Realization Calculus

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1 \section{Exact finite and countable packet coordinates}
2 
3 
4 
5 \begin{readerroadmap}
6 Packets give exact coordinates through idempotent maps.  For finitely many packets, the reconstruction is algebraic.  For countably many packets, we must also specify how the series converges and bound the coordinate maps.  We begin with finite reconstruction, then consider packet support, tensor products, duality, and the universal mapping property before turning to convergence.
7 \end{readerroadmap}
8 \subsection{Finite exact packets}
9 
10 \begin{definition}[Finite packet system]
11 \label{def:packet-system}
12 A finite packet system on a vector space $X$ over $\bbk$ is a family
13 \[
14 \Pcal=(P_\alpha)_{\alpha\in I}\subseteq\End_{\bbk}(X)
15 \]
16 satisfying
17 \[
18 P_\alpha^2=P_\alpha,
19 \qquad
20 P_\alpha P_\beta=0\quad(\alpha\neq\beta),
21 \qquad
22 \sum_{\alpha\in I}P_\alpha=I_X.
23 \]
24 Set $X_\alpha=P_\alpha X$.  Zero packet spaces may be removed without changing any identity.  No orthogonality is assumed unless stated.
25 \end{definition}
26 \RecordID{PSC-I-D-PACKET-SYSTEM}
27 
28 \begin{definition}[Packet support and packet pure vectors]
29 \label{def:packet-support}
30 For a finite packet system \(\Pcal=(P_\alpha)_{\alpha\in I}\) and \(x\in X\), define
31 \[
32 \Supp_{\Pcal}(x)
33 :=
34 \{\alpha\in I:P_\alpha x\neq0\}.
35 \]
36 The vector \(x\) is \emph{packet pure} if \(|\Supp_{\Pcal}(x)|\leq1\).  A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.
37 \end{definition}
38 \RecordID{PSC-I-D-PACKET-SUPPORT}
39 
40 \begin{interpretation}
41 Packet support records which packet coordinates are present.  A seed may lie in one packet, have coordinates in several packets, or be zero.  Later, when we count the seeds needed for generation, this distinction matters because a seed with several nonzero coordinates can meet several graph sources at once.
42 \end{interpretation}
43 
44 
45 \begin{lemma}[Finite packet aggregation]
46 \label{lem:packet-aggregation}
47 Let $\sigma:I\twoheadrightarrow J$ be a surjection and define
48 \[
49 Q_j:=\sum_{\sigma(\alpha)=j}P_\alpha.
50 \]
51 Then $(Q_j)_{j\in J}$ is a finite packet system and
52 \[
53 Q_jX=\bigoplus_{\sigma(\alpha)=j}X_\alpha.
54 \]
55 \end{lemma}
56 \RecordID{PSC-I-L-PACKET-AGGREGATION}
57 \Depends{PSC-I-D-PACKET-SYSTEM.}
58 
59 \begin{proof}
60 Pairwise annihilation of the $P_\alpha$ gives
61 \[
62 Q_j^2=Q_j,
63 \qquad
64 Q_jQ_k=0\quad(j\neq k).
65 \]
66 Summing over $j$ recovers $\sum_\alpha P_\alpha=I$.  The range statement follows from the directness forced by pairwise annihilation.
67 \end{proof}
68 
69 \begin{definition}[Packet analysis and synthesis]
70 \label{def:analysis-synthesis}
71 For a finite packet system, define
72 \[
73 \mathcal A_{\Pcal}:X\longrightarrow\bigoplus_{\alpha\in I}X_\alpha,
74 \qquad
75 x\longmapsto(P_\alpha x)_\alpha,
76 \]
77 and
78 \[
79 \mathcal S_{\Pcal}:\bigoplus_{\alpha\in I}X_\alpha\longrightarrow X,
80 \qquad
81 (x_\alpha)_\alpha\longmapsto\sum_\alpha x_\alpha.
82 \]
83 \end{definition}
84 \RecordID{PSC-I-D-ANALYSIS-SYNTHESIS}
85 
86 \begin{theorem}[Exact packet coordinate isomorphism]
87 \label{thm:exact-packet-coordinates}
88 For every finite packet system,
89 \[
90 \boxed{
91 \mathcal S_{\Pcal}\mathcal A_{\Pcal}=I_X,
92 \qquad
93 \mathcal A_{\Pcal}\mathcal S_{\Pcal}=I_{\oplus X_\alpha}.
94 }
95 \]
96 Hence
97 \[
98 \boxed{X\cong\bigoplus_{\alpha\in I}X_\alpha.}
99 \]
100 \end{theorem}
101 \RecordID{PSC-I-T-EXACT-PACKET-COORDINATES}
102 \Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-ANALYSIS-SYNTHESIS.}
103 \Exports{Every exact packet reconstruction in the series.}
104 
105 \begin{proof}
106 For $x\in X$,
107 \[
108 \mathcal S_{\Pcal}\mathcal A_{\Pcal}x
109 =
110 \sum_\alpha P_\alpha x
111 =x.
112 \]
113 If $x_\alpha\in X_\alpha$, then $P_\beta x_\alpha=\delta_{\alpha\beta}x_\alpha$, so
114 \[
115 \mathcal A_{\Pcal}\mathcal S_{\Pcal}(x_\alpha)_\alpha
116 =
117 \left(P_\beta\sum_\alpha x_\alpha\right)_\beta
118 =(x_\beta)_\beta.
119 \]
120 Thus analysis and synthesis are inverse linear maps.
121 \end{proof}
122 
123 \begin{keyidea}
124 Analysis records every packet coordinate and synthesis adds them back.  The theorem gives an exact change of coordinates.  Its proof uses the idempotent identities and does not require orthogonality.  Questions about norms, energy, and conditioning require further structure.
125 \end{keyidea}
126 
127 
128 \begin{corollary}[Packetwise equality]
129 \label{cor:packetwise-equality}
130 For $x,y\in X$,
131 \[
132 x=y
133 \quad\Longleftrightarrow\quad
134 P_\alpha x=P_\alpha y\text{ for every }\alpha.
135 \]
136 \end{corollary}
137 \RecordID{PSC-I-C-PACKETWISE-EQUALITY}
138 \Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}
139 
140 \begin{proof}
141 The forward implication is immediate.  For the reverse implication, sum the packet equalities and use $\sum_\alpha P_\alpha=I$.
142 \end{proof}
143 
144 
145 
146 \begin{corollary}[Universal mapping property of finite packet coordinates]
147 \label{cor:packet-universal-property}
148 Let \(\Pcal=(P_\alpha)_{\alpha\in I}\) be a finite packet system on \(X\), and let \(Y\) be a vector space over \(\bbk\).
149 \begin{enumerate}
150 \item Every family of linear maps \(f_\alpha:X_\alpha\to Y\) determines a unique linear map \(f:X\to Y\) with \(f|_{X_\alpha}=f_\alpha\), namely
151 \[
152 f=\sum_\alpha f_\alpha P_\alpha.
153 \]
154 \item Every family of linear maps \(g_\alpha:Y\to X_\alpha\) determines a unique linear map \(g:Y\to X\) with \(P_\alpha g=g_\alpha\), namely
155 \[
156 g=\sum_\alpha g_\alpha.
157 \]
158 \end{enumerate}
159 Equivalently,
160 \[
161 \Hom(X,Y)\cong\prod_\alpha\Hom(X_\alpha,Y),
162 \qquad
163 \Hom(Y,X)\cong\prod_\alpha\Hom(Y,X_\alpha),
164 \]
165 where the products are finite.
