block09_section.tex
Stratified Universality
block09_section.tex
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Included manuscript sections
1 \section{Holonomic Ore charts, multisection modules, and a Fuchsian packet theorem}
2 \label{sec:block09-holonomic}
3
4 Holonomic atlas develops the next native stratum in the declared work program. The general holonomic problem has two logically separate parts. First, a D-finite series and its P-recursive coefficient sequence require normalized operator charts that remember the finite initial data lost by tail recurrences. Second, arithmetic multisection preserves holonomicity but can change minimal order through root-orbit aliasing, resonance, and address-dependent cancellation. The first part is closed here on the ordinary-point locus. The second is organized in general and solved exactly for a nontrivial regular-singular family.
5
6 Throughout this section, $K$ is a characteristic-zero field. When local monodromy is discussed, $K$ is embedded in $\mathbb C$ and all local exponents are taken over an algebraic closure.
7
8 \subsection{Ore annihilators and normalized holonomic charts}
9 \label{subsec:block09-ore}
10
11 Let
12 \[
13 \mathcal D_K=K(z)\langle D\rangle,
14 \qquad
15 Dq=qD+q',
16 \]
17 be the differential Ore algebra, and let
18 \[
19 \mathcal S_K=K(n)\langle S\rangle,
20 \qquad
21 Sq(n)=q(n+1)S,
22 \]
23 be the forward-shift Ore algebra. Both are Euclidean skew-polynomial rings in their operator variable, so their nonzero left ideals have unique monic generators \cite{Ore1933}.
24
25 \begin{definition}[Minimal differential annihilator]
26 For a nonzero D-finite series $F\in K[[z]]$, define
27 \[
28 \operatorname{Ann}_{\mathcal D}(F)
29 =
30 \{L\in\mathcal D_K:LF=0\}.
31 \]
32 Its unique monic generator is denoted
33 \[
34 \boxed{L_F.}
35 \]
36 The differential order
37 \[
38 r_D(F)=\operatorname{ord}_D L_F
39 \]
40 is the minimal scalar differential order of $F$ over $K(z)$.
41 \end{definition}
42
43 \begin{definition}[Minimal tail recurrence]
44 For a sequence $a=(a_n)_{n\geq0}$, define its tail annihilator by
45 \[
46 \operatorname{Ann}_{\rm tail}(a)
47 =
48 \{R\in\mathcal S_K:(Ra)_n=0\text{ for all sufficiently large }n\}.
49 \]
50 For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted
51 \[
52 \boxed{R_a.}
53 \]
54 If the sequence is eventually zero, the tail generator is $1$ and has order zero.
55 \end{definition}
56
57 Write
58 \[
59 R_a
60 =
61 S^s+\sum_{j=0}^{s-1}r_j(n)S^j.
62 \]
63 Let $\nu_a$ be the least index from which $R_a$ annihilates the sequence, and define $N_a$ to be the least integer $N\geq\nu_a$ larger than every nonnegative pole of the rational functions $r_j(n)$. Then
64 \[
65 \mathfrak R(a)
66 =
67 \bigl(R_a,N_a,a_0,\ldots,a_{N_a+s-1}\bigr)
68 \]
69 is a canonical complete recurrence chart. The operator controls the tail, while the finite prefix crosses every singular index and fixes the exact sequence rather than only its eventual equivalence class.
70
71 \begin{proposition}[Derivative-module rank]
72 \label{prop:block09-derivative-rank}
73 For a nonzero D-finite series $F$,
74 \[
75 \boxed{
76 r_D(F)
77 =
78 \dim_{K(z)}\operatorname{span}_{K(z)}
79 \{F,DF,D^2F,\ldots\}.}
80 \]
81 \end{proposition}
82
83 \begin{proof}
84 A differential operator of order $r$ gives a linear dependence among $F,DF,\ldots,D^rF$. Minimality of $L_F$ says that no dependence exists among the first $r$ derivative vectors. Therefore they form a basis of the derivative span.
85 \end{proof}
86
87 \subsection{The exact coefficient bridge}
88 \label{subsec:block09-coefficient-bridge}
89
90 The classical equivalence between P-recursive sequences and D-finite ordinary generating functions is effective in both directions \cite{Stanley1980}. The following formulas make the finite residual explicit.
91
92 Let
93 \[
94 F(z)=\sum_{n\geq0}a_nz^n
95 \]
96 and
97 \[
98 L
99 =
100 \sum_{j=0}^{r}q_j(z)D^j,
101 \qquad
102 q_j(z)=\sum_{m=0}^{d_j}q_{j,m}z^m.
