Realization Calculus

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1 \part{Nonlinear packet calculus, obstruction complexes, and holonomy}\label{part:nonlinear}
2 \noindent This part moves from exact coordinates for arbitrary functions to differentiable interactions, specified correction problems, and finite or smooth transport. Each ordinary proof retains its stated hypotheses. These results are not asserted to belong to the finite Lean contract of Part~I.
3 
4 \medskip\noindent The notational change from \(\cA,\cS\) to \(\Acal,\Scal\) is typographical: both denote analysis and synthesis for the declared packet system. A connection remains additional data. Smooth canonical transport is constructed from a moving projector family, whereas the algebraic exact coordinate transport in Part~I compares fixed exact descriptions of one ambient object.
5 \section{Introduction}\label{sec:introduction}
6 
7 A finite family of pairwise annihilating idempotents gives exact coordinates.  If
8 \[
9 P_\alpha P_\beta=0
10 \quad\text{for }\alpha\ne\beta,
11 \qquad
12 \sum_\alpha P_\alpha=I,
13 \]
14 then every vector is the sum of its packet components.  Linear operators become block matrices in these coordinates.  A nonlinear map cannot be reconstructed from its diagonal restrictions, since mixed interactions between packet components are usually present.  It can still be represented exactly if its input is the complete packet tuple.
15 
16 This part develops nonlinear packet representation and the obstruction theory that follows from it.  Exact coordinates preserve a nonlinear problem, but they do not solve it.  Once the equation is written in packet coordinates, its derivatives, compositions, correction maps, refinement failures, and gluing defects can be separated without discarding mixed terms.
17 
18 The first stage of Part~II treats the local nonlinear calculus.  Packet analysis and synthesis conjugate arbitrary maps.  Fr\'echet derivatives split into multilinear packet jets.  The higher chain rule then becomes a Fa\`a di Bruno formula whose terms carry intermediate packet labels.  Standard background on Fr\'echet calculus and the higher chain rule may be found in \cite{AbrahamMarsdenRatiu1988,ConstantineSavits1996,Deimling1985}.
19 
20 The next stage studies nonlinear iteration.  Repeated substitution is indexed by rooted trees rather than operator words.  A scalar majorant gives an absolute convergence theorem for polynomial fixed-point equations.  The quadratic case recovers the Catalan series.  Rooted tree combinatorics are discussed in \cite{Stanley2012,LodayVallette2012,HairerLubichWanner2006}.
21 
22 A linear correction map
23 \[
24 D:U\longrightarrow V
25 \]
26 places the obstruction to solving $Du=-r$ in the canonical quotient $\coker D$.  This quotient has a universal property and behaves naturally under commuting squares.  Mapping cones extend the same construction to cochain complexes.  In a Hilbert space with closed range, the quotient is represented by the orthogonal projection onto $\Ker D^*$.  The Moore--Penrose inverse then gives the least norm correction.  Relevant background appears in \cite{Weibel1994,Penrose1955}.
27 
28 A vanishing linear obstruction is only the first step.  We give a contraction theorem for nonlinear correction, followed by a local Lyapunov--Schmidt reduction.  In a complete filtered problem, every lifting step has its own cokernel class.  Vanishing at all depths and compatible choices of corrections produce an exact inverse limit solution.  A filtered correction complex carries a spectral sequence.  Its first page records graded obstruction candidates, while later differentials record interactions that can cancel them.  We use the standard spectral sequence of a filtered complex from \cite{Weibel1994,McCleary2001}.
29 
30 The final stage studies elimination and gluing.  The implicit function theorem gives an exact Schur derivative after hidden variables are removed.  Finite transport systems reduce, after a spanning tree gauge, to one holonomy element for each fundamental cycle.  The same idea extends to smooth projector families through Kato transport.  We use standard material from \cite{Deimling1985,GrossTucker1987,Kato1950,Kato1995,KobayashiNomizu1963}.
31 
32 The chapter does not claim that every nonlinear problem has a finite obstruction space.  The cokernel is canonical only after a correction map has been specified.  It may be infinite dimensional.  A first-order obstruction may vanish while a higher-order obstruction remains.  The purpose of the calculus is to identify the exact object in which each failure lives and to show how those objects change under packets, refinement, filtration, and transport.
33 
34 
35 \section{A map of the nonlinear calculus}\label{sec:map}
36 
37 Four questions organize the nonlinear calculus: representation, interaction, correction, and gluing.
38 
39 The first question is representation.  Given a nonlinear map $F$, can its packet form be written without approximation?  The answer is yes.  Analysis and synthesis are inverse coordinate maps, so conjugation preserves the full equation.
40 
41 The second question is interaction.  Which input packets influence a given output packet, and at what derivative order?  Packet jets answer this question.  They form a directed hypergraph rather than an ordinary block matrix because one nonlinear interaction can have several inputs.
42 
43 The third question is correction.  Suppose an approximate solution leaves a defect $r$.  A declared linearized correction map $D$ removes the part of $r$ lying in $\im D$.  The quotient class
44 \[
45 [r]\in\coker D
46 \]
47 is the part that no allowed first-order correction can remove.  This quotient class is the obstruction.  Its vanishing must be supplemented by a nonlinear theorem.
48 
49 The fourth question is gluing.  Local coordinate changes can solve every edge problem and still fail around a loop.  The remaining cycle product is holonomy.  It measures the failure of a global frame even when each local equation has been solved.
50 
51 These four layers fit into the sequence
52 \[
53 \begin{gathered}
54 \text{exact coordinates}
55 \longrightarrow
56 \text{packet interactions}
57 \longrightarrow
58 \text{correction classes}
59 \\
60 \longrightarrow
61 \text{global gluing}.
62 \end{gathered}
63 \]
64 Each arrow needs hypotheses.  Exact coordinates need a complete packet system.  Differential interactions need regularity.  A cokernel obstruction needs a specified correction map.  Holonomy needs invertible local transports.
65 
66 A second distinction concerns scale.  A refinement defect compares a fine nonlinear model with a coarse one.  It is a point in the coarse output space.  An obstruction class is the image of such a defect, or of another residual, in a quotient by allowed corrections.  The correction map must be declared before the defect determines an obstruction class.  A small defect can have a nonzero obstruction, and a large defect can have zero obstruction if it lies entirely in the correction range.
67 
68 The chapter uses finite packet systems in the coordinate and graph sections.  Banach space hypotheses enter only where norms, contractions, or Fr\'echet derivatives are used.  Module and complex language is used for cokernels, filtered lifting, and cohomological obstructions.  This separation keeps each result at the level of generality supported by its proof.
69 
70 \subsection*{Standing conventions}
71 All packet systems are finite unless a statement says otherwise.  In a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator-valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map
72 \[
73 X\longrightarrow\varprojlim_m X/F^mX
74 \]
75 is an isomorphism and separatedness means $\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.
76 
77 \section{Exact nonlinear packet coordinates}\label{sec:nonlinear_coordinates}
78 
79 Let $X$ be a module over a unital ring.  A finite packet system on $X$ is a family
80 \[
81 \Pcal=(P_\alpha)_{\alpha\in I}\subseteq\End(X)
82 \]
83 of nonzero idempotents such that
84 \[
85 P_\alpha P_\beta=0
86 \quad\text{when }\alpha\ne\beta,
87 \qquad
88 \sum_{\alpha\in I}P_\alpha=I_X.
