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Special Functions

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19 \newcommand{\A}{A_{a,\beta,Q}}
20 \newcommand{\Z}{Z_Q}
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22 \begin{document}
23 \MonographCover
24 \frontmatter
25 \MonographTitleLeaf
26 \MonographPreface
27 \gdef\CurrentChapter{Contents}\tableofcontents
28 \clearpage
29 \FrontChapter{Reading map}\label{reading-map}
30 \section*{Five families, one mathematical question}
31 This monograph investigates structural invariants in five families of special functions. Feedback scales, arithmetic frequencies, signed prime masses, rotation orbits, and coupled channels provide distinct settings. Their connections are established through specific formulas, invariants, and obstructions, not assumed from a shared vocabulary.
32 
33 The argument starts with entire functions defined by contracted nonlinear feedback. It then studies arithmetic generating functions and prime-sieve measures, before passing to orbit spectral zeta functions and exact coefficient laboratories. Coupled entire systems connect several of these ideas, while a resolution atlas measures the difference between finite observation and global mathematical structure.
34 
35 The main chapters give defining formulas, stated hypotheses, mathematical consequences, and computational figures. Extended proofs follow as technical appendices. The final notes distinguish analytic proofs, finite checks, classical inputs, and questions that remain open.
36 
37 \section*{Four starting constructions}
38 The first four families lead to distinct results: an entire-growth law, an obstruction to finite differential equations, a positivity boundary, and a rotation-invariant spectral coordinate. A coupled fifth family is developed in Part IV.
39 
40 \renewcommand{\arraystretch}{1.45}
41 \begin{tabular}{@{}>{\raggedright\arraybackslash}P{0.21\linewidth}>{\raggedright\arraybackslash}P{0.40\linewidth}>{\raggedright\arraybackslash}P{0.33\linewidth}@{}}
42 \toprule
43 Family & Structure worked out & Status of the construction\\
44 \midrule
45 Asymmetric dilation & Entire solutions; exact growth balance;
46 off-diagonal nonrescaling; intrinsic feedback banks & Candidate extension of a
47 known one-dilation tangent analog\\
48 Complex arithmetic depth & Exact exponential type; coupled shift calculus;
49 strict positive kernel; exact parameter recovery; no polynomial-coefficient linear ODE & Candidate
50 extension of the project's integer arithmetic energy\\
51 Prime-sieve packets & Exact moment field; real Hankel singularities;
52 positivity-margin collapse; multiple-prime geometry; arithmetic resolvent & Candidate arithmetic
53 specialization of classical orthogonal-polynomial machinery\\
54 Orbit spectral zeta & Global meromorphic continuation; rational integer values;
55 regularized product; multiplicity-aware orbit quotient & Worked specialization of known
56 Mittag--Leffler seeds and spectral-zeta methods\\
57 \bottomrule
58 \end{tabular}
59 
60 \claim{An established diagonal antecedent}
61 The equation $f_q'(z)=1+f_q(qz)^2$, $f_q(0)=0$, is already present in
62 M\'endez~\cite[Example~2.4]{Mendez2024}. It remains an
63 important baseline, but its definition is not a discovery of this project.
64 The asymmetric extension keeps two separate contractions and permits a
65 proved distinction from every amplitude/argument rescaling of that baseline.
66 
67 \claim{How to read the evidence}
68 Theorem statements below concern exact infinite objects under the displayed
69 hypotheses. Plots and decimal roots illustrate them. Finite assertions check
70 implementations within recorded ranges. None of these three forms of evidence
71 establishes historical priority by itself.
72 
73 The source archive preserves extended arguments, earlier review records, negative controls, and open questions. The evidence register distinguishes those records from the checks performed for this edition.
74 
75 
76 \mainmatter
77 \MonographPart{I}{Entire functions and their growth}{Contractions, coefficients and nonlinear feedback.}
78 \chapter{Contracted nonlinear feedback}\label{ch:feedback}
79 \pageheading{FAMILY 1}{Two scales inside the function}
80 For real $0<q,r<1$, define the normalized odd solution
81 \[
82 \boxed{T_{q,r}'(z)=1+T_{q,r}(qz)T_{q,r}(rz),\qquad T_{q,r}(0)=0.}
83 \]
84 The contractions are part of the defining feedback. The function is symmetric
85 in $q,r$, but unequal scales retain left/right address information.