103 \]
104
105 \begin{theorem}[Differential-to-recurrence coefficient formula]
106 \label{thm:block09-diff-to-rec}
107 For every $n\geq0$,
108 \[
109 \boxed{
110 [z^n]LF
111 =
112 \sum_{j=0}^{r}
113 \sum_{m=0}^{\min(d_j,n)}
114 q_{j,m}
115 (n-m+j)^{\underline{j}}
116 a_{n-m+j},}
117 \]
118 where
119 \[
120 x^{\underline j}=x(x-1)\cdots(x-j+1).
121 \]
122 Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.
123 \end{theorem}
124
125 \begin{proof}
126 The coefficient of $z^{n-m}$ in $D^jF$ is
127 \[
128 (n-m+j)^{\underline j}a_{n-m+j}.
129 \]
130 Multiplication by $q_{j,m}z^m$ shifts that contribution to degree $n$. Summing over $j$ and $m$ gives the formula.
131 \end{proof}
132
133 Conversely, suppose
134 \[
135 \sum_{s=0}^{q}p_s(n)a_{n+s}=0
136 \qquad(n\geq N),
137 \]
138 where $p_s\in K[n]$, and put
139 \[
140 \theta=zD.
141 \]
142
143 \begin{theorem}[Recurrence residual as an Euler-operator polynomial]
144 \label{thm:block09-rec-to-diff}
145 Define
146 \[
147 \boxed{
148 Q
149 =
150 \sum_{s=0}^{q}
151 z^{q-s}p_s(\theta-s).}
152 \]
153 Then
154 \[
155 \boxed{
156 [z^{n+q}]QF
157 =
158 \sum_{s=0}^{q}p_s(n)a_{n+s}.}
159 \]
160 Hence $QF$ is a polynomial of degree less than $N+q$, and
161 \[
162 \boxed{D^{N+q}QF=0.}
163 \]
164 \end{theorem}
165
166 \begin{proof}
167 The operator $p_s(\theta-s)$ multiplies the coefficient of $z^k$ by $p_s(k-s)$. After multiplication by $z^{q-s}$, the coefficient at $z^{n+q}$ is $p_s(n)a_{n+s}$. The recurrence kills every coefficient from degree $N+q$ onward. Differentiating more times than the residual degree gives a homogeneous annihilator.
168 \end{proof}
169
170 The polynomial residual is not noise. It is the exact finite-prefix certificate that distinguishes a recurrence valid only in the tail from an identity in the full formal-series chart.
171
172 \subsection{The ordinary-point holonomic atlas}
173 \label{subsec:block09-ordinary-atlas}
174
175 \begin{definition}[Ordinary-point locus]
176 A nonzero D-finite series $F\in K[[z]]$ belongs to
177 \[
178 \mathbf{Hol}^{\rm ord}_K
179 \]
180 when its monic minimal operator can be written
181 \[
182 L_F
183 =
184 D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
185 \]
186 with every $\ell_j$ regular at $z=0$.
187 \end{definition}
188
189 The normalized jet is
190 \[
191 J_0(F)
192 =
193 \bigl(F(0),F'(0),\ldots,F^{(r-1)}(0)\bigr).
194 \]
195
196 \begin{theorem}[Ordinary-point reconstruction]
197 \label{thm:block09-ordinary-reconstruction}
198 Let
199 \[
200 L=D^r+\sum_{j=0}^{r-1}\ell_j(z)D^j
201 \]
202 with all $\ell_j$ regular at zero. For every jet
203 \[
204 (y_0,\ldots,y_{r-1})\in K^r
205 \]
206 there is a unique formal series $F\in K[[z]]$ satisfying
207 \[
208 LF=0,
209 \qquad
210 F^{(j)}(0)=y_j.
211 \]
212 \end{theorem}
213
214 \begin{proof}
215 Write $F(z)=\sum c_nz^n$. At coefficient degree $n$, the monic term $D^rF$ contributes
216 \[
217 \frac{(n+r)!}{n!}c_{n+r},
218 \]
219 while every lower derivative involves only $c_0,\ldots,c_{n+r-1}$. Characteristic zero makes the displayed coefficient invertible. The equation therefore determines $c_{n+r}$ recursively from the jet.
220 \end{proof}
221
222 \begin{theorem}[Normalized ordinary holonomic atlas]
223 \label{thm:block09-ordinary-atlas}
224 On the ordinary-point locus, the following normalized presentations determine one another exactly:
225 \[
226 \boxed{
227 \begin{aligned}
228 &F\in K[[z]],\\
229 &(L_F,J_0(F)),\\
230 &\operatorname{span}_{K(z)}\{D^jF:j\geq0\}
231 \text{ with cyclic vector }F,\\
232 &(R_a,N_a,a_0,\ldots,a_{N_a+s-1}),
233 \qquad a_n=[z^n]F.
234 \end{aligned}}
235 \]
236 With equality morphisms, these charts form equivalent normalized groupoids, and every fiber over a fixed semantic series is contractible.