89 \]
90 Write
91 \[
92 X_\alpha=P_\alpha X,
93 \qquad
94 X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
95 \]
96 The analysis and synthesis maps are
97 \[
98 \Acal_{\Pcal}x=(P_\alpha x)_\alpha,
99 \qquad
100 \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
101 \]
102 They are inverse isomorphisms.
103 
104 \begin{definition}[Nonlinear packet coordinate map]\label{def:nonlinear_packet_map}
105 Let $\Pcal$ be a packet system on $X$ and let $\Qcal$ be a packet system on $Y$.  For any map $F:X\to Y$, define
106 \[
107 F_{\Qcal\leftarrow\Pcal}
108 =
109 \Acal_{\Qcal}F\Scal_{\Pcal}.
110 \]
111 Thus
112 \[
113 F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha)
114 =
115 \left(
116 Q_\gamma F\left(\sum_\alpha x_\alpha\right)
117 \right)_\gamma.
118 \]
119 \end{definition}
120 
121 \begin{theorem}[Exact nonlinear coordinate theorem]\label{thm:exact_nonlinear_coordinates}
122 For every map $F:X\to Y$,
123 \[
124 \Scal_{\Qcal}F_{\Qcal\leftarrow\Pcal}\Acal_{\Pcal}=F.
125 \]
126 If $G:Y\to Z$ and $\Rcal$ is a packet system on $Z$, then
127 \[
128 (G\circ F)_{\Rcal\leftarrow\Pcal}
129 =
130 G_{\Rcal\leftarrow\Qcal}
131 \circ
132 F_{\Qcal\leftarrow\Pcal}.
133 \]
134 No linearity, continuity, or differentiability assumption is needed.
135 \end{theorem}
136 
137 \begin{proof}
138 The identities $\Scal_{\Pcal}\Acal_{\Pcal}=I_X$ and $\Scal_{\Qcal}\Acal_{\Qcal}=I_Y$ give
139 \[
140 \Scal_{\Qcal}F_{\Qcal\leftarrow\Pcal}\Acal_{\Pcal}
141 =
142 \Scal_{\Qcal}\Acal_{\Qcal}F\Scal_{\Pcal}\Acal_{\Pcal}
143 =F.
144 \]
145 For composition,
146 \[
147 G_{\Rcal\leftarrow\Qcal}F_{\Qcal\leftarrow\Pcal}
148 =
149 \Acal_{\Rcal}G\Scal_{\Qcal}\Acal_{\Qcal}F\Scal_{\Pcal}
150 =
151 \Acal_{\Rcal}(G\circ F)\Scal_{\Pcal}.
152 \]
153 \end{proof}
154 
155 \begin{corollary}[Dynamical equivalence]\label{cor:dynamical_equivalence}
156 Suppose $X=Y$ and the same packet system is used on both sides.  Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility.  In particular,
157 \[
158 F(x)=x
159 \quad\Longleftrightarrow\quad
160 F_{\Pcal\leftarrow\Pcal}(\Acal_{\Pcal}x)=\Acal_{\Pcal}x.
161 \]
162 \end{corollary}
163 
164 \begin{proof}
165 The packet map is conjugate to $F$ through the isomorphism $\Acal_{\Pcal}$.  Every listed property is invariant under conjugacy.
166 \end{proof}
167 
168 \begin{proposition}[Change of packet coordinates]\label{prop:change_packet_coordinates}
169 Let $\Pcal$ and $\Pcal'$ be packet systems on $X$.  Let $\Qcal$ and $\Qcal'$ be packet systems on $Y$.  For any two packet systems $\Ucal$ and $\Vcal$ on the same space, write
170 \[
171 J_{\Ucal\leftarrow\Vcal}
172 =
173 \Acal_{\Ucal}\Scal_{\Vcal}.
174 \]
175 Then
176 \[
177 F_{\Qcal'\leftarrow\Pcal'}
178 =
179 J_{\Qcal'\leftarrow\Qcal}
180 F_{\Qcal\leftarrow\Pcal}
181 J_{\Pcal\leftarrow\Pcal'}.
182 \]
183 \end{proposition}
184 
185 \begin{proof}
186 Insert $\Scal_{\Qcal}\Acal_{\Qcal}=I_Y$ and $\Scal_{\Pcal}\Acal_{\Pcal}=I_X$ into the definition of $F_{\Qcal'\leftarrow\Pcal'}$.
187 \end{proof}
188 
189 \begin{keyidea}[title={Exactness by cancellation}]
190 Analysis and synthesis undo each other on the complete packet tuple. In a composition of packet descriptions, the intermediate synthesis and analysis maps cancel. This compares exact descriptions of the same ambient object. A connection adds data for transport along parameters or paths, with its own hypotheses.
191 \end{keyidea}
192 
193 \begin{example}[A mixed quadratic map]\label{ex:mixed_quadratic}
194 Let $X=Y=\bbk^2$ with coordinate packets.  For
195 \[
196 F(x_1,x_2)=(x_1x_2,x_1^2+x_2),
197 \]
198 the first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.
199 \end{example}
200 
201 \begin{remark}\label{rem:coordinates_not_independence}
202 Exact packet coordinates do not imply packet independence.  They retain the complete input tuple.  Every output packet may depend on every input packet.
203 \end{remark}
204 
205 \section{Packet jets and higher composition}\label{sec:packet_jets}
206 
207 Let $X$ and $Y$ be Banach spaces over the same field $\mathbb R$ or $\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\to Y$ is $m$ times continuously Fr\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\le r\le m$, the derivative
208 \[
209 D^rF(x):X^r\longrightarrow Y
210 \]
211 is a continuous symmetric multilinear map.
212 
213 The range $X_\alpha$ is closed because
214 $X_\alpha=\ker(I-P_\alpha)$. It is therefore Banach with its inherited norm. The same holds for every $Y_\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,
215 $\|\Acal_{\Pcal}\|\le\max_\alpha\|P_\alpha\|$ and
216 $\|\Scal_{\Pcal}\|\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.
217 
218 \begin{definition}[Packet jet]\label{def:packet_jet}
219 For $1\le r\le m$, define
220 \[
221 J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
222 =
223 Q_\gamma D^rF(x)
224 \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
225 \]
226 This is a continuous multilinear map
227 \[
228 X_{\alpha_1}\times\cdots\times X_{\alpha_r}
229 \longrightarrow
230 Y_\gamma.
231 \]
232 \end{definition}
233 
234 \begin{theorem}[Exact reconstruction of derivatives]\label{thm:jet_reconstruction}
235 For $h_1,\ldots,h_r\in X$,
236 \[
237 D^rF(x)[h_1,\ldots,h_r]
238 =
239 \sum_{\gamma}
240 \sum_{\alpha_1,\ldots,\alpha_r}
241 J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
242 [P_{\alpha_1}h_1,\ldots,P_{\alpha_r}h_r].
243 \]
244 \end{theorem}
245 
246 \begin{proof}
247 Insert $I_Y=\sum_\gamma Q_\gamma$ on the output and $I_X=\sum_\alpha P_\alpha$ in every input slot.  Multilinearity expands the finite sums.
