Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates

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1 % A visual appendix of worked examples to the preserved Paper I v1.2.1 manuscript.
2 % Every page starts explicitly, making overflow detectable by the build audit.
3 
4 \vpage{1}{What can one seed actually do?}{PAPER I / A VISUAL APPENDIX OF WORKED EXAMPLES}{Main text: \cref{def:seed-rank,prop:acting-algebra-monotonicity,thm:exact-packet-coordinates}}
5 A seed does not carry a fixed amount of generative power by itself. The answer depends on the operators that are allowed to act on it. Paper I makes that dependence part of the mathematical object rather than leaving it implicit.
6 
7 Use the same space of dimension six throughout the first comparison:
8 \[
9 X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad
10 s=e_0+f_0.
11 \]
12 Here each $J_3(0)$ moves a basis vector forward along a chain of three basis vectors. The projectors $P_e,P_f$ select the two chains. Their formulas and the proofs of the minimum seed counts appear on \pagerefV{3} and \pagerefV{4}.
13 
14 \vfig[.92]{grammar_budget}
15 \atlasCaption{A count of minimum generators, not a timing comparison. The ambient space is unchanged. Enlarging the acting algebra changes what one seed can separate.}
16 
17 \head{Three questions to keep apart}
18 \begin{tabularx}{\linewidth}{@{}p{.18\linewidth}Y@{}}
19 \textbf{Generation} & Can the fixed grammar reach every vector by finite linear combinations of operator words?\\[4pt]
20 \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt]
21 \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology?
22 \end{tabularx}
23 
24 A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure.
25 
26 \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''}
27 
28 \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}}
29 Most of the paper can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix.
30 
31 {\small
32 \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule
33 \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule
34 $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\
35 $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\
36 $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\
37 $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\
38 $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\
39 $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\
40 $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\
41 $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\
42 $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\
43 $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\
44 $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule
45 \end{tabularx}}
46 
47 \head{Read products from right to left}
48 In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column.
49 
50 \head{Where to begin}
51 For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}.
52 
53 \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$}
54 Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define
55 \[
56 Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0,
57 \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
58 \]
59 Starting with $s=e_0+f_0$, the two chains advance together:
60 \[
61 s=e_0+f_0,\qquad Ns=e_1+f_1,\qquad N^2s=e_2+f_2,\qquad N^3s=0.
62 \]
63 
64 \vfig[.94]{linked_chains}
65 \atlasCaption{The vertical dotted pairings record equal coefficients, not an additional allowed operator. Every polynomial in $N$ uses the same coefficients on both chains.}
66 
67 Since $N^3=0$, every polynomial orbit vector has the form
68 \[
69 p(N)s=a(e_0+f_0)+b(e_1+f_1)+c(e_2+f_2).
70 \]
71 The generated hull therefore has dimension three, even though the seed has a nonzero component in each packet of dimension three. The Krylov ranks are $1,2,3,3,3,3$.
72 
73 \head{What changes when a projector becomes admissible?}
74 Let $P_e$ keep the coordinates along the $e$ chain and $P_f$ those along the $f$ chain. If these are merely measurement maps, the preceding hull is unchanged. If they become allowed generators, then
75 \[
76 P_es=e_0,\qquad P_fs=f_0,
77 \]
78 and powers of $N$ now recover all six basis vectors. The same seed becomes cyclic for the larger algebra $\QQ\langle N,P_e,P_f\rangle$.
79 
80 \takeaway{Nonzero components in every packet are not enough. One must also be able to separate the independent directions inside and across those packets using the allowed operations.}
81 
82 \vpage{4}{A quotient proves two seeds are necessary}{MODEL A / A CERTIFICATE FOR THE LOWER BOUND}{Main text: \cref{prop:scalar-quotient,thm:single-operator}}
83 The failure of the particular seed on \pagerefV{3} does not prove that \emph{every} single seed fails. A certificate for the lower bound does.
84 
85 Define a surjective linear map
86 \[
87 q:X\longrightarrow\QQ^2,\qquad
88 q\Bigl(\sum a_je_j+\sum b_jf_j\Bigr)=(a_0,b_0).
89 \]
90 
91 \vfig[.78]{quotient_heads}
92 \atlasCaption{An actual $2\times6$ quotient matrix. It forgets the advanced positions in both chains and retains two independent head coordinates.}
93 
94 Because $qN=0$, one has, for every polynomial $p$,
95 \[
96 q\bigl(p(N)x\bigr)=p(0)q(x).
97 \]
98 Thus every operator acts as a \emph{scalar} on the quotient. The image of the orbit of one seed stays on one line in $\QQ^2$. It cannot span this quotient of dimension two.
