Realization Calculus

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1 \section{Coefficient conventions, logical status, and scope}
2 \begin{readerroadmap}
3 This section sets out the conventions used throughout the chapter.  It explains when a statement concerns algebra, when topology enters the argument, which results are taken as classical input, and how the stable record IDs connect this chapter to later work.  Keeping these settings distinct makes the hypotheses of each result easier to follow.
4 \end{readerroadmap}
5 \subsection{Ambient regimes and theorem records}
6 
7 \begin{definition}[Coefficient and module regimes]
8 \label{def:coefficient-regime}
9 All modules are left modules unless stated otherwise.  The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$.  When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\bbk$, where $\bbk$ is a field, and $M$ is a vector space over $\bbk$.  The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators.  Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.
10 \end{definition}
11 \RecordID{PSC-I-D-COEFFICIENT-REGIME}
12 \paragraph{Conventions for empty families.} For finite families of nonnegative integers we use $\max\varnothing=0$. The height of an empty finite partially ordered set is $0$, as is the height of a nonempty poset with no strict comparable pair. Thus the height in the theorem for the support preorder counts strict steps.
13 
14 \begin{definition}[Canonical theorem record]
15 \label{def:theorem-record}
16 A canonical theorem record is the tuple
17 \[
18 (\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement},
19 \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
20 \]
21 A later chapter cites the stable record ID.  A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
22 \end{definition}
23 \RecordID{PSC-I-D-THEOREM-RECORD}
24 
25 \begin{boundary}[Symbolic scope]
26 \label{bnd:symbolic-scope}
27 This chapter proves the algebraic and topological statements printed below.  It does not assert historical novelty, independent external certification, automatic discovery of useful packet systems, nonlinear obstruction theory, multilevel refinement, arithmetic congruences, quantum gravity, automorphy, or local or global Langlands reciprocity.
28 \end{boundary}
29 \RecordID{PSC-I-B-SYMBOLIC-SCOPE}
30 
31 \section{Typed operator grammars, realizations, and seed systems}
32 \begin{readerroadmap}
33 An operator grammar describes which compositions are allowed and which relations they satisfy.  A realization gives this syntax a concrete action on vector spaces.  Its realized algebra consists of the operators that actually occur in that action.  We can then ask which vectors serve as seeds.
34 \end{readerroadmap}
35 \subsection{Typed syntax and module realizations}
36 
37 \begin{definition}[Typed operator grammar]
38 \label{def:typed-grammar}
39 Let $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set.  A typed operator grammar over $\bbk$ is a pair
40 \[
41 \mathfrak G=(Q,I),
42 \]
43 where $I\triangleleft \bbk Q$ is an ideal on both sides generated by linear combinations over $\bbk$ of paths having common source and common target.  Write
44 \[
45 \Lambda_{\mathfrak G}:=\bbk Q/I.
46 \]
47 A realization of $\mathfrak G$ is a family $(X_v)_{v\in Q_0}$ and linear maps
48 \[
49 X_a:X_{s(a)}\longrightarrow X_{t(a)}
50 \qquad(a\in Q_1)
51 \]
52 that annihilate every relation in $I$.  A morphism $f:X\to Y$ of realizations is a family of linear maps $f_v:X_v\to Y_v$ satisfying
53 \[
54 f_{t(a)}X_a=Y_af_{s(a)}
55 \qquad(a\in Q_1).
56 \]
57 \end{definition}
58 \RecordID{PSC-I-D-TYPED-GRAMMAR}
59 
60 \begin{definition}[Realization on a single space]
61 \label{def:collapsed-realization}
62 For a typed realization $X$, define its realization on a single space by
63 \[
64 |X|:=\bigoplus_{v\in Q_0}X_v.
65 \]
66 Every vertex idempotent acts as a coordinate projector on $|X|$, and every path acts as a block operator from its source space to its target space.
67 \end{definition}
68 \RecordID{PSC-I-D-COLLAPSED-REALIZATION}
69 
70 \begin{theorem}[Typed realization and module equivalence]
71 \label{thm:typed-module-equivalence}
72 The category of realizations of $\mathfrak G$ is equivalent to the category of unital left modules over $\Lambda_{\mathfrak G}$.  Explicitly, a realization $X$ determines the collapsed space $|X|$ of \cref{def:collapsed-realization}
73 with the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target.  Conversely, a unital module $M$ over $\Lambda_{\mathfrak G}$ determines the realization
74 \[
75 M_v=e_vM,
76 \qquad
77 M_a(x)=ax.
78 \]
79 The two constructions are inverse up to canonical natural isomorphism.
80 \end{theorem}
81 \RecordID{PSC-I-T-TYPED-MODULE-EQUIVALENCE}
82 \Depends{PSC-I-D-TYPED-GRAMMAR.}
83 
84 \begin{proof}
85 Because $Q_0$ is finite, $1=\sum_{v\in Q_0}e_v$ in $\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\Lambda_{\mathfrak G}$.  Given a realization, define the action of a path by composition of its arrow maps and extend linearly.  Every element of $I$ acts as zero, so the action factors through $\Lambda_{\mathfrak G}$.  The vertex idempotents give
86 \[
87 |X|=\bigoplus_v e_v|X|=\bigoplus_vX_v.
88 \]
89 Conversely, if $M$ is unital, then
90 \[
91 M=1M=\sum_ve_vM.
92 \]
93 Orthogonality makes the sum direct.  For an arrow $a:v\to w$, the identities $a=e_wae_v$ imply
94 \[
95 a(e_vM)\subseteq e_wM.
96 \]
97 The relations in $I$ act as zero because $M$ is a module over $\Lambda_{\mathfrak G}$.  Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps.  These identifications commute with morphisms, giving an equivalence of categories.
98 \end{proof}
99 
100 \begin{interpretation}
101 The equivalence does not claim that a quiver presentation is unique.  It says that once a grammar is fixed, working with typed arrows and working with a module over the quotient path algebra carry the same information.  The typed description is often easier for constructing an operator system.  The module description is often easier for proving generation and quotient theorems.
102 \end{interpretation}
103 
104 \begin{microexample}[title={A grammar with two vertices}]
105 Take two vertices and one arrow \(a:1\to2\).  A realization is a pair of spaces and a linear map
106 \[
107 X_1\xrightarrow{\,X_a\,}X_2.
108 \]
109 After collapse, \(|X|=X_1\oplus X_2\), the vertex idempotents are the two coordinate projectors, and \(a\) is a single block outside the diagonal.  This elementary case already contains the viewpoint of packets and transfer blocks used later.
110 \end{microexample}
111 
112 
113 \subsection{The realized action and faithful reduction}
114 \begin{definition}[Realized operator algebra]
115 \label{def:realized-algebra}
116 For a realization $X$, the realized operator algebra is
117 \[
118 \A_X:=\im\!\left(\Lambda_{\mathfrak G}\longrightarrow\End_{\bbk}(|X|)\right).