248 \end{proof}
249 
250 \begin{figure}[htbp]
251 \centering
252 \includegraphics[width=.96\linewidth]{packet_jet_worked_example.pdf}
253 \caption{Second packet jets for $F(x,y)=(x+xy,y+x^2)$ at the origin. Coordinate axis projectors give $DF(0)=I$. The first output has mixed second derivatives equal to one, while the second has a repeated first input derivative equal to two. Each grid records the ordered pair of input packets. The mixed entries retain an interaction that diagonal restrictions would lose.}
254 \label{fig:packet-jet-example}
255 \end{figure}
256 \clearpage
257 
258 \begin{proposition}[Jet norm bound]\label{prop:jet_norm_bound}
259 Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate
260 \[
261 \left\|J^r_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)\right\|
262 \le \|Q_\gamma\|\,\|D^rF(x)\|.
263 \]
264 Its ambient extension inserts $P_{\alpha_j}$ in the $j$th slot and has norm at most
265 $\|Q_\gamma\|\,\|D^rF(x)\|\prod_j\|P_{\alpha_j}\|$.
266 Consequently the operator norm of the restricted packet jet also satisfies
267 \[
268 \left\|J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)\right\|
269 \le
270 \|Q_\gamma\|\,\|D^rF(x)\|
271 \prod_{j=1}^r\|P_{\alpha_j}\|.
272 \]
273 When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full derivative.
274 \end{proposition}
275 
276 \begin{proof}
277 For $h_j\in X_{\alpha_j}$, the inherited norm gives
278 \[
279 \|Q_\gamma D^rF(x)[h_1,\ldots,h_r]\|
280 \le \|Q_\gamma\|\,\|D^rF(x)\|\prod_j\|h_j\|.
281 \]
282 Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace $h_j$ by $P_{\alpha_j}h_j$ and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its range. The displayed product bound therefore also follows for the restricted jet. If a zero packet is retained as a redundant index, multilinearity makes any jet with that input or output packet zero, so both estimates remain valid.
283 \end{proof}
284 
285 \begin{corollary}[Packet Taylor formula]\label{cor:packet_taylor}
286 Assume $F$ is $m+1$ times continuously Fr\'echet differentiable on an open set containing the segment $x+th$ for $0\le t\le1$.  Then
287 \[
288 \begin{aligned}
289 Q_\gamma F(x+h)
290 ={}&Q_\gamma F(x)
291 +
292 \sum_{r=1}^{m}\frac1{r!}
293 \sum_{\alpha_1,\ldots,\alpha_r}
294 J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)
295 [P_{\alpha_1}h,\ldots,P_{\alpha_r}h]\\
296 &+Q_\gamma R_{m+1}(x,h),
297 \end{aligned}
298 \]
299 where the Bochner integral remainder is
300 \[
301 R_{m+1}(x,h)
302 =
303 \frac1{m!}
304 \int_0^1(1-t)^m
305 D^{m+1}F(x+th)[h,\ldots,h]\,dt.
306 \]
307 \end{corollary}
308 
309 \begin{proof}
310 Set $g(t)=F(x+th)$ on the open preimage of the domain under $t\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives
311 $g^{(r)}(t)=D^rF(x+th)[h,\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put
312 \[
313 T_j=\frac1{j!}\int_0^1(1-t)^j g^{(j+1)}(t)\,dt
314 \qquad(0\le j\le m).
315 \]
316 The fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\ge1$, integration by parts gives
317 $T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.
318 
319 Apply $Q_\gamma$ to the identity and use \cref{thm:jet_reconstruction}. Pairwise annihilation and idempotence retain exactly the output packet $\gamma$. Boundedness of $Q_\gamma$ allows it to pass through the Bochner integral. In particular, if
320 $\|D^{m+1}F(x+th)\|\le M$ along the segment, then
321 \[
322 \|Q_\gamma R_{m+1}(x,h)\|
323 \le \frac{\|Q_\gamma\|M\|h\|^{m+1}}{(m+1)!}.
324 \]
325 For $m=0$ the Taylor sum is empty. For $h=0$, multilinearity makes every positive-order term and the remainder zero.
326 \end{proof}
327 
328 The relation to derivatives of the complete coordinate representation is also literal. For an open domain $U\subseteq X$, put
329 $\widehat F=\Acal_{\Qcal}\circ F\circ\Scal_{\Pcal}$ on the open coordinate set $\Acal_{\Pcal}(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\le r\le m$,
330 \[
331 D^r\widehat F(u)[v_1,\ldots,v_r]
332 =\Acal_{\Qcal}\bigl(D^rF(\Scal_{\Pcal}u)
333 [\Scal_{\Pcal}v_1,\ldots,\Scal_{\Pcal}v_r]\bigr).
334 \]
335 Indeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\alpha_j$, synthesis includes that slot into $X$. Its $\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.
336 
337 At a fixed base point, the nonzero packet jets form a directed hypergraph.  An order $r$ interaction has $r$ input packet labels and one output packet label.  The support therefore records interactions relative to the chosen packet system.
338 
339 For a block $B=\{i_1<\cdots<i_s\}$ and an intermediate packet $\gamma$, write
340 \[
341 G_{\gamma,B}(x\mid h)
342 =
343 J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x)
344 [h_{i_1},\ldots,h_{i_s}].
345 \]
346 
347 \begin{interpretation}[title={Arity of packet jets}]
348 A packet jet has one output slot and as many input slots as its derivative order. In the composition formula, each partition block collects inputs for one derivative of the inner map. The number of blocks determines the arity of the derivative of the outer map. Intermediate packet labels record the outputs passed between them.
349 \end{interpretation}
350 
351 \begin{theorem}[Packet Fa\`a di Bruno formula]\label{thm:packet_faa}
352 Let $n\ge1$, let $X,Y,Z$ be Banach spaces over the same real or complex field, and let $G:X\to Y$ and $F:Y\to Z$ be $n$ times continuously Fr\'echet differentiable. Let $\Pcal$, $\Qcal$, and $\Rcal$ be packet systems on $X$, $Y$, and $Z$.  Fix input packets $\alpha_1,\ldots,\alpha_n$, an output packet $\delta$, and directions $h_i\in X_{\alpha_i}$.  Then
353 \[
354 \begin{aligned}
355 &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x)
356 [h_1,\ldots,h_n]\\
357 &=
358 \sum_{\pi=\{B_1,\ldots,B_k\}}
359 \sum_{\gamma_1,\ldots,\gamma_k}
360 J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx)
361 \big[
362 G_{\gamma_1,B_1}(x\mid h),\ldots,
363 G_{\gamma_k,B_k}(x\mid h)
364 \big].
365 \end{aligned}
366 \]
367 The first sum runs over all set partitions of $\{1,\ldots,n\}$.
368 \end{theorem}
369 
370 \begin{proof}
371 First derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\{n+1\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\{1,\ldots,n+1\}$ arises exactly once: if $\{n+1\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.
372 
373 Insert $I_Y=\sum_\gamma Q_\gamma$ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply $R_\delta$ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block in increasing index order and blocks in increasing least-element order. Symmetry of the higher derivative makes the final value independent of the block ordering.
374 \end{proof}
375 
376 \begin{proposition}[Number of labeled composition terms]\label{prop:faa_term_count}
377 For $n\ge1$, if $Y$ has $M$ packets, then the number of packet labeled terms at order $n$ before zero jets are removed is
378 \[
379 N_{n,M}
380 =
381 \sum_{k=1}^{n}S(n,k)M^k,
382 \]
383 where $S(n,k)$ is a Stirling number of the second kind.  Thus $N_{n,M}$ is the $n$th Touchard polynomial evaluated at $M$.