99 
100 \head{Lower and upper bounds meet}
101 If $r$ seeds generated $X$, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, $r\ge2$. Conversely, the two seeds $e_0,f_0$ generate their respective chains, so
102 \[
103 \boxed{\srank_{\QQ[N]}(X)=2.}
104 \]
105 This proof uses the displayed operator and quotient. It does not ask a computer to search every possible seed, and it does not need an existence theorem for Jordan form.
106 
107 \head{A general pattern worth recognizing}
108 A surjective intertwiner $q:X\twoheadrightarrow Y$ turns any generating family of $X$ into a generating family of $Y$. Therefore a quotient on which the grammar loses mixing power can certify a lower bound in the original, much larger space.
109 
110 \takeaway{To rule out one seed, find a quotient in which its entire orbit is forced onto one line while the quotient still has at least two independent directions.}
111 
112 \vpage{5}{Relations identify words, not just outputs}{PRESENTATIONS / WHY A LEFT IDEAL APPEARS}{Main text: \cref{thm:free-presentation,cor:one-seed-presentation,prop:two-sided-annihilator}}
113 Let $F=\kk\langle z_1,\ldots,z_r\rangle$ be the algebra of noncommuting operator words. Evaluation at a fixed seed is
114 \[
115 \Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
116 \]
117 Two words represent the same generated vector exactly when their difference lies in $I_s=\Ker\Phi_s$. Consequently
118 \[
119 F/I_s\cong\Hull_F(s)
120 \]
121 as \emph{left modules}. The relation $f(T)s=0$ implies $g(T)f(T)s=0$ for every word $g$, which explains closure under multiplication on the left.
122 
123 \head{The distinction is visible in two dimensions}
124 Use $X=\QQ^2$, seed $e_1=(1,0)^\mathsf T$, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column:
125 \[
126 I_{e_1}=\left\{\mat{0&a\\0&b}:a,b\in\QQ\right\}.
127 \]
128 
129 \vfig[.90]{left_ideal}
130 \atlasCaption{The relation $Ce_1=0$ is destroyed by multiplying $C$ on the right by $B$. The product $CB$ sends the seed to itself.}
131 
132 For arbitrary $D$, $(DC)e_1=D(Ce_1)=0$. But $(CB)e_1=e_1\ne0$. Thus $I_{e_1}$ is a left ideal, but it is not stable under multiplication on the right. The quotient retains the action on generated vectors. Calling it a ring would lose the distinction that makes the presentation correct.
133 
134 \head{Several seeds give a module of relations}
135 For $(s_1,\ldots,s_m)$, evaluation is $(f_1,\ldots,f_m)\mapsto\sum_j f_j(T)s_j$. Its kernel is a submodule of $F^m$ and can include relations \emph{between} seeds, not only relations annihilating each seed separately.
136 
137 \takeaway{A presentation says which operator expressions are identified in a particular realization. A seed annihilator is not automatically an algebra relation ideal.}
138 
139 \vpage{6}{A reached type can still hide a direction}{TYPED MODEL / $U=\QQ^2$ AND $V=\QQ^3$}{Main text: \cref{prop:typed-reachability,def:directed-seed-rank}}
140 The paper's typed model has a rotation $R:U\to U$, an inclusion $A:U\to V$, and a retraction $B:V\to U$:
141 \[
142 R=\mat{0&-1\\1&0},\qquad
143 A=\mat{1&0\\0&1\\0&0},\qquad
144 B=\mat{1&0&0\\0&1&0}.
145 \]
146 The typed relation is $BA=I_U$. In the opposite order,
147 \[
148 AB=\diag(1,1,0)\ne I_V.
149 \]
150 The missing direction is $\Ker B=\QQ v_3$, not an entirely missing vertex.
151 
152 \vfig[.96]{typed_spaces}
153 \atlasCaption{Dark slots indicate independent directions in the generated hull. The same two vertex types are present in every row, but their generated subspaces differ.}
154 
155 \head{One seed supported at $U$}
156 The vectors $u_1,Ru_1$ span $U$. Their images under $A$ span $\QQ v_1+\QQ v_2$. Every path from $U$ to $V$ ends with $A$, so it cannot supply $v_3$. The hull has dimension four. Adding the pure seed $v_3$ gives all five dimensions.
157 
158 \head{Why this does not prove an unrestricted minimum of two}
159 The bound path algebra includes the vertex idempotents $P_U,P_V$. A mixed seed
160 \[
161 s=u_1+v_3\in U\oplus V
162 \]
163 satisfies $P_Us=u_1$ and $P_Vs=v_3$. It therefore generates the entire representation by itself. The paper proves that no seed \emph{supported at $U$} is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.