119 \]
120 It records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.
121 \end{definition}
122 \RecordID{PSC-I-D-REALIZED-ALGEBRA}
123 
124 \begin{proposition}[Action factorization and hull invariance]
125 \label{prop:action-factorization}
126 Let $S\subseteq |X|$.  The generated submodule under the grammar algebra equals the generated submodule under the realized operator algebra:
127 \[
128 \Lambda_{\mathfrak G}S=\A_XS.
129 \]
130 Consequently, every statement about seed rank depends only on the realized action, not on redundant relations in a chosen presentation of the grammar.
131 \end{proposition}
132 \RecordID{PSC-I-P-ACTION-FACTORIZATION}
133 \Depends{PSC-I-D-REALIZED-ALGEBRA.}
134 
135 \begin{proof}
136 The action map $\rho:\Lambda_{\mathfrak G}\twoheadrightarrow\A_X$ is surjective by definition and satisfies $\lambda s=\rho(\lambda)s$.  Hence every grammar orbit vector lies in $\A_XS$.  The reverse inclusion holds because every element of $\A_X$ is $\rho(\lambda)$ for some $\lambda$.
137 \end{proof}
138 
139 \begin{keyidea}
140 Replacing the abstract grammar algebra by its image \(\A_X\) removes syntax that acts as zero.  This is a faithful reduction of the action.  It does not change any orbit, hull, or computation of seed rank.
141 \end{keyidea}
142 
143 
144 \subsection{Seeds, addressed hulls, and natural presentations}
145 \begin{definition}[Linear seed system]
146 \label{def:seed-system}
147 A linear seed system is a tuple
148 \[
149 \mathfrak S=(X,\A,S,\Pcal,\tau),
150 \]
151 where $X$ is a unital left module over $\A$, $\A$ is fixed independently of the target element, $S=(s_1,\ldots,s_r)$ is a finite distinguished seed family, $\Pcal$ is either empty or a declared packet system, and $\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
152 \end{definition}
153 \RecordID{PSC-I-D-SEED-SYSTEM}
154 
155 \begin{definition}[Generated hull]
156 \label{def:generated-hull}
157 For $S\subseteq X$, define
158 \[
159 \Hull_{\A}(S):=\sum_{s\in S}\A s.
160 \]
161 In the topological regime define
162 \[
163 \Hull_{\A,\tau}(S):=\cl{\Hull_{\A}(S)}^{\,\tau}.
164 \]
165 For a singleton $S=\{s\}$, write $\Hull_{\A}(s)$.
166 \end{definition}
167 \RecordID{PSC-I-D-GENERATED-HULL}
168 
169 The algebraic hull uses finite sums of addressed seed vectors.  In the topological setting, taking its closure may add limits that no finite expression reaches.  Thus algebraic generation and dense generation answer different questions, even when they use the same seeds and operators.
170 
171 \begin{definition}[Natural presentation]
172 \label{def:natural-presentation}
173 A seed presentation is natural relative to a declared grammar when the primitive operators and relations are fixed independently of the target, the relation system is finite or effectively enumerable, the topology is intrinsic to the realization, no primitive operator directly stores the complete target, and the presentation exposes a reusable invariant, theorem, or computation not encoded by a primitive chosen for the target.
174 \end{definition}
175 \RecordID{PSC-I-D-NATURAL-PRESENTATION}
176 
177 \begin{proposition}[Algebra generation and module generation are distinct]
178 \label{prop:algebra-module-generation}
179 Every nonzero unital algebra $A$, regarded as a left module over itself, has
180 \[
181 \srank_A(A)=1,
182 \]
183 with cyclic seed $1_A$.  This conclusion is independent of the number of algebra generators required to present $A$ as an algebra.
184 \end{proposition}
185 \RecordID{PSC-I-P-ALGEBRA-MODULE-GENERATION}
186 \Depends{PSC-I-D-GENERATED-HULL.}
187 
188 \begin{proof}
189 For every $a\in A$, one has $a=a1_A$.  Hence $A1_A=A$.  The number of operations needed to generate $A$ as an algebra is a different invariant and does not enter the module calculation.
190 \end{proof}
191 
192 \begin{boundary}[Operators chosen for a target]
193 \label{bnd:target-dependent-operators}
194 For a fixed seed $s$ and target $x$, the existence of a specially defined operator $A_x$ satisfying $A_xs=x$ does not establish a natural seed calculus unless $A_x$ belongs to a grammar that is fixed independently of the target and satisfies \cref{def:natural-presentation}.
195 \end{boundary}
196 \RecordID{PSC-I-B-TARGET-DEPENDENT-OPERATORS}
197 
198 \begin{warningbox}
199 An operator may be chosen because it sends the current seed to a desired target.  Such a choice proves only an existence statement for that one target unless the operator already belongs to the fixed algebra.  The algebra must be declared before the question of seed rank is asked.  Otherwise the acting system changes with the answer one wants to obtain.
200 \end{warningbox}
201 
202 
203 \begin{figure}[htbp]
204 \centering
205 \includegraphics[width=0.96\textwidth]{01_seed_operator_grammar.pdf}
206 \caption{Schematic organization of the seed, operator, relation, and reconstruction data. This is a guide to the presentation, not an additional formal theorem.}
207 \label{fig:grammar}
208 \end{figure}
209 
210 \section{Free presentations by seeds and relations}
211 \begin{readerroadmap}
212 A seed family gives a canonical map from a finite free module.  The generated hull is exactly its image, and the relations among addressed seeds form the kernel.  This converts orbit generation into a presentation problem.
213 \end{readerroadmap}
214 \subsection{Finite free covers}
215 \begin{definition}[Seed evaluation map]
216 \label{def:seed-evaluation}
217 Let $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\ldots,s_r)\in M^r$.  Define the linear map over $A$
218 \[
219 \varepsilon_S:A^{\oplus r}\longrightarrow M,
220 \qquad
221 (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
222 \]
223 \end{definition}
224 \RecordID{PSC-I-D-SEED-EVALUATION}
225 
226 \begin{theorem}[Finite free presentation]
227 \label{thm:free-presentation}
228 For the map in \cref{def:seed-evaluation},
229 \[
230 \im\varepsilon_S=\Hull_A(S),
231 \qquad
232 N_S:=\Ker\varepsilon_S\leq A^{\oplus r},
233 \]
234 and the induced map is an isomorphism of left modules over $A$
235 \[
236 \boxed{A^{\oplus r}/N_S\cong\Hull_A(S).}
237 \]
238 Consequently, $S$ generates $M$ if and only if $\varepsilon_S$ is surjective.
239 \end{theorem}
240 \RecordID{PSC-I-T-FREE-PRESENTATION}
241 \Depends{PSC-I-D-SEED-EVALUATION and the first isomorphism theorem.}
242 \Exports{All later free seed presentations.}
243 
244 \begin{proof}
245 The image consists exactly of finite sums $\sum_ja_js_j$, which is $\Hull_A(S)$.  Because $\varepsilon_S$ is linear over $A$, its kernel is a left submodule.  The first isomorphism theorem gives
246 \[
247 A^{\oplus r}/N_S\cong\im\varepsilon_S=\Hull_A(S).
248 \]
249 The final equivalence is the equality $\im\varepsilon_S=M$.
250 \end{proof}
251 
252 \begin{interpretation}
253 The theorem converts the geometric language of addressed seed orbits into the algebraic language of generators and relations.  A word in the operators is represented by an element of the free module.  Two addressed expressions represent the same vector exactly when their difference lies in the kernel.  The generated hull is therefore not merely spanned by the free module.  It is its canonical quotient.