237 \end{theorem}
238
239 \begin{proof}
240 The series gives its unique minimal Ore operators and finite data. The derivative module recovers $L_F$ as the first monic dependence among consecutive derivatives. The operator and jet recover $F$ by \cref{thm:block09-ordinary-reconstruction}. The recurrence chart reconstructs every coefficient after its singular prefix. The two explicit coefficient formulas transport between the differential and recurrence channels. Minimal monic normalization removes coordinate automorphisms, so each normalized fiber has only the identity.
241 \end{proof}
242
243 \begin{figure}[htbp]
244 \centering
245 \includegraphics[width=.97\textwidth]{holonomic_ore_atlas.pdf}
246 \caption{The ordinary holonomic atlas keeps the semantic series, minimal differential operator and jet, cyclic derivative module, and minimal tail recurrence with singular prefix as separate normalized charts.}
247 \label{fig:block09-ore-atlas}
248 \end{figure}
249
250 \subsection{General arithmetic multisection}
251 \label{subsec:block09-general-multisection}
252
253 For $N\geq1$ and $0\leq j<N$, define
254 \[
255 \mathfrak M_{N,j}F(x)
256 =
257 \sum_{m\geq0}a_{Nm+j}x^m.
258 \]
259 If $\omega_N$ is a primitive $N$th root of unity and $x=t^N$, then
260 \[
261 \boxed{
262 t^j\mathfrak M_{N,j}F(t^N)
263 =
264 \frac1N
265 \sum_{q=0}^{N-1}
266 \omega_N^{-jq}F(\omega_N^qt).}
267 \]
268 This is the character projector for the cyclic rotation action.
269
270 \begin{theorem}[Holonomic multisection closure]
271 \label{thm:block09-multisection-closure}
272 If $F$ is D-finite, then every $\mathfrak M_{N,j}F$ is D-finite. Equivalently, arithmetic sections of P-recursive sequences are P-recursive.
273 \end{theorem}
274
275 \begin{proof}
276 The root-filter formula uses finitely many algebraic substitutions and a finite linear combination. D-finite series are closed under those operations. The resulting section has coefficients in the original base field, so the operator descends.
277 \end{proof}
278
279 Closure does not determine the smallest operator. A separate exact state construction gives a useful bound.
280
281 \begin{theorem}[Nonautonomous state jump]
282 \label{thm:block09-state-jump}
283 Suppose a P-recursive sequence is represented, beyond its singular prefix, by
284 \[
285 v_{n+1}=C(n)v_n,
286 \qquad
287 a_n=\lambda(n)v_n,
288 \]
289 where $v_n\in K^s$ and $C(n)$ is rational. Then the $j$th $N$-section has the jump system
290 \[
291 \boxed{
292 w_{m+1}=J_{N,j}(m)w_m,}
293 \]
294 where
295 \[
296 \boxed{
297 J_{N,j}(m)
298 =
299 C(Nm+j+N-1)\cdots C(Nm+j).}
300 \]
301 After denominator clearing, every scalar output has a polynomial recurrence of order at most $s$.
302 \end{theorem}
303
304 \begin{proof}
305 Compose the $N$ consecutive one-step transfers. The $s+1$ consecutive scalar output rows are linearly dependent over $K(m)$, producing a recurrence of order at most $s$.
306 \end{proof}
307
308 The root filter also gives a construction-independent order ledger before the algebraic descent from $t$ to $x=t^N$. Work over a finite extension $E/K$ containing $\omega_N$. Put
309 \[
310 F_q(t)=F(\omega_N^qt),
311 \qquad
312 G_{N,j}(t)=\frac1N\sum_{q=0}^{N-1}\omega_N^{-jq}F_q(t).
313 \]
314 Let $L_q\in E(t)\langle D_t\rangle$ be the monic minimal annihilator of $F_q$. Rotation is an Ore-algebra automorphism, so every $L_q$ has order $r=\operatorname{ord}L_F$. Let
315 \[
316 L_N^{\rm orb}=\operatorname{LCLM}(L_0,\ldots,L_{N-1}),
317 \qquad
318 r_N^{\rm orb}=\operatorname{ord}L_N^{\rm orb},
319 \]
320 and, when $G_{N,j}\neq0$, let $L_{N,j}^{\rm char}$ be its monic minimal annihilator with order $r_{N,j}^{\rm char}$. For a zero character packet put $r_{N,j}^{\rm char}=0$.
321
322 \begin{definition}[Root-filter alias and address-cancellation defects]
323 \label{def:block09-general-defects}
324 Define
325 \[
326 \mathfrak a_N(F)=Nr-r_N^{\rm orb},
327 \qquad
328 \mathfrak c_{N,j}(F)=r_N^{\rm orb}-r_{N,j}^{\rm char}.