164 
165 \takeaway{Keep three notions separate: reaching a vertex, filling its entire vector space, and minimizing seeds when each seed must lie in a single packet. They answer different questions.}
166 
167 \vpage{7}{Exact does not mean orthogonal}{PACKET COORDINATES / AN OBLIQUE SPLITTING}{Main text: \cref{thm:exact-packet-coordinates,cor:adjoint-reversal}}
168 In $\RR^2$, take
169 \[
170 P=\mat{1&-1\\0&0},\qquad Q=\mat{0&1\\0&1}.
171 \]
172 These maps satisfy $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.
173 
174 \vfig[.68]{oblique_vectors}
175 \atlasCaption{Actual Euclidean vector geometry for $x=(3,2)$. The parallelogram reconstructs $x$ from $Px=(1,0)$ and $Qx=(2,2)$. Neither component is an orthogonal projection of $x$.}
176 
177 Analysis records $(Px,Qx)$ and synthesis adds the two vectors. For a packet tuple $(u,v)\in\im P\times\im Q$,
178 \[
179 P(u+v)=u,\qquad Q(u+v)=v.
180 \]
181 Both inverse identities follow from annihilation and completeness, without any angle condition.
182 
183 \head{What the algebra does not preserve}
184 For the displayed $x$,
185 \[
186 \norm{x}^2=13,\qquad \norm{Px}^2+\norm{Qx}^2=1+8=9.
187 \]
188 There is no Parseval identity here. The cross term is nonzero because the packet directions are not perpendicular. Likewise, the simple formula for reversing adjoints of packet blocks requires selfadjoint projectors. It is not justified for this oblique pair.
189 
190 \takeaway{Exact reconstruction is an algebraic inverse statement. Energy preservation and adjoint identities require additional geometric hypotheses.}
191 
192 \vpage{8}{A transfer block has an input and an output}{OPERATOR COORDINATES / MULTIPLICATION AS A SUM}{Main text: \cref{thm:operator-reconstruction,thm:block-multiplication,cor:block-powers}}
193 Use the three coordinate packets of dimension one in $\QQ^3$. The block $T_{ij}=P_iTP_j$ sends input packet $j$ into output packet $i$. For this example, its nonzero scalar is the $(i,j)$ matrix entry.
194 
195 \vfig[.94]{block_square}
196 \atlasCaption{All displayed entries are exact integers. The square is computed from the operator, not from a thresholded graph or a matrix of block norms.}
197 
198 \head{Reconstruction retains every block}
199 The identity $P_1+P_2+P_3=I$ gives
200 \[
201 T=\sum_{i,j=1}^3 P_iTP_j.
202 \]
203 Selecting only the diagonal would change the operator. The entries $2,3,4$ carry transfers between different packets.
204 
205 \head{Composition sums over intermediate packets}
206 The $(1,2)$ entry of the square has two contributing routes:
207 \[
208 (T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32}
209 =1\cdot2+2\cdot1+0=4.
210 \]
211 For input $3$ and output $1$, only the route through packet $2$ survives:
212 \[
213 (T^2)_{13}=T_{12}T_{23}=2\cdot3=6.
214 \]
215 For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
216 
217 \takeaway{An edge says a block is nonzero. The block itself carries the coefficient, sign, rank and action. A transfer graph alone does not retain those data.}
218 
219 \vpage{9}{Two visible routes can cancel exactly}{REACHABILITY / NECESSARY IS NOT SUFFICIENT}{Main text: \cref{cor:block-powers,prop:reachability-obstruction}}
220 Let a single operator $D$ on $\QQ^4$ act by
221 \[
222 De_1=e_2+e_3,\qquad De_2=e_4,\qquad De_3=-e_4,\qquad De_4=0.
223 \]
224 Every vertex of its transfer graph for the coordinate packets is reachable from vertex $1$. Nevertheless,
225 \[
226 D^2e_1=e_4-e_4=0,\qquad D^2=0.
227 \]
228 
229 \vfig[.84]{cancelled_paths}
230 \atlasCaption{The diamond is the actual weighted action of $D$. The two paths of length two exist, but their contributions to the same transfer block sum to zero.}
231 
232 \head{What the seed generates}
233 With the fixed algebra $\QQ[D]$ and seed $e_1$,
234 \[
235 \Hull_{\QQ[D]}(e_1)=\Span\{e_1,e_2+e_3\}.