254 \end{interpretation}
255 
256 \begin{microexample}[title={A cyclic nilpotent chain}]
257 Let \(A=\bbk[t]\) and \(M=\bbk[t]/(t^3)\).  The residue class \(s=[1]\) is one seed and
258 \[
259 \varepsilon_s:A\to M,
260 \qquad
261 f(t)\mapsto f(t)[1]
262 \]
263 has kernel \((t^3)\).  Hence \(M\cong A/(t^3)\), and the addressed vectors \(s,ts,t^2s\) are the visible orbit chain.
264 \end{microexample}
265 
266 
267 \begin{corollary}[Characterization by minimal free covers]
268 \label{cor:minimal-free-cover}
269 For every finitely generated left module $M$ over $A$,
270 \[
271 \boxed{
272 \srank_A(M)=\min\{r\geq0:\text{there exists a surjection }A^{\oplus r}\twoheadrightarrow M\}.
273 }
274 \]
275 \end{corollary}
276 \RecordID{PSC-I-C-MINIMAL-FREE-COVER}
277 \Depends{PSC-I-T-FREE-PRESENTATION.}
278 
279 \begin{proof}
280 A seed family of size $r$ gives a surjection by \cref{thm:free-presentation}.  Conversely, the images of the standard basis vectors under any surjection $A^{\oplus r}\twoheadrightarrow M$ form a generating family of size at most $r$.
281 \end{proof}
282 
283 \begin{corollary}[Presentation with one seed]
284 \label{cor:one-seed-presentation}
285 For $s\in M$, the annihilator
286 \[
287 \operatorname{Ann}_A(s):=\{a\in A:as=0\}
288 \]
289 is a left ideal and
290 \[
291 \boxed{A/\operatorname{Ann}_A(s)\cong\Hull_A(s).}
292 \]
293 \end{corollary}
294 \RecordID{PSC-I-C-ONE-SEED-PRESENTATION}
295 \Depends{PSC-I-T-FREE-PRESENTATION.}
296 
297 \begin{proof}
298 The evaluation map for one seed is $a\mapsto as$.  Its kernel is $\operatorname{Ann}_A(s)$, which is therefore a left ideal, and \cref{thm:free-presentation} gives the quotient isomorphism.
299 \end{proof}
300 
301 \subsection{One seed and the distinction between module and algebra relations}
302 \begin{proposition}[Criterion for the annihilator to be an ideal on both sides]
303 \label{prop:two-sided-annihilator}
304 For $s\in M$, the left ideal $L=\operatorname{Ann}_A(s)$ is an ideal on both sides if and only if
305 \[
306 L\,\Hull_A(s)=0.
307 \]
308 If $s$ is cyclic, these conditions are equivalent to
309 \[
310 \operatorname{Ann}_A(s)=\operatorname{Ann}_A(M).
311 \]
312 \end{proposition}
313 \RecordID{PSC-I-P-ANNIHILATOR-TWO-SIDED}
314 \Depends{PSC-I-C-ONE-SEED-PRESENTATION.}
315 
316 \begin{proof}
317 The left ideal $L$ is an ideal on both sides exactly when $ab\in L$ for every $a\in L$ and $b\in A$.  This is equivalent to
318 \[
319 0=(ab)s=a(bs)
320 \qquad(a\in L,\ b\in A).
321 \]
322 By additivity, the last identity is equivalent to $L\Hull_A(s)=0$.
323 
324 If $s$ is cyclic, then $\Hull_A(s)=M$.  Hence the preceding condition is $L\subseteq\operatorname{Ann}_A(M)$.  The reverse inclusion always holds because an operator annihilating all of $M$ annihilates $s$.  Thus equality follows.  Conversely, $\operatorname{Ann}_A(M)$ is an ideal on both sides, so equality implies that $L$ is an ideal on both sides.
325 \end{proof}
326 
327 \begin{microexample}[title={Why a seed annihilator need not be an ideal on both sides}]
328 Let \(A=M_2(\bbk)\) act on \(M=\bbk^2\), and let \(s=e_1\).  Then \(\operatorname{Ann}_A(s)\) consists of matrices whose first column is zero.  It is a left ideal because left multiplication preserves a zero first column.  It is not generally a right ideal.  Thus the natural quotient for one seed is a quotient of the left regular module, not automatically a quotient algebra.
329 \end{microexample}
330 
331 
332 \subsection{Topological completion of a presentation}
333 \begin{theorem}[Completion of a dense free presentation]
334 \label{thm:completed-presentation}
335 Let $X$ be a complete Hausdorff locally convex space, let $\A\subseteq\End(X)$ act continuously, and let $S=(s_1,\ldots,s_r)$.  Put
336 \[
337 E:=\A^{\oplus r}/N_S
338 \]
339 algebraically and let $j:E\to X$ be the injective map induced by $\varepsilon_S$.  Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by
340 \[
341 \widetilde p(e):=p(j(e))
342 \]
343 for the continuous seminorms $p$ on $X$.  Every $a\in\A$ acts continuously on $E$ in the transported topology and therefore extends to $\widehat E$.  If $\Hull_{\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\A$
344 \[
345 \boxed{\widehat E\cong X.}
346 \]
347 \end{theorem}
348 \RecordID{PSC-I-T-COMPLETED-PRESENTATION}
349 \Depends{PSC-I-T-FREE-PRESENTATION and uniqueness of Hausdorff completion.}
350 
351 \begin{proof}
352 By \cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\A$ from $E$ onto $\Hull_{\A}(S)$.  The transported seminorms make it a topological isomorphism onto that subspace.  For $a\in\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\widehat E$.  The subspace is dense by hypothesis.  A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete.  This defines a continuous extension $\widehat j:\widehat E\to X$.  For any $x\in X$, choose a net $(e_\lambda)$ in $E$ with $j(e_\lambda)\to x$.  The transported topology makes $(e_\lambda)$ Cauchy, so it converges to some $\xi\in\widehat E$, and continuity gives $\widehat j(\xi)=x$.  Thus $\widehat j$ is surjective.  If $\widehat j(\xi)=0$, represent $\xi$ by a Cauchy net $(e_\lambda)$.  Then $j(e_\lambda)\to0$, so every defining seminorm $\widetilde p(e_\lambda)$ tends to zero.  Thus $e_\lambda\to0$ in the completion and $\xi=0$.  Hence $\widehat j$ is bijective.  Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.