329 \]
330 The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the $j$th character projector after the rotated orbit has been assembled.
331 \end{definition}
332
333 \begin{theorem}[Exact two-stage root-filter defect identity]
334 \label{thm:block09-general-defect-identity}
335 The defects in \cref{def:block09-general-defects} are nonnegative and satisfy
336 \[
337 \boxed{
338 Nr-r_{N,j}^{\rm char}
339 =
340 \mathfrak a_N(F)+\mathfrak c_{N,j}(F).}
341 \]
342 Thus orbit aliasing occurs before address selection, while address cancellation occurs after the Fourier character has been applied.
343 \end{theorem}
344
345 \begin{proof}
346 The least common left multiple has order at most the sum of the orders of its inputs, hence $r_N^{\rm orb}\leq Nr$. It annihilates every rotated series and therefore annihilates their character sum $G_{N,j}$. By minimality, $L_N^{\rm orb}$ belongs to the principal left annihilator ideal generated by $L_{N,j}^{\rm char}$ whenever the character sum is nonzero, so $r_{N,j}^{\rm char}\leq r_N^{\rm orb}$. The zero case is immediate. The displayed identity is the telescoping decomposition
347 \[
348 Nr-r_{N,j}^{\rm char}
349 =
350 (Nr-r_N^{\rm orb})+(r_N^{\rm orb}-r_{N,j}^{\rm char}).
351 \]
352 \end{proof}
353
354 \begin{boundarybox}
355 \textbf{Root-filter order versus packet order.}
356 The preceding theorem is exact in the rotated $t$-coordinate over $E(t)$. The arithmetic packet is the series $\mathfrak M_{N,j}F(x)$ over the base field, related by
357 \[
358 G_{N,j}(t)=t^j\mathfrak M_{N,j}F(t^N).
359 \]
360 Algebraic pullback, base extension, gauge multiplication, and descent may change the minimal scalar operator visible in the $x$-chart. That comparison is a separately typed descent problem. The exact binomial--Fuchsian family below computes both sides and therefore resolves this additional step in a nontrivial model case.
361 \end{boundarybox}
362
363 Let $Q^{\rm cl}_{N,j}$ be an annihilator produced by a declared closure or elimination pipeline, and let $L_{N,j}$ be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by $L_{N,j}$,
364 \[
365 Q^{\rm cl}_{N,j}=A_{N,j}L_{N,j}
366 \]
367 for some Ore operator $A_{N,j}$. The nonnegative quantity
368 \[
369 \delta^{\rm cl}_{N,j}
370 =
371 \operatorname{ord}Q^{\rm cl}_{N,j}
372 -
373 \operatorname{ord}L_{N,j}
374 \]
375 measures cancellation relative to that construction. It is not intrinsic until the construction has been fixed.
376
377 \subsection{Operator singularities and local exponent packets}
378 \label{subsec:block09-local-exponents}
379
380 Multiply $L_F$ by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative
381 \[
382 \widetilde L_F
383 =
384 \sum_{j=0}^{r}q_j(z)D^j.
385 \]
386 Its operator singular support is
387 \[
388 \Sigma_{\rm op}(F)
389 =
390 \{q_r(z)=0\}\cup\{\infty\}.
391 \]
392 Apparent singularities remain part of this operator coordinate until an explicit desingularization has been performed.
393
394 Let $\alpha$ be a finite point and put
395 \[
396 v_j=v_\alpha(q_j),
397 \qquad
398 \mu=\min_j(v_j-j).
399 \]
400 Write the first nonzero local coefficient of $q_j$ as $q_{j,v_j}$. The indicial polynomial is
401 \[
402 \boxed{
403 I_{\alpha,L}(\rho)
404 =
405 \sum_{v_j-j=\mu}
406 q_{j,v_j}\rho^{\underline j}.}
407 \]
408 At a regular singular point, its roots form the local exponent packet. Over $\mathbb C$, the corresponding local monodromy eigenvalues are
409 \[
410 e^{2\pi i\rho}.
411 \]
412 Integer differences between exponents signal resonance. Exponent eigenvalues alone do not determine logarithmic extension data or the full monodromy conjugacy class.
413
414 \begin{proposition}[Singularity orbit under multisection]
415 \label{prop:block09-singularity-orbit}
416 Assume $F$ is an analytic germ at zero with analytic continuation away from a finite set $\Sigma$. Then every finite nonzero analytic singularity of $\mathfrak M_{N,j}F$ lies in
417 \[
418 \boxed{
419 \Sigma^{[N]}
420 =
421 \{\alpha^N:\alpha\in\Sigma\}.}
422 \]
423 Distinct singularities in one fiber of $\alpha\mapsto\alpha^N$ are aliased by the multisection map. A character projection may remove some aliased local contributions.