86 \fig{01_asymmetric_functions}{Computed Taylor curves. Each entire curve uses
87 terms through $z^{321}$; positive-coefficient Cauchy estimates bound the omitted
88 tail on each displayed interval. The black dashed curve is ordinary tangent.}
89 For $T_{q,r}(z)=\sum_{n\ge0}c_nz^{2n+1}$, $c_0=1$, coefficient matching gives
90 \[
91 c_n=\frac{1}{2n+1}\sum_{i=0}^{n-1}
92 q^{2i+1}r^{2(n-1-i)+1}c_ic_{n-1-i},\qquad n\ge1.
93 \]
94 In particular,
95 \[
96 T_{q,r}(z)=z+\frac{qr}{3}z^3+
97 \frac{q^2r^2(q^2+r^2)}{15}z^5+O(z^7).
98 \]
99 The coefficients are positive and bounded by tangent coefficients, giving a
100 convergent germ. If the solution is known on $|z|<R$, its integral equation
101 extends it to $|z|<R/\max(q,r)$. Repetition proves that it is entire.
102 
103 The parameters can be recovered, up to swapping, from $c_1,c_2$. Put
104 $p=3c_1$ and $u=15c_2/p^2$. Then $q,r$ are the roots of
105 $x^2-\sqrt{u+2p}\,x+p=0$.
106 
107 \pageheading{FAMILY 1\; /\; GROWTH}{A scale-balance atlas}
108 \[
109 \boxed{\rho(T_{q,r})=\rho\quad\text{where}\quad q^\rho+r^\rho=1.}
110 \]
111 Here $M_f(R)=\max_{|z|=R}|f(z)|$ and
112 $\rho(f)=\limsup_{R\to\infty}\log\log M_f(R)/\log R$ is the order of an entire function. Moreover $\log M_{T}(R)=O(R^\rho)$: the upper type is finite.
113 \smallfig{02_order_atlas}{The displayed values numerically solve the proved
114 characteristic equation. Colors saturate at order 7; the upper corner contains
115 larger orders. The dashed diagonal is the previously known one-scale family.}
116 \begin{tabular}{@{}lll@{}}\toprule
117 Scales & Order & Interpretation\\\midrule
118 $q=r=1/4$ & $1/2$ & Slow entire growth\\
119 $q+r=1$ & $1$ & Many distinct functions at one order\\
120 $q=r=1/\sqrt2$ & $2$ & Quadratic logarithmic growth scale\\
121 $q=r=3/4$ & $\log2/\log(4/3)$ & Approximately $2.4094208397$\\
122 \bottomrule\end{tabular}
123 
124 \claim{Why this equation appears}
125 Nonnegative coefficients give $M(R)=T_{q,r}(R)$. The integral equation implies
126 \[
127 M(R)\le R[1+M(qR)M(rR)],
128 \]
129 \[
130 M(R)\ge(1-\alpha)R M(\alpha qR)M(\alpha rR),\quad0<\alpha<1.
131 \]
132 Taking logarithms turns product feedback into a sum over contracted scales.
133 Comparison with $CR^\rho$ proves the upper bound; comparison with
134 $cR^{\rho_\alpha}$, where $(\alpha q)^{\rho_\alpha}+
135 (\alpha r)^{\rho_\alpha}=1$, proves the lower bound. Letting $\alpha\uparrow1$
136 gives the same order. The complete interval-induction proof is given in Appendix~\ref{app:growth}.
137 
138 \input{sections/dilation_growth.tex}
139 
140 \pageheading{FAMILY 1\; /\; SEPARATION}{More than a rescaled one-scale seed}
141 \claim{Nonrescaling theorem}
142 If $q\ne r$, then $T_{q,r}$ cannot equal $A f_s(Bz)$ for nonzero constants
143 $A,B$ and any real $0<s<1$, where $f_s'=1+f_s(sz)^2$ is the known family.
144 
145 Suppose such an equality held. Normalization gives $AB=1$. Write $p=qr$,
146 $u=q^2+r^2$, and let $\rho$ be the common order. The first two nontrivial
147 coefficients and the growth law force
148 \[
149 B^2s^2=p,\qquad s^2=u/2,\qquad 2s^\rho=q^\rho+r^\rho=1.
150 \]
151 Consequently
152 \[
153 2\left(\frac{q^2+r^2}{2}\right)^{\rho/2}=q^\rho+r^\rho.