166 \end{corollary}
167 \RecordID{PSC-I-C-PACKET-UNIVERSAL-PROPERTY}
168 \Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}
169 \Exports{Assembly of packetwise operators, channels, corrections, and local transport data.}
170 
171 \begin{proof}
172 For \(x=\sum_\alpha P_\alpha x\), any extension of the \(f_\alpha\) must satisfy
173 \[
174 f(x)=\sum_\alpha f_\alpha(P_\alpha x),
175 \]
176 which proves both existence and uniqueness of the first formula.  For the second, the sum \(g=\sum_\alpha g_\alpha\) satisfies
177 \[
178 P_\beta g
179 =\sum_\alpha P_\beta g_\alpha
180 =g_\beta
181 \]
182 because \(g_\alpha(Y)\subseteq X_\alpha\).  Uniqueness follows from \(\sum_\alpha P_\alpha=I\).
183 \end{proof}
184 
185 
186 \subsection{Tensor products and dual packets}
187 
188 \begin{theorem}[Tensor product packet systems]
189 \label{thm:tensor-packet-system}
190 Let \(\Pcal=(P_\alpha)_{\alpha\in I}\) be a finite packet system on \(X\), and let \(\Qcal=(Q_\gamma)_{\gamma\in J}\) be a finite packet system on \(Y\).  Then
191 \[
192 \Pcal\boxtimes\Qcal
193 :=
194 (P_\alpha\otimes Q_\gamma)_{(\alpha,\gamma)\in I\times J}
195 \]
196 is a finite packet system on the algebraic tensor product \(X\otimes_{\bbk}Y\), with packet spaces
197 \[
198 (P_\alpha\otimes Q_\gamma)(X\otimes Y)
199 =X_\alpha\otimes Y_\gamma.
200 \]
201 Hence
202 \[
203 \boxed{
204 X\otimes Y
205 \cong
206 \bigoplus_{(\alpha,\gamma)\in I\times J}
207 X_\alpha\otimes Y_\gamma.
208 }
209 \]
210 \end{theorem}
211 \RecordID{PSC-I-T-TENSOR-PACKET-SYSTEM}
212 \Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}
213 \Exports{Tensor refinement for quantum, signal, arithmetic, and multivariate realizations.}
214 
215 \begin{proof}
216 For every pair of packet indices,
217 \[
218 (P_\alpha\otimes Q_\gamma)^2
219 =P_\alpha^2\otimes Q_\gamma^2
220 =P_\alpha\otimes Q_\gamma.
221 \]
222 If \((\alpha,\gamma)\neq(\beta,\delta)\), then either \(P_\alpha P_\beta=0\) or \(Q_\gamma Q_\delta=0\), so the corresponding tensor idempotents annihilate each other.  Finally,
223 \[
224 \sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma
225 =
226 \left(\sum_\alpha P_\alpha\right)
227 \otimes
228 \left(\sum_\gamma Q_\gamma\right)
229 =I_X\otimes I_Y.
230 \]
231 The range is generated by pure tensors \(P_\alpha x\otimes Q_\gamma y\), hence equals \(X_\alpha\otimes Y_\gamma\).  Exact reconstruction follows from \cref{thm:exact-packet-coordinates}.
232 \end{proof}
233 
234 \begin{proposition}[Dual packet systems]
235 \label{prop:dual-packet-system}
236 Let \(X\) be finite dimensional and let \(\Pcal=(P_\alpha)\) be a finite packet system.  The algebraic dual maps
237 \[
238 P_\alpha^*:X^*\to X^*,
239 \qquad
240 P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
241 \]
242 form a finite packet system on \(X^*\).  Its packet with index \(\alpha\) is canonically isomorphic to \(X_\alpha^*\), and
243 \[
244 X^*\cong\bigoplus_\alpha X_\alpha^*.
245 \]
246 \end{proposition}
247 \RecordID{PSC-I-P-DUAL-PACKET-SYSTEM}
248 \Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-T-EXACT-PACKET-COORDINATES.}
249 \Exports{Dual observables, covectors, trace pairings, and adjoint packet calculations.}
250 
251 \begin{proof}
252 Because dualization reverses products,
253 \[
254 P_\alpha^*P_\beta^*=(P_\beta P_\alpha)^*
255 =\delta_{\alpha\beta}P_\alpha^*.
256 \]
257 Also \(\sum_\alpha P_\alpha^*=I_{X^*}\).  Thus the dual maps form a packet system.  Restriction gives a map from \(P_\alpha^*X^*\) to \(X_\alpha^*\).  It is injective because a covector in the range of \(P_\alpha^*\) vanishes on every other packet, and it is surjective because any covector on \(X_\alpha\) extends to \(X\) through the direct sum decomposition.
258 \end{proof}
259 
260 \begin{microexample}[title={Tensoring the oblique packet split}]
261 Tensor the two oblique packets in Running Example R3 with the coordinate packets of \(\bbk^3\).  The result has six exact packets.  No orthogonality is created by tensoring, but the analysis and synthesis identities remain exact because they are idempotent identities.
262 \end{microexample}
263 
264 \subsection{Banach stability and coordinate conditioning}
265 Exact reconstruction answers whether the coordinates retain the whole vector.  Stability asks how errors in those coordinates affect the reconstructed vector.  A finite packet system answers the first question through its idempotent identities.  On a Banach space, boundedness of analysis and synthesis answers the second.  The chosen norm on the coordinate space matters, so we fix it before comparing operator sizes.
266 
267 \begin{theorem}[Finite Banach packet isomorphism]
268 \label{thm:banach-packets}
269 Let $X$ be Banach and let $(P_\alpha)_{\alpha\in I}$ be a finite family of bounded pairwise annihilating idempotents with sum $I_X$.  Then every range $X_\alpha=P_\alpha X$ is closed.  For any finite product norm equivalent to the maximum norm on $\bigoplus_\alpha X_\alpha$, packet analysis and synthesis are bounded inverse operators.
270 \end{theorem}
271 \RecordID{PSC-I-T-BANACH-PACKET-ISOMORPHISM}
272 \Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}
273 
274 \begin{proof}
275 The range of a bounded idempotent is its space of fixed points,
276 \[
277 P_\alpha X=\Ker(I-P_\alpha),
278 \]
279 which is closed.  Because $I$ is finite, analysis and synthesis are bounded for the maximum and sum norms, respectively, and therefore for every equivalent finite product norm.  Their inverse identities are algebraic and were proved in \cref{thm:exact-packet-coordinates}.
280 \end{proof}
281 
282 \begin{definition}[Packet coordinate condition number]
283 \label{def:packet-condition-number}
284 Fix a finite product norm on $\bigoplus_\alpha X_\alpha$ equivalent to the maximum norm, as in \cref{thm:banach-packets}. Define
285 \[
286 \kappa(\Pcal):=
287 \|\mathcal A_{\Pcal}\|\,
288 \|\mathcal S_{\Pcal}\|.
289 \]
290 \end{definition}
291 \RecordID{PSC-I-D-PACKET-CONDITION-NUMBER}
292 
293 \begin{proposition}[Conditioned coordinate transport]
294 \label{prop:conditioned-coordinates}
295 For a finite Banach packet system with $X\ne0$,
296 \[
297 \kappa(\Pcal)\geq1.
298 \]
299 If $T\in\mathcal B(X)$ and
300 \[
301 [T]_{\Pcal}:=\mathcal A_{\Pcal}T\mathcal S_{\Pcal},
302 \]
303 then
304 \[
305 \boxed{
306 \|[T]_{\Pcal}\|
307 \leq\kappa(\Pcal)\|T\|,
308 \qquad
309 \|T\|
310 \leq\kappa(\Pcal)\|[T]_{\Pcal}\|.