353 \end{proof}
354 
355 The topology in this completion theorem comes from the realized image.  It is not an independently chosen topology on the free algebra.  This choice ensures that the completion recovers the given space, while leaving separate the question of whether another natural topology on the grammar produces the same completion.
356 
357 \begin{sectionrecap}
358 A seed family always gives a free presentation of its hull.  The kernel stores all linear relations among addressed seed words.  One seed gives a cyclic module, but only special annihilators permit an algebra quotient.  Dense generation is handled by completing the presented image rather than by silently replacing algebraic span with closure.
359 \end{sectionrecap}
360 
361 \section{Seed rank and functorial calculus}
362 
363 
364 
365 \begin{readerroadmap}
366 For a finitely generated module, seed rank is the smallest number of seeds needed to generate it.  We first examine how this number behaves when the acting algebra changes or the module is replaced by a quotient.  We then give bounds for exact sequences, direct sums, and tensor products.  Quotients with simple actions provide useful witnesses for lower bounds.
367 \end{readerroadmap}
368 \subsection{Definition, cyclicity, and monotonicity}
369 
370 \begin{definition}[Seed rank]
371 \label{def:seed-rank}
372 For a unital left module $M$ over $A$, define
373 \[
374 \srank_A(M):=\min\{|S|:AS=M\}.
375 \]
376 For a topological module $(X,\tau)$, define
377 \[
378 \srank_{\A,\tau}(X):=\min\{|S|:\cl{\A S}^{\,\tau}=X\}.
379 \]
380 Set $\srank_A(0)=0$.  When the value is finite, it is the minimal number of module generators.
381 \end{definition}
382 \RecordID{PSC-I-D-SEED-RANK}
383 
384 \begin{proposition}[Cyclic characterization]
385 \label{prop:cyclic-characterization}
386 For $M\neq0$,
387 \[
388 \srank_A(M)=1
389 \quad\Longleftrightarrow\quad
390 M\text{ contains a cyclic vector}.
391 \]
392 \end{proposition}
393 \RecordID{PSC-I-P-CYCLIC-CHARACTERIZATION}
394 \Depends{PSC-I-D-SEED-RANK.}
395 
396 \begin{proof}
397 Both conditions say that there is $m\in M$ with $Am=M$.
398 \end{proof}
399 
400 \begin{proposition}[Monotonicity under enlargement of the acting algebra]
401 \label{prop:acting-algebra-monotonicity}
402 Let $B\subseteq A\subseteq\End_{\bbk}(X)$ be unital operator algebras with the same identity.  Then
403 \[
404 \boxed{\srank_A(X)\leq\srank_B(X).}
405 \]
406 The same statement holds for topological seed rank when both algebras act continuously.
407 \end{proposition}
408 \RecordID{PSC-I-P-ACTING-ALGEBRA-MONOTONICITY}
409 \Depends{PSC-I-D-SEED-RANK.}
410 
411 \begin{proof}
412 For every $S\subseteq X$, one has $BS\subseteq AS$.  Hence every generating family over $B$ is also a generating family over $A$.  Taking closures gives the topological statement.
413 \end{proof}
414 
415 \begin{interpretation}
416 Enlarging the acting algebra can only create more orbit directions, so it can only lower the number of seeds required.  This is why every statement about seed rank must name the acting algebra.  The same vector space may be cyclic for a large algebra and require many seeds for a smaller one.
417 \end{interpretation}
418 
419 
420 \subsection{Quotients, isomorphisms, and exact sequences}
421 \begin{theorem}[Quotient monotonicity]
422 \label{thm:quotient-monotonicity}
423 Let $\varphi:A\to B$ be a unital ring homomorphism and let
424 \[
425 \Phi:M\twoheadrightarrow N
426 \]
427 be a surjective additive map satisfying
428 \[
429 \Phi(am)=\varphi(a)\Phi(m)
430 \qquad(a\in A,\ m\in M).
431 \]
432 Then
433 \[
434 \boxed{\srank_B(N)\leq\srank_A(M).}
435 \]
436 If $M$ and $N$ are topological modules, $\Phi$ is continuous and surjective, and the actions are continuous, then
437 \[
438 \srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
439 \]
440 \end{theorem}
441 \RecordID{PSC-I-T-QUOTIENT-MONOTONICITY}
442 \Depends{PSC-I-D-GENERATED-HULL and PSC-I-D-SEED-RANK.}
443 \Exports{All quotient lower bounds and realization inequalities.}
444 
445 \begin{proof}
446 Let $S$ generate $M$.  For every $n\in N$, choose $m\in M$ with $\Phi(m)=n$ and write $m=\sum_ia_is_i$.  Then
447 \[
448 n=\sum_i\varphi(a_i)\Phi(s_i),
449 \]
450 so $\Phi(S)$ generates $N$.
451 
452 For the topological statement, let $S$ generate $M$ densely.  Given $n=\Phi(m)$, choose a net $m_\lambda\in AS$ converging to $m$.  Continuity gives
453 \[
454 \Phi(m_\lambda)\longrightarrow n,
455 \]
456 and every $\Phi(m_\lambda)$ lies in $B\Phi(S)$.  Thus $B\Phi(S)$ is dense in $N$.
457 \end{proof}
458 
459 \begin{microexample}[title={A quotient can strictly lower seed rank}]
460 Let \(A=\bbk\) act by scalars on \(M=\bbk^3\).  Then \(\srank_A(M)=3\).  Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is \(2\).  Quotient monotonicity is therefore often strict.
461 \end{microexample}
462 
463 
464 \begin{corollary}[Isomorphism invariance]
465 \label{cor:isomorphism-invariance}
466 An intertwining module isomorphism preserves algebraic seed rank.  A topological intertwining isomorphism preserves topological seed rank.
467 \end{corollary}
468 \RecordID{PSC-I-C-ISOMORPHISM-INVARIANCE}
469 \Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}
470 
471 \begin{proof}
472 Apply \cref{thm:quotient-monotonicity} to the isomorphism and its inverse.
473 \end{proof}
474 
475 \begin{theorem}[Short exact sequence bounds]
476 \label{thm:exact-sequence-bounds}
477 Let
478 \[
479 0\longrightarrow L\xrightarrow{\iota}M\xrightarrow{\pi}N\longrightarrow0
480 \]
481 be a short exact sequence of left modules over $A$.  Then
482 \[
483 \boxed{
484 \srank_A(N)\leq\srank_A(M)
485 \leq\srank_A(L)+\srank_A(N).