154 \]
155 For $\rho\ne2$, strict convexity or concavity of $x^{\rho/2}$ forces $q=r$.
156 For the exceptional case $\rho=2$, we have $u=1$, $s^2=1/2$, $B^2=2p$.
157 The two $z^7$ coefficients become
158 \[
159 \frac{p^3(3-p^2)}{315}\quad\text{and}\quad\frac{11p^3}{1260}.
160 \]
161 Their equality forces $p=1/2$ and again $(q-r)^2=u-2p=0$.
162 This completes the exceptional-order case.
163 
164 \claim{A worked order-one member}
165 \[
166 T_{1/3,2/3}(z)=z+\frac{2}{27}z^3+
167 \frac{4}{2187}z^5+\frac{232}{11160261}z^7+O(z^9).
168 \]
169 It has order one, and $T_{1/3,2/3}(1)\approx1.0759239897408249$.
170 The diagonal point $q=r=1/2$ gives
171 $f_{1/2}(z)=\sqrt2\sinh(z/\sqrt2)$. Every off-diagonal member is separated
172 from all rescaled diagonal members by the theorem above.
173 
174 \claim{A combinatorial address formula}
175 Sum over full ordered binary trees with $n$ internal vertices. A vertex with
176 $i$ internal vertices in its left subtree and $j$ in its right contributes
177 \[
178 \frac{q^{2i+1}r^{2j+1}}{2(i+j+1)+1}.
179 \]
180 Multiply these factors over internal vertices, then sum over trees. Root
181 splitting reproduces the coefficient recurrence exactly. Unequal contractions
182 therefore record left and right path lengths separately.
183 
184 This is a precise separation result from one authenticated antecedent.
185 It does not prove that no earlier author studied the asymmetric equation.
186 
187 \pageheading{FAMILY 1\; /\; TAPS}{Feedback banks and weak-channel sensitivity}\label{sec:banks}
188 Choose finitely many channels $j=1,\ldots,J$, with $h_j>0$, $\sum_jh_j=1$,
189 and $0<q_j,r_j<1$. Define
190 \[
191 F_\Theta'(z)=1+\sum_jh_jF_\Theta(q_jz)F_\Theta(r_jz),\qquad F_\Theta(0)=0.
192 \]
193 These taps act inside the nonlinear definition. With
194 $\mu_1=\sum_jh_jq_jr_j$ and
195 $\mu_2=\sum_jh_jq_jr_j(q_j^2+r_j^2)$,
196 \[
197 F_\Theta(z)=z+\frac{\mu_1}{3}z^3+\frac{\mu_1\mu_2}{15}z^5+O(z^7).
198 \]
199 If $q_j^{\rho_j}+r_j^{\rho_j}=1$, then
200 \[
201 \boxed{\rho(F_\Theta)=\max_{j\text{ active}}\rho_j.}
202 \]
203 The positive weights alter local coefficients. The most expansive active
204 channel determines the asymptotic order. A zero-weight channel is removed
205 before taking the maximum.
206 \fig{07_weak_tap}{A two-channel example: scales $(1/4,1/4)$ with weight
207 $1-\varepsilon$, and $(3/4,3/4)$ with weight $\varepsilon$.
208 The function varies continuously on every fixed compact set, while its growth
209 order jumps when the second channel becomes active.}
210 For $0<\varepsilon\le1$ the order is $\log2/\log(4/3)$; at $\varepsilon=0$
211 it is $1/2$. Coefficients depend continuously on $\varepsilon$ and are
212 dominated by those of the entire $f_{3/4}$. Dominated convergence of the Taylor
213 series therefore proves uniform convergence on compact sets as
214 $\varepsilon\downarrow0$.
215 
216 This gives a useful design distinction: finite-window similarity does not
217 control entire growth order. For a bank with weights $1/4,3/4$ at these scales,
218 \[
219 F(z)=z+\frac7{48}z^3+\frac{427}{30720}z^5+
220 \frac{176681}{188743680}z^7+O(z^9).