236 \]
237 It has dimension two, not four. Both the independent difference $e_2-e_3$ and the last coordinate $e_4$ are missing.
238 
239 \head{Why projecting in the middle changes the experiment}
240 If $P_2$ becomes an admissible primitive, the word $DP_2D$ satisfies
241 \[
242 DP_2De_1=DP_2(e_2+e_3)=De_2=e_4.
243 \]
244 But $P_2$ was not an allowed generator of the original $\QQ[D]$ grammar. Using it to argue that the original seed already reached $e_4$ would silently enlarge the algebra.
245 
246 \takeaway{Unreachable nonzero packets obstruct generation. Reachability by itself does not prove generation: coefficients can cancel, blocks can have small images, and intermediate projectors may not be admissible.}
247 
248 \vpage{10}{Which zero patterns survive multiplication?}{SUPPORT ALGEBRAS / A UNIVERSAL TEST}{Main text: \cref{thm:support-pattern}. All packet spaces are nonzero}
249 Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let $E_{ij}$ denote the matrix that sends $e_j$ to $e_i$ and kills the other basis vectors.
250 
251 \vfig[.96]{support_closure}
252 \atlasCaption{Here $1$ means ``allowed,'' not that every admissible operator has that numerical value. The middle matrix is an actual product that violates the original pattern.}
253 
254 The first pattern contains the diagonal and permits $1\to2$ and $2\to3$, but forbids $1\to3$. Yet
255 \[
256 E_{32}E_{21}=E_{31}\ne0.
257 \]
258 Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
259 
260 \head{Reflexivity and transitivity have different jobs}
261 The identity has a nonzero diagonal block on every nonzero packet, so including the identity requires every packet to be related to itself. Multiplicative closure requires every composite pair. Consequently, for arbitrary algebraic block maps on nonzero packet spaces,
262 \[
263 \boxed{\text{unital universal support algebra}
264 \quad\Longleftrightarrow\quad
265 \text{reflexive, transitive support relation}.}
266 \]
267 The transitive closure in the figure is the lower triangular pattern. It does define a unital algebra.
268 
269 \head{Why this does not contradict cancellation}
270 On \pagerefV{9}, one particular operator has cancelling paths. The support theorem asks whether multiplication works for \emph{every} assignment of allowed block maps. Choosing two suitable maps of rank one removes the possibility of hiding a failed composite behind special cancellation.
271 
272 \takeaway{A zero pattern is a structural rule only when it survives all permitted block assignments. A coincidence in one matrix is not a closure theorem.}
273 
274 \vpage{11}{Move the packets with the coordinates}{CONJUGACY / STRUCTURE VERSUS RAW SPARSITY}{Main text: \cref{thm:coordinate-transport,cor:isomorphism-invariance,thm:naturality}}
275 Start with $T=\diag(1,2)$ and the two coordinate projections. Apply the invertible shear $U$. The transformed operator is $T^U=UTU^{-1}$, and the transformed projectors are $Q_i=UP_iU^{-1}$.
276 
277 \vfig[.76]{transported_packets}
278 \atlasCaption{The transformed operator has a nonzero entry away from the diagonal in the standard basis. Relative to its transported, oblique packets, however, both blocks between distinct packets remain zero.}
279 
280 Indeed,
281 \[
282 Q_1T^UQ_2=Q_2T^UQ_1=0,
283 \qquad
284 Q_iT^UQ_j=U(P_iTP_j)U^{-1}.
285 \]
286 A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.
287 
288 \head{When the map is not invertible}
289 For an intertwiner $\Phi A=\varphi(A)\Phi$, the hull identity uses the \emph{image algebra}:
290 \[
291 \Phi\bigl(\Hull_{\AA}(S)\bigr)=\Hull_{\varphi(\AA)}(\Phi S).
292 \]
293 Replacing $\varphi(\AA)$ with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.
294 
295 \takeaway{Transport the space, operators and packet maps together. The invariant is the collection of commuting algebraic relationships, rather than the appearance of the matrix.}
296 
297 \vpage{12}{Shared control and independent control differ}{DIRECT SUMS / ONE SEED CAN CARRY TWO COMPONENTS}{Main text: \cref{prop:shared-direct-sum,thm:independent-direct-sum}}
298 Take $X=\kk\oplus\kk$ and $s=(1,1)$. With one shared scalar acting on both coordinates, every orbit vector is $(a,a)$. With independent scalars, the orbit contains every $(a,b)$.