311 }
312 \]
313 \end{proposition}
314 \RecordID{PSC-I-P-CONDITIONED-COORDINATES}
315 \Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}
316 
317 \begin{proof}
318 Since $I=\mathcal S_{\Pcal}\mathcal A_{\Pcal}$,
319 \[
320 1=\|I\|\leq\|\mathcal S_{\Pcal}\|\|\mathcal A_{\Pcal}\|.
321 \]
322 The first displayed estimate follows from submultiplicativity.  The inverse identity
323 \[
324 T=\mathcal S_{\Pcal}[T]_{\Pcal}\mathcal A_{\Pcal}
325 \]
326 gives the second estimate.
327 \end{proof}
328 
329 \begin{interpretation}
330 The quantity \(\kappa(\Pcal)\) measures how strongly the chosen packet coordinates can amplify perturbations.  An orthogonal Hilbert decomposition has the best possible behavior in the natural \(\ell^2\) sum norm.  An oblique packet system can reconstruct every vector exactly and still have a large condition number.
331 \end{interpretation}
332 
333 
334 \begin{corollary}[Completion preserves a finite packet split]
335 \label{cor:completion-packet-split}
336 Let $M$ be a dense normed subspace of a Banach space $X$.  Suppose every bounded $P_\alpha$ leaves $M$ invariant and the restricted projectors form a packet system on $M$.  Then
337 \[
338 P_\alpha X=\cl{P_\alpha M}
339 \]
340 and the packet decomposition of $M$ extends uniquely to the completion $X$.
341 \end{corollary}
342 \RecordID{PSC-I-C-COMPLETION-PACKET-SPLIT}
343 \Depends{PSC-I-T-BANACH-PACKET-ISOMORPHISM.}
344 
345 \begin{proof}
346 For $x\in X$, choose $m_n\in M$ with $m_n\to x$.  Continuity gives $P_\alpha m_n\to P_\alpha x$, so $P_\alpha X\subseteq\cl{P_\alpha M}$.  The reverse inclusion follows because $P_\alpha X$ is closed and contains $P_\alpha M$.  The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.
347 \end{proof}
348 
349 \subsection{Countable packets and unconditional convergence}
350 For a finite family, synthesis adds a fixed number of coordinates.  For a countable family, it must also take a limit.  The algebraic idempotent identities still separate the coordinates, but they do not by themselves guarantee that the sum converges.  The next definition includes unconditional convergence in the coordinate space itself, so synthesis is defined on precisely the sequences whose packet sums can be assembled in this way.
351 
352 \begin{definition}[Unconditional packet sum space]
353 \label{def:unconditional-packet-space}
354 Let $(X_n)_{n\geq1}$ be closed subspaces of a real or complex Banach space $X$.  Define
355 \[
356 \bigoplus_{n\geq1}^{u}X_n
357 \]
358 to be the space of sequences $x=(x_n)$ with $x_n\in X_n$ for which $\sum_nx_n$ converges unconditionally in $X$.  Equip it with
359 \[
360 \|x\|_u
361 :=
362 \sup_{F\Subset\mathbb N}
363 \sup_{|a_n|\leq1}
364 \left\|\sum_{n\in F}a_nx_n\right\|.
365 \]
366 \end{definition}
367 \RecordID{PSC-I-D-UNCONDITIONAL-PACKET-SPACE}
368 
369 \begin{lemma}[Completeness of the unconditional packet sum]
370 \label{lem:unconditional-completeness}
371 The normed space $\bigoplus_{n\geq1}^{u}X_n$ in \cref{def:unconditional-packet-space} is Banach.
372 \end{lemma}
373 \RecordID{PSC-I-L-UNCONDITIONAL-COMPLETENESS}
374 \Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE.}
375 
376 \begin{proof}
377 Let $x^{(k)}=(x_n^{(k)})$ be Cauchy in $\|\cdot\|_u$.  For every fixed $n$,
378 \[
379 \|x_n^{(k)}-x_n^{(\ell)}\|
380 \leq
381 \|x^{(k)}-x^{(\ell)}\|_u,
382 \]
383 so $x_n^{(k)}$ converges to some $x_n\in X_n$ because $X_n$ is closed.  Given $\varepsilon>0$, choose $k_0$ such that
384 \[
385 \|x^{(k)}-x^{(\ell)}\|_u<\varepsilon
386 \qquad(k,\ell\geq k_0).
387 \]
388 For finite $F$ and $|a_n|\leq1$, pass $\ell\to\infty$ in the finite sum to obtain
389 \[
390 \left\|\sum_{n\in F}a_n(x_n^{(k)}-x_n)\right\|
391 \leq\varepsilon
392 \qquad(k\geq k_0).
393 \]
394 Hence $\|x^{(k)}-x\|_u\leq\varepsilon$ once $x$ is known to be unconditionally summable.
395 
396 Choose such a $k\geq k_0$.  Since $x^{(k)}$ is unconditionally summable, there is a finite $F_0$ such that every finite $F\subseteq\mathbb N\setminus F_0$ and every $|a_n|\leq1$ satisfy
397 \[
398 \left\|\sum_{n\in F}a_nx_n^{(k)}\right\|<\varepsilon.
399 \]
400 The preceding uniform estimate then gives
401 \[
402 \left\|\sum_{n\in F}a_nx_n\right\|<2\varepsilon.
403 \]
404 Thus the tails of $\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion.  Therefore $x\in\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\to x$ in $\|\cdot\|_u$.
405 \end{proof}
406 
407 \begin{theorem}[Countable unconditional packet isomorphism]
408 \label{thm:countable-packets}
409 Let $X$ be a real or complex Banach space and let $(P_n)_{n\geq1}\subseteq\mathcal B(X)$ satisfy
410 \[
411 P_nP_m=\delta_{nm}P_n
412 \]
413 and
414 \[
415 x=\sum_{n\geq1}P_nx
416 \]
417 unconditionally for every $x\in X$.  Assume
418 \[
419 K:=
420 \sup_{F\Subset\mathbb N}
421 \sup_{|a_n|\leq1}
422 \left\|\sum_{n\in F}a_nP_n\right\|<\infty.
423 \]
424 Then analysis and synthesis define inverse Banach space isomorphisms
425 \[
426 \boxed{
427 X\cong\bigoplus_{n\geq1}^{u}P_nX.
428 }
429 \]
430 More precisely,
431 \[
432 \|\mathcal A_{\Pcal}\|\leq K,
433 \qquad
434 \|\mathcal S_{\Pcal}\|\leq1.
435 \]
436 \end{theorem}
437 \RecordID{PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS}
438 \Depends{PSC-I-D-UNCONDITIONAL-PACKET-SPACE and PSC-I-L-UNCONDITIONAL-COMPLETENESS.}
439 \Exports{Countable analytic packet decompositions under explicit convergence control.}
440 
441 \begin{proof}
442 Each $P_nX=\Ker(I-P_n)$ is closed.  For $x\in X$,
443 \[
444 \|\mathcal A_{\Pcal}x\|_u
445 =
446 \sup_{F,a}
447 \left\|\sum_{n\in F}a_nP_nx\right\|
448 \leq K\|x\|.
449 \]
450 For $y=(y_n)\in\bigoplus^uP_nX$, define
451 \[
452 \mathcal S_{\Pcal}y=\sum_ny_n.
