Stratified Universality

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1 \section{Nonlinear polynomial fixed points, compressed packet composition, and finite tree signatures}
2 \label{sec:block11-nonlinear-tree}
3 
4 Nonlinear tree atlas develops the nonlinear and rooted-tree stratum after the algebraic milestone.  The existing nonlinear sources already prove exact packet jets, the packet Fa\`a di Bruno law, colored operadic composition, refinement coherence, rooted-tree reconstruction, analytic majorant estimates, and local near-identity inversion \cite{P3,RT,NLO}.  The remaining problem is not whether nonlinear maps can be expanded.  It is whether a declared nonlinear family admits a normalized finite presentation and a finite signature that controls a named packet complexity.
5 
6 This section gives three restricted answers.  First, a marked polynomial fixed-point system is organized into equivalent system, packet-hypergraph, operadic-tree, and exact packet-equation charts.  Second, repeated-direction packet Fa\`a di Bruno sums admit an exact orbit compression indexed by colored integer partitions rather than all labeled set partitions.  Third, simply generated plane-tree equations possess a finite arity-gcd signature that determines every packet-support pattern and, under the classical smooth implicit schema, the complete dominant-singularity orbit after arithmetic packetization.  A dependency condensation isolates the exact cyclic nonlinear core of a marked packet system.  None of these results is promoted to a universal nonlinear normal form.
7 
8 Throughout the analytic part, $X$ is a Banach space over $\mathbb R$ or $\mathbb C$.  A finite packet system on $X$ is a family of bounded projections
9 \[
10  P_iP_j=\delta_{ij}P_i,
11  \qquad
12  \sum_{i\in I}P_i=I_X.
13 \]
14 All sums over packet addresses are finite.  The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.
15 
16 \subsection{Marked polynomial fixed-point systems}
17 \label{subsec:block11-marked-systems}
18 
19 Let
20 \[
21  A_m:X^m\longrightarrow X,
22  \qquad 2\le m\le D,
23 \]
24 be bounded symmetric multilinear maps, and let $s\in X$.  Consider the marked equation
25 \[
26  \boxed{
27  u=s+\Phi(u),
28  \qquad
29  \Phi(u)=\sum_{m=2}^{D}A_m[u,\ldots,u].}
30  \tag{11.1}
31 \]
32 The word \emph{marked} means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation.  The fixed point $u$ alone is not assumed to reconstruct those data.
33 
34 Put
35 \[
36  a=\lVert s\rVert,
37  \qquad
38  b_m=\lVert A_m\rVert,
39  \qquad
40  \varphi(t)=\sum_{m=2}^{D}b_mt^m.
41 \]
42 For a radius $R>0$, define
43 \[
44  \kappa_R=\varphi'(R)
45  =\sum_{m=2}^{D}m b_mR^{m-1}.
46 \]
47 
48 \begin{theorem}[Selected polynomial fixed point and tree expansion]
49 \label{thm:block11-polynomial-fixed-point}
50 Assume
51 \[
52  a+\varphi(R)\le R,
53  \qquad
54  \kappa_R<1.
55 \]
56 Then equation \textup{(11.1)} has a unique solution in the closed ball $\overline B_R$.  Moreover, that solution is the absolutely convergent sum
57 \[
58  \boxed{
59  u=\sum_{T\in\mathcal T_{\mathcal A}}\operatorname{ev}(T),}
60  \tag{11.2}
61 \]
62 where $\mathcal A=\{m:A_m\neq0\}$, the leaves of $T$ evaluate to $s$, and an internal vertex of arity $m$ evaluates by $A_m$ applied to the ordered values of its children.
63 \end{theorem}
64 
65 \begin{proof}
66 The map
67 \[
68  \Psi(x)=s+\Phi(x)
69 \]
70 preserves $\overline B_R$ because
71 \[
72  \lVert\Psi(x)\rVert
73  \le a+\sum_{m=2}^{D}b_mR^m
74  \le R.
75 \]
76 For $x,y\in\overline B_R$, multilinearity gives
77 \[
78  A_m[x^m]-A_m[y^m]
79  =
80  \sum_{r=1}^{m}
81  A_m[y^{r-1},x-y,x^{m-r}],
82 \]
83 so
84 \[
85  \lVert\Psi(x)-\Psi(y)\rVert
86  \le
87  \kappa_R\lVert x-y\rVert.