384 \end{proposition}
385 
386 \begin{proof}
387 A set partition with $k$ blocks receives one independent intermediate packet label on each block.  It therefore gives $M^k$ labeled terms.  There are $S(n,k)$ such partitions.
388 \end{proof}
389 
390 \begin{corollary}[Structural pruning]\label{cor:structural_pruning}
391 For a map $H$ with fixed input and output packet systems, let $\supp_r H(x)$ be the set of external profiles $(\delta;\alpha_1,\ldots,\alpha_r)$ for which the restricted jet $J^r_{\delta\mid\alpha_1,\ldots,\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\pi=\{B_1,\ldots,B_k\}$ and intermediate labels $\gamma_1,\ldots,\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \cref{thm:packet_faa} are nonzero. Then $\supp_n(F\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.
392 \end{corollary}
393 
394 \begin{proof}
395 If a profile has no admissible decorated partition, every summand in \cref{thm:packet_faa} has a zero constituent and therefore vanishes. The composed jet is then the zero multilinear map. This proves the support inclusion by contraposition. The inclusion can be strict. Nonzero constituent maps may compose to zero, and nonzero summands may cancel. Thus the admissible set is a structural upper bound rather than a claim of equality.
396 \end{proof}
397 
398 
399 \begin{example}[The second-order composition law]\label{ex:second_order_composition}
400 At order two, the partition formula has two shapes.  For packet directions $h_1$ and $h_2$,
401 \[
402 \begin{aligned}
403 R_\delta D^2(F\circ G)(x)[h_1,h_2]
404 ={}&
405 \sum_{\gamma_1,\gamma_2}
406 J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx)
407 [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\
408 &+
409 \sum_\gamma
410 J^1_{\delta\mid\gamma}F(Gx)
411 [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]].
412 \end{aligned}
413 \]
414 The first line records two first-order changes entering a quadratic interaction of $F$.  The second records a genuine second-order interaction created inside $G$ before $F$ is applied.
415 \end{example}
416 
417 \begin{remark}\label{rem:jet_coordinate_dependence}
418 Jet sparsity is not invariant under an arbitrary change of coordinates.  A useful packet system normally comes from symmetry, scale, support, grading, valuation, or another structure that is already present in the problem.  Every sparsity claim should therefore name the packet system with respect to which it is made.
419 \end{remark}
420 
421 \section{Rooted tree expansions}\label{sec:trees}
422 
423 Repeated nonlinear substitution is indexed by rooted trees.  We begin with a polynomial fixed point equation on a Banach space.
424 
425 Let $X$ be a Banach space.  Fix $s\in X$ and continuous multilinear maps
426 \[
427 B_r:X^r\longrightarrow X,
428 \qquad
429 2\le r\le d.
430 \]
431 Set
432 \[
433 a=\|s\|,
434 \qquad
435 b_r=\|B_r\|,
436 \qquad
437 \phi(t)=a+\sum_{r=2}^{d}b_rt^r.
438 \]
439 An ordered rooted tree has leaves labeled by $s$ and every internal vertex of arity $r$ labeled by $B_r$.  Its value is obtained by evaluating from the leaves toward the root.
440 
441 \begin{theorem}[Polynomial tree expansion]\label{thm:polynomial_tree_expansion}
442 Suppose there is $R>0$ such that
443 \[
444 \phi(R)\le R
445 \]
446 and
447 \[
448 q=
449 \sum_{r=2}^{d}r b_rR^{r-1}<1.
450 \]
451 Then the map
452 \[
453 \Phi(x)=s+\sum_{r=2}^{d}B_r(x,\ldots,x)
454 \]
455 has a unique fixed point in the closed ball $\overline B_R(0)$.  The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.
456 \end{theorem}
457 
458 \begin{proof}
459 For $x,y\in\overline B_R(0)$, multilinearity and telescoping in the $r$ input positions give
460 \[
461 \|B_r(x,\ldots,x)-B_r(y,\ldots,y)\|
462 \le
463 r b_rR^{r-1}\|x-y\|.
464 \]
465 Hence $\Phi$ is $q$-Lipschitz on the ball.  The inequality $\phi(R)\le R$ shows that $\Phi$ preserves the ball.  The contraction mapping theorem gives a unique fixed point $u$ in that ball.
466 
467 Let $\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\tau$ has arity $r$ and ordered subtrees $\tau_1,\ldots,\tau_r$, define
468 \[
469 B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
470 \]
471 Induction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\Phi(u_m)$ satisfies
472 \[
473 u_m=\sum_{\tau\in\mathfrak T_m}B_\tau(s).
474 \]
475 Define a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\ldots,t_r)\mapsto b_rt_1\cdots t_r$ at a vertex of arity $r$.  A second induction gives
476 \[
477 \sum_{\tau\in\mathfrak T_m}\|B_\tau(s)\|
478 \le
479 \sum_{\tau\in\mathfrak T_m}w(\tau)
480 =t_m,
481 \]
482 where $t_0=0$ and $t_{m+1}=\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\le R$ and $\phi(R)\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.
483 \end{proof}
484 
485 \begin{corollary}[Quadratic Catalan expansion]\label{cor:quadratic_tree_expansion}
486 Let $B:X\times X\to X$ be bounded and consider
487 \[
488 u=s+B(u,u).
489 \]
490 Put $a=\|s\|$ and $b=\|B\|$.  If $4ab<1$, then
491 \[
492 u=
493 \sum_{n\ge1}
494 \sum_{\substack{\tau\text{ full ordered binary tree}\\\text{with }n\text{ leaves}}}
495 B_\tau(s)
496 \]
497 converges absolutely.  It also satisfies
498 \[
499 \|u\|
500 \le
501 \frac{1-\sqrt{1-4ab}}{2b}
502 \]
503 when $b>0$.
504 \end{corollary}
505 
506 \begin{proof}
507 If $b=0$, the equation is $u=s$ and the tree series contains only its one leaf.  If $a=0$, every tree value is zero.  Choose any $R$ with $0<R<1/(2b)$ and apply \cref{thm:polynomial_tree_expansion} to obtain uniqueness of the zero fixed point in that ball.
508 
509 Assume now that $a>0$ and $b>0$.  Take
510 \[
511 R=
512 \frac{1-\sqrt{1-4ab}}{2b}.
513 \]
514 Then $a+bR^2=R$ and $2bR<1$.  Apply \cref{thm:polynomial_tree_expansion}.  The number of full ordered binary trees with $n$ leaves is the Catalan number $C_{n-1}$, which gives the displayed majorant.
515 \end{proof}
516 
517 \begin{theorem}[Packet labeled tree reconstruction]\label{thm:packet_tree_reconstruction}
518 Let $X$ carry a packet system $\Pcal$.  At each internal vertex of a tree, insert the output identity and one input identity for every child.  Summing over all internal packet labels reconstructs the original tree value exactly.
519 \end{theorem}
520 
521 \begin{proof}
522 At a vertex labeled by an $r$ multilinear map $B_r$, insert
523 \[
524 I_X=\sum_\beta P_\beta
525 \]
526 on the output and in every input slot.  Multilinearity gives an exact finite sum of packet blocks at that vertex.  Apply this identity successively from the leaves toward the root.  Since the tree and the packet set are finite, all sums are finite and their total reconstructs the original tree value.