424 \end{proposition}
425
426 \begin{proof}
427 Away from $\Sigma^{[N]}$ and zero, choose a local analytic branch $t=x^{1/N}$. Every rotated argument $\omega_N^qt$ avoids $\Sigma$, so the root-filter expression is analytic. A singularity can therefore occur only when one rotated preimage meets $\Sigma$.
428 \end{proof}
429
430 At nonzero regular singular points, the map $t\mapsto t^N$ is unramified. It preserves local exponents before character projection. At zero and infinity it is ramified, so exponent scaling must be recorded separately.
431
432 \subsection{An exact binomial--Fuchsian packet family}
433 \label{subsec:block09-binomial-family}
434
435 The general minimal multisection problem remains open. The family
436 \[
437 \boxed{
438 F_{\lambda,\alpha}(z)
439 =
440 (1-\alpha z)^{-\lambda}}
441 \]
442 provides a nontrivial exact model that contains irreducible Fuchsian packets, rational collapse, finite-support aliasing, and zero-address cancellation in one parameter space.
443
444 Its coefficients are
445 \[
446 a_n
447 =
448 \frac{(\lambda)_n}{n!}\alpha^n.
449 \]
450 For $0\leq j<N$, put
451 \[
452 G_{N,j}^{\lambda,\alpha}(x)
453 =
454 \sum_{m\geq0}
455 \frac{(\lambda)_{Nm+j}}{(Nm+j)!}
456 \alpha^{Nm+j}x^m.
457 \]
458 Define the upper and lower parameter packets
459 \[
460 A_{N,j}(\lambda)
461 =
462 \left\{
463 \frac{\lambda+j+r}{N}:0\leq r<N
464 \right\},
465 \]
466 \[
467 B_{N,j}
468 =
469 \left\{
470 \frac{j+1+r}{N}:0\leq r<N
471 \right\}\setminus\{1\}.
472 \]
473
474 \begin{theorem}[Exact hypergeometric packet form]
475 \label{thm:block09-hypergeom-packet}
476 If $(\lambda)_j\neq0$, then
477 \[
478 \boxed{
479 G_{N,j}^{\lambda,\alpha}(x)
480 =
481 \frac{(\lambda)_j\alpha^j}{j!}
482 {}_NF_{N-1}
483 \!\left(
484 \begin{matrix}A_{N,j}(\lambda)\\ B_{N,j}\end{matrix}
485 ;\alpha^Nx
486 \right).}
487 \]
488 Its coefficients satisfy the first-order polynomial recurrence
489 \[
490 \boxed{
491 \prod_{r=0}^{N-1}(Nm+j+1+r)u_{m+1}
492 =
493 \alpha^N
494 \prod_{r=0}^{N-1}(Nm+j+\lambda+r)u_m.}
495 \]
496 The series is annihilated by
497 \[
498 \boxed{
499 \mathcal H_{N,j}^{\lambda,\alpha}
500 =
501 \theta
502 \prod_{b\in B_{N,j}}(\theta+b-1)
503 -
504 \alpha^Nx
505 \prod_{a\in A_{N,j}(\lambda)}(\theta+a),}
506 \]
507 where $\theta=xD_x$.
508 \end{theorem}
509
510 \begin{proof}
511 Split the rising factorials and ordinary factorials into their $N$ residue factors. The multiplication formula gives the displayed hypergeometric coefficients. Taking the ratio of consecutive packet coefficients gives the recurrence. Coefficient comparison with the standard ${}_NF_{N-1}$ operator gives the differential equation.
512 \end{proof}
513
514 \begin{theorem}[Minimal differential order phase diagram]
515 \label{thm:block09-order-phase}
516 Let
517 \[
518 r_D(N,j;\lambda)
519 =
520 r_D\bigl(G_{N,j}^{\lambda,\alpha}\bigr),
521 \qquad \alpha\neq0.
522 \]
523 Then:
524
525 \begin{enumerate}
526 \item If $\lambda\notin\mathbb Z$, then
527 \[
528 \boxed{r_D(N,j;\lambda)=N.}
529 \]
530 \item If $\lambda=L\in\mathbb Z_{>0}$, then every packet is a nonzero rational function with exact denominator
531 \[
532 (1-\alpha^Nx)^L,
533 \]
534 and
535 \[
536 \boxed{r_D(N,j;L)=1.}
537 \]
538 \item If $\lambda=-M\in\mathbb Z_{\leq0}$, then $G_{N,j}^{-M,\alpha}$ is zero when $j>M$. Otherwise it is a nonzero polynomial of degree
539 \[
540 \left\lfloor\frac{M-j}{N}\right\rfloor
541 \]
542 and
543 \[
544 \boxed{
545 r_D(N,j;-M)
546 =
547 \begin{cases}
548 1,&j\leq M,\\
549 0,&j>M.