88 \]
89 The contraction theorem gives the unique fixed point.
90 
91 Let $u^{(0)}=0$ and $u^{(h+1)}=\Psi(u^{(h)})$.  Expanding recursively shows that $u^{(h)}$ is the sum of the evaluations of all allowed plane rooted trees of height at most $h-1$.  If $r_0=0$ and
92 \[
93  r_{h+1}=a+\varphi(r_h),
94 \]
95 then the sum of the norms of those tree evaluations is at most $r_h$.  The scalar iterates increase and remain bounded by $R$, hence converge.  Monotone convergence of the nonnegative norm majorant proves absolute convergence of the full tree sum.  Regrouping trees by the arity of the root gives equation \textup{(11.1)}.
96 \end{proof}
97 
98 \begin{remark}[Formal filtered version]
99 If $X$ is replaced by a complete filtered module, $s$ begins in positive filtration, and every $A_m$ raises total filtration, then the same tree expansion is coefficientwise finite at each filtration depth.  It therefore defines the unique formal solution without a Banach norm.  The analytic and formal doctrines are related but are not silently identified.
100 \end{remark}
101 
102 \begin{boundary}[The selected object is a marked system]
103 Two different polynomial systems can have the same fixed point.  Conversely, a single polynomial system can have several fixed points outside the contraction ball.  The normalized object in this section is therefore
104 \[
105  (s,A_2,\ldots,A_D;\mathcal P;\text{certificate}),
106 \]
107 not an unmarked semantic vector $u$.
108 \end{boundary}
109 
110 \subsection{Packet jets, hypergraphs, and the marked nonlinear atlas}
111 \label{subsec:block11-packet-atlas}
112 
113 For $i,j_1,\ldots,j_m\in I$, define the resolved multilinear block
114 \[
115  A_{m;i\mid j_1,\ldots,j_m}
116  =
117  P_iA_m(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
118 \]
119 Completeness of the packet family gives the exact reconstruction
120 \[
121  \boxed{
122  A_m
123  =
124  \sum_{i,j_1,\ldots,j_m}
125  A_{m;i\mid j_1,\ldots,j_m}.}
126  \tag{11.3}
127 \]
128 Write
129 \[
130  u_i=P_iu,
131  \qquad
132  s_i=P_is.
133 \]
134 Then equation \textup{(11.1)} is equivalent to the complete packet system
135 \[
136  \boxed{
137  u_i
138  =
139  s_i+
140  \sum_{m=2}^{D}
141  \sum_{j_1,\ldots,j_m}
142  A_{m;i\mid j_1,\ldots,j_m}
143  [u_{j_1},\ldots,u_{j_m}].}
144  \tag{11.4}
145 \]
146 
147 The active nonlinear transfer hypergraph has packet addresses as colors and one typed hyperedge
148 \[
149  (j_1,\ldots,j_m)\longrightarrow i
150 \]
151 for every nonzero block $A_{m;i\mid j_1,\ldots,j_m}$.  Rooted substitution of these hyperedges is precisely the colored symmetric operad already present in the nonlinear core sources \cite{RT,NLO}.
152 
153 \begin{theorem}[Normalized atlas of a marked polynomial system]
154 \label{thm:block11-marked-atlas}
155 Fix a finite packet system and a certificate satisfying the hypotheses of \cref{thm:block11-polynomial-fixed-point}, or a declared complete filtration satisfying the formal version.  The following presentations determine one another exactly:
156 \[
157  \boxed{
158  \begin{aligned}
159  &\text{the marked polynomial system }(s,A_2,\ldots,A_D),\\
160  &\text{the complete family of packet blocks }A_{m;i\mid j_1,\ldots,j_m},\\
161  &\text{the colored operadic tree grammar with its actual vertex operations},\\
162  &\text{the exact packet fixed-point system \textup{(11.4)}}.
163  \end{aligned}}
164 \]
165 Each chart reconstructs the same selected fixed point.  With literal normalized data and equality morphisms, every semantic fiber is contractible.