453 \]
454 Since the finite partial sums with every coefficient equal to one are among those used in $\|y\|_u$,
455 \[
456 \|\mathcal S_{\Pcal}y\|
457 \leq\|y\|_u.
458 \]
459 The assumed expansion gives $\mathcal S_{\Pcal}\mathcal A_{\Pcal}x=x$.  Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:
460 \[
461 P_m\mathcal S_{\Pcal}y
462 =
463 \sum_nP_my_n
464 =y_m.
465 \]
466 Thus $\mathcal A_{\Pcal}\mathcal S_{\Pcal}=I$.  Completeness of the coordinate space is \cref{lem:unconditional-completeness}.
467 \end{proof}
468 
469 \begin{remark}[Uniform boundedness]
470 \label{rem:uniform-boundedness}
471 If every expansion $\sum_nP_nx$ is unconditional, the operators obtained from finite partial sums with signs form a pointwise bounded family.  The uniform boundedness principle therefore yields a finite constant $K$.  We retain this constant explicitly in \cref{thm:countable-packets} because later estimates use it to bound the coordinate map.
472 \end{remark}
473 \RecordID{PSC-I-R-UNIFORM-BOUNDEDNESS}
474 
475 \begin{boundary}[Infinite packet sums]
476 \label{bnd:infinite-packet-topology}
477 For an infinite family, an identity such as $\sum_\alpha P_\alpha=I$ requires a declared convergence mode and the corresponding uniform boundedness statement.  Pointwise, strong, norm, weak, and unconditional convergence are not interchangeable.
478 \end{boundary}
479 \RecordID{PSC-I-B-INFINITE-PACKET-TOPOLOGY}
480 
481 \begin{sectionrecap}
482 Finite packet systems give exact coordinates in a direct sum, with no orthogonality requirement.  Their universal property assembles arbitrary packetwise linear maps uniquely.  Tensor products and duals preserve packet structure.  In Banach spaces, the analysis and synthesis norms control conditioning.  The countable theory developed here uses unconditional convergence and a uniform bound on finite partial sums with signs.
483 \end{sectionrecap}
484 
485 
486 \section{Transfer block algebra and support patterns}
487 \begin{readerroadmap}
488 Once vectors are resolved into packets, every operator becomes a matrix of maps between packet spaces.  To multiply operators, we sum over the possible intermediate packets.  The blocks permitted to be nonzero define a relation on the packet indices.  When that relation is a preorder, the resulting algebra has a triangular structure and a nilpotent ideal of strict transfers.
489 \end{readerroadmap}
490 \subsection{Exact operator coordinates}
491 
492 \begin{definition}[Transfer block]
493 \label{def:transfer-block}
494 For $T\in\End_{\bbk}(X)$ and a finite packet system $\Pcal$, define
495 \[
496 T_{\beta\alpha}:=P_\beta TP_\alpha
497 \in\Hom_{\bbk}(X_\alpha,X_\beta).
498 \]
499 \end{definition}
500 \RecordID{PSC-I-D-TRANSFER-BLOCK}
501 
502 \begin{theorem}[Exact operator reconstruction]
503 \label{thm:operator-reconstruction}
504 Every $T\in\End_{\bbk}(X)$ satisfies
505 \[
506 \boxed{T=\sum_{\alpha,\beta\in I}T_{\beta\alpha}.}
507 \]
508 \end{theorem}
509 \RecordID{PSC-I-T-OPERATOR-RECONSTRUCTION}
510 \Depends{PSC-I-T-EXACT-PACKET-COORDINATES.}
511 
512 \begin{proof}
513 Insert the finite packet identity on both sides:
514 \[
515 T=
516 \left(\sum_\beta P_\beta\right)
517 T
518 \left(\sum_\alpha P_\alpha\right)
519 =
520 \sum_{\alpha,\beta}P_\beta TP_\alpha.
521 \]
522 \end{proof}
523 
524 \begin{interpretation}
525 Operator reconstruction inserts the identity on both sides of \(T\).  The source index \(\alpha\) records where a vector begins, and the target index \(\beta\) records where its image lands.  The resulting matrix is exact, and each entry is itself a linear map.
526 \end{interpretation}
527 
528 
529 \begin{theorem}[Packet matrix multiplication]
530 \label{thm:block-multiplication}
531 For $S,T\in\End_{\bbk}(X)$,
532 \[
533 \boxed{
534 (ST)_{\gamma\alpha}
535 =
536 \sum_{\beta\in I}S_{\gamma\beta}T_{\beta\alpha}.
537 }
538 \]
539 \end{theorem}
540 \RecordID{PSC-I-T-BLOCK-MULTIPLICATION}
541 \Depends{PSC-I-D-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}
542 \Exports{All packet path expansions.}
543 
544 \begin{proof}
545 Insert $I=\sum_\beta P_\beta$ between $S$ and $T$:
546 \[
547 P_\gamma STP_\alpha
548 =
549 \sum_\beta
550 P_\gamma SP_\beta TP_\alpha.
551 \]
552 \end{proof}
553 
554 \begin{keyidea}
555 The intermediate packet index in block multiplication plays the role of a summed matrix index.  Repeated multiplication therefore becomes a sum over packet paths.  Chapter II adds weights and filtrations to this path expansion.  Chapter III adds nonlinear trees.
556 \end{keyidea}
557 
558 
559 \begin{corollary}[Packet powers and paths]
560 \label{cor:block-powers}
561 For $m\geq1$,
562 \[
563 (T^m)_{\beta_m\beta_0}
564 =
565 \sum_{\beta_1,\ldots,\beta_{m-1}}
566 T_{\beta_m\beta_{m-1}}
567 \cdots
568 T_{\beta_1\beta_0}.
569 \]
570 The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order.
571 \end{corollary}
572 \RecordID{PSC-I-C-BLOCK-POWERS}
573 \Depends{PSC-I-T-BLOCK-MULTIPLICATION.}
574 
575 \begin{proof}
576 Induct on $m$ using \cref{thm:block-multiplication}.  For a product of different operators, use the same induction with the specified operator at each step.
577 \end{proof}
578 
579 \begin{corollary}[Adjoint reversal]
580 \label{cor:adjoint-reversal}
581 If $X$ is Hilbert, every $P_\alpha$ is orthogonal, and $T\in\mathcal B(X)$, then
582 \[
583 \boxed{(T_{\beta\alpha})^*=(T^*)_{\alpha\beta}.}
584 \]
585 \end{corollary}
586 \RecordID{PSC-I-C-ADJOINT-REVERSAL}
587 \Depends{PSC-I-D-TRANSFER-BLOCK.}
588 
589 \begin{proof}
590 Because $P_\alpha^*=P_\alpha$,
591 \[
592 (P_\beta TP_\alpha)^*=P_\alpha T^*P_\beta.