550 \end{cases}}
551 \]
552 \end{enumerate}
553 \end{theorem}
554
555 \begin{proof}
556 For nonintegral $\lambda$, no difference between an upper parameter and a completed lower parameter is an integer. The classical irreducibility criterion for generalized hypergeometric operators therefore makes $\mathcal H_{N,j}^{\lambda,\alpha}$ irreducible of order $N$ \cite{BeukersHeckman1989}. Since the packet is nonzero, its minimal operator has order $N$.
557
558 For $L>0$, the coefficient is a polynomial of degree $L-1$ in $m$ times $\alpha^{Nm+j}$. Its generating function is rational with denominator $(1-\alpha^Nx)^L$, and the leading polynomial term prevents cancellation of the highest denominator power. Every nonconstant rational function has a first-order logarithmic-derivative annihilator.
559
560 For $\lambda=-M$, the source series is the polynomial $(1-\alpha z)^M$. Selecting one residue class leaves exactly the displayed finite support. A nonzero polynomial has differential order one, while the zero function has order zero.
561 \end{proof}
562
563 The tail recurrence order and differential order are different invariants. A nonzero finite-support coefficient sequence has tail recurrence order zero but its polynomial generating function has differential order one.
564
565 \subsection{Orbit aliasing and address cancellation}
566 \label{subsec:block09-alias-cancellation}
567
568 Let $\omega_N$ be primitive and consider the rotated orbit
569 \[
570 \mathcal O_N(F_{\lambda,\alpha})
571 =
572 \left
573 \{(1-\alpha\omega_N^rt)^{-\lambda}:0\leq r<N\right\}.
574 \]
575 Let $r_{\rm orb}$ be its dimension over the constant field.
576
577 \begin{theorem}[Exact orbit rank]
578 \label{thm:block09-orbit-rank}
579 For the binomial family,
580 \[
581 \boxed{
582 r_{\rm orb}(N;\lambda)
583 =
584 \begin{cases}
585 N,&\lambda\notin\mathbb Z_{\leq0},\\
586 \min(N,M+1),&\lambda=-M\in\mathbb Z_{\leq0}.
587 \end{cases}}
588 \]
589 \end{theorem}
590
591 \begin{proof}
592 If $\lambda$ is nonintegral, each rotated branch has a branch singularity at a different point. If $\lambda$ is a positive integer, each has a pole at a different point. A constant linear relation would therefore have zero coefficient at every branch.
593
594 If $\lambda=-M$, expand
595 \[
596 (1-\alpha\omega_N^rt)^M
597 =
598 \sum_{n=0}^{M}
599 \binom Mn(-\alpha)^n\omega_N^{rn}t^n.
600 \]
601 The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely $\min(N,M+1)$.
602 \end{proof}
603
604 Define the alias and cancellation deficits by
605 \[
606 \boxed{
607 A_N(\lambda)=N-r_{\rm orb}(N;\lambda),}
608 \]
609 \[
610 \boxed{
611 C_{N,j}(\lambda)
612 =
613 r_{\rm orb}(N;\lambda)
614 -r_D(N,j;\lambda).}
615 \]
616 Then
617 \[
618 \boxed{
619 N-r_D(N,j;\lambda)
620 =
621 A_N(\lambda)+C_{N,j}(\lambda).}
622 \]
623
624 \begin{corollary}[Exact defect formulas]
625 \label{cor:block09-defect-formulas}
626 The three parameter regimes have the following signatures.
627
628 \begin{enumerate}
629 \item If $\lambda\notin\mathbb Z$, then
630 \[
631 A_N(\lambda)=0,
632 \qquad
633 C_{N,j}(\lambda)=0.
634 \]
635 \item If $\lambda=L>0$, then
636 \[
637 A_N(L)=0,
638 \qquad
639 C_{N,j}(L)=N-1.
640 \]
641 \item If $\lambda=-M\leq0$, then
642 \[
643 A_N(-M)=\max(0,N-M-1),
644 \]
645 and
646 \[
647 C_{N,j}(-M)
648 =
649 \begin{cases}
650 \min(N,M+1)-1,&j\leq M,\\
651 \min(N,M+1),&j>M.
652 \end{cases}
653 \]
654 \end{enumerate}
655 \end{corollary}
656
657 This separates two mechanisms that are easily conflated. Orbit aliasing occurs before an address is selected. Address cancellation occurs after the Fourier projector has been applied.