527 \end{proof}
528 
529 For an infinite tree expansion, the finite sum of packet decorations for each underlying tree is kept together. The unlabeled tree majorant does not imply absolute convergence when all decorations are summed separately for arbitrary oblique projectors. Finite tree reconstruction and the grouped interpretation remain exact.
530 
531 \begin{interpretation}[title={Arity in a tree}]
532 At each internal vertex, the number of children is the number of input slots of its multilinear map. Their order is retained because symmetry is not assumed. Packet labels resolve each finite tree into a finite sum. In an infinite expansion these sums stay grouped by tree, as required by the convergence statement above.
533 \end{interpretation}
534 
535 \begin{corollary}[Tree pruning]\label{cor:tree_pruning}
536 A packet labeled tree vanishes if one internal vertex carries a zero packet block of the corresponding multilinear map.  Removing all such zero terms leaves every finite tree sum unchanged and therefore leaves the grouped infinite series unchanged.
537 \end{corollary}
538 
539 \begin{proof}
540 A zero vertex map makes the value of the full tree zero.
541 \end{proof}
542 
543 
544 The tree theorem separates combinatorial growth from analytic convergence.  The number of trees may grow rapidly, while multilinear bounds control their values. The scalar map $\phi$ controls the total weight of the expansion.  Packet pruning changes the combinatorics but not the proof.  It removes trees whose values are already zero before any norm estimate is used.
545 
546 \begin{example}[Catalan parity packets]\label{ex:catalan_parity}
547 Over a field of characteristic zero, the formal equation
548 \[
549 u=z+u^2
550 \]
551 has the branch
552 \[
553 u(z)=\frac{1-\sqrt{1-4z}}2.
554 \]
555 Write $u=u_0+u_1$, where $u_0$ contains the even powers of $z$ and $u_1$ contains the odd powers.  Exact packet projection gives
556 \[
557 u_0=u_0^2+u_1^2,
558 \qquad
559 u_1=z+2u_0u_1.
560 \]
561 The two equations remain coupled.  Their sum is the original fixed point equation.
562 \end{example}
563 
564 \section{Graded nonlinear operations}\label{sec:graded_operations}
565 
566 Let $G$ be a finite abelian group and let
567 \[
568 A=\bigoplus_{g\in G}A_g
569 \]
570 be a $G$ graded module.
571 
572 \begin{definition}[Grade respecting operation]\label{def:grade_respecting}
573 An $r$ multilinear map $\mu:A^r\to A$ respects the grading when
574 \[
575 \mu(A_{g_1},\ldots,A_{g_r})
576 \subseteq
577 A_{g_1+\cdots+g_r}.
578 \]
579 \end{definition}
580 
581 \begin{theorem}[Graded packet convolution]\label{thm:graded_convolution}
582 If $a_j=\sum_{g\in G}a_{j,g}$ and $\mu$ respects the grading, then
583 \[
584 P_k\mu(a_1,\ldots,a_r)
585 =
586 \sum_{g_1+\cdots+g_r=k}
587 \mu(a_{1,g_1},\ldots,a_{r,g_r}).
588 \]
589 \end{theorem}
590 
591 \begin{proof}
592 Expand every input into its homogeneous components.  Each term lands in the component whose label is the sum of its input labels.  Projection onto $A_k$ retains exactly the displayed terms.
593 \end{proof}
594 
595 \begin{corollary}[Cyclic product formula]\label{cor:cyclic_product}
596 For a cyclically graded algebra
597 \[
598 A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j,
599 \qquad
600 A_iA_j\subseteq A_{i+j},
601 \]
602 one has
603 \[
604 P_k(ab)=
605 \sum_{i+j=k}a_ib_j.
606 \]
607 \end{corollary}
608 
609 \begin{proof}
610 Apply \cref{thm:graded_convolution} to multiplication.
611 \end{proof}
612 
613 \begin{theorem}[Packet Maurer Cartan equations]\label{thm:packet_mc}
614 Let
615 \[
616 \Lcal=\bigoplus_{g\in G}\Lcal_g
617 \]
618 be a differential graded Lie algebra over a ring in which two is invertible.  Assume
619 \[
620 d\Lcal_g\subseteq\Lcal_g,
621 \qquad
622 [\Lcal_g,\Lcal_h]\subseteq\Lcal_{g+h}.
623 \]
624 For a degree one element $\eta=\sum_g\eta_g$, the Maurer Cartan equation
625 \[
626 d\eta+\frac12[\eta,\eta]=0
627 \]
628 is equivalent to
629 \[
630 d\eta_k
631 +
632 \frac12
633 \sum_{g+h=k}[\eta_g,\eta_h]
634 =0
635 \]
636 for every $k\in G$.
637 \end{theorem}
638 
639 \begin{proof}
640 The differential preserves the group label and the bracket adds labels.  Apply the projector onto $\Lcal_k$ and use exact reconstruction.
641 \end{proof}
642 
643 The same argument applies to higher brackets.
644 
645 \begin{proposition}[Packet equations for $L_\infty$ operations]\label{prop:packet_l_infinity}
646 Let $\Lcal$ be complete and carry operations $\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series
647 \[
648 \sum_{r\ge1}\frac1{r!}\ell_r(\eta,\ldots,\eta)
649 \]
650 converges and the required factorials are invertible.  Then its packet of label $k$ is
651 \[
652 \sum_{r\ge1}\frac1{r!}
653 \sum_{g_1+\cdots+g_r=k}
654 \ell_r(\eta_{g_1},\ldots,\eta_{g_r}).
655 \]
656 The full Maurer Cartan equation holds if and only if every packet equation vanishes.
657 \end{proposition}
658 
659 \begin{proof}
660 Apply \cref{thm:graded_convolution} to each operation $\ell_r$.  Continuity of the packet projectors permits the convergent sums to be projected term by term.
661 \end{proof}
662 
663 \begin{definition}[Divided square]\label{def:divided_square}
664 A divided square on degree one elements is a map $q$ satisfying
665 \[
666 q(x+y)=q(x)+q(y)+[x,y],
667 \qquad
668 q(ax)=a^2q(x),
669 \]
670 whenever the expressions are defined.
671 \end{definition}
672 
673 \begin{proposition}[Divided square packet equations]\label{prop:divided_square_packets}
674 Assume $q(\Lcal_g)\subseteq\Lcal_{2g}$.  Choose an ordering of the finite label set.  The equation
675 \[
676 d\eta+q(\eta)=0
677 \]
678 is equivalent to
679 \[
680 d\eta_k
681 +
682 \sum_{2g=k}q(\eta_g)
683 +
684 \sum_{\substack{g<h\\g+h=k}}[\eta_g,\eta_h]
685 =0
686 \]
687 for every $k$.
688 \end{proposition}
689 
690 \begin{proof}
691 Repeated use of the polarization identity for $q$ gives
692 \[
693 q\left(\sum_g\eta_g\right)
694 =
695 \sum_gq(\eta_g)
696 +
697 \sum_{g<h}[\eta_g,\eta_h].
698 \]
699 Project onto the component of label $k$.
700 \end{proof}
701 
702 
703 \begin{example}[Matrix difference grading]\label{ex:matrix_difference_grading}
704 Let $A=M_N(\bbk)$ and assign the matrix unit $E_{rs}$ the label $r-s$ modulo $N$.  Since
705 \[
706 E_{rs}E_{uv}=\delta_{su}E_{rv},
707 \]
708 the output label is
709 \[
710 (r-s)+(u-v)=r-v
711 \]
712 whenever the product is nonzero.  Matrix multiplication therefore obeys the cyclic convolution law exactly.  The same grading resolves commutators and polynomial matrix equations.