593 \]
594 \end{proof}
595 
596 \subsection{The full block algebra and intertwining equations}
597 
598 \begin{corollary}[Tensor product transfer blocks]
599 \label{cor:tensor-transfer-blocks}
600 Let \(S\in\End(X)\) and \(T\in\End(Y)\), with packet systems \(\Pcal\) and \(\Qcal\).  Relative to the tensor product packet system of \cref{thm:tensor-packet-system},
601 \[
602 \boxed{
603 (S\otimes T)_{(\beta,\delta),(\alpha,\gamma)}
604 =
605 S_{\beta\alpha}\otimes T_{\delta\gamma}.
606 }
607 \]
608 \end{corollary}
609 \RecordID{PSC-I-C-TENSOR-TRANSFER-BLOCKS}
610 \Depends{PSC-I-T-TENSOR-PACKET-SYSTEM and PSC-I-D-TRANSFER-BLOCK.}
611 \Exports{Tensor network, quantum circuit, and multivariate packet calculations.}
612 
613 \begin{proof}
614 By definition,
615 \begin{align*}
616 (S\otimes T)_{(\beta,\delta),(\alpha,\gamma)}
617 &=(P_\beta\otimes Q_\delta)(S\otimes T)(P_\alpha\otimes Q_\gamma)\\
618 &=(P_\beta SP_\alpha)\otimes(Q_\delta TQ_\gamma)\\
619 &=S_{\beta\alpha}\otimes T_{\delta\gamma}.
620 \end{align*}
621 \end{proof}
622 
623 \begin{theorem}[Blockwise intertwining equations]
624 \label{thm:block-intertwining}
625 Let \(T\in\End(X)\), \(S\in\End(Y)\), let \(\Pcal=(P_\alpha)_{\alpha\in I}\) and \(\Qcal=(Q_\gamma)_{\gamma\in J}\) be finite packet systems, and let \(U:X\to Y\) be linear.  Define
626 \[
627 T_{\beta\alpha}=P_\beta TP_\alpha,
628 \qquad
629 S_{\gamma\delta}=Q_\gamma SQ_\delta,
630 \qquad
631 U_{\gamma\alpha}=Q_\gamma UP_\alpha.
632 \]
633 Then
634 \[
635 \boxed{UT=SU}
636 \]
637 if and only if, for every \(\gamma\in J\) and \(\alpha\in I\),
638 \[
639 \boxed{
640 \sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha}
641 =
642 \sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}.
643 }
644 \]
645 \end{theorem}
646 \RecordID{PSC-I-T-BLOCK-INTERTWINING}
647 \Depends{PSC-I-T-OPERATOR-RECONSTRUCTION and PSC-I-T-BLOCK-MULTIPLICATION.}
648 \Exports{Packetwise equivalence, gluing, symmetry, and local representation comparison.}
649 
650 \begin{proof}
651 Apply \(Q_\gamma\) on the left and \(P_\alpha\) on the right.  Inserting \(I_X=\sum_\beta P_\beta\) gives
652 \[
653 Q_\gamma UTP_\alpha
654 =
655 \sum_\beta Q_\gamma UP_\beta TP_\alpha
656 =
657 \sum_\beta U_{\gamma\beta}T_{\beta\alpha}.
658 \]
659 Inserting \(I_Y=\sum_\delta Q_\delta\) gives
660 \[
661 Q_\gamma SUP_\alpha
662 =
663 \sum_\delta Q_\gamma SQ_\delta UP_\alpha
664 =
665 \sum_\delta S_{\gamma\delta}U_{\delta\alpha}.
666 \]
667 Thus \(UT=SU\) implies every block equation.  Conversely, if every block equation holds, then all blocks of \(UT-SU\) vanish.  Exact operator reconstruction yields \(UT-SU=0\).
668 \end{proof}
669 
670 \begin{interpretation}
671 A global intertwiner is equivalent to a coupled finite system of packet equations.  Later arguments about holonomy and representations use this theorem after the local blocks have been constructed.  The equations state the exact compatibility those blocks must satisfy.
672 \end{interpretation}
673 
674 
675 \begin{theorem}[Full block algebra representation]
676 \label{thm:block-algebra}
677 The map
678 \[
679 \Theta_{\Pcal}:\End_{\bbk}(X)
680 \longrightarrow
681 \prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta),
682 \qquad
683 T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
684 \]
685 is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators.
686 \end{theorem}
687 \RecordID{PSC-I-T-BLOCK-ALGEBRA}
688 \Depends{\nolinkurl{PSC-I-T-OPERATOR-RECONSTRUCTION} and
689 \nolinkurl{PSC-I-T-BLOCK-MULTIPLICATION}.}
690 
691 \begin{proof}
692 Injectivity follows from exact reconstruction.  Given arbitrary linear maps $B_{\beta\alpha}:X_\alpha\to X_\beta$, define
693 \[
694 T\left(\sum_\alpha x_\alpha\right)
695 :=
696 \sum_{\alpha,\beta}B_{\beta\alpha}x_\alpha.
697 \]
698 The packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps.  Thus $\Theta_{\Pcal}$ is surjective.  Compatibility with multiplication is \cref{thm:block-multiplication}.
699 \end{proof}
700 
701 
702 
703 \begin{corollary}[Propagation of packet support]
704 \label{cor:packet-support-propagation}
705 Let \(T\in\End(X)\) and \(x\in X\).  Then
706 \[
707 \boxed{
708 \Supp_{\Pcal}(Tx)
709 \subseteq
710 \left\{\beta:\exists\alpha\in\Supp_{\Pcal}(x)
711 \text{ with }T_{\beta\alpha}\neq0\right\}.
712 }
713 \]
714 Equality holds whenever, for every target \(\beta\) in the set on the right, the active contributions
715 \[
716 \bigl(T_{\beta\alpha}P_\alpha x:
717 \alpha\in\Supp_{\Pcal}(x),\ T_{\beta\alpha}\neq0\bigr)
718 \]
719 have nonzero sum.  In particular, equality holds if at most one active source packet feeds each such \(\beta\) and that source coordinate does not lie in the kernel of the corresponding block.
720 \end{corollary}
721 \RecordID{PSC-I-C-PACKET-SUPPORT-PROPAGATION}
722 \Depends{PSC-I-D-PACKET-SUPPORT and PSC-I-T-OPERATOR-RECONSTRUCTION.}
723 \Exports{Graph reachability, leakage bounds, and support pruning.}
724 
725 \begin{proof}
726 For every target packet \(\beta\),
727 \[
728 P_\beta Tx
729 =
730 \sum_{\alpha\in\Supp_{\Pcal}(x)}T_{\beta\alpha}P_\alpha x.
731 \]
732 If every block in this sum is zero, then the coordinate with index \(\beta\) vanishes.  This proves the inclusion.  Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right.  The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.
733 \end{proof}
734 
735 
736 \begin{corollary}[Dimension count]
737 \label{cor:dimension-count}
738 If $X$ is finite dimensional and $d_\alpha=\dim X_\alpha$, then
739 \[
740 \dim\End_{\bbk}(X)
741 =
742 \sum_{\alpha,\beta}d_\alpha d_\beta
743 =
744 \left(\sum_\alpha d_\alpha\right)^2.
745 \]
746 \end{corollary}
747 \RecordID{PSC-I-C-DIMENSION-COUNT}
748 \Depends{PSC-I-T-BLOCK-ALGEBRA.}
749 
750 \begin{proof}
751 Take dimensions in the block algebra isomorphism and use $\dim\Hom(X_\alpha,X_\beta)=d_\alpha d_\beta$.
752 \end{proof}
753 
754 \begin{proposition}[Block diagonal projection and packet commutant]
755 \label{prop:block-diagonal-projection}
756 Define
757 \[
758 \Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
759 \]
760 Then $\Delta_{\Pcal}$ is a linear idempotent on $\End(X)$ whose range is the block diagonal subalgebra.  For $T\in\End(X)$, the following are equivalent:
761 \begin{enumerate}[label=(\roman*)]
762 \item $TP_\alpha=P_\alpha T$ for every $\alpha$.
763 \item $T_{\beta\alpha}=0$ whenever $\beta\neq\alpha$.
764 \item $\Delta_{\Pcal}(T)=T$.