166 \end{theorem}
167 
168 \begin{proof}
169 Equation \textup{(11.3)} reconstructs each multilinear operation from its complete block family.  The active hypergraph together with the actual block maps is simply another indexing of the same data.  Rooted operadic substitution generates exactly the tree evaluations of \cref{thm:block11-polynomial-fixed-point}.  Applying the output projections to the fixed-point equation gives \textup{(11.4)}, and synthesis recovers the original equation.  Once the packet system, block maps, source term, and certificate are normalized literally, an automorphism fixing the semantic object and all marked data is the identity.
170 \end{proof}
171 
172 \begin{figure}[htbp]
173 \centering
174 \includegraphics[width=.98\textwidth]{nonlinear_marked_atlas.pdf}
175 \caption{The marked system, its complete packet jets, its operation-labeled tree grammar, and its exact packet equations are reversible charts.  The fixed point by itself is directional because it forgets the generating system.}
176 \label{fig:block11-marked-atlas}
177 \end{figure}
178 
179 \subsection{Exact diagonal packet Fa\`a di Bruno compression}
180 \label{subsec:block11-faa-compression}
181 
182 The packet Fa\`a di Bruno theorem from the nonlinear source expands an $n$th derivative over all set partitions and all intermediate packet labels \cite{P3,RT,NLO}.  When all directions are equal, the symmetric derivative tensors make many labeled terms identical.  They can be grouped exactly by block-size and packet-color multiplicities.
183 
184 Let $F:Y\to Z$ and $G:X\to Y$ be $n$ times differentiable.  Fix an input packet $\beta$, an output packet $\delta$, a point $x$, and a direction $h\in P_\beta X$.  For every intermediate packet $\gamma$ and every $r\ge1$, put
185 \[
186  g_{r,\gamma}
187  =
188  Q_\gamma D^rG(x)[h,\ldots,h].
189 \]
190 A colored partition profile is a finite family
191 \[
192  \mu=(\mu_{r,\gamma})_{r\ge1,\,\gamma\in\Gamma}
193 \]
194 of nonnegative integers satisfying
195 \[
196  \sum_{r,\gamma}r\mu_{r,\gamma}=n.
197 \]
198 Let
199 \[
200  k(\mu)=\sum_{r,\gamma}\mu_{r,\gamma}.
201 \]
202 
203 \begin{theorem}[Compressed diagonal packet Fa\`a di Bruno formula]
204 \label{thm:block11-compressed-faa}
205 With the notation above,
206 \[
207 \boxed{
208 \begin{aligned}
209 &P_\delta D^n(F\circ G)(x)[h,\ldots,h]\\
210 &=\sum_{\mu}
211 \frac{n!}
212 {\displaystyle\prod_{r,\gamma}
213  (r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!}
214 \,P_\delta D^{k(\mu)}F(Gx)
215 \bigl[
216  (g_{r,\gamma})^{\mu_{r,\gamma}}
217 \bigr]_{r,\gamma}.
218 \end{aligned}}
219 \tag{11.5}
220 \]
221 Here the last line means that $g_{r,\gamma}$ is inserted $\mu_{r,\gamma}$ times; symmetry makes the insertion order irrelevant.
222 \end{theorem}
223 
224 \begin{proof}
225 Start with the packet Fa\`a di Bruno sum over a set partition $\pi$ of $\{1,\ldots,n\}$ and a packet color attached to every block.  Because every input direction is $h$, the value of one term depends only on the number $\mu_{r,\gamma}$ of blocks of size $r$ carrying color $\gamma$.  The symmetric group acts transitively on all colored set partitions with one fixed profile.  The stabilizer permutes the $r$ entries inside each block and permutes the indistinguishable blocks of a fixed type $(r,\gamma)$.  Its order is
226 \[
227  \prod_{r,\gamma}(r!)^{\mu_{r,\gamma}}\mu_{r,\gamma}!,
228 \]
229 so the orbit has the coefficient printed in \textup{(11.5)}.  Summing one representative per orbit gives the result.