486 }
487 \]
488 \end{theorem}
489 \RecordID{PSC-I-T-EXACT-SEQUENCE-BOUNDS}
490 \Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}
491 
492 \begin{proof}
493 The lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\ell_i)_{i\in I}$ of $L$ and $(n_j)_{j\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\in M$ of each $n_j$. For any $m\in M$, a finite subset $F\subseteq J$ and coefficients $a_j\in A$ give
494 \[
495 \pi(m)=\sum_{j\in F}a_jn_j.
496 \]
497 The vector $m-\sum_{j\in F}a_jm_j$ belongs to $\iota(L)$, so it is a finite linear combination of the $\iota(\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.
498 \end{proof}
499 
500 \begin{hypothesisguide}
501 For a short exact sequence, a generating family for \(M\) always maps to a generating family for the quotient.  It need not generate the submodule \(L\).  For the upper bound, choose generators of \(L\) separately and combine them with lifts of quotient generators.
502 \end{hypothesisguide}
503 
504 
505 \subsection{Direct sums under shared and independent actions}
506 \begin{proposition}[Direct sum bounds for a shared action]
507 \label{prop:shared-direct-sum}
508 If one algebra $A$ acts diagonally on $M\oplus N$, then
509 \[
510 \boxed{
511 \max\{\srank_A(M),\srank_A(N)\}
512 \leq\srank_A(M\oplus N)
513 \leq\srank_A(M)+\srank_A(N).
514 }
515 \]
516 \end{proposition}
517 \RecordID{PSC-I-P-SHARED-DIRECT-SUM}
518 \Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}
519 
520 \begin{proof}
521 The coordinate projections are intertwining quotients and give the lower bound.  For the upper bound, embed minimal generating families of $M$ as $(m_i,0)$ and of $N$ as $(0,n_j)$. Their union generates by finite sums and has cardinality at most the sum of the two seed ranks.
522 \end{proof}
523 
524 \begin{theorem}[Direct sum formula for independent actions]
525 \label{thm:independent-direct-sum}
526 Let $M$ be a left module over $A$ and $N$ a left module over $B$.  Under the action of the product algebra
527 \[
528 (a,b)(m,n)=(am,bn),
529 \]
530 one has
531 \[
532 \boxed{
533 \srank_{A\times B}(M\oplus N)
534 =
535 \max\{\srank_A(M),\srank_B(N)\}.
536 }
537 \]
538 \end{theorem}
539 \RecordID{PSC-I-T-INDEPENDENT-DIRECT-SUM}
540 \Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}
541 
542 \begin{proof}
543 The coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\in I}$ and $(n_i)_{i\in I}$. For any $m\in M$ and $n\in N$, their finite expressions involve a common finite subset $F\subseteq I$ after zero coefficients have been added. Thus
544 \[
545 m=\sum_{i\in F}a_im_i,
546 \qquad
547 n=\sum_{i\in F}b_in_i,
548 \]
549 and
550 \[
551 (m,n)=\sum_{i\in F}(a_i,b_i)(m_i,n_i).
552 \]
553 The family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.
554 \end{proof}
555 
556 \subsection{Tensor generation and witnesses for lower bounds}
557 \begin{proposition}[Tensor product bound]
558 \label{prop:tensor-bound}
559 Let $A$ and $B$ be unital algebras over $\bbk$, let $M$ be a left module over $A$, and let $N$ be a left module over $B$.  Under the natural $A\otimes_{\bbk}B$ action,
560 \[
561 \boxed{
562 \srank_{A\otimes B}(M\otimes_{\bbk}N)
563 \leq
564 \srank_A(M)\srank_B(N).
565 }
566 \]
567 \end{proposition}
568 \RecordID{PSC-I-P-TENSOR-BOUND}
569 \Depends{PSC-I-D-SEED-RANK.}
570 
571 \begin{proof}
572 Choose minimal generating families $(m_i)_{i\in I}$ for $M$ and $(n_j)_{j\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms
573 \[
574 (am_i)\otimes(bn_j)=(a\otimes b)(m_i\otimes n_j).
575 \]
576 Simple tensors span the algebraic tensor product, so the family $(m_i\otimes n_j)_{(i,j)\in I\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.
577 \end{proof}
578 
579 \begin{interpretation}
580 The tensor product bound is constructive.  Tensor the chosen seed families and act independently in the two factors.  Equality can fail because a single tensor seed may contain several useful components, or because the tensor product algebra acquires additional relations.  Later quantum applications should therefore cite the bound, not assume multiplicativity.
581 \end{interpretation}
582 
583 
584 \begin{boundary}[Tensor rank is not generally multiplicative]
585 \label{bnd:tensor-nonequality}
586 The inequality in \cref{prop:tensor-bound} can be strict.  Equality requires additional hypotheses on the acting algebras and their module multiplicities and is not exported as a universal law.
587 \end{boundary}
588 \RecordID{PSC-I-B-TENSOR-NONEQUALITY}
589 
590 
591 
592 \begin{proposition}[Monotonicity under extension of scalars]
593 \label{prop:scalar-extension}
594 Let \(K/\bbk\) be a field extension, let \(A\) be a unital algebra over \(\bbk\), and let \(M\) be a finitely generated left module over \(A\).  Set
595 \[
596 A_K:=K\otimes_{\bbk}A,
597 \qquad
598 M_K:=K\otimes_{\bbk}M.
599 \]
600 Then
601 \[
602 \boxed{
603 \srank_{A_K}(M_K)\leq\srank_A(M).
604 }
605 \]
606 Every generating family \(S\subseteq M\) extends to the generating family \(1\otimes S\subseteq M_K\).
607 \end{proposition}
608 \RecordID{PSC-I-P-SCALAR-EXTENSION-MONOTONICITY}
609 \Depends{PSC-I-D-SEED-RANK.}
610 \Exports{Extension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.}
611 
612 \begin{proof}
613 Let \(S=\{s_1,\ldots,s_r\}\) generate \(M\).  Every pure tensor \(\lambda\otimes m\in M_K\) has an expression
614 \[
615 m=\sum_{i=1}^r a_is_i,
616 \]
617 so
618 \[
619 \lambda\otimes m
620 =
621 \sum_{i=1}^r(\lambda\otimes a_i)(1\otimes s_i).
622 \]
623 Pure tensors span \(M_K\), hence \(1\otimes S\) generates \(M_K\) over \(A_K\).  Taking the minimum over \(S\) proves the inequality.
624 \end{proof}
625 
626 \begin{hypothesisguide}
627 The proposition gives an inequality in one direction.  A smaller generating family found after extending scalars need not descend to the original field.  Descent requires additional structure.  The chapter therefore exports monotonicity, not an unconditional equality.
628 \end{hypothesisguide}
629 
630 
631 \subsection{Quotient witnesses for lower bounds}
632 \begin{definition}[Witness for seed rank]
633 \label{def:seed-rank-witness}
634 A surjective intertwiner $\Phi:M\twoheadrightarrow N$ is a witness for seed rank of order $r$ when $\srank(N)=r$.  It certifies $\srank(M)\geq r$.