713 \end{example}
714 
715 \begin{remark}\label{rem:graded_boundary}
716 These results begin with a graded differential or higher algebra.  They do not assert that every deformation problem has such a controlling object.  Differential graded and infinity algebra methods are developed in \cite{LodayVallette2012}.
717 \end{remark}
718 
719 \section{Nonlinear refinement defects}\label{sec:nonlinear_defects}
720 
721 Let
722 \[
723 F_f:X_f\longrightarrow Y_f,
724 \qquad
725 F_c:X_c\longrightarrow Y_c
726 \]
727 be fine and coarse maps.  Let
728 \[
729 C_X:X_f\longrightarrow X_c,
730 \qquad
731 C_Y:Y_f\longrightarrow Y_c
732 \]
733 be linear coarsening maps.  Whenever a norm, a Lipschitz constant, or a Fr\'echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.
734 
735 \begin{definition}[Nonlinear refinement defect]\label{def:nonlinear_defect}
736 The pointwise defect of $F$ is
737 \[
738 \Delta_F(x)
739 =
740 C_YF_f(x)-F_c(C_Xx).
741 \]
742 The two models are exactly compatible when $\Delta_F$ vanishes everywhere.
743 \end{definition}
744 
745 
746 The defect is a vector in $Y_c$.  It determines an obstruction class after a correction map into $Y_c$ has been chosen and the defect has been passed to the corresponding cokernel.  Norm estimates measure the size of the defect.  The quotient determines whether an allowed linear correction can remove it.
747 
748 \begin{theorem}[Exact defect under composition]\label{thm:defect_composition_identity}
749 Let $G_f:Y_f\to Z_f$ and $G_c:Y_c\to Z_c$, with coarsening $C_Z:Z_f\to Z_c$.  Then
750 \[
751 \Delta_{G\circ F}(x)
752 =
753 \Delta_G(F_f(x))
754 +
755 G_c\big(F_c(C_Xx)+\Delta_F(x)\big)
756 -
757 G_c\big(F_c(C_Xx)\big).
758 \]
759 If $G_c$ is $L$-Lipschitz on the relevant set, then
760 \[
761 \|\Delta_{G\circ F}(x)\|
762 \le
763 \|\Delta_G(F_f(x))\|
764 +
765 L\|\Delta_F(x)\|.
766 \]
767 \end{theorem}
768 
769 \begin{proof}
770 Add and subtract $G_c(C_YF_f(x))$.  Use
771 \[
772 C_YF_f(x)=F_c(C_Xx)+\Delta_F(x).
773 \]
774 The norm estimate follows from the Lipschitz bound.
775 \end{proof}
776 
777 \begin{corollary}[Linear product rule]\label{cor:linear_defect_product}
778 If all maps in \cref{thm:defect_composition_identity} are linear, then
779 \[
780 \Delta_{GF}
781 =
782 \Delta_G F_f+G_c\Delta_F.
783 \]
784 \end{corollary}
785 
786 \begin{proof}
787 Linearity turns the last difference in \cref{thm:defect_composition_identity} into $G_c\Delta_F(x)$.
788 \end{proof}
789 
790 \begin{theorem}[Iterated orbit defect]\label{thm:iterated_orbit_defect}
791 Let $F_f:X_f\to X_f$ and $F_c:X_c\to X_c$, with coarsening $C:X_f\to X_c$.  Suppose $F_c$ is $q$-Lipschitz.  For a fine orbit starting at $x$, assume
792 \[
793 \|\Delta_F(F_f^j x)\|\le\varepsilon_j.
794 \]
795 Then
796 \[
797 \left\|CF_f^n(x)-F_c^n(Cx)\right\|
798 \le
799 \sum_{j=0}^{n-1}q^{n-1-j}\varepsilon_j.
800 \]
801 If $0\le q<1$ and every $\varepsilon_j\le\varepsilon$, then
802 \[
803 \left\|CF_f^n(x)-F_c^n(Cx)\right\|
804 \le
805 \varepsilon\frac{1-q^n}{1-q}.
806 \]
807 \end{theorem}
808 
809 \begin{proof}
810 Let
811 \[
812 e_n=\left\|CF_f^n(x)-F_c^n(Cx)\right\|.
813 \]
814 The definition of the defect and the Lipschitz property give
815 \[
816 e_{n+1}\le\varepsilon_n+qe_n.
817 \]
818 Iterating this recurrence proves the first estimate.  The second is the geometric sum.
819 \end{proof}
820 
821 \begin{theorem}[Fixed point stability]\label{thm:fixed_point_stability}
822 Assume $F_c$ is a contraction with constant $q<1$.  Let $x_f$ and $x_c$ be fixed points of $F_f$ and $F_c$.  If
823 \[
824 \|\Delta_F(x_f)\|\le\varepsilon,
825 \]
826 then
827 \[
828 \|Cx_f-x_c\|
829 \le
830 \frac{\varepsilon}{1-q}.
831 \]
832 When the defect at $x_f$ is zero, the fixed points are compatible.
833 \end{theorem}
834 
835 \begin{proof}
836 Using the fixed-point equations,
837 \[
838 \begin{aligned}
839 \|Cx_f-x_c\|
840 &\le
841 \|CF_f(x_f)-F_c(Cx_f)\|
842 +
843 \|F_c(Cx_f)-F_c(x_c)\|\\
844 &\le
845 \varepsilon+q\|Cx_f-x_c\|.
846 \end{aligned}
847 \]
848 Rearrange.
849 \end{proof}
850 
851 \begin{corollary}[Several refinement levels]\label{cor:multilevel_fixed_points}
852 Let $C_\ell:X_{\ell+1}\to X_\ell$ for $0\le\ell<m$.  Suppose $x_\ell$ is a fixed point at level $\ell$, the coarse map at level $\ell$ has contraction constant $q_\ell<1$, and the adjacent defect at $x_{\ell+1}$ is at most $\varepsilon_\ell$.  Write
853 \[
854 C_{a,b}=C_aC_{a+1}\cdots C_{b-1}
855 \quad\text{for }a<b,
856 \qquad
857 C_{a,a}=I.
858 \]
859 Then
860 \[
861 \|C_{0,m}x_m-x_0\|
862 \le
863 \sum_{\ell=0}^{m-1}
864 \|C_{0,\ell}\|
865 \frac{\varepsilon_\ell}{1-q_\ell}.
866 \]
867 \end{corollary}
868 
869 \begin{proof}
870 Insert the intermediate vectors $C_{0,\ell}x_\ell$ and telescope.  Apply \cref{thm:fixed_point_stability} to every adjacent pair.
871 \end{proof}
872 
873 \begin{proposition}[Derivative defects]\label{prop:derivative_defects}
874 Assume $F_f$ and $F_c$ are $r$ times Fr\'echet differentiable and the coarsening maps are bounded and linear.  Then
875 \[
876 D\Delta_F(x)
877 =
878 C_YDF_f(x)-DF_c(C_Xx)C_X.
879 \]
880 For $r\ge2$,
881 \[
882 D^r\Delta_F(x)[h_1,\ldots,h_r]
883 =
884 C_YD^rF_f(x)[h_1,\ldots,h_r]
885 -
886 D^rF_c(C_Xx)[C_Xh_1,\ldots,C_Xh_r].
887 \]
888 \end{proposition}
889 
890 \begin{proof}
891 Differentiate the defining identity for $\Delta_F$.  The coarsening maps are linear, so their higher derivatives vanish.