221 \]
222 
223 \input{sections/dilation_parameters.tex}
224 
225 
226 \MonographPart{II}{Arithmetic depth and signed measures}{Complex parameters, positive kernels and the boundaries of positivity.}
227 \chapter{Arithmetic at complex depth}\label{ch:arithmetic}
228 \pageheading{FAMILY 2}{Arithmetic energy at complex depth}
229 For real $a>1/2$, $\beta>0$, and integer $Q\ge1$, define
230 \[
231 \boxed{\A(s)=\sum_{\substack{n\ge2\\(n,Q)=1}}
232 \mu(n)^2n^{-2a}(1-n^{-\beta})^{2s},\qquad s\in\mathbb C.}
233 \]
234 The complex power uses the real logarithm of $1-n^{-\beta}>0$.
235 The atom $n=1$ is excluded because its logarithm is undefined. At positive
236 integer depths its omitted contribution would be zero. At depth zero the
237 corresponding mass-one atom is absent. In particular, $A(0)=Z_Q(2a)-1$, where
238 \[
239 Z_Q(v)=\sum_{\substack{n\ge1\\(n,Q)=1}}\mu(n)^2n^{-v}
240 =\frac{\zeta(v)}{\zeta(2v)}\prod_{p\mid Q}(1+p^{-v})^{-1},\qquad v>1.
241 \]
242 \fig{03_complex_depth}{Left: $a=\beta=2$ at real depths, using the exact
243 zeta--Newton representation at 90-digit working precision. Right: initial
244 squarefree frequencies and weights for $Q=1$. Only finitely many atoms are
245 shown; the proof uses the infinite support.}
246 Let $\mathcal S_Q=\{n\ge2:\mu(n)^2=1,\ (n,Q)=1\}$ and
247 $\lambda_n=-2\log(1-n^{-\beta})$. Then
248 \[
249 A(s)=\sum_{n\in\mathcal S_Q}w_ne^{-\lambda_ns},\qquad w_n=n^{-2a}.
250 \]
251 The positive weights are summable, and the distinct support frequencies decrease to zero.
252 If $p_*$ is the least prime not dividing $Q$, then
253 \[
254 \tau=-2\log(1-p_*^{-\beta}),\qquad M=\Z(2a)-1,
255 \]
256 \[
257 |A(s)|\le M e^{\tau\max(-\Re s,0)}.
258 \]
259 This proves local uniform convergence and entire continuation. The first atom
260 dominates $A(-t)$ as $t\to\infty$, so the entire function has exact order one
261 and exponential type $\tau$.
262 
263 \pageheading{FAMILY 2\; /\; CALCULUS}{Closed shifts, infinite differential complexity}
264 The Euler product for $Z_Q$ shows that only
265 $\operatorname{rad}(Q)=\prod_{p\mid Q}p$ matters. Expansion of the depth factor gives
266 \[
267 \boxed{\A(s+1)=\A(s)-2A_{a+\beta/2,\beta,Q}(s)
268 +A_{a+\beta,\beta,Q}(s).}
269 \]
270 The fractional transfer and Newton formulas are
271 \[
272 \A(s+t)=\sum_{j\ge0}(-1)^j\binom{2t}{j}A_{a+j\beta/2,\beta,Q}(s),
273 \]
274 \[
275 \A(t)=\sum_{j\ge0}(-1)^j\binom{2t}{j}[\Z(2a+j\beta)-1].
276 \]
277 They converge absolutely and locally uniformly: binomial coefficients have
278 at most polynomial growth on compact parameter sets, while the shifted sums
279 decay geometrically. The subtraction of one is essential at noninteger depth.
280 
281 \claim{No polynomial-coefficient linear ODE}
282 Suppose $\sum_{j=0}^{r}P_j(s)A^{(j)}(s)=0$, where the $P_j$ are polynomials and are not all zero.
283 At $s=-t$, divide by the exponential of the first support atom. The remaining
284 tail is exponentially smaller, so the polynomial
285 $\sum_jP_j(-t)(-\lambda_{p_*})^j$ tends to zero exponentially and must vanish
286 identically. Remove that atom and repeat. Every support frequency therefore
287 satisfies $\sum_jP_j(s)(-\lambda_n)^j=0$ for all $s$.
288 For fixed $s$, this polynomial in $\lambda$ has infinitely many distinct
289 roots. All its coefficients vanish. Thus all $P_j$ are zero, a contradiction.
290 
291 The function is \textbf{not D-finite in depth}. This excludes identification
292 in the depth variable with fixed finite-parameter hypergeometric or Meijer
293 G functions and ordinary finite D-finite combinations. It does not exclude
294 nonalgebraic substitutions or representations by known non-D-finite functions.
295 
296 \claim{Strict positive complex kernel}
297 For distinct complex points $s_i$, $K_{ij}=A(s_i+\overline{s_j})$ is strictly
298 positive definite. A zero quadratic form would force a finite exponential
299 sum to vanish at every $\lambda_n$. Their accumulation at zero invokes the
300 identity theorem; derivatives at zero then give a nonsingular Vandermonde
301 system. On the real axis, $(-1)^jA^{(j)}(s)>0$ for every $j\ge0$.