230 \end{proof}
231 
232 \begin{corollary}[Colored integer-partition term count]
233 \label{cor:block11-compressed-count}
234 If $Y$ has $M$ intermediate packets, the number of compressed diagonal profiles at order $n$ is
235 \[
236  \boxed{
237  C_{n,M}
238  =[x^n]\prod_{r\ge1}(1-x^r)^{-M}.}
239  \tag{11.6}
240 \]
241 Before compression, the number of packet-labeled set-partition terms is
242 \[
243  N_{n,M}=\sum_{k=1}^{n}S(n,k)M^k.
244 \]
245 \end{corollary}
246 
247 \begin{proof}
248 For each block size $r$, the multiplicities among $M$ colors contribute
249 \[
250  (1-x^r)^{-M}.
251 \]
252 Multiplying over all $r$ and taking the coefficient of $x^n$ counts exactly the profiles satisfying the weighted-size constraint.  The raw count is the existing packet term-count formula, grouped by the number of blocks.
253 \end{proof}
254 
255 \input{generated/block11_faa_counts_table.tex}
256 
257 \begin{figure}[htbp]
258 \centering
259 \includegraphics[width=.88\textwidth]{faa_di_bruno_compression.pdf}
260 \caption{For repeated directions, packet-labeled set partitions collapse to colored integer-partition profiles.  The compression is exact, not asymptotic.}
261 \label{fig:block11-faa-compression}
262 \end{figure}
263 
264 \begin{boundary}[Scope of the compression theorem]
265 Equation \textup{(11.5)} closes the repeated-direction, single-input-packet case.  For fully mixed directions, blocks carry the types of their actual input subsets; the relevant orbit type is finer and can approach the full set-partition expansion.  No universal mixed-direction compression ratio is asserted.
266 \end{boundary}
267 
268 \subsection{The cyclic nonlinear core of a packet system}
269 \label{subsec:block11-dependency-core}
270 
271 For the packet system \textup{(11.4)}, define the directed dependency graph $G_\Phi$ on $I$ by placing an edge
272 \[
273  j\longrightarrow i
274 \]
275 when some active block contributing to the equation for $u_i$ contains $u_j$ as an input.  Let
276 \[
277  \operatorname{SCC}(G_\Phi)
278 \]
279 be its strongly connected components.  A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.
280 
281 \begin{definition}[Address-preserving cyclic dimension]
282 The cyclic packet dimension of the marked system is
283 \[
284  \boxed{
285  \operatorname{cdim}_{\mathcal P}(\Phi)
286  =
287  \sum_{C\text{ cyclic SCC}}|C|.}
288  \tag{11.7}
289 \]
290 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.
291 \end{definition}
292 
293 \begin{theorem}[Exact SCC reduction]
294 \label{thm:block11-scc-reduction}
295 Order the SCC condensation graph topologically.  Every acyclic singleton component can be substituted exactly once its predecessors have been determined.  Repeating this elimination gives a solution-preserving bijection between:
296 \begin{enumerate}
297 \item solutions of the full packet system, and
298 \item solutions of the subsystem supported on the cyclic SCCs, followed by deterministic reconstruction along the condensation graph.
299 \end{enumerate}
300 Consequently, the coupled nonlinear core has exactly $\operatorname{cdim}_{\mathcal P}(\Phi)$ packet variables.
301 \end{theorem}
302 
303 \begin{proof}
304 A singleton SCC without a self-loop has an equation in which its own variable does not occur.  Once all predecessor components are known, that equation computes the variable directly.  Substituting this expression into every successor equation is an algebraic identity and gives a bijection of solution sets.  The condensation graph is acyclic, so repeated topological elimination terminates.  Components containing a directed cycle cannot be removed by this one-pass address-preserving substitution because each contains a variable depending, directly or transitively, on itself.
305 \end{proof}
306 
307 \begin{figure}[htbp]
308 \centering
309 \includegraphics[width=.94\textwidth]{nonlinear_dependency_core.pdf}
310 \caption{Condensation separates exact one-pass reconstruction from the cyclic nonlinear core.  The resulting dimension is relative to the declared packet coordinates.}
311 \label{fig:block11-dependency-core}
312 \end{figure}
313 
314 \begin{boundary}[Not a universal nonlinear state minimum]
315 The quantity \(\operatorname{cdim}_{\mathcal P}\) is invariant under relabeling of the declared packet system and exact elimination within its dependency grammar.  A nonlinear change of variables, hidden-variable introduction, algebraic elimination, or another packet family can change it.  It is therefore a certified address-preserving system dimension, not a universal minimum over all nonlinear presentations.
