Stratified Universality

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1 \section{Selected algebraic branches, packet Weil restriction, and collision invariants}
2 \label{sec:block10-algebraic}
3 
4 Algebraic atlas develops the algebraic stratum after the fixed-radix, product, local, and holonomic milestones.  The central issue is not algebraicity alone.  A polynomial equation may have several formal branches, and a packet elimination can contain branch tuples that do not belong to the selected semantic germ.  A normalized algebraic chart must therefore retain both a minimal polynomial and a finite branch selector.  Arithmetic packets are then compiled as a finite polynomial system.  The generic packet scheme is a Weil restriction of the source branch scheme, which gives a uniform degree bound.  A separate collision ledger explains why the degree of one selected packet may be smaller than that bound.
5 
6 Throughout this section, $K$ is a characteristic-zero field and $\overline K$ is an algebraic closure.  Algebraic degree always means degree over the displayed rational-function field.  All assertions about formal branches are made inside $\overline K[[z]]$ unless a different local field is named.
7 
8 \subsection{Minimal polynomials and finite branch selectors}
9 \label{subsec:block10-branch-selector}
10 
11 Let $F\in K[[z]]$ be algebraic over $K(z)$.  Its monic irreducible polynomial is
12 \[
13  P_F(z,Y)\in K(z)[Y],
14  \qquad
15  P_F(z,F(z))=0.
16 \]
17 Write
18 \[
19  d(F)=\deg_Y P_F.
20 \]
21 After multiplying by a common denominator and dividing by the greatest common divisor of the polynomial coefficients, one obtains a primitive representative in $K[z,Y]$, unique up to a nonzero scalar.  We normalize that scalar by fixing the first nonzero coefficient in graded lexicographic order to be one.
22 
23 A polynomial alone does not select a branch.  Let
24 \[
25  \operatorname{Br}_0(P_F)
26  =
27  \{G\in\overline K[[z]]:P_F(z,G(z))=0\}.
28 \]
29 This set has cardinality at most $d(F)$.
30 
31 \begin{definition}[Branch-separation index]
32 For $F\in\operatorname{Br}_0(P_F)$, define
33 \[
34  \sigma(P_F,F)
35  =
36  \begin{cases}
37  0,&\operatorname{Br}_0(P_F)=\{F\},\\[3pt]
38  1+\displaystyle\max_{G\in\operatorname{Br}_0(P_F),\,G\neq F}
39  \operatorname{ord}_z(F-G),&\text{otherwise}.
40  \end{cases}
41 \]
42 The corresponding branch selector is the finite coefficient jet
43 \[
44  \boxed{
45  J_{\rm alg}(F)
46  =
47  (a_0,a_1,\ldots,a_{\sigma-1}),
48  \qquad
49  F(z)=\sum_{n\geq0}a_nz^n.}
50 \]
51 When $\sigma=0$, the selector is empty.
52 \end{definition}
53 
54 \begin{theorem}[Finite branch separation]
55 \label{thm:block10-finite-branch-separation}
56 The pair
57 \[
58  \boxed{(P_F,J_{\rm alg}(F))}
59 \]
60 selects $F$ uniquely among the formal power-series branches of $P_F$ at the origin.
61 \end{theorem}
62 
63 \begin{proof}
64 There are at most $d(F)$ formal branches.  The difference of two distinct formal series has finite $z$-adic order.  By definition of $\sigma$, every other branch differs from $F$ in one of the coefficients recorded by $J_{\rm alg}(F)$.
65 \end{proof}
66 
67 The simplest case is the ordinary implicit-function locus.
68 
69 \begin{corollary}[Simple-branch normalization]
70 \label{cor:block10-simple-branch}
71 If
72 \[
73  P_F(0,a_0)=0,
74  \qquad
75  \partial_Y P_F(0,a_0)\neq0,
76 \]
77 then $a_0$ alone selects a unique formal branch.  Equivalently, $\sigma(P_F,F)\leq1$.
78 \end{corollary}
79 
80 \begin{proof}
81 Coefficient comparison in $P_F(z,F(z))=0$ determines each next coefficient linearly, with diagonal coefficient $\partial_Y P_F(0,a_0)$.
82 \end{proof}
83 
84 The finite selector is semantic data.  It cannot be dropped merely because the polynomial is irreducible over $K(z)$.  Irreducibility controls the generic function-field extension, while the jet chooses one local embedding of that extension into formal series.