299 
300 \vfig[.94]{shared_independent}
301 \atlasCaption{The geometric picture uses $\kk=\RR$. On the right, shading shows a finite window of the whole plane. On the left, only the diagonal line is generated. The algebraic distinction holds over any field.}
302 
303 The corresponding minimum seed numbers are
304 \[
305 \srank_{\kk I}(\kk^2)=2,
306 \qquad
307 \srank_{\kk\times\kk}(\kk\oplus\kk)=1.
308 \]
309 In the independent case, the algebra contains $(1,0)$ and $(0,1)$, which recover the two coordinates of $s$ separately.
310 
311 \head{This is why the independent formula uses a maximum}
312 For independent actions of $\AA$ on $X$ and $\BB$ on $Y$, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving
313 \[
314 \srank_{\AA\times\BB}(X\oplus Y)
315 =\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
316 \]
317 With a shared action, this selection need not be available. The general bound is the sum.
318 
319 \head{One operator can supply independent control}
320 Over $\QQ$, $T=\diag(1,2)$ has polynomial projectors $2I-T$ and $T-I$. Thus $\QQ[T]$ already separates the two eigenspaces. If the eigenvalues coincide, these formulas no longer separate them. Distinct primary labels play the same role on \pagerefV{14}.
321 
322 \takeaway{Counting occupied components is not the same as counting seeds. One mixed seed may generate several components when the algebra can act on them independently.}
323 
324 \vpage{13}{How much row space can one seed supply?}{SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULA}{Main text: \cref{thm:semisimple-seed-rank,cor:algebraically-closed-formula}}
325 Let $M_2(\QQ)$ act on $X=M_{2\times5}(\QQ)$ by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a $2\times5$ matrix $V$.
326 
327 Every row of $AV$ is a linear combination of the two rows of $V$. Hence one seed exposes at most two independent directions in the multiplicity space of dimension five. Two seeds expose at most four. At least three are necessary.
328 
329 \vfig[.98]{row_budget}
330 \atlasCaption{The three fixed seeds cover multiplicity directions $1,2$, then $3,4$, then $5$. The exact dimensions of their cumulative matrix orbits are $4,8,10$.}
331 
332 \head{An explicit synthesis, not just dimension counting}
333 For
334 \[
335 W=\mat{1&2&3&4&5\\6&7&8&9&10},
336 \]
337 choose $A_1$ from columns $1,2$ of $W$, $A_2$ from columns $3,4$, and $A_3$ with first column equal to column $5$ of $W$ and second column zero. Then
338 \[
339 A_1V_1+A_2V_2+A_3V_3=W.
340 \]
341 The same construction works for arbitrary entries of $W$, so the lower bound is attained:
342 \[
343 \boxed{\srank_{M_2(\QQ)}M_{2\times5}(\QQ)=\lceil5/2\rceil=3.}
344 \]
345 
346 \head{Several split simple blocks}
347 If the acting algebra and module are supplied in compatible product coordinates, the row budget is computed separately in each block and the formula for independent actions takes the maximum. For block data $(d,m)=(2,5)$ and $(3,4)$, the result is $\max(3,2)=3$, not $3+2$.
348 
349 \takeaway{The formula measures multiplicity directions per seed. The formal development constructs compatible semisimple decompositions; the split formula uses the displayed algebra and module equivalences.}
350 
351 \vpage{14}{Distinct labels let one polynomial separate chains}{PRIMARY DATA / AN ACTUAL EXTRACTION CERTIFICATE}{Main text: \cref{thm:single-operator,cor:invariant-factor-count}}
352 Keep the model in six dimensions, but compare two different operators:
353 \[
354 N=J\oplus J,\qquad T=J\oplus(2I+J),\qquad J=J_3(0).
355 \]
356 For the same mixed seed $s=e_0+f_0$, the Krylov matrices are different:
357 
358 \vfig[.89]{primary_krylov}
359 \atlasCaption{Exact rational calculations. Changing the primary label from $0$ to $2$ changes the operator, not merely its coordinates. The determinant on the right is $512\ne0$.}
360 
361 \head{The missing separating operation is now a polynomial}
362 Define
363 \[
364 p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
365 \]
366 These polynomials satisfy
367 \[
368 p_2\equiv0\pmod{t^3},\qquad p_2\equiv1\pmod{(t-2)^3}.
369 \]
370 Consequently $p_0(T)=P_e$ and $p_2(T)=P_f$. Applying them to $s$ extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on \pagerefV{3}, the separating maps already belong to the polynomial grammar.