316 \end{boundary}
317 
318 \subsection{A finite residue signature for simply generated trees}
319 \label{subsec:block11-tree-signature}
320 
321 Consider the scalar marked equation
322 \[
323  \boxed{
324  u(z)=z+\Phi(u(z)),
325  \qquad
326  \Phi(y)=\sum_{m\in\mathcal A}w_my^m,}
327  \tag{11.8}
328 \]
329 where $\mathcal A\subseteq\{2,3,\ldots\}$ is finite and every $w_m$ is a positive integer.  Its unique formal solution belongs to $z\mathbb Z[[z]]$ and counts plane rooted trees whose internal arities lie in $\mathcal A$, weighted by the $w_m$.
330 
331 Define the arity period
332 \[
333  \boxed{
334  d_{\mathcal A}
335  =
336  \gcd\{m-1:m\in\mathcal A\}.}
337  \tag{11.9}
338 \]
339 Every tree with internal vertices $v$ has leaf count
340 \[
341  \ell(T)=1+\sum_{v}(\deg(v)-1).
342 \]
343 
344 \begin{theorem}[Finite tree-address signature]
345 \label{thm:block11-tree-address}
346 Fix a packet width $N$ and put
347 \[
348  g_N=\gcd(N,d_{\mathcal A}).
349 \]
350 For the arithmetic packets
351 \[
352  U_j(x)=\sum_{r\ge0}[z^{Nr+j}]u(z)\,x^r,
353 \]
354 the following are equivalent:
355 \begin{enumerate}
356 \item $U_j\neq0$;
357 \item $j-1$ belongs to the subgroup of $\mathbb Z/N\mathbb Z$ generated by $\{m-1:m\in\mathcal A\}$;
358 \item
359 \[
360  \boxed{j\equiv1\pmod{g_N}.}
361 \]
362 \end{enumerate}
363 Hence the exact number of active packet addresses is
364 \[
365  \boxed{
366  \#\operatorname{Supp}_N(u)=\frac{N}{g_N}.}
367  \tag{11.10}
368 \]
369 \end{theorem}
370 
371 \begin{proof}
372 Replacing one leaf by an $m$-ary vertex increases the leaf count by $m-1$.  Starting from the one-leaf tree, every nonnegative sum of allowed increments is realized by a sequence of leaf replacements.  Positivity of the weights prevents cancellation, so the coefficient support is exactly
373 \[
374  1+\left\langle m-1:m\in\mathcal A\right\rangle_{\mathbb N}.
375 \]
376 Modulo $N$, the monoid generated by the increments equals the subgroup they generate because every element of a finite cyclic subgroup has a nonnegative representative.  That subgroup is $g_N\mathbb Z/N\mathbb Z$, which proves the congruence and the count.
377 \end{proof}
378 
379 \begin{corollary}[Finite bottom-up residue automaton]
380 Label every leaf by $1\in\mathbb Z/N\mathbb Z$ and every $m$-ary vertex by the sum of its child labels.  The reachable state set is exactly
381 \[
382  \{j:j\equiv1\pmod{g_N}\}.
383 \]
384 Thus the finite pair
385 \[
386  \boxed{
387  \mathfrak T_N(\mathcal A)
388  =
389  (d_{\mathcal A},g_N,N/g_N)}
390  \tag{11.11}
391 \]
392 classifies packet support for the marked simply generated family.
393 \end{corollary}
394 
395 \begin{figure}[htbp]
396 \centering
397 \includegraphics[width=.72\textwidth]{tree_residue_signature.pdf}
398 \caption{For arities three and five, the increment gcd is two.  At width twelve, exactly the six addresses congruent to one modulo two are active.}
399 \label{fig:block11-tree-residue}
400 \end{figure}
401 
402 \subsection{Dominant singularity orbits}
403 \label{subsec:block11-tree-singularity}
404 
405 Assume now that the coefficients $w_m$ are positive real numbers.  The classical smooth implicit-function schema for simply generated trees applies when there is a positive solution $\tau$ of
406 \[
407  \Phi'(\tau)=1
408 \]
409 and
410 \[
411  \rho=\tau-\Phi(\tau)>0.