658
659 \begin{figure}[htbp]
660 \centering
661 \includegraphics[width=.97\textwidth]{holonomic_multisection_pipeline.pdf}
662 \caption{The declared holonomic multisection pipeline. General closure constructs an annihilator but does not prove minimality. The binomial--Fuchsian family admits an exact orbit-rank and address-cancellation decomposition.}
663 \label{fig:block09-multisection-pipeline}
664 \end{figure}
665
666 \begin{figure}[htbp]
667 \centering
668 \includegraphics[width=.96\textwidth]{holonomic_order_phase.pdf}
669 \caption{Minimal differential order in the binomial packet family. Nonintegral exponents retain order $N$, positive integers collapse to rational order one, and nonpositive integers terminate with explicit aliasing and zero-address packets.}
670 \label{fig:block09-order-phase}
671 \end{figure}
672
673 \subsection{Local exponent and monodromy packet}
674 \label{subsec:block09-fuchsian-exponents}
675
676 For $\lambda\notin\mathbb Z$, the hypergeometric operator in \cref{thm:block09-hypergeom-packet} is Fuchsian with singularities
677 \[
678 \boxed{
679 0,
680 \qquad
681 \alpha^{-N},
682 \qquad
683 \infty.}
684 \]
685 Its exponent packets are
686 \[
687 \boxed{
688 E_0
689 =
690 \{0\}\cup\{1-b:b\in B_{N,j}\},}
691 \]
692 \[
693 \boxed{
694 E_{\alpha^{-N}}
695 =
696 \{0,1,\ldots,N-2,-\lambda\},}
697 \]
698 and
699 \[
700 \boxed{
701 E_\infty
702 =
703 A_{N,j}(\lambda).}
704 \]
705 Modulo integers, $E_0$ is the complete $N$-torsion set
706 \[
707 \left\{0,\frac1N,\ldots,\frac{N-1}{N}\right\},
708 \]
709 and the pairwise differences in $E_\infty$ are also $N$-torsion.
710
711 \begin{proposition}[Local monodromy ledger]
712 \label{prop:block09-local-monodromy}
713 For nonintegral rational $\lambda$:
714
715 \begin{enumerate}
716 \item the monodromy eigenvalues at zero form the complete group of $N$th roots of unity;
717 \item the ratios of monodromy eigenvalues at infinity are $N$th roots of unity;
718 \item the distinguished nontrivial eigenvalue at $\alpha^{-N}$ is
719 \[
720 e^{-2\pi i\lambda}.
721 \]
722 \end{enumerate}
723 \end{proposition}
724
725 The local eigenvalues are finite-order when $\lambda\in\mathbb Q$. This does not imply that the global monodromy group is finite. Connection matrices and logarithmic extension data remain additional invariants.
726
727 \begin{figure}[htbp]
728 \centering
729 \includegraphics[width=.94\textwidth]{holonomic_exponent_packet.pdf}
730 \caption{The exact Fuchsian exponent packet of a nonintegral binomial multisection. Root-orbit aliasing moves $N$ rotated source singularities into one finite singularity while the complete $N$-torsion exponent sets remain visible at zero and infinity.}
731 \label{fig:block09-exponent-packet}
732 \end{figure}
733
734
735 \subsection{Marked normalized packet atlas}
736 \label{subsec:block09-marked-atlas}
737
738 The order theorem is most naturally a theorem about marked packets. Without the mark, distinct choices of source exponent, scale, width, and address can represent the same zero packet or the same rational function.
739
740 \begin{definition}[Marked binomial--Fuchsian packet]
741 \label{def:block09-marked-packet}
742 A marked packet is a tuple
743 \[
744 \mathfrak b=(\alpha,\lambda,N,j;G),
745 \]
746 where $\alpha\neq0$, $N\geq1$, $0\leq j<N$, and
747 \[
748 G(x)=G_{N,j}^{\lambda,\alpha}(x).
749 \]
750 The parameters $(\alpha,\lambda,N,j)$ are part of the semantic object rather than metadata to be forgotten.
751 \end{definition}
752
753 For a nonzero marked packet, consider the following normalized charts:
754 \begin{enumerate}[label=\textup{(\roman*)}]
755 \item the marked coefficient series $G$;
756 \item the primitive first-order tail recurrence obtained from the exact coefficient ratio, together with its finite initial record;
757 \item the monic minimal differential operator from \cref{thm:block09-order-phase}, together with the distinguished power-series solution selected by the marked coefficient record;
758 \item the hypergeometric parameter packet $\bigl(A_{N,j}(\lambda),B_{N,j};\alpha^N\bigr)$ and leading coefficient;
759 \item the primitive minimal-operator singular and indicial record
760 \[
761 \mathscr E(G)
762 =
763 \left(
764 \Sigma_{\rm op}(G),
765 \bigl(I_{\xi,L_G}\bigr)_{\xi\in\Sigma_{\rm op}(G)}
766 \right),
767 \]
768 together with the orbit-rank and defect data
769 \[
770 \left(r_{\rm orb},r_D,A_N,C_{N,j}\right).