371 
372 \takeaway{Repeated labels can lock chains together. Distinct primary labels can admit polynomial extraction. Here the block data and extractor are explicit. No theorem asserting the existence of Jordan form is inferred from this example.}
373 
374 \vpage{15}{Walk counts make an endpoint cyclic}{GRAPH MODEL / THE PATH CERTIFICATE}{Main text: Worked graph model. Endpoint Krylov proof for every path size}
375 Let $A$ be the adjacency matrix of the path with five vertices. The $(i,j)$ entry of
376 \[
377 K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]
378 \]
379 counts walks of length $j-1$ from vertex $1$ to vertex $i$.
380 
381 \vfig[.65]{path_walks}
382 \atlasCaption{The entries are walk counts. At each new power, a new furthest vertex appears with coefficient one. The dark graph vertex is the seed.}
383 
384 A walk cannot reach vertex $i$ in fewer than $i-1$ steps. Therefore entries below the diagonal vanish. There is exactly one walk of length $i-1$ from the first vertex to vertex $i$: the shortest path. The diagonal entries are all one, so $\det K=1$.
385 
386 \head{The same reason works at every size}
387 For a path with any $n\ge1$, the endpoint Krylov matrix is upper triangular with unit diagonal. This is a combinatorial proof of cyclicity over \emph{every field}. It does not rely on numerical eigenvalues. The appendix checks the finite cases $1\le n\le12$ as implementation examples, separately from the paper's proof for every path size.
388 
389 \takeaway{The endpoint distinguishes distances. Each extra operator power reveals one genuinely new coordinate before the finite space is exhausted.}
390 
391 \vpage{16}{Closing a path changes the seed obstruction}{GRAPH MODEL / A CYCLE WITH SIX VERTICES}{Main text: Worked graph model. Proof of the minimum seed count for complex cycles}
392 For the adjacency matrix $A$ of $C_6$, the distinct eigenvalues are $2,1,-1,-2$, with multiplicities $1,2,2,1$. A single vertex seed spans only four directions. Two adjacent vertex seeds span all six.
393 
394 \vfig[.86]{cycle_multiplicity}
395 \atlasCaption{The two dark vertices are the displayed generating pair. The bars show exact eigenspace dimensions, not estimated spectral peaks.}
396 
397 \head{A lower bound without a numerical eigensolver}
398 The two rows of the following matrix are independent and obey $qA=q$:
399 \[
400 q=\mat{2&1&-1&-2&-1&1\\0&1&1&0&-1&-1}.
401 \]
402 Thus $q:\QQ^6\twoheadrightarrow\QQ^2$ is a scalar quotient for the eigenvalue $1$. Its extension over $\CC$ gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.
403 
404 \head{The matching construction}
405 From adjacent vertex vectors $e_0,e_1$, the identity
406 \[
407 e_{j+1}=Ae_j-e_{j-1}
408 \]
409 recovers successive vertices around the cycle. The lower and upper bounds therefore give two seeds for this example. The paper proves the corresponding minimum for every complex simple cycle of size at least three.
410 
411 \takeaway{Connectedness is not cyclicity for the adjacency grammar. If a directed cyclic shift were allowed instead, one vertex would generate the whole cycle. That would be a different operator algebra.}
412 
413 \vpage{17}{A relation becomes a packet product rule}{COORDINATE ALGEBRA / NORMAL FORM RATHER THAN A CURVE SKETCH}{Main text: Worked coordinate ring model. Normal form and exact multiplication}
414 Work over $\QQ$ in
415 \[
416 A=\QQ[x,y]/(y^2-x^3+x).
417 \]
418 Because the relation is monic of degree two in $y$, each element has one unique form $a(x)+yb(x)$. The two packet coefficients are polynomials, not scalars.
419 
420 \vfig[.97]{normal_form}
421 \atlasCaption{A finite window of the two infinite coefficient lanes. Higher powers of $y$ are rewritten using the relation. They are not simply discarded. No geometric genus or smoothness claim is encoded by this picture.}
422 
423 \head{Multiplication by $y$ transfers between the lanes}
424 In the ordered coefficient coordinates $(a,b)$,
425 \[
426 M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
427 \]
428 The entries denote multiplication maps on $\QQ[x]$. For $f=a_0+ya_1$ and $g=b_0+yb_1$,
429 \[
430 fg=\bigl(a_0b_0+(x^3-x)a_1b_1\bigr)
431 +y\bigl(a_0b_1+a_1b_0\bigr).
432 \]
433 In particular,
434 \[
435 (x^2+y)(x-y)=x+(x-x^2)y.
436 \]
437 
438 \head{One module seed does not mean one algebra generator}
439 Starting from $1$, the fixed multiplication operators $M_x,M_y$ generate every monomial in normal form. Thus the algebra is cyclic as a module over its multiplication algebra. This does not remove either address operator, and does not say that the algebra is generated by one algebra element.