635 \end{definition}
636 \RecordID{PSC-I-D-SEED-RANK-WITNESS}
637 
638 \begin{proposition}[Scalar quotient witness]
639 \label{prop:scalar-quotient}
640 Let $M$ be a module over $A$ with coefficient field $\bbk$.  Suppose $M$ has a quotient $N$ of dimension $r$ on which $A$ acts through a character
641 \[
642 \chi:A\longrightarrow\bbk.
643 \]
644 Then
645 \[
646 \boxed{\srank_A(M)\geq r.}
647 \]
648 \end{proposition}
649 \RecordID{PSC-I-P-SCALAR-QUOTIENT-WITNESS}
650 \Depends{PSC-I-T-QUOTIENT-MONOTONICITY.}
651 
652 \begin{proof}
653 For every $n\in N$, the cyclic submodule $An$ is contained in the scalar span $\bbk n$.  Therefore $q$ seed images span a subspace of dimension at most $q$.  Generating the quotient of dimension $r$ requires at least $r$ seeds.  Quotient monotonicity transfers the lower bound to $M$.
654 \end{proof}
655 
656 \begin{sectionrecap}
657 Seed rank is functorial under quotients and isomorphisms, subadditive across extensions, given by the maximum for independent product actions, and bounded above under tensor products.  Lower bounds come from quotients on which the action simplifies.  Scalar extension cannot increase seed rank, but descent of a reduced seed family is a separate problem.
658 \end{sectionrecap}
659 
660 \section{Generator number modulo the Jacobson radical}
661 
662 
663 
664 \begin{readerroadmap}
665 For a finitely generated module, Nakayama's lemma allows us to count generators after passing to the quotient by the Jacobson radical.  When the algebra has finite dimension, this quotient is semisimple and its multiplicities give an explicit answer.  We begin with the radical reduction and then calculate those multiplicities.
666 \end{readerroadmap}
667 \subsection{Reduction to the module top}
668 
669 \begin{classicalinput}[Nakayama lemma]
670 \label{ci:nakayama}
671 Let $A$ be a unital ring, let $J\subseteq\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  If $M=N+JM$ for a submodule $N\leq M$, then $M=N$.
672 \end{classicalinput}
673 \RecordID{PSC-I-CI-NAKAYAMA}
674 
675 \begin{theorem}[Generator number is detected modulo the Jacobson radical]
676 \label{thm:module-top}
677 Let $A$ be a unital ring, let $J=\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  A family $S\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$.  When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top.  No semisimplicity of $A/J$ is required for the general equality.  Consequently,
678 \[
679 \boxed{
680 \srank_A(M)=\srank_{A/J}(M/JM).
681 }
682 \]
683 \end{theorem}
684 \RecordID{PSC-I-T-MODULE-TOP}
685 \Depends{PSC-I-CI-NAKAYAMA and PSC-I-T-QUOTIENT-MONOTONICITY.}
686 \Exports{Generator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modules.}
687 
688 \begin{proof}
689 If $S$ generates $M$, its image generates every quotient.  Conversely, suppose the image of $S$ generates $M/JM$ and put $N=AS$.  Then
690 \[
691 M=N+JM.
692 \]
693 Nakayama's lemma gives $M=N$.  Thus the same finite families generate before and after passage to the top, and the minimal cardinalities agree.
694 \end{proof}
695 
696 \begin{interpretation}
697 The theorem says that nilpotent or radical directions can lengthen orbit chains but cannot reduce the number of independent directions in the module top that must be seeded.  Once the images of the seeds generate \(M/JM\), Nakayama's lemma forces the entire module to be generated.
698 \end{interpretation}
699 
700 \begin{microexample}[title={A local nilpotent algebra}]
701 Let \(A=\bbk[\varepsilon]/(\varepsilon^3)\) and \(M=A^{\oplus2}\).  The radical is \(J=(\varepsilon)\), and
702 \[
703 M/JM\cong\bbk^2.
704 \]
705 Therefore two seeds are necessary and sufficient.  The extra nilpotent layers \(\varepsilon M\) and \(\varepsilon^2M\) are generated automatically once the two top directions are seeded.
706 \end{microexample}
707 
708 
709 \begin{corollary}[Local algebra formula]
710 \label{cor:local-algebra-formula}
711 Let $A$ be a local ring with Jacobson radical $J$ and residue division ring $D=A/J$.  For every finitely generated left module $M$ over $A$,
712 \[
713 \boxed{
714 \srank_A(M)=\dim_D(M/JM),
715 }
716 \]
717 where the dimension is that of a left vector space over $D$.
718 \end{corollary}
719 \RecordID{PSC-I-C-LOCAL-ALGEBRA-FORMULA}
720 \Depends{PSC-I-T-MODULE-TOP.}
721 
722 \begin{proof}
723 The module $M/JM$ over $A/J$ is a left vector space over $D$.  Its minimal number of generators is its dimension over $D$.
724 \end{proof}
725 
726 \subsection{Semisimple multiplicity calculus}
727 \begin{classicalinput}[Wedderburn and Artin decomposition]
728 \label{ci:wedderburn}
729 Every semisimple Artinian ring is isomorphic to a finite product
730 \[
731 A\cong\prod_{i=1}^{q}M_{d_i}(D_i)
732 \]
733 of full matrix rings over division rings.
734 \end{classicalinput}
735 \RecordID{PSC-I-CI-WEDDERBURN-ARTIN}
736 
737 \begin{lemma}[Multiplicities in the regular module]
738 \label{lem:regular-multiplicities}
739 Let
740 \[
741 A=\prod_{i=1}^{q}M_{d_i}(D_i)
742 \]
743 and let $S_i=D_i^{d_i}$ denote the natural simple left module for the $i$th factor, extended by zero across the other factors.  Then, as a left module over $A$,
744 \[
745 \boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
746 \]
747 \end{lemma}
748 \RecordID{PSC-I-L-REGULAR-MODULE-MULTIPLICITIES}
749 \Depends{PSC-I-CI-WEDDERBURN-ARTIN.}
750 
751 \begin{proof}
752 For one factor $M_d(D)$, let $e_{jj}$ be the diagonal matrix units.  The left regular module decomposes as
753 \[
754 M_d(D)=\bigoplus_{j=1}^{d}M_d(D)e_{jj}.
755 \]
756 Each minimal left ideal $M_d(D)e_{jj}$ is isomorphic to the natural column module $D^d$.  Taking the direct product of factors gives the stated decomposition.
757 \end{proof}
758 
759 \begin{theorem}[Semisimple seed rank formula over arbitrary division rings]
760 \label{thm:semisimple-seed-rank}
761 Let
762 \[
763 A\cong\prod_{i=1}^{q}M_{d_i}(D_i)
764 \]
765 be semisimple Artinian and let
766 \[
767 M\cong\bigoplus_{i=1}^{q}S_i^{\oplus m_i}
768 \]
769 be a finitely generated left module over $A$.  Then
770 \[
771 \boxed{
772 \srank_A(M)
773 =
774 \max_{1\leq i\leq q}
775 \left\lceil\frac{m_i}{d_i}\right\rceil.
