moving packets carry a canonical connection and curvature. Channels and measurements act on the corresponding operator packets, while the conditions for error correction become statements about compression to the code. Exact elimination retains the influence of hidden sectors through self-energy and memory.
1311 
1312 The finite theory therefore provides a coherent language for
1313 \[
1314 \boxed{
1315 \begin{gathered}
1316 \text{seed reachability}
1317 +
1318 \text{symmetry resolution}
1319 +
1320 \text{circuit paths}\\
1321 +
1322 \text{effective dynamics}
1323 +
1324 \text{geometric transport}
1325 +
1326 \text{constraint diagnostics}
1327 \end{gathered}
1328 }
1329 \]
1330 A physical continuum theorem remains open. The next required results are anomaly-free refinement, uniform control of effective operators, Lorentzian reconstruction, the Einstein infrared limit, and a positive massless spin-two sector. Each requires its own hypotheses and proof. Agreement in a finite numerical model cannot replace those arguments.
1331 
1332 

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