440 
441 \takeaway{The relation controls an exact weighted transfer between coefficient lanes. Packet reconstruction is $a+yb$, while packet multiplication retains the weight $x^3-x$.}
442 
443 \vpage{18}{Density is not finite generation of each vector}{INFINITE MODEL / THE BILATERAL SHIFT}{Main text: \cref{prop:shift-seed,thm:completed-presentation}}
444 On $\ell^2(\ZZ)$, declare every integer shift
445 \[
446 (S_mx)[n]=x[n-m].
447 \]
448 The seed $\delta_0$ generates $\delta_m=S_m\delta_0$. Finite linear combinations give precisely the sequences with finite support. Their norm closure is all of $\ell^2(\ZZ)$.
449 
450 \vfig[.93]{shift_tail}
451 \atlasCaption{A finite window of the actual sequence $x[n]=2^{-|n|}$. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation $x^{(3)}$, not a periodic wraparound.}
452 
453 This $x$ cannot be a finite linear combination of shifted impulses: it has infinite support. But if $x^{(N)}$ agrees with $x$ on $|n|\le N$ and is zero elsewhere, then
454 \[
455 \norm{x-x^{(N)}}_2^2
456 =2\sum_{n>N}4^{-n}
457 =\frac{2}{3}\,4^{-N}\longrightarrow0.
458 \]
459 The exact squared errors for $N=1,3,5$ are $1/6$, $1/96$, and $1/1536$. These are identities from a geometric series, not floating residuals.
460 
461 \head{What the topology contributes}
462 The statement is that \emph{one seed has dense hull}. It is not that every vector is a finite word expression. The completion theorem additionally specifies a normed presentation, a dense linear isometry into a Banach target, and the bounded intertwining needed to extend operators.
463 
464 \takeaway{Completion adds limits in a declared topology. A finite numerical window can illustrate the approximants, but cannot be substituted for the infinite object or its convergence proof.}
465 
466 \vpage{19}{Residue packets retain an address carry}{INFINITE MODEL / THREE EXACT SUBSEQUENCES}{Main text: \cref{thm:banach-packets}. Residue packets}
467 For $j=0,1,2$, let $P_j$ keep the entries with $n\equiv j\pmod3$. These are bounded orthogonal projectors whose sum is the identity. The packet coordinates can be written as
468 \[
469 x_j[k]=x[3k+j],\qquad k\in\ZZ.
470 \]
471 
472 \vfig[.98]{residue_carry}
473 \atlasCaption{Each occupied tile represents one integer address in a displayed window. The three rows partition the addresses exactly. Negative indices use the same residue convention.}
474 
475 \head{What one shift does in these coordinates}
476 Let $S=S_1$ and $z=Sx$. Substituting $3k+j$ into the shift definition gives
477 \[
478 z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
479 \]
480 The passage from phase $2$ to phase $0$ carries a shift of the coarse address. If $Lh[k]=h[k-1]$, the exact packet matrix is
481 \[
482 S_{\mathrm{packets}}=\mat{0&0&L\\I&0&0\\0&I&0}.
483 \]
484 Replacing $L$ by a scalar $1$ would preserve the phase graph but would lose the operator's action on the infinite sequences.
485 
486 \head{Synthesis keeps the positions}
487 To reconstruct $x$, place $x_j[k]$ at its original address $3k+j$. In the ambient space formulation, this is the finite sum $x=P_0x+P_1x+P_2x$. Boundedness makes the same formulas valid after taking the $\ell^2$ completion.
488 
489 \takeaway{A cyclic packet graph is only the routing pattern. The block at the phase boundary still carries an actual shift operator.}
490 
491 \vpage{20}{Stop when the subspace stops growing}{EXACT COMPUTATION / AN INVARIANCE CERTIFICATE}{Main text: \cref{con:orbit-hull-algorithm}}
492 For a fixed finite operator family $(O_i)$, start from $W_0=\Span S$ and iterate
493 \[
494 W_{n+1}=W_n+\sum_i O_iW_n.
495 \]
496 This is an ascending chain. When two consecutive subspaces agree, every generator maps the stable subspace into itself. The stable value is therefore the entire hull of operator words, not just the span of the words tested so far.
497 
498 \vfig[.58]{hull_iteration}
499 \atlasCaption{Exact dimensions for the same mixed seed in Model A. The dashed curve allows only $N$. The solid curve also admits the two chain projectors. Plateau values are repeated to show the certified stop.}
500 
501 \head{Why the process stops in finite dimensions}
502 If an inclusion is strict, the dimension rises by at least one. It cannot rise beyond $\dim X$. At equality, invariance and containment of the seeds prove both inclusions between the stable space and the least invariant hull.