412 \]
413 Under its standard nondegeneracy hypotheses, the reduced aperiodic series has a square-root singularity at its positive radius \cite{FlajoletSedgewick2009}.  The arity period restores the complete source orbit.
414 
415 \begin{theorem}[Complete dominant-orbit signature]
416 \label{thm:block11-dominant-orbit}
417 Under the smooth implicit schema, the dominant singularities of $u(z)$ form
418 \[
419  \boxed{
420  \rho\,\mu_{d_{\mathcal A}},}
421  \tag{11.12}
422 \]
423 where $\mu_d$ is the group of $d$th roots of unity.  After packetization with $x=z^N$, the union of dominant singularities of the complete packet tuple is
424 \[
425  \boxed{
426  \rho^N\,\mu_{h_N},
427  \qquad
428  h_N=\frac{d_{\mathcal A}}{\gcd(N,d_{\mathcal A})}.}
429  \tag{11.13}
430 \]
431 Therefore the same finite arity signature determines both the number of active packet addresses and the size of the complete dominant-singularity orbit.
432 \end{theorem}
433 
434 \begin{proof}
435 By \cref{thm:block11-tree-address}, the coefficient support is contained in $1+d_{\mathcal A}\mathbb N$ and the reduced exponent set is aperiodic by maximality of the gcd.  Hence
436 \[
437  u(z)=zV(z^{d_{\mathcal A}})
438 \]
439 with $V$ aperiodic.  The smooth implicit schema gives one positive dominant singularity for $V$, and pulling back through $z^{d_{\mathcal A}}$ gives \textup{(11.12)}.  The map $z\mapsto z^N$ sends $\mu_{d_{\mathcal A}}$ onto a root group of cardinality
440 \[
441  d_{\mathcal A}/\gcd(N,d_{\mathcal A}).
442 \]
443 No packet can introduce a new dominant image outside this set.  Conversely, if every packet were analytic at one image, the exact synthesis
444 \[
445  u(z)=\sum_{j=0}^{N-1}z^jU_j(z^N)
446 \]
447 would make $u$ analytic at every source point above that image, contradicting \textup{(11.12)}.  Thus the union of packet singularities is exactly \textup{(11.13)}.
448 \end{proof}
449 
450 \begin{figure}[htbp]
451 \centering
452 \includegraphics[width=.94\textwidth]{tree_singularity_orbit.pdf}
453 \caption{Arithmetic packetization maps the source root-of-unity singularity orbit through $z\mapsto z^N$.  Distinct source singularities can alias to one packet-variable singularity.}
454 \label{fig:block11-tree-singularity}
455 \end{figure}
456 
457 \begin{boundary}[Individual-packet cancellation]
458 The theorem identifies the union of dominant singularities of the complete packet tuple.  One individual packet can lose some orbit points through address cancellation.  The finite signature does not, by itself, determine every individual local amplitude or cancellation coefficient.
459 \end{boundary}
460 
461 \subsection{\texorpdfstring{The sharp full \(q\)-ary family}{The sharp full q-ary family}}
462 \label{subsec:block11-qary}
463 
464 Take
465 \[
466  \boxed{u=z+u^q,\qquad q\ge2.}
467  \tag{11.14}
468 \]
469 Lagrange inversion gives the exact coefficient formula
470 \[
471  \boxed{
472  [z^n]u
473  =
474  \begin{cases}
475  \displaystyle
476  \frac{1}{(q-1)m+1}\binom{qm}{m},
477  &n=1+(q-1)m,\\[9pt]
478  0,&n\not\equiv1\pmod{q-1}.
479  \end{cases}}
480  \tag{11.15}
481 \]
482 The nonzero coefficients are the Fuss--Catalan numbers of full plane $q$-ary trees.
483 
484 Write
485 \[
486  u(z)=\sum_{j=0}^{N-1}z^jU_j(z^N),
487  \qquad x=z^N.