765 \end{enumerate}
766 \end{proposition}
767 \RecordID{PSC-I-P-BLOCK-DIAGONAL-COMMUTANT}
768 \Depends{PSC-I-T-OPERATOR-RECONSTRUCTION.}
769 \Exports{Symmetry sector invariance and packet leakage decompositions.}
770 
771 \begin{proof}
772 Pairwise annihilation gives $\Delta_{\Pcal}^2=\Delta_{\Pcal}$ and identifies its range with diagonal block families.  If $T$ commutes with all $P_\alpha$, then for $\beta\neq\alpha$,
773 \[
774 P_\beta TP_\alpha=P_\beta P_\alpha T=0.
775 \]
776 Thus (i) implies (ii).  Exact reconstruction shows (ii) implies (iii).  If $T=\sum_\alpha P_\alpha TP_\alpha$, then
777 \[
778 TP_\gamma=P_\gamma TP_\gamma=P_\gamma T,
779 \]
780 so (iii) implies (i).
781 \end{proof}
782 
783 \begin{microexample}[title={Lower triangular support}]
784 Take three packets of dimension one.  Permit the diagonal blocks and transfers from a source with a lower index to a target with a higher index.  The support relation is a total preorder.  The blocks outside the diagonal form a nilpotent ideal, and every product of three strict transfers vanishes.  This finite algebraic example anticipates the weighted triangular inverses developed in Chapter II.
785 \end{microexample}
786 
787 
788 \subsection{Support patterns, preorders, and triangular structure}
789 \begin{definition}[Support relation and support space]
790 \label{def:support-relation}
791 For a relation $R\subseteq I\times I$, with $(\beta,\alpha)$ interpreted as an allowed transfer $\alpha\to\beta$, define
792 \[
793 \mathfrak M_R
794 :=
795 \{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
796 \]
797 \end{definition}
798 \RecordID{PSC-I-D-SUPPORT-RELATION}
799 
800 \begin{theorem}[Support pattern algebra criterion]
801 \label{thm:support-pattern}
802 Assume $X_\alpha\neq0$ for every $\alpha\in I$.  Then:
803 \begin{enumerate}[label=(\roman*)]
804 \item $I_X\in\mathfrak M_R$ if and only if $R$ is reflexive.
805 \item $\mathfrak M_R\mathfrak M_R\subseteq\mathfrak M_R$ if and only if $R$ is transitive.
806 \item $\mathfrak M_R$ is a unital subalgebra if and only if $R$ is a preorder.
807 \end{enumerate}
808 \end{theorem}
809 \RecordID{PSC-I-T-SUPPORT-PATTERN-ALGEBRA}
810 \Depends{PSC-I-T-BLOCK-MULTIPLICATION.}
811 
812 \begin{proof}
813 The identity has diagonal blocks $I_{X_\alpha}$ and no blocks outside the diagonal, so it belongs to $\mathfrak M_R$ exactly when $(\alpha,\alpha)\in R$ for every $\alpha$.
814 
815 Assume $R$ is transitive.  If a summand
816 \[
817 S_{\gamma\beta}T_{\beta\alpha}
818 \]
819 in the $(\gamma,\alpha)$ block of $ST$ is nonzero, then $(\gamma,\beta)$ and $(\beta,\alpha)$ lie in $R$.  Transitivity gives $(\gamma,\alpha)\in R$, so every forbidden block of $ST$ vanishes.
820 
821 Conversely, suppose transitivity fails.  Choose $\alpha,\beta,\gamma$ with
822 \[
823 (\beta,\alpha)\in R,
824 \qquad
825 (\gamma,\beta)\in R,
826 \qquad
827 (\gamma,\alpha)\notin R.
828 \]
829 Choose nonzero vectors $u\in X_\alpha$, $v\in X_\beta$, and $w\in X_\gamma$.  Choose linear functionals $f\in X_\alpha^*$ and $g\in X_\beta^*$ with $f(u)=g(v)=1$.  Let $T$ have the sole block $x\mapsto f(x)v$ from $X_\alpha$ to $X_\beta$, and let $S$ have the sole block $y\mapsto g(y)w$ from $X_\beta$ to $X_\gamma$.  Then $S,T\in\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\gamma,\alpha)$ block.  This proves necessity.  Combining reflexivity and transitivity gives (iii).
830 \end{proof}
831 
832 \begin{corollary}[Bounded support patterns closed under $*$]
833 \label{cor:star-support}
834 Let $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\alpha\neq0$ for every $\alpha\in I$, and put
835 \[
836 \mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
837 \]
838 Then $\mathfrak M_R^{\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric.  Hence $\mathfrak M_R^{\mathrm b}$ is a unital $*$ subalgebra of $\mathcal B(X)$ if and only if $R$ is an equivalence relation.
839 \end{corollary}
840 \RecordID{PSC-I-C-SUPPORT-PATTERN-STAR}
841 \Depends{PSC-I-C-ADJOINT-REVERSAL and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}
842 
843 \begin{proof}
844 Adjoint reversal sends an allowed $(\beta,\alpha)$ block to an $(\alpha,\beta)$ block.  Symmetry is therefore sufficient.  If symmetry fails, choose a nonzero map of rank one in an allowed nonsymmetric block.  Its adjoint occupies the forbidden reverse block, proving necessity.  A reflexive, transitive, symmetric relation is an equivalence relation.
845 \end{proof}
846 
847 \begin{theorem}[Preorder support algebra decomposition]
848 \label{thm:preorder-decomposition}
849 Let $R$ be a preorder on the finite set $I$.  Define
850 \[
851 \alpha\sim\beta
852 \quad\Longleftrightarrow\quad
853 (\beta,\alpha)\in R\text{ and }(\alpha,\beta)\in R.
854 \]
855 Let $\overline I=I/{\sim}$, which carries the induced partial order, and put
856 \[
857 X_C:=\bigoplus_{\alpha\in C}X_\alpha.
858 \]
859 Let $\mathfrak N_R\subseteq\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class.  Then:
860 \begin{enumerate}[label=(\roman*)]
861 \item $\mathfrak N_R$ is a nilpotent ideal on both sides.
862 \item If $h$ is the maximal number of strict steps in a chain of $\overline I$, then
863 \[
864 \mathfrak N_R^{\,h+1}=0.
865 \]
866 \item The map to the diagonal classes induces an algebra isomorphism
867 \[
868 \boxed{
869 \mathfrak M_R/\mathfrak N_R
870 \cong
871 \prod_{C\in\overline I}\End_{\bbk}(X_C).
872 }
873 \]
874 \end{enumerate}
875 \end{theorem}
876 \RecordID{PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION}
877 \Depends{PSC-I-T-BLOCK-MULTIPLICATION and PSC-I-T-SUPPORT-PATTERN-ALGEBRA.}
878 \Exports{Filtered transfer algebras and triangular packet inverses in Paper II.}
879 
880 \begin{proof}
881 A nonzero block in $\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order.  The block multiplication law shows that products concatenate such moves.  Therefore left or right multiplication by an element of $\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\mathfrak N_R$ is an ideal on both sides.  A product of $h+1$ elements of $\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\overline I$, which does not exist.  This proves nilpotence.
882 
883 Let $P_C=\sum_{\alpha\in C}P_\alpha$.  Define
884 \[
885 \rho:\mathfrak M_R\longrightarrow\prod_C\End(X_C),
886 \qquad
887 \rho(T)=(P_CTP_C)_C.