85 
86 \subsection{The normalized selected-branch atlas}
87 \label{subsec:block10-branch-atlas}
88 
89 Let
90 \[
91  A_F=K(z)[Y]/(P_F),
92 \]
93 and let $\bar Y$ denote the residue class of $Y$.  In the basis
94 \[
95  1,\bar Y,\ldots,\bar Y^{d-1},
96 \]
97 multiplication by $\bar Y$ is represented by the companion matrix $M_F$.  Its characteristic and minimal polynomials are both $P_F$.
98 
99 \begin{theorem}[Normalized selected-branch algebraic atlas]
100 \label{thm:block10-branch-atlas}
101 The following normalized presentations determine one another exactly:
102 \[
103  \boxed{
104  \begin{aligned}
105  &F\in K[[z]],\\
106  &(P_F,J_{\rm alg}(F)),\\
107  &(A_F,\bar Y,J_{\rm alg}(F)),\\
108  &(M_F,J_{\rm alg}(F)).
109  \end{aligned}}
110 \]
111 With equality morphisms, these presentations form equivalent normalized groupoids.  Every fiber over a fixed semantic branch is contractible.
112 \end{theorem}
113 
114 \begin{proof}
115 The semantic branch gives its unique monic minimal polynomial and its finite branch selector.  The polynomial gives the finite field algebra and its companion multiplication matrix.  Conversely, the companion matrix recovers $P_F$ as its minimal polynomial.  The branch selector then recovers the selected formal root by \cref{thm:block10-finite-branch-separation}.  Every normalized presentation is determined literally, so a normalized automorphism fixing the semantic branch is the identity.
116 \end{proof}
117 
118 \begin{figure}[htbp]
119 \centering
120 \includegraphics[width=.98\textwidth]{algebraic_branch_atlas.pdf}
121 \caption{The selected algebraic branch, its minimal polynomial and finite selector, its finite quotient algebra, and its complete packet system are reversible normalized charts.  The Ore chart is directional because an annihilator can forget branch data.}
122 \label{fig:block10-branch-atlas}
123 \end{figure}
124 
125 \subsection{The exact residue-packet polynomial compiler}
126 \label{subsec:block10-packet-compiler}
127 
128 Fix $N\geq1$.  For
129 \[
130  F(z)=\sum_{n\geq0}a_nz^n,
131 \]
132 define the arithmetic packets
133 \[
134  U_j(x)
135  =
136  \sum_{m\geq0}a_{Nm+j}x^m,
137  \qquad
138  0\leq j<N.
139 \]
140 They satisfy the exact synthesis identity
141 \[
142  \boxed{
143  F(z)
144  =
145  \sum_{j=0}^{N-1}z^jU_j(z^N).}
146 \]
147 
148 Let $P(z,Y)\in K[z,Y]$ be the normalized primitive representative of $P_F$.  Introduce indeterminates
149 \[
150  \mathbf U=(U_0,\ldots,U_{N-1})
151 \]
152 and substitute
153 \[
154  Y=\sum_{j=0}^{N-1}z^jU_j.
155 \]
156 Every polynomial in $z$ has a unique decomposition modulo the relation $z^N=x$:
157 \[
158  P\!\left(z,\sum_{j=0}^{N-1}z^jU_j\right)
159  =
160  \sum_{r=0}^{N-1}z^r\Phi_{N,r}(x,\mathbf U).
161 \]
162 The polynomials
163 \[
164  \boxed{
165  \Phi_{N,r}(x,\mathbf U)\in K[x,\mathbf U]}
166 \]
167 are the algebraic packet equations.
168 
169 \begin{theorem}[Exact algebraic packet system]
170 \label{thm:block10-exact-packet-system}
171 Let $d=\deg_Y P_F$.  Then:
172 \begin{enumerate}
173 \item the semantic packet tuple $(U_0,\ldots,U_{N-1})$ satisfies
174 \[
175  \Phi_{N,r}(x,\mathbf U)=0
176  \qquad(0\leq r<N);
177 \]
178 \item every $\Phi_{N,r}$ has total degree at most $d$ in the packet variables;
179 \item a formal packet tuple satisfying all $N$ equations synthesizes a formal branch of $P_F$;
180 \item the packet tuple selected by the transported coefficient jet reconstructs $F$ uniquely.