776 }
777 \]
778 If $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\Jac(A)M$ over $A/\Jac(A)$ computes $\srank_A(M)$.
779 \end{theorem}
780 \RecordID{PSC-I-T-SEMISIMPLE-SEED-RANK}
781 \Depends{PSC-I-T-MODULE-TOP, PSC-I-CI-WEDDERBURN-ARTIN, PSC-I-L-REGULAR-MODULE-MULTIPLICITIES, and PSC-I-C-MINIMAL-FREE-COVER.}
782 \Exports{Every calculation of seed rank for semisimple algebras of finite dimension in the series.}
783 
784 \begin{proof}
785 By \cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\oplus r}$.  By \cref{lem:regular-multiplicities},
786 \[
787 A^{\oplus r}\cong\bigoplus_iS_i^{\oplus rd_i}.
788 \]
789 Every short exact sequence of modules over a semisimple ring splits.  Therefore a quotient isomorphic to $M$ exists if and only if
790 \[
791 m_i\leq rd_i
792 \qquad\text{for every }i.
793 \]
794 The smallest such $r$ is the displayed maximum.  For a nonsemisimple algebra of finite dimension, \cref{thm:module-top} reduces the calculation to the semisimple module $M/\Jac(A)M$ over $A/\Jac(A)$.
795 \end{proof}
796 
797 \begin{interpretation}
798 For the factor \(M_{d_i}(D_i)\), one free generator contributes \(d_i\) copies of the corresponding simple module.  If the top of \(M\) contains \(m_i\) copies, at least \(\lceil m_i/d_i\rceil\) free generators are required.  The worst simple type controls the global answer.
799 
800 This count concerns copies of a simple module, rather than the number of coordinates in a chosen basis.  A single seed can reach several copies through the matrix action.  The regular module records precisely how many copies one free generator can supply.
801 \end{interpretation}
802 
803 \begin{microexample}[title={Multiplicity consumes seed capacity}]
804 For \(A=M_2(\bbk)\) acting on \((\bbk^2)^{\oplus3}\), the regular module contributes two copies of the natural simple module per generator.  One generator cannot cover all three multiplicity directions.  Two can, giving seed rank \(2\).
805 \end{microexample}
806 
807 
808 \Needspace{16\baselineskip}
809 \begin{corollary}[Algebraically closed multiplicity formula]
810 \label{cor:algebraically-closed-formula}
811 Let $\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\bbk$, put $J=\Jac(A)$, and let $M$ be a finitely generated left module over $A$.  Assume compatible decompositions indexed by an integer $q\geq0$:
812 \[
813 A/J\cong\prod_{i=1}^{q}M_{d_i}(\bbk),
814 \qquad
815 M/JM\cong\bigoplus_{i=1}^{q}(\bbk^{d_i})^{\oplus m_i},
816 \]
817 where the $d_i$ are positive integers and the $m_i$ are nonnegative integers.  The first isomorphism is an isomorphism of algebras over $\bbk$.  The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors.  Then
818 \[
819 \srank_A(M)=\max_{1\leq i\leq q}\left\lceil\frac{m_i}{d_i}\right\rceil,
820 \]
821 with $\max\varnothing=0$ when $q=0$.
822 \end{corollary}
823 \RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA}
824 \Depends{PSC-I-T-MODULE-TOP and PSC-I-T-SEMISIMPLE-SEED-RANK.}
825 
826 \begin{proof}
827 By \cref{thm:module-top}, $\srank_A(M)=\srank_{A/J}(M/JM)$.  Apply \cref{thm:semisimple-seed-rank} to the displayed decompositions of $A/J$ and $M/JM$, with every $D_i=\bbk$.  This gives the stated maximum.  If $q=0$, the unital $A/J$-module $M/JM$ is zero, and \cref{thm:module-top} gives seed rank zero.
828 \end{proof}
829 
830 \begin{corollary}[Full matrix algebra]
831 \label{cor:full-matrix}
832 For the natural module $S=D^d$ over $M_d(D)$,
833 \[
834 \srank_{M_d(D)}(S)=1.
835 \]
836 More strongly, every nonzero vector of $S$ is cyclic.
837 \end{corollary}
838 \RecordID{PSC-I-C-FULL-MATRIX}
839 \Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}
840 
841 \begin{proof}
842 The formula gives $\lceil1/d\rceil=1$.  Directly, let $v\neq0$ and choose a coordinate $v_j\neq0$.  For any target $w=(w_i)$, the matrix whose $j$th column has entries $w_iv_j^{-1}$ and whose other columns vanish sends $v$ to $w$.
843 \end{proof}
844 
845 \begin{corollary}[Multiplicity obstruction]
846 \label{cor:multiplicity-obstruction}
847 For the diagonal action of $M_d(D)$ on $S^{\oplus m}$,
848 \[
849 \boxed{
850 \srank_{M_d(D)}(S^{\oplus m})
851 =
852 \left\lceil\frac{m}{d}\right\rceil.
853 }
854 \]
855 \end{corollary}
856 \RecordID{PSC-I-C-MULTIPLICITY-OBSTRUCTION}
857 \Depends{PSC-I-T-SEMISIMPLE-SEED-RANK.}
858 
859 \begin{proof}
860 Apply \cref{thm:semisimple-seed-rank} to the single simple type with multiplicity $m$.
861 \end{proof}
862 
863 \begin{sectionrecap}
864 For finitely generated modules, the minimal number of seeds is detected on the quotient by the Jacobson radical.  In the semisimple top, each generator has a fixed multiplicity capacity \(d_i\) in simple type \(i\), and the largest capacity ratio determines the exact seed rank.
865 \end{sectionrecap}
866 
867 \begin{figure}[htbp]
868 \centering
869 \includegraphics[width=0.96\textwidth]{04_semisimple_seed_rank.pdf}
870 \caption{The semisimple multiplicity formula for one simple block. A seed can expose at most $d$ independent multiplicity directions, so $m$ copies require $\lceil m/d\rceil$ seeds.}
871 \label{fig:semisimple}
872 \end{figure}
873 
874 \section{Modules for a single operator over arbitrary fields}
875 
876 
877 
878 \begin{readerroadmap}
879 One operator is equivalent to a module over \(\bbk[t]\).  Primary decomposition isolates the irreducible polynomial sectors, and the quotient by \(p(T)\) counts how many cyclic chains begin in each sector.
880 \end{readerroadmap}
881 \subsection{Primary components and generator counts}
882 
883 \begin{classicalinput}[Primary decomposition for polynomial modules]
884 \label{ci:primary-decomposition}
885 Every module $X$ over $\bbk[t]$ with finite dimension over the coefficient field decomposes as
886 \[
887 X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
888 \]
889 where $\Irr(\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\bbk[t]/(p^e)$.