888 \]
889 If an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\to D$ and $D\to C$, forcing $C=D$ in the quotient poset.  Hence
890 \[
891 P_CSTP_C=(P_CSP_C)(P_CTP_C),
892 \]
893 so $\rho$ is an algebra homomorphism.  Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\rho$ is surjective.  Its kernel is exactly $\mathfrak N_R$.  The first isomorphism theorem gives the quotient formula.
894 \end{proof}
895 
896 \begin{sectionrecap}
897 Every operator has an exact packet matrix.  Products sum over intermediate packet indices, tensor products factor blockwise, and global intertwiners are equivalent to coupled block equations.  A support pattern is closed under multiplication precisely when its relation is transitive.  For a preorder pattern, the quotient is block diagonal and the ideal of strict transfers is nilpotent.
898 \end{sectionrecap}
899 
900 \begin{figure}[htbp]
901 \centering
902 \includegraphics[width=0.96\textwidth]{02_transfer_block_norms.pdf}
903 \caption{An operator represented by the norms of its blocks between packets. The diagonal entries preserve packets. Entries outside the diagonal describe transfers between packets.}
904 \label{fig:transfer-blocks}
905 \end{figure}
906 
907 \section{Transfer graphs and directed packet generation}
908 
909 
910 
911 \begin{readerroadmap}
912 The transfer graph records which blocks between packets occur in a declared generating family.  Reachability constrains every word in that family.  To obtain an exact seed count from the graph, we also need hypotheses that let generation propagate along its edges.
913 \end{readerroadmap}
914 \subsection{Graphs generated by declared operators}
915 
916 \begin{definition}[Transfer graph of a family generating the algebra]
917 \label{def:transfer-graph}
918 Let $\Pcal=(P_\alpha)_{\alpha\in I}$ be a finite packet system on $X$ with every $P_\alpha\in\A$.  Let $\mathcal G\subseteq\A$ satisfy
919 \[
920 \A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
921 \]
922 where $\alg_{\bbk}(E)$ denotes the unital subalgebra over $\bbk$ generated by $E$.  The transfer graph $\Gamma(\mathcal G,\Pcal)$ has vertex set $I$ and an arrow
923 \[
924 \alpha\longrightarrow\beta
925 \]
926 when
927 \[
928 P_\beta GP_\alpha\neq0
929 \]
930 for at least one $G\in\mathcal G$.  Reachability includes paths of length zero.
931 \end{definition}
932 \RecordID{PSC-I-D-TRANSFER-GRAPH}
933 
934 \begin{theorem}[Reachability support of the generated algebra]
935 \label{thm:algebra-reachability-support}
936 For every $A\in\A$,
937 \[
938 P_\beta AP_\alpha=0
939 \qquad\text{whenever }\beta\notin\Reach(\alpha).
940 \]
941 Equivalently,
942 \[
943 \A\subseteq\mathfrak M_{R_\Gamma},
944 \qquad
945 R_\Gamma:=\{(\beta,\alpha):\beta\in\Reach(\alpha)\}.
946 \]
947 \end{theorem}
948 \RecordID{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT}
949 \Depends{PSC-I-D-TRANSFER-GRAPH and PSC-I-C-BLOCK-POWERS.}
950 
951 \begin{proof}
952 It is enough to treat a word in the generators and packet projectors.  Insert
953 \[
954 I=\sum_{\gamma\in I}P_\gamma
955 \]
956 between consecutive nonprojector generators.  Every resulting $(\beta,\alpha)$ block is a finite sum of products
957 \[
958 P_\beta G_mP_{\gamma_{m-1}}
959 G_{m-1}\cdots
960 P_{\gamma_1}G_1P_\alpha.
961 \]
962 A nonzero product requires every consecutive block to be nonzero, hence gives a directed path
963 \[
964 \alpha\to\gamma_1\to\cdots\to\gamma_{m-1}\to\beta.
965 \]
966 Packet projectors only insert stationary steps.  Thus a word has no block from $\alpha$ to an unreachable $\beta$.  Linear combinations preserve the same zero pattern.
967 \end{proof}
968 
969 \begin{interpretation}
970 The graph belongs to the declared generating family within \(\End(X)\).  Multiplying its generators concatenates paths, so a word in them cannot create a block outside graph reachability.  Reachability alone gives less information in the other direction.  A nonzero edge block may reach only a proper subspace of its target packet.
971 \end{interpretation}
972 
973 
974 A path records a possible sequence of transfers.  It does not guarantee that the composed transfer is nonzero.  The image of one block may lie in the kernel of the next, and different path contributions may cancel when they are added.  The reachability theorem therefore gives a restriction on support.  To prove that a target packet is generated, we must also examine what the blocks do to vectors.  Surjectivity along edges will provide that additional information below.
975 
976 \subsection{Reachability obstructions for seeds}
977 \begin{proposition}[Reachability obstruction for generated hulls]
978 \label{prop:reachability-obstruction}
979 Let $I_0\subseteq I$ and let
980 \[
981 S\subseteq\bigoplus_{\alpha\in I_0}X_\alpha.
982 \]
983 Then
984 \[
985 \boxed{
986 \Hull_{\A}(S)
987 \subseteq
988 \bigoplus_{\beta\in\Reach(I_0)}X_\beta.
989 }
990 \]
991 In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component.
992 \end{proposition}
993 \RecordID{PSC-I-P-REACHABILITY-OBSTRUCTION}
994 \Depends{PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT.}
995 
996 \begin{proof}
997 For $s\in S$, write $s=\sum_{\alpha\in I_0}P_\alpha s$.  If $\beta$ is unreachable from $I_0$, then for every $A\in\A$,
998 \[
999 P_\beta As
1000 =
1001 \sum_{\alpha\in I_0}P_\beta AP_\alpha s
1002 =0
1003 \]
1004 by \cref{thm:algebra-reachability-support}.  The same holds for every finite linear combination of orbit vectors.
1005 \end{proof}
1006 
1007 \subsection{Packet pure generation and source components}
1008 A strongly connected component groups packets that can reach one another.  The condensation graph replaces each such group by a single vertex.  It has no directed cycles between distinct components.  A source component has no incoming edge from another component, so generation there must begin with seed coordinates already present in that component.  The rank below records the effect of requiring each seed to lie in a single packet.
1009 
1010 \Needspace{15\baselineskip}
1011 \begin{definition}[Packet pure seed rank]
1012 \label{def:directed-seed-rank}
1013 A vector is packet pure when it is zero or lies in one packet space $X_\alpha$, equivalently when its packet support has cardinality at most one.  Define
1014 \[
1015 \psrank_{\A,\Pcal}(X)
1016 :=
1017 \min\{|S|:S\text{ is packet-pure and }\A S=X\}.
1018 \]
1019 For $C\subseteq I$, put
1020 \[
1021 P_C:=\sum_{\alpha\in C}P_\alpha,
1022 \qquad
1023 X_C:=P_CX,
1024 \qquad
1025 \A_C:=P_C\A P_C,
1026 \]
1027 and define the local packet pure rank
1028 \[
1029 r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
1030 \]
1031 \end{definition}
1032 \RecordID{PSC-I-D-DIRECTED-SEED-RANK}
1033 
1034 \begin{proposition}[Basic packet pure rank bounds]
1035 \label{prop:packet-pure-rank-bounds}
1036 For every $A$ module resolved into packets with $P_\alpha\in A$,
1037 \[
1038 \boxed{
1039 \srank_A(X)
1040 \leq
1041 \psrank_{A,\Pcal}(X)
1042 \leq
1043 \sum_{\alpha\in I}
1044 \srank_{P_\alpha A P_\alpha}(X_\alpha).