181 \end{enumerate}
182 \end{theorem}
183 
184 \begin{proof}
185 Substituting the semantic synthesis formula into $P_F(z,F(z))=0$ gives the first statement.  Substitution does not increase degree in the $U_j$, so the second follows.  If all residue polynomials vanish, then the synthesized series satisfies $P_F=0$.  The source branch selector is distributed among the packets by residue class.  The resulting packet jet synthesizes the original jet, so \cref{thm:block10-finite-branch-separation} identifies the selected branch.
186 \end{proof}
187 
188 If the source selector records $a_0,\ldots,a_{\sigma-1}$, the transported packet selector records precisely
189 \[
190  [x^m]U_j=a_{Nm+j}
191  \qquad
192  \text{for }Nm+j<\sigma.
193 \]
194 No information is added or lost.  The coefficient prefix is merely rewritten in packet addresses.
195 
196 \subsection{Weil-restricted packet degree}
197 \label{subsec:block10-weil-degree}
198 
199 Put
200 \[
201  R=K(x),
202  \qquad
203  E_N=R[z]/(z^N-x).
204 \]
205 Characteristic zero makes $z^N-x$ separable over $R$.  Interpret the monic polynomial $P_F(z,Y)$ as a polynomial in $E_N[Y]$, and define
206 \[
207  B_{N,F}=E_N[Y]/(P_F).
208 \]
209 The packet polynomial system is the coordinate presentation of the finite Weil restriction
210 \[
211  \operatorname{Res}_{E_N/R}\operatorname{Spec}B_{N,F}.
212 \]
213 
214 \begin{theorem}[Algebraic packet degree bound]
215 \label{thm:block10-packet-degree-bound}
216 Let $F$ have algebraic degree $d$ over $K(z)$.  For every packet width $N$:
217 \begin{enumerate}
218 \item the generic geometric degree of the complete packet scheme is
219 \[
220  \boxed{d^N;}
221 \]
222 \item the joint field generated by a selected complete packet satisfies
223 \[
224  \boxed{
225  [K(x)(U_0,\ldots,U_{N-1}):K(x)]\leq d^N;}
226 \]
227 \item every individual packet satisfies
228 \[
229  \boxed{
230  [K(x)(U_j):K(x)]\leq d^N.}
231 \]
232 \end{enumerate}
233 \end{theorem}
234 
235 \begin{proof}
236 Over an algebraic closure of $K(x)$, the equation $z^N=x$ has $N$ distinct roots.  The algebra $E_N$ therefore splits into a product of $N$ copies of the algebraic closure.  At each address, the separable degree-$d$ polynomial $P_F$ has $d$ geometric roots.  The Weil-restricted fiber is their Cartesian product and has $d^N$ points.  A selected packet tuple is one point of this finite scheme, so its residue field and every coordinate subfield have degree at most the scheme degree.
237 \end{proof}
238 
239 This bound is constructive.  The quotient of
240 \[
241  K(x)[U_0,\ldots,U_{N-1}]
242 \]
243 by the packet equations is a finite algebra of generic rank $d^N$.  Multiplication by $U_j$ is therefore a matrix of size $d^N$.  Its characteristic polynomial annihilates $U_j$ by Cayley--Hamilton.
244 
245 \begin{corollary}[Constructive packet eliminant]
246 \label{cor:block10-constructive-eliminant}
247 For each $j$, one can compute an annihilating polynomial
248 \[
249  Q_{N,j}(x,T)\in K(x)[T]
250 \]
251 with
252 \[
253  \deg_TQ_{N,j}\leq d^N
254 \]
255 by multiplication matrices, resultants, or a Gr\"obner elimination of the exact packet system.
256 \end{corollary}
257 
258 The bound is a universal degree ceiling for the minimal-polynomial grammar.  It does not assert that every coordinate attains degree $d^N$ or that one elimination algorithm returns the minimal polynomial without factor selection.
259 
260 \begin{figure}[htbp]
261 \centering
262 \includegraphics[width=.98\textwidth]{algebraic_packet_weil_restriction.pdf}
263 \caption{After the address extension $x=z^N$ splits, each of the $N$ addresses carries $d$ algebraic branches.  The complete packet scheme is their Cartesian product and has generic degree $d^N$.}
264 \label{fig:block10-weil-restriction}
265 \end{figure}
266 
267 \subsection{Tuple collision and selected-orbit loss}
268 \label{subsec:block10-collision-ledger}
269 
270 Let $\Omega$ be a normal splitting field of the generic packet scheme over $K(x)$, and let
271 \[
272  \Gamma=\operatorname{Gal}(\Omega/K(x)).