892 \end{proof}
893 
894 \begin{example}[Sharp affine bound]\label{ex:sharp_affine_bound}
895 Let $F_c(y)=qy+b$ with $0<q<1$.  Choose a fine fixed point whose coarsening has constant defect $\varepsilon$ in the direction of the scalar state.  The fixed point mismatch is exactly $\varepsilon/(1-q)$.  Thus the constant in \cref{thm:fixed_point_stability} cannot be improved without more information.
896 \end{example}
897 
898 \section{Canonical obstruction objects}\label{sec:obstruction_objects}
899 
900 We begin by isolating the part of a defect that cannot be removed by an allowed linear correction.
901 
902 \begin{definition}[Linear correction datum]\label{def:linear_correction_datum}
903 A linear correction datum consists of a homomorphism
904 \[
905 D:U\longrightarrow V
906 \]
907 and a defect $r\in V$.  A correction is an element $u\in U$ satisfying
908 \[
909 Du=-r.
910 \]
911 The obstruction class is
912 \[
913 \Ob_D(r)=[r]\in\coker D=V/\im D.
914 \]
915 \end{definition}
916 
917 \begin{theorem}[Cokernel obstruction criterion]\label{thm:cokernel_obstruction}
918 A linear correction exists if and only if
919 \[
920 \Ob_D(r)=0.
921 \]
922 The quotient map
923 \[
924 q_D:V\longrightarrow\coker D
925 \]
926 has the following universal property.  If $\varphi:V\to W$ is linear and $\varphi D=0$, then there is a unique linear map
927 \[
928 \overline\varphi:\coker D\longrightarrow W
929 \]
930 with $\varphi=\overline\varphi q_D$.
931 \end{theorem}
932 
933 \begin{proof}
934 The class of $r$ is zero exactly when $r\in\im D$, which is equivalent to solvability of $Du=-r$.  If $\varphi D=0$, then $\varphi$ vanishes on $\im D$.  It therefore descends uniquely to the quotient.
935 \end{proof}
936 
937 The universal property explains why the cokernel is the canonical obstruction object relative to $D$.  Any linear obstruction test that vanishes on correctable defects factors through it.
938 
939 \begin{keyidea}[title={Canonical obstruction data}]
940 The residual, the correction map, and the obstruction class are distinct data. The residual is a vector. The correction map specifies which changes are allowed, and its cokernel records the remaining class. Canonicality is relative to that declared correction map. A scalar or orthogonal representative of the class may require further choices or hypotheses.
941 \end{keyidea}
942 
943 \begin{figure}[htbp]
944 \centering
945 \includegraphics[width=.96\linewidth]{correction_cokernel_plane.pdf}
946 \caption{A specified correction range in $\mathbb R^2$. For $L(a,b)=(a,0)$, the image is the horizontal axis. The defect $(2,3)$ splits into $(2,0)$ in that image and the remaining representative $(0,3)$. Subtracting the image component leaves the same cokernel class. The functional $(x,y)\mapsto y$ vanishes on the correction range and detects the value three. The correction equation $Lu=-(2,3)$ therefore has no solution.}
947 \label{fig:correction-cokernel-example}
948 \end{figure}
949 \clearpage
950 
951 \begin{theorem}[Naturality of obstruction classes]\label{thm:obstruction_naturality}
952 Suppose
953 \[
954 \begin{array}{ccc}
955 U & \xrightarrow{D} & V\\
956 \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\
957 U' & \xrightarrow{D'} & V'
958 \end{array}
959 \]
960 is a commuting square.  Then there is a unique induced map
961 \[
962 \overline b:\coker D\longrightarrow\coker D'
963 \]
964 satisfying
965 \[
966 \overline b\,\Ob_D(r)=\Ob_{D'}(br).
967 \]
968 \end{theorem}
969 
970 \begin{proof}
971 The relation $bD=D'a$ shows that the quotient map $q_{D'}b$ vanishes on $\im D$.  Apply the universal property in \cref{thm:cokernel_obstruction}.
972 \end{proof}
973 
974 \begin{theorem}[Packet decomposition of an obstruction]\label{thm:packet_obstruction_decomposition}
975 Suppose
976 \[
977 U=\bigoplus_{\alpha\in I}U_\alpha,
978 \qquad
979 V=\bigoplus_{\alpha\in I}V_\alpha,
980 \]
981 and $D(U_\alpha)\subseteq V_\alpha$ for every $\alpha$.  Let
982 \[
983 D_\alpha:U_\alpha\longrightarrow V_\alpha
984 \]
985 be the restriction.  Then
986 \[
987 \coker D
988 \cong
989 \bigoplus_{\alpha\in I}\coker D_\alpha.
990 \]
991 Under this isomorphism, the obstruction class of $r=\sum_\alpha r_\alpha$ is the tuple
992 \[
993 \big([r_\alpha]\big)_\alpha.
994 \]
995 Hence $r$ is correctable if and only if every packet obstruction vanishes.
996 \end{theorem}
997 
998 \begin{proof}
999 The grading assumption gives
1000 \[
1001 \im D=\bigoplus_\alpha\im D_\alpha.
1002 \]
1003 Taking the quotient of the direct sum by this direct sum of images gives the displayed isomorphism.
1004 \end{proof}
1005 
1006 \begin{remark}\label{rem:block_diagonal_needed}
1007 The conclusion of \cref{thm:packet_obstruction_decomposition} requires the correction map to preserve the packets.  Off-diagonal correction blocks can couple packet defects.  In that case the full cokernel remains canonical, but it need not split into the separate packet cokernels.
1008 \end{remark}
1009 
1010 \begin{proposition}[Projectors on the obstruction quotient]\label{prop:obstruction_quotient_projectors}
1011 Let $(Q_\alpha)_{\alpha\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume
1012 \[
1013 Q_\alpha(\im D)\subseteq\im D
1014 \]
1015 for every $\alpha$.  Then each $Q_\alpha$ induces an idempotent $\overline Q_\alpha$ on $\coker D$, and
1016 \[
1017 \coker D
1018 =
1019 \bigoplus_{\alpha\in I}\overline Q_\alpha(\coker D).
1020 \]
1021 For every defect $r\in V$,
1022 \[
1023 \overline Q_\alpha\Ob_D(r)
1024 =
1025 [Q_\alpha r].
1026 \]
1027 No packet decomposition of the correction space $U$ is required.
1028 \end{proposition}
1029 
1030 \begin{proof}
1031 If $v-v'\in\im D$, then
1032 \[
1033 Q_\alpha(v-v')\in\im D.
1034 \]
1035 Hence $[v]\mapsto[Q_\alpha v]$ is well defined on the quotient.  The idempotent, annihilation, and reconstruction identities descend from $V$ to $V/\im D$.  The formula for an obstruction class follows from the definition.
1036 \end{proof}
1037 
1038 \subsection{Relative obstruction under coarsening}
1039 
1040 Coarsening can erase an obstruction that remains present in the fine problem.  The next exact sequence identifies the part removed by coarsening.
1041 
1042 \begin{theorem}[Relative obstruction exact sequence]\label{thm:relative_obstruction_sequence}
1043 Suppose the following diagram of modules commutes and has exact rows:
1044 \[
1045 \begin{array}{ccccccccc}
1046 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt]
1047 &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt]
1048 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0.