1045 }
1046 \]
1047 The corner algebra on the right is unital on $X_\alpha$ with identity $P_\alpha$.
1048 \end{proposition}
1049 \RecordID{PSC-I-P-PACKET-PURE-RANK-BOUNDS}
1050 \Depends{PSC-I-D-SEED-RANK and PSC-I-D-DIRECTED-SEED-RANK.}
1051 
1052 \begin{proof}
1053 Every packet pure generating family is an unrestricted generating family, which gives the first inequality.  For each $\alpha$, choose a minimal generating family $S_\alpha\subseteq X_\alpha$ for the corner action $P_\alpha A P_\alpha$.  The union $S=\bigcup_\alpha S_\alpha$ is packet pure.  Since every $x\in X$ has the unique decomposition $x=\sum_\alpha x_\alpha$ and every $x_\alpha$ lies in $(P_\alpha A P_\alpha)S_\alpha\subseteq AS$, the union generates $X$.  Its cardinality is at most the displayed sum.
1054 \end{proof}
1055 
1056 \begin{definition}[Packet system surjective along edges]
1057 \label{def:edge-surjective}
1058 The module resolved into packets is surjective along edges relative to $\Gamma(\mathcal G,\Pcal)$ when, for every graph edge $\alpha\to\beta$, there exists $B\in\A$ such that the pure block
1059 \[
1060 P_\beta BP_\alpha:X_\alpha\twoheadrightarrow X_\beta
1061 \]
1062 is surjective.
1063 \end{definition}
1064 \RecordID{PSC-I-D-EDGE-SURJECTIVE}
1065 
1066 \begin{theorem}[Source component formula for packet pure seed rank]
1067 \label{thm:source-component-rank}
1068 Assume $I$ is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges.  Let $\mathfrak C_0$ be the set of source strongly connected components of the condensation graph of $\Gamma(\mathcal G,\Pcal)$.  Then
1069 \[
1070 \boxed{
1071 \psrank_{\A,\Pcal}(X)
1072 =
1073 \sum_{C\in\mathfrak C_0}r_C.
1074 }
1075 \]
1076 \end{theorem}
1077 \RecordID{PSC-I-T-SOURCE-COMPONENT-SEED-RANK}
1078 \Depends{PSC-I-P-REACHABILITY-OBSTRUCTION and PSC-I-D-EDGE-SURJECTIVE.}
1079 \Exports{Residual transport forests and directed source rank calculations.}
1080 
1081 \begin{proof}
1082 Let $S$ be a packet pure generating family for $X$.  Discard zero seeds, which do not change the generated hull, and fix a source component $C$.  Seeds outside $C$ cannot reach $C$ by \cref{prop:reachability-obstruction}.  Moreover, for $s\in X_C$ and $A\in\A$,
1083 \[
1084 P_CAs=(P_CAP_C)s\in\A_Cs.
1085 \]
1086 Thus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\A_C$, and there are at least $r_C$ of them.  Summing over disjoint source components gives
1087 \[
1088 \psrank_{\A,\Pcal}(X)
1089 \geq
1090 \sum_{C\in\mathfrak C_0}r_C.
1091 \]
1092 
1093 For the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\A_C$, and put $S=\bigcup_CS_C$.  Then every source component lies in $\A S$.  Traverse the finite condensation graph in a topological order.  Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\A S$.  Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\alpha\to\beta$ with $\alpha$ already generated and $\beta\in D$.  Surjectivity along edges gives a pure block
1094 \[
1095 B_{\beta\alpha}:X_\alpha\twoheadrightarrow X_\beta.
1096 \]
1097 Because $X_\alpha\subseteq\A S$ and $\A S$ is invariant under $\A$, surjectivity implies $X_\beta\subseteq\A S$.  For any other vertex $\gamma\in D$, choose a directed path inside the strongly connected component from $\beta$ to $\gamma$.  Successive surjective edge blocks propagate the full packet space along the path, so $X_\gamma\subseteq\A S$.  Hence $X_D\subseteq\A S$.  Induction over the condensation graph gives $X=\A S$, proving the upper bound.
1098 \end{proof}
1099 
1100 \begin{hypothesisguide}
1101 The formula for source components uses local generation inside those components and surjectivity along the edges.  Local generation supplies the starting packet spaces.  Surjectivity then propagates their full spaces through the condensation graph.  Without it, a path may exist while its blocks reach only a proper subspace.
1102 \end{hypothesisguide}
1103 
1104 
1105 \begin{definition}[Packet propagating system]
1106 \label{def:packet-propagating}
1107 A packet system that is surjective along edges is called packet propagating when every packet space $X_\alpha$ is cyclic under its diagonal corner algebra
1108 \[
1109 P_\alpha\A P_\alpha.
1110 \]
1111 \end{definition}
1112 \RecordID{PSC-I-D-PACKET-PROPAGATING}
1113 
1114 \begin{corollary}[Unweighted source component count]
1115 \label{cor:source-component-count}
1116 Assume the module resolved into packets is packet propagating and $X_\alpha\neq0$ for every $\alpha\in I$.  If $c_0$ denotes the number of source strongly connected components, then
1117 \[
1118 \boxed{
1119 \psrank_{\A,\Pcal}(X)=c_0.
1120 }
1121 \]
1122 \end{corollary}
1123 \RecordID{PSC-I-C-SOURCE-COMPONENT-COUNT}
1124 \Depends{PSC-I-T-SOURCE-COMPONENT-SEED-RANK.}
1125 
1126 \begin{proof}
1127 Fix a source component $C$ and a vertex $\alpha\in C$.  Choose a cyclic vector $s\in X_\alpha$ for $P_\alpha\A P_\alpha$.  It generates all of $X_\alpha$.  Strong connectivity and surjectivity along edges then propagate the whole packet space to every vertex of $C$.  Thus $r_C=1$ for every source component.  Apply \cref{thm:source-component-rank}.
1128 \end{proof}
1129 
1130 \begin{boundary}[Mixed support seeds can merge source requirements]
1131 \label{bnd:packet-pure-boundary}
1132 The source component formulas concern packet pure seeds.  They do not give the unrestricted seed rank.  For example, let
1133 \[
1134 \A=\bbk\times\bbk,
1135 \qquad
1136 X=\bbk\oplus\bbk
1137 \]
1138 with the coordinate action and the two coordinate packet projectors.  The transfer graph has two source components and
1139 \[
1140 \psrank_{\A,\Pcal}(X)=2,
1141 \]
1142 but the mixed vector $(1,1)$ is cyclic, so
1143 \[
1144 \srank_{\A}(X)=1.
1145 \]
1146 \end{boundary}
1147 \RecordID{PSC-I-B-PACKET-PURE-BOUNDARY}
1148 
1149 \begin{microexample}[title={Why mixed seeds change the answer}]
1150 Let the acting algebra be diagonal on \(X=X_1\oplus X_2\), with no transfer between the packets.  Generation by packet pure seeds requires one seed in each source component, so \(\psrank=2\).  A single mixed vector \(x_1+x_2\) may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be \(1\).
1151 \end{microexample}
1152 
1153 \begin{sectionrecap}
1154 Graph reachability restricts which blocks can be nonzero in the algebra generated by the declared operators.  Source components give the exact rank for packet pure seeds under the stated local generation and edge propagation hypotheses.  Seeds with several nonzero packet coordinates have a different minimum count and may meet several source requirements at once.
1155 \end{sectionrecap}
1156 
1157 

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