273 \]
274 After splitting $E_N$, the geometric packet branches form a set
275 \[
276  \mathcal B_N
277 \]
278 of cardinality $d^N$.  Let
279 \[
280  \pi_j:\mathcal B_N\to\Omega
281 \]
282 be the $j$th packet coordinate, and set
283 \[
284  \mathcal C_{N,j}=\pi_j(\mathcal B_N).
285 \]
286 For the selected semantic packet $U_j$, separability gives
287 \[
288  [K(x)(U_j):K(x)]
289  =
290  |\Gamma\cdot U_j|.
291 \]
292 
293 \begin{definition}[Algebraic packet collision defects]
294 Define
295 \[
296  \mathfrak c^{\rm tuple}_{N,j}
297  =
298  d^N-|\mathcal C_{N,j}|,
299 \]
300 and
301 \[
302  \mathfrak s^{\rm Gal}_{N,j}
303  =
304  |\mathcal C_{N,j}|-|\Gamma\cdot U_j|.
305 \]
306 The first records collisions caused by projection from complete branch tuples to one packet coordinate.  The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.
307 \end{definition}
308 
309 \begin{theorem}[Exact algebraic degree-defect identity]
310 \label{thm:block10-degree-defect}
311 The two defects are nonnegative and satisfy
312 \[
313  \boxed{
314  d^N-[K(x)(U_j):K(x)]
315  =
316  \mathfrak c^{\rm tuple}_{N,j}
317  +
318  \mathfrak s^{\rm Gal}_{N,j}.}
319 \]
320 \end{theorem}
321 
322 \begin{proof}
323 The coordinate image $\mathcal C_{N,j}$ has at most $d^N$ values.  The Galois orbit of the selected value is contained in that image.  The displayed identity is the telescoping decomposition
324 \[
325  d^N-|\Gamma\cdot U_j|
326  =
327  (d^N-|\mathcal C_{N,j}|)
328  +
329  (|\mathcal C_{N,j}|-|\Gamma\cdot U_j|).
330 \]
331 \end{proof}
332 
333 This signature separates two mechanisms that a raw resultant merges.  Repeated projected values are a coordinate collision.  A smaller selected orbit is a descent or stabilizer phenomenon.  They are not the same obstruction.
334 
335 \begin{figure}[htbp]
336 \centering
337 \includegraphics[width=.96\textwidth]{algebraic_collision_defects.pdf}
338 \caption{The universal degree $d^N$ can drop first because several branch tuples have the same packet coordinate, then because the selected coordinate has a nontrivial Galois stabilizer.}
339 \label{fig:block10-collision-ledger}
340 \end{figure}
341 
342 \subsection{A sharp Catalan parity theorem}
343 \label{subsec:block10-catalan}
344 
345 Let
346 \[
347  C(z)=1+zC(z)^2
348 \]
349 be the Catalan branch with $C(0)=1$, and write
350 \[
351  C(z)=E(z^2)+zO(z^2).
352 \]
353 Put $x=z^2$.
354 
355 \begin{theorem}[Exact Catalan packet system]
356 \label{thm:block10-catalan-system}
357 The parity packets satisfy
358 \[
359  \boxed{
360  E=1+2xEO,
361  \qquad
362  O=E^2+xO^2.}
363 \]
364 Elimination gives
365 \[
366  \boxed{4xE^4-E^2+1=0}
367 \]
368 and
369 \[
370  \boxed{4x^3O^4-8x^2O^3+5xO^2-O+1=0.}
371 \]
372 Both packets have algebraic degree four over $K(x)$.  Hence the universal bound
373 \[
374  d^N=2^2=4
375 \]
376 is sharp.
377 \end{theorem}
378 
379 \begin{proof}
380 Substitute $C=E+zO$ into $C=1+zC^2$ and separate even and odd powers.  The two resultants give the displayed quartics.
381 
382 For the even packet, put $T=E^2$.  Then
383 \[
384  4xT^2-T+1=0.