503 
504 \head{What the companion code actually does}
505 The new checker retains exact column bases using rational row reduction. It then verifies that adjoining the images of those bases under every generator does not increase rank. These examples use neither a cutoff on singular values nor a numerical tolerance.
506 
507 The Lean statement is a mathematical termination and correctness specification. It is not a claim that this Python implementation for maintaining a basis was extracted from Lean or that a particular runtime bound was proved. The original numerical companion and its tolerances remain separate.
508 
509 \takeaway{``No new directions'' is a stopping certificate only after checking the full declared operator family. Stopping after a convenient selection of words is not the same test.}
510 
511 \vpage{21}{Tensor products require every cross term}{OPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONAL}{Main text: \cref{prop:tensor-bound}. Algebraic tensor products}
512 If $(x_i)$ generates $X$ under $\AA$ and $(y_j)$ generates $Y$ under $\BB$, then the candidate tensor seeds are \emph{all} $x_i\otimes y_j$. For finite expansions,
513 \[
514 x=\sum_i A_ix_i,\qquad y=\sum_j B_jy_j,
515 \]
516 bilinearity gives the required double sum:
517 \[
518 x\otimes y=\sum_i\sum_j(A_i\otimes B_j)(x_i\otimes y_j).
519 \]
520 
521 \vfig[.95]{tensor_rectangle}
522 \atlasCaption{The exact expansion of $(2e_1+3e_2)\otimes(5f_1+7f_2)$. The terms away from the diagonal $14$ and $15$ cannot be omitted.}
523 
524 Keeping only the tensors with matching indices $e_1\otimes f_1$ and $e_2\otimes f_2$ spans a subspace of dimension two in a tensor product of dimension four. It misses both cross directions. Under scalar actions on $\QQ^2$ and $\QQ^2$, all four basis tensors are necessary and sufficient.
525 
526 \head{The resulting upper bound}
527 Since simple tensors span the algebraic tensor product, the complete rectangular family proves
528 \[
529 \boxed{\srank_{\AA\otimes\BB}(X\otimes Y)
530 \le\srank_{\AA}(X)\,\srank_{\BB}(Y).}
531 \]
532 This is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebraic argument itself establish a theorem for completed topological tensor products.
533 
534 \takeaway{The generator construction is indexed by a Cartesian product. Its proof must retain the finite double sum and the hypotheses on scalar actions that make the tensor action well defined.}
535 
536 \vpage{22}{What the pictures certify}{PROOF BOUNDARY / RETURN TO THE MANUSCRIPT}{Main text: Reading the theorem hypotheses and companion evidence}
537 The mathematical statements in the main text have explicit hypotheses and written proofs. The Lean guide separately identifies the checked declarations, hypotheses, and coverage. The machine native paper identifies the 108 stable foundation records and the checks executed on their structured representations. These are distinct forms of evidence.
538 
539 {\small\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.23\linewidth}Y>{\raggedright\arraybackslash}p{.25\linewidth}@{}}\toprule
540 \textbf{Topic} & \textbf{Mathematical content} & \textbf{What must be retained}\\\midrule
541 Presentations and hulls & Evaluation, module quotients, and generation under a fixed algebra. & The grammar must be fixed independently of the target.\\
542 Finite packets & Idempotence, annihilation, completeness and exact finite synthesis. & Orthogonality requires an additional hypothesis.\\
543 Semisimple formula & Multiplicities over division rings determine the minimal free cover. & The stated classical inputs and decomposition data.\\
544 Primary and graph formulas & Primary components and scalar quotients give sharp generator counts. & The coefficient field and multiplicity conventions.\\
545 Paths and cycles & Formulas for paths of every size and for complex cycles have separate proofs. & Finite tests illustrate rather than prove claims for every size.\\
546 Coordinate algebra & Normal form, multiplication, packet identities and a unit seed. & Smooth projective geometry needs additional classical input.\\
547 Completion & Density and bounded extraction in the stated topology. & Infinite sequences are distinct from periodic finite arrays.\\\bottomrule\end{tabularx}}
548 
549 \head{Evidence travels with the examples}
550 The release includes mathematical source, exact model definitions, deterministic graphics, Python checks, Lean source and theorem mapping, and the machine native record archive. The visual models use integer, rational, symbolic, or explicitly analytic calculations. The separate verification record distinguishes fresh executions from archived evidence.
551 
552 \takeaway{The pictures make the hypotheses visible. Read the traditional proof, the formal contract, and the executable record at their respective stated scopes.}

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