488 \]
489 
490 \begin{theorem}[Exact $q$-ary packet phase theorem]
491 \label{thm:block11-qary-phase}
492 Put
493 \[
494  g=\gcd(N,q-1),
495  \qquad
496  h=\frac{q-1}{g}.
497 \]
498 Then:
499 \begin{enumerate}
500 \item the active packet addresses are exactly
501 \[
502  \boxed{j\equiv1\pmod g;}
503 \]
504 \item the complete packet system is
505 \[
506 \boxed{
507  U_j(x)
508  =
509  \delta_{j,1}
510  +
511  \sum_{\substack{0\le r_1,\ldots,r_q<N\\
512  r_1+\cdots+r_q\equiv j\ (\mathrm{mod}\ N)}}
513  x^{(r_1+\cdots+r_q-j)/N}
514  \prod_{a=1}^{q}U_{r_a}(x);}
515  \tag{11.16}
516 \]
517 \item the smooth critical values are
518 \[
519  \tau=q^{-1/(q-1)},
520  \qquad
521  \rho=(q-1)q^{-q/(q-1)};
522 \]
523 \item the source dominant orbit has $q-1$ points, while the complete packet orbit has exactly $h$ points.
524 \end{enumerate}
525 The phase is therefore determined by the finite tuple
526 \[
527  \boxed{(q,N,g,N/g,h).}
528  \tag{11.17}
529 \]
530 \end{theorem}
531 
532 \begin{proof}
533 Formula \textup{(11.15)} follows from Lagrange inversion applied to
534 \[
535  z=u(1-u^{q-1}).
536 \]
537 The support and active-address statements are \cref{thm:block11-tree-address} with $d_{\mathcal A}=q-1$.  Expanding $u^q$ in complete packets and collecting the exponent carry
538 \[
539  r_1+\cdots+r_q=j+Nc
540 \]
541 gives \textup{(11.16)}.  The critical equations
542 \[
543  q\tau^{q-1}=1,
544  \qquad
545  \rho=\tau-\tau^q
546 \]
547 give the printed values, and \cref{thm:block11-dominant-orbit} gives the orbit count.
548 \end{proof}
549 
550 \input{generated/block11_qary_phase_table.tex}
551 
552 \subsection{Verification, status, and exact boundary}
553 \label{subsec:block11-validation}
554 
555 The Nonlinear tree atlas verifier performs four independent exact audits.  It enumerates colored integer-partition profiles and compares their orbit-weighted Fa\`a di Bruno values with the full set-partition expansion.  It constructs exact fixed-point prefixes for marked polynomial tree families and checks the finite arity-gcd support theorem over many packet widths.  It verifies Fuss--Catalan formulas and the complete carry-form packet equations for the full $q$-ary family.  Finally, it checks dependency-graph reductions against exhaustive bounded graph instances.
556 
557 \input{generated/block11_validation_table.tex}
558 
559 \begin{resultbox}
560 \textbf{Section result.}
561 Marked polynomial fixed-point systems admit exact packet, hypergraph, operadic-tree, and packet-equation charts.  The repeated-direction packet Fa\`a di Bruno expansion compresses exactly from colored set partitions to colored integer-partition profiles.  The packet dependency graph has an exact SCC reduction whose cyclic vertices form the address-preserving nonlinear core.  For positive simply generated tree systems, the arity gcd is a finite signature determining every arithmetic packet-support pattern and the complete dominant-singularity orbit.  The full $q$-ary family supplies a closed Fuss--Catalan phase theorem and exact packet equations.
562 \end{resultbox}
563 
564 \begin{boundarybox}
565 \textbf{Exact boundary of Nonlinear tree atlas.}
566 The atlas classifies marked polynomial systems, not semantic fixed points independently of their generators.  The compression theorem treats repeated directions; arbitrary mixed-direction jets can require a finer orbit type.  The SCC dimension is relative to the declared packet coordinates.  The tree signature classifies support and the complete dominant-singularity orbit for the marked simply generated family.  It does not classify algebraic degree, universal minimal system dimension, every individual packet amplitude, arbitrary sign-canceling weights, noncontractive global dynamics, or all nonlinear operadic systems.
567 \end{boundarybox}

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