385 \]
386 Its discriminant $1-16x$ is not a square in $K(x)$, so $[K(x)(T):K(x)]=2$.  The norm of $T$ from this quadratic field is $1/(4x)$, which is not a square in $K(x)$.  Therefore $T$ is not a square in its quadratic field and $E$ has degree four.
387 
388 The first packet equation gives
389 \[
390  O=\frac{E-1}{2xE},
391  \qquad
392  E=\frac{1}{1-2xO}.
393 \]
394 Thus $K(x)(E)=K(x)(O)$, and the odd packet also has degree four.
395 \end{proof}
396 
397 The Catalan parity packet has no tuple-collision or selected-orbit defect.  Its one-coordinate degree already sees all four geometric branch tuples of the quadratic source over the two addresses.
398 
399 \subsection{A quadratic-radical collision phase diagram}
400 \label{subsec:block10-quadratic-radical}
401 
402 Let
403 \[
404  F_{\alpha,\beta}(z)
405  =
406  \sqrt{(1-\alpha z)(1-\beta z)},
407  \qquad
408  F_{\alpha,\beta}(0)=1,
409 \]
410 and assume first that $\alpha\neq\beta$, so the normalized source degree is two.  Write
411 \[
412  F_{\alpha,\beta}(z)=E(x)+zO(x),
413  \qquad
414  x=z^2,
415 \]
416 and put
417 \[
418  s=\alpha+\beta,
419  \qquad
420  p=\alpha\beta.
421 \]
422 
423 \begin{theorem}[Quadratic-radical parity packets]
424 \label{thm:block10-radical-packets}
425 The packets satisfy
426 \[
427  \boxed{
428  E^2+xO^2=1+px,
429  \qquad
430  2EO=-s.}
431 \]
432 Their eliminants are
433 \[
434  \boxed{
435  E^4-(1+px)E^2+\frac{s^2x}{4}=0}
436 \]
437 and
438 \[
439  \boxed{
440  xO^4-(1+px)O^2+\frac{s^2}{4}=0.}
441 \]
442 If
443 \[
444  s\neq0,
445  \qquad
446  \alpha^2\neq\beta^2,
447 \]
448 then both selected packets have degree four over $K(x)$.
449 \end{theorem}
450 
451 \begin{proof}
452 The two equations follow by adding and subtracting the squares of
453 \[
454  F(z)=E+zO,
455  \qquad
456  F(-z)=E-zO.
457 \]
458 Elimination gives the quartics.  Their quadratic discriminant is
459 \[
460  \Delta(x)
461  =
462  (1-\alpha^2x)(1-\beta^2x),
463 \]
464 which is not a square under the stated hypotheses.  Hence $E^2$ generates a quadratic extension.  Its norm is $s^2x/4$, which is not a square in $K(x)$, so $E$ has degree four.  Since $EO=-s/2$, the fields generated by $E$ and $O$ coincide.
465 \end{proof}
466 
467 The symmetric case produces an exact collision.
468 
469 \begin{corollary}[Antisymmetric branch collision]
470 \label{cor:block10-antisymmetric-collision}
471 If $\beta=-\alpha\neq0$, then
472 \[
473  \boxed{O=0,
474  \qquad
475  E^2=1-\alpha^2x.}
476 \]
477 For the even coordinate,
478 \[
479  |\mathcal B_2|=4,
480  \qquad
481  |\mathcal C_{2,0}|=3,
482  \qquad
483  |\Gamma\cdot E|=2.
484 \]
485 Thus
486 \[
487  \mathfrak c^{\rm tuple}_{2,0}=1,
488  \qquad
489  \mathfrak s^{\rm Gal}_{2,0}=1.
490 \]
491 For the selected odd coordinate,
492 \[
493  |\mathcal C_{2,1}|=3,
494  \qquad
495  |\Gamma\cdot O|=1,
496 \]
497 so
498 \[
499  \mathfrak c^{\rm tuple}_{2,1}=1,
500  \qquad
501  \mathfrak s^{\rm Gal}_{2,1}=2.
502 \]
503 \end{corollary}
504 
505 \begin{proof}
506 Now $F(z)=F(-z)=\sqrt{1-\alpha^2z^2}$ for the selected branch, so the odd packet vanishes.  Over the split branch tuple set, the two signs are chosen independently.  The even projection has values $E,0,-E$, while the odd projection has values $0,E/z,-E/z$.  The selected Galois orbits have sizes two and one, respectively.