771 \]
772 In the nonintegral Fuchsian regime, \(\mathscr E(G)\) is the explicit three-point packet
773 \[
774 \left(E_0,E_{\alpha^{-N}},E_\infty\right)
775 \]
776 computed above. In the integral regimes it is computed from the corresponding minimal first-order operator. The zero packets in the terminated regime are retained as separately marked zero objects with the zero-operator convention.
777 \end{enumerate}
778
779 \begin{theorem}[Effective marked binomial--Fuchsian atlas]
780 \label{thm:block09-marked-descent}
781 The five normalized presentations above determine one another on the declared marked family. With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively. The package
782 \[
783 \boxed{
784 \mathcal J_{\rm BF}
785 =
786 \left(
787 \alpha,\lambda,N,j,
788 R_{N,j},L_{N,j},
789 A_{N,j}(\lambda),B_{N,j},
790 \mathscr E(G),
791 r_{\rm orb},r_D,A_N,C_{N,j}
792 \right)}
793 \]
794 is complete on this marked category.
795 \end{theorem}
796
797 \begin{proof}
798 The mark constructs the coefficient packet directly. Consecutive coefficient ratios give the normalized recurrence chart, and the multiplication formula gives the hypergeometric chart. \Cref{thm:block09-order-phase} identifies the monic minimal differential operator in every parameter regime. The primitive polynomial representative of that operator determines \(\mathscr E(G)\) by the indicial construction in \cref{subsec:block09-local-exponents}. In the nonintegral regime, \cref{prop:block09-local-monodromy} gives its explicit Fuchsian eigenvalue ledger. \Cref{thm:block09-orbit-rank,cor:block09-defect-formulas} compute the orbit and cancellation data in every regime. Conversely, the displayed invariant retains the complete mark and therefore reconstructs the source packet and every normalized chart. Uniqueness of monic Ore normalization and of the marked coefficient record leaves no nontrivial automorphism in a fiber.
799 \end{proof}
800
801 \subsection{Examples and exact verification}
802 \label{subsec:block09-verification}
803
804 The parity sections of
805 \[
806 \frac{2}{1-z^2}
807 \]
808 give the smallest singularity-alias example. The source singularities $1$ and $-1$ both map to $1$ under squaring. The even address is $2/(1-x)$, while the odd address vanishes.
809
810 For
811 \[
812 2\cosh z,
813 \]
814 the even packet is nonzero and the odd packet is zero. There is no finite singularity, so the cancellation occurs at the level of the character-projected solution rather than by finite singularity collision.
815
816 For the Catalan sequence, every arithmetic section remains hypergeometric in the packet index. Its recurrence order is one, while its ordinary generating function remains a second-order algebraic D-finite function. This again shows that recurrence order and scalar differential order are different coordinates.
817
818 For the Ap\'ery numbers, the known two-dimensional nonautonomous state jumps to every arithmetic section. The jump theorem gives an exact order-two recurrence bound, but the present holonomic theorem does not claim a closed minimality formula for all Ap\'ery addresses.
819
820 \input{generated/block09_examples_table.tex}
821 \input{generated/block09_validation_table.tex}
822
823 The Holonomic atlas verifier checks the binomial packet recurrence, the generalized hypergeometric differential coefficient identity, nonresonance of the irreducible parameter packets, zero and infinity exponent torsion classes, positive-integral rational collapse, nonpositive-integral orbit rank and address support, and root-filter orthogonality. It records 174,940 exact checks with no failures.
824
825 \begin{boundarybox}
826 \textbf{Exact holonomic boundary.}
827 The holonomic section proves normalized minimal Ore charts on the ordinary-point locus and an exact multisection minimality theorem for the binomial--Fuchsian packet family. It does not prove a universal formula for minimal multisection operators of arbitrary D-finite series. Irregular singularities, Stokes data, apparent-singularity removal, resonant logarithmic extensions, general differential-module cyclicity after character projection, and a finite all-address signature for unrestricted holonomic systems remain open.
828 \end{boundarybox}
829
830 \begin{resultbox}
831 \textbf{Section result.}
832 P-recursive and D-finite presentations now form an explicit normalized atlas on the ordinary-point locus, with minimal Ore annihilators, finite prefix and jet certificates, and exact coefficient transport. General arithmetic multisection has a typed root-filter and state-jump construction, together with an explicit construction-relative minimality defect. The binomial--Fuchsian family supplies the first exact holonomic packet phase diagram: nonintegral exponents retain differential order $N$, positive integers collapse to rational order one, nonpositive integers terminate with a closed orbit-alias and address-cancellation formula, and the nonintegral local exponent packet exposes the complete finite torsion at zero and infinity.
833 \end{resultbox}
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