890 \end{classicalinput}
891 \RecordID{PSC-I-CI-PID-PRIMARY-DECOMPOSITION}
892 
893 \begin{theorem}[Seed rank for a single operator]
894 \label{thm:single-operator}
895 Let $X$ have finite dimension over $\bbk$ and let $T\in\End_{\bbk}(X)$.  Regard $X$ as a module over $\bbk[t]$ by $t\cdot x=Tx$.  For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set
896 \[
897 \mu_p(T):=
898 \dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
899 \]
900 Then
901 \[
902 \boxed{
903 \srank_{\bbk[T]}(X)=\max_p\mu_p(T).
904 }
905 \]
906 Moreover, $\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.
907 \end{theorem}
908 \RecordID{PSC-I-T-SINGLE-OPERATOR}
909 \Depends{\nolinkurl{PSC-I-CI-PID-PRIMARY-DECOMPOSITION},
910 \nolinkurl{PSC-I-C-LOCAL-ALGEBRA-FORMULA}, and
911 \nolinkurl{PSC-I-T-INDEPENDENT-DIRECT-SUM}.}
912 \Exports{Companion, recurrence, and systems of subsequences in residue classes.}
913 
914 \begin{proof}
915 Let the minimal polynomial be
916 \[
917 m_T(t)=\prod_{p}p(t)^{e_p}.
918 \]
919 The acting algebra is
920 \[
921 \bbk[T]\cong\bbk[t]/(m_T)
922 \cong\prod_pR_p,
923 \qquad
924 R_p:=\bbk[t]/(p^{e_p}),
925 \]
926 by the Chinese remainder theorem.  The factor $R_p$ acts on $X_p$ and annihilates every other primary component.  Hence the formula for a direct sum with independent actions gives
927 \[
928 \srank_{\bbk[T]}(X)=\max_p\srank_{R_p}(X_p).
929 \]
930 Each $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\bbk[t]/(p)$.  By \cref{cor:local-algebra-formula},
931 \[
932 \srank_{R_p}(X_p)
933 =
934 \dim_{K_p}X_p/p(T)X_p
935 =
936 \mu_p(T).
937 \]
938 Finally, every summand $\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.
939 \end{proof}
940 
941 \begin{interpretation}
942 For one operator, the quotient \(X_p/p(T)X_p\) counts the number of independent primary chains for the irreducible polynomial \(p\).  Over an algebraic closure these become Jordan chains.  The seed rank is the largest chain count among the primary sectors, not the largest chain length.
943 \end{interpretation}
944 
945 \begin{microexample}[title={Block count versus block size}]
946 A nilpotent operator with Jordan block sizes \((5,1,1)\) has seed rank \(3\), even though its largest block has length \(5\).  Three seeds are needed because there are three independent chains.  Once their initial vectors are chosen, repeated application of \(T\) generates every position in those chains.
947 \end{microexample}
948 
949 
950 \subsection{Invariant factors, Jordan blocks, and cyclicity}
951 \begin{corollary}[Invariant factor count]
952 \label{cor:invariant-factor-count}
953 If $X\neq0$ and the rational canonical decomposition is written with nonconstant invariant factors
954 \[
955 X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j),
956 \qquad
957 f_1\mid f_2\mid\cdots\mid f_r,
958 \]
959 then
960 \[
961 \boxed{\srank_{\bbk[T]}(X)=r.}
962 \]
963 \end{corollary}
964 \RecordID{PSC-I-C-INVARIANT-FACTOR-COUNT}
965 \Depends{PSC-I-T-SINGLE-OPERATOR.}
966 
967 \begin{proof}
968 For each irreducible $p$, the number of invariant factors divisible by $p$ equals the number of elementary divisors for $p$.  Because $f_1\mid\cdots\mid f_r$ and $f_1\neq1$ after zero summands are removed, some irreducible divisor of $f_1$ divides all $r$ invariant factors.  Hence $\max_p\mu_p(T)=r$.
969 \end{proof}
970 
971 \begin{corollary}[Formula for geometric multiplicity]
972 \label{cor:geometric-multiplicity}
973 If $\bbk$ is algebraically closed, then
974 \[
975 \boxed{
976 \srank_{\bbk[T]}(X)
977 =
978 \max_{\lambda\in\operatorname{Spec}(T)}
979 \dim_{\bbk}\Ker(T-\lambda I).
980 }
981 \]
982 \end{corollary}
983 \RecordID{PSC-I-C-ALGEBRAICALLY-CLOSED-SINGLE-OPERATOR}
984 \Depends{PSC-I-T-SINGLE-OPERATOR.}
985 
986 \begin{proof}
987 The irreducibles are $p=t-\lambda$.  On the primary component for $\lambda$, the quotient by $(T-\lambda I)X_\lambda$ has dimension equal to the number of Jordan blocks, which is also $\dim\Ker(T-\lambda I)$.
988 \end{proof}
989 
990 \begin{corollary}[Criterion for a cyclic matrix]
991 \label{cor:cyclic-matrix}
992 For $X\ne0$ of finite dimension, the following are equivalent:
993 \begin{enumerate}[label=(\roman*)]
994 \item $T$ has a cyclic vector.
995 \item $\srank_{\bbk[T]}(X)=1$.
996 \item Every nonzero primary component for $p$ is cyclic.
997 \item The rational canonical form has one invariant factor.
998 \item The minimal polynomial equals the characteristic polynomial.
999 \item After scalar extension to an algebraic closure, every eigenvalue has geometric multiplicity one.
1000 \end{enumerate}
1001 \end{corollary}
1002 \RecordID{PSC-I-C-CYCLIC-MATRIX}
1003 \Depends{PSC-I-T-SINGLE-OPERATOR, PSC-I-C-INVARIANT-FACTOR-COUNT, and PSC-I-P-CYCLIC-CHARACTERIZATION.}
1004 
1005 \begin{proof}
1006 The equivalence of (i) and (ii) is \cref{prop:cyclic-characterization}.  By \cref{thm:single-operator}, (ii) holds exactly when every $\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii).  Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv).  In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\Leftrightarrow$(v).  Scalar extension preserves the equality of minimal and characteristic polynomials.  Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).
1007 \end{proof}
1008 
1009 \begin{sectionrecap}
1010 A single operator is cyclic exactly when each primary sector has one cyclic chain.  Over an algebraic closure this is equivalent to one Jordan block per eigenvalue and to equality of the minimal and characteristic polynomials.
1011 \end{sectionrecap}
1012 
1013 \begin{figure}[htbp]
1014 \centering
1015 \includegraphics[width=0.96\textwidth]{03_cyclic_rank_profiles.pdf}
1016 \caption{Krylov rank profiles for a cyclic companion operator and for two equal primary blocks. The second orbit stalls before reaching the ambient dimension because one seed cannot separate the repeated blocks.}
1017 \label{fig:krylov}
1018 \end{figure}
1019 

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