507 \end{proof}
508 
509 If $\alpha=\beta$, the quadratic equation
510 \[
511  Y^2-(1-\alpha z)^2=0
512 \]
513 is not the normalized source polynomial.  The selected branch is rational:
514 \[
515  F_{\alpha,\alpha}(z)=1-\alpha z,
516 \]
517 so the correct source degree is one.  This is a negative control against using a nonminimal defining equation in the degree ledger.
518 
519 \begin{figure}[htbp]
520 \centering
521 \includegraphics[width=.98\textwidth]{algebraic_quadratic_phase.pdf}
522 \caption{Catalan parity packets attain the universal quadratic bound.  Generic radicals do the same, while the antisymmetric radical separates tuple collision from selected-orbit loss.  A repeated linear factor must first be reduced to its minimal source polynomial.}
523 \label{fig:block10-quadratic-phase}
524 \end{figure}
525 
526 \subsection{The directional algebraic-to-holonomic bridge}
527 \label{subsec:block10-algebraic-holonomic}
528 
529 Algebraic functions are D-finite, but this bridge is directional unless branch data is retained.
530 
531 \begin{proposition}[Algebraic derivative closure]
532 \label{prop:block10-algebraic-dfinite}
533 Let $F$ have separable minimal polynomial $P_F$ of degree $d$.  Then
534 \[
535  F'
536  =
537  -\frac{\partial_zP_F(z,F)}{\partial_YP_F(z,F)}
538  \in
539  K(z,F).
540 \]
541 Every higher derivative lies in the same degree-$d$ function field.  Consequently, $F$ has a differential annihilator of order at most $d$.
542 \end{proposition}
543 
544 \begin{proof}
545 Differentiate $P_F(z,F(z))=0$.  Separability makes $\partial_YP_F(z,F)$ nonzero in the function field.  The field is closed under the induced derivation.  The $d+1$ elements $F,F',\ldots,F^{(d)}$ are linearly dependent over $K(z)$.
546 \end{proof}
547 
548 The annihilator can forget both the polynomial relation and the selected branch.  For example,
549 \[
550  \sqrt{1+z}
551  \quad\text{and}\quad
552  -\sqrt{1+z}
553 \]
554 satisfy the same first-order differential operator.  Their initial values distinguish them.  More generally, a D-finite equation may have transcendental solutions, so an Ore chart does not reconstruct an algebraic minimal polynomial without additional recognition data.
555 
556 \begin{boundarybox}
557 \textbf{Exact algebraic boundary.}
558 Algebraic atlas proves finite selected-branch normalization, an exact packet polynomial compiler, the generic $d^N$ Weil-restriction degree, a constructive coordinate eliminant, and a two-stage collision invariant.  It proves sharp packet degrees for Catalan parity and a complete collision phase for a marked quadratic-radical family.  It does not prove that every elimination output is minimal without factor selection, one canonical minimal polynomial-system dimension across arbitrary multivariate grammars, an optimal packet-degree bound using all source symmetries, or unrestricted equivalence between algebraic and D-finite presentations.
559 \end{boundarybox}
560 
561 \subsection{Exact examples and finite verification}
562 \label{subsec:block10-verification}
563 
564 The verifier independently constructs packet equations by substitution and reduction modulo $z^N=x$.  It checks Catalan, Motzkin, and marked radical branches at several packet widths.  It verifies the sharp Catalan quartics, the quadratic-radical trace and norm systems, the generic and antisymmetric collision phases, the rejection of nonminimal quadratic equations, and finite branch-selector certificates.
565 
566 \begin{resultbox}
567 \textbf{Section result.}
568 A selected algebraic branch has a canonical minimal polynomial and a finite branch-separating jet.  Complete arithmetic packets are the solutions of an explicit $N$-equation polynomial system obtained by residue reduction.  The generic packet scheme is a finite Weil restriction of degree $d^N$, giving a constructive degree bound for every packet coordinate.  Degree loss splits exactly into tuple projection collisions and Galois stabilizer loss.  Catalan parity attains the universal quadratic bound, while the marked radical family exhibits generic sharpness, antisymmetric cancellation, and rational collapse.  The algebraic-to-holonomic bridge remains directional unless the minimal polynomial and branch selector are retained.
569 \end{resultbox}
570 
571 \input{generated/block10_examples_table.tex}
572 \input{generated/block10_validation_table.tex}
573 

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