1049 \end{array}
1050 \]
1051 There is a natural exact sequence
1052 \[
1053 \begin{aligned}
1054 0&\longrightarrow\Ker D_0
1055 \longrightarrow\Ker D_f
1056 \longrightarrow\Ker D_c
1057 \xrightarrow{\delta}\coker D_0\\
1058 &\longrightarrow\coker D_f
1059 \longrightarrow\coker D_c
1060 \longrightarrow0.
1061 \end{aligned}
1062 \]
1063 For $u_c\in\Ker D_c$, choose $u_f\in U_f$ with $p_Uu_f=u_c$.  There is a unique $v_0\in V_0$ such that
1064 \[
1065 D_fu_f=i_Vv_0,
1066 \]
1067 and the connecting map is
1068 \[
1069 \delta(u_c)=[v_0].
1070 \]
1071 \end{theorem}
1072 
1073 \begin{proof}
1074 Since $D_cu_c=0$, commutativity gives
1075 \[
1076 p_VD_fu_f=D_cp_Uu_f=0.
1077 \]
1078 Exactness of the lower row gives a unique $v_0$ with $D_fu_f=i_Vv_0$.  A different lift has the form $u_f+i_Uu_0$.  The corresponding element of $V_0$ changes by $D_0u_0$, so $[v_0]$ is independent of the lift.
1079 
1080 The map $\Ker D_0\to\Ker D_f$ is injective because $i_U$ is injective.  If $u_f\in\Ker D_f$ maps to zero in $U_c$, then $u_f=i_Uu_0$ for a unique $u_0$.  The identity
1081 \[
1082 i_VD_0u_0=D_fi_Uu_0=0
1083 \]
1084 and injectivity of $i_V$ show that $u_0\in\Ker D_0$.  This proves exactness at $\Ker D_f$.
1085 
1086 If $u_c$ comes from an element of $\Ker D_f$, one may choose that element as its lift and obtain $\delta(u_c)=0$.  Conversely, if $\delta(u_c)=0$, then $v_0=D_0u_0$ for some $u_0$.  Replacing $u_f$ by $u_f-i_Uu_0$ gives an element of $\Ker D_f$ that maps to $u_c$.  This proves exactness at $\Ker D_c$.
1087 
1088 The class $[v_0]\in\coker D_0$ maps to zero in $\coker D_f$ whenever it equals $\delta(u_c)$, since the chosen lift satisfies $i_Vv_0=D_fu_f$.  Conversely, if $i_Vv_0=D_fu_f$, then $u_c=p_Uu_f$ lies in $\Ker D_c$ and $\delta(u_c)=[v_0]$.  This proves exactness at $\coker D_0$.
1089 
1090 If a class in $\coker D_f$ comes from $\coker D_0$, its image in $\coker D_c$ is zero.  Conversely, suppose $[v_f]$ maps to zero in $\coker D_c$.  Choose $u_c$ with $D_cu_c=p_Vv_f$ and lift it to $u_f\in U_f$.  Then
1091 \[
1092 p_V(v_f-D_fu_f)=0,
1093 \]
1094 so $v_f-D_fu_f=i_Vv_0$ for some $v_0\in V_0$.  Thus $[v_f]$ comes from $[v_0]$.  Surjectivity at the last term follows from surjectivity of $p_V$.
1095 \end{proof}
1096 
1097 \begin{corollary}[Coarse vanishing and fine vanishing]\label{cor:coarse_fine_obstruction}
1098 Let $[r_f]\in\coker D_f$.  If its image in $\coker D_c$ is zero, then $[r_f]$ comes from $\coker D_0$.  Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening.  It does not imply $[r_f]=0$.
1099 \end{corollary}
1100 
1101 \begin{proof}
1102 This is exactness of the sequence in \cref{thm:relative_obstruction_sequence} at $\coker D_f$.
1103 \end{proof}
1104 
1105 \begin{example}[A fine obstruction erased by coarsening]\label{ex:coarse_cancellation}
1106 Take all three correction spaces to be zero.  Let
1107 \[
1108 V_f=\bbk^2,
1109 \qquad
1110 V_c=\bbk,
1111 \qquad
1112 p_V(x,y)=x+y,
1113 \]
1114 and let
1115 \[
1116 V_0=\Ker p_V=\Span\{(1,-1)\}.
1117 \]
1118 The fine defect $(1,-1)$ is a nonzero class in $\coker D_f=V_f$, while its coarse image is zero.  The class comes from $\coker D_0=V_0$, exactly as \cref{cor:coarse_fine_obstruction} predicts.
1119 \end{example}
1120 
1121 The same construction has a cohomological form.
1122 
1123 \begin{definition}[Mapping cone]\label{def:mapping_cone}
1124 Let $D:C^\bullet\to E^\bullet$ be a cochain map.  Its mapping cone is
1125 \[
1126 \Cone(D)^n=E^n\oplus C^{n+1}
1127 \]
1128 with differential
1129 \[
1130 d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
1131 \]
1132 \end{definition}
1133 
1134 \begin{theorem}[Cohomological obstruction through the cone]\label{thm:cone_obstruction}
1135 Let $e\in E^n$ satisfy $d_Ee=0$.  The class of $(e,0)$ in
1136 \[
1137 H^n(\Cone(D))
1138 \]
1139 vanishes if and only if the class $[e]\in H^n(E)$ lies in the image of
1140 \[
1141 H^n(D):H^n(C)\longrightarrow H^n(E).
1142 \]
1143 When $C$ and $E$ are concentrated in degree zero,
1144 \[
1145 H^0(\Cone(D))\cong\coker D.
1146 \]
1147 \end{theorem}
1148 
1149 \begin{proof}
1150 The class of $(e,0)$ vanishes exactly when there are $e'\in E^{n-1}$ and $c\in C^n$ with
1151 \[
1152 (e,0)=d_{\Cone}(e',c).
1153 \]
1154 This means
1155 \[
1156 e=d_Ee'+Dc,
1157 \qquad
1158 d_Cc=0.
1159 \]
1160 Equivalently, $[e]=H^n(D)[c]$.  The degree zero statement follows directly from the definition of the cone differential.
1161 \end{proof}
1162 
1163 \begin{corollary}[Packet obstruction complexes]\label{cor:packet_obstruction_complexes}
1164 If $C^\bullet$ and $E^\bullet$ split into finite packet subcomplexes and $D$ preserves those packets, then $\Cone(D)$ splits into packet subcomplexes and
1165 \[
1166 H^n(\Cone(D))
1167 \cong
1168 \bigoplus_\alpha H^n(\Cone(D_\alpha)).
1169 \]
1170 \end{corollary}
1171 
1172 \begin{proof}
1173 Both the cone and its cohomology commute with finite direct sums.
1174 \end{proof}
1175 
1176 \begin{example}[A two component obstruction]\label{ex:two_component_obstruction}
1177 Let
1178 \[
1179 D:\bbk\longrightarrow\bbk^2,
1180 \qquad
1181 D(t)=(t,t).
1182 \]
1183 The cokernel is one dimensional.  The defect $(a,b)$ is correctable exactly when $a=b$.  The obstruction can be represented by $a-b$, but this scalar representative depends on a choice of isomorphism from the cokernel to $\bbk$.  The quotient class itself is canonical.
1184 \end{example}
1185 

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