302 
303 \pageheading{FAMILY 2\; /\; ASYMPTOTICS}{An arithmetic sector law}
304 The negative real direction is governed by the first admissible prime. The
305 positive direction is governed by the density of infinitely many squarefree
306 integers. These are different pieces of the same entire function.
307 
308 Let
309 \[
310 \alpha=\frac{2a-1}{\beta},\qquad
311 \delta_Q=\frac1{\zeta(2)}\prod_{p\mid Q}\frac{p}{p+1}.
312 \]
313 As $|s|\to\infty$ in any closed sector
314 $|\arg s|\le\pi/2-\epsilon$ with $0<\epsilon<\pi/2$,
315 \[
316 \boxed{\A(s)=\frac{\delta_Q}{\beta}B(\alpha,2s+1)
317 +O_{a,\beta,Q,\epsilon}\left(|s|^{-(2a-1/2)/\beta}\right).}
318 \]
319 In particular,
320 \[
321 \A(s)\sim\frac{\delta_Q\Gamma(\alpha)}{\beta}(2s)^{-\alpha}.
322 \]
323 The principal power is used in the right half-plane. The beta expression is
324 an asymptotic continuous-density model, not an identity with the arithmetic sum.
325 
326 \claim{Proof mechanism}
327 Squarefree inclusion--exclusion gives
328 \[
329 \sum_{\substack{n\le x\\(n,Q)=1}}\mu(n)^2
330 =\delta_Q x+O_Q(\sqrt{x}).
331 \]
332 Integrate against $g_s(x)=x^{-2a}(1-x^{-\beta})^{2s}$ on $[1,\infty)$.
333 The density term becomes the beta integral after $y=x^{-\beta}$. Partial
334 integration bounds the error by
335 $C_Q\int_1^\infty\sqrt{x}|g_s'(x)|\dd x$.
336 The two derivative terms are beta integrals with decay exponent
337 $(2a-1/2)/\beta$; $|s|/\Re s$ stays bounded in the stated sector.
338 This gives the error term and then the leading asymptotic.
339 
340 \claim{One complex example}
341 At $a=\beta=2$, $Q=1$,
342 \[
343 A(1/2+i/4)\approx
344 0.06033183289343401791658
345 -0.007412384053785436251981\,i.
346 \]
347 Direct finite sums and a zeta--Newton evaluation agree within the recorded
348 omitted-tail envelope. This is an ordinary high-precision example, not an
349 interval certificate.
350 
351 Mixed Dirichlet-character weights provide matrix-valued positive kernels.
352 Individual off-diagonal entries may be signed or complex, so complete
353 monotonicity is asserted only for the positive principal family.
354 
355 \input{sections/arithmetic_inverse.tex}
356 
357 \chapter{Prime sieves and signed measures}\label{ch:sieves}
358 \pageheading{FAMILY 3}{An infinite prime shelf changes the packets}
359 For a prime $p$ and $\lambda\ge-1$, let
360 \[
361 x_n=(1-n^{-2})^2,\qquad
362 \nu_{p,\lambda}=\sum_{n\ge1}\mu(n)^2n^{-4}
363 (1+\lambda\mathbf1_{p\mid n})\delta_{x_n}.
364 \]
365 Define monic orthogonal polynomials $P_m^{(k;p,\lambda)}$ for the measure
366 $x^k\dd\nu_{p,\lambda}$, $k\ge0$ integer. The zero atom has mass one at
367 $k=0$ and is annihilated when $k\ge1$.
368 \fig{04_prime_polynomials}{Left: degree-two packets for $p=2$, depth zero.
369 Right: initial atom masses. Strength $-1$ deletes every even squarefree atom;
370 strength $0$ is the original measure; strength $1$ doubles that shelf.}
371 The exact moments are finite zeta stencils:
372 \[
373 M_j^{p,\lambda}=\sum_{r=0}^{2j}(-1)^r\binom{2j}{r}
374 \frac{\zeta(4+2r)}{\zeta(8+4r)}
375 \left(1+\frac{\lambda}{p^{4+2r}+1}\right).
376 \]
377 All polynomial coefficients belong to $\mathbb Q(\pi^{-2},\lambda)$.
378 With $\Delta=M_0M_2-M_1^2$,
379 \[
380 P_1=x-\frac{M_1}{M_0},\qquad
381 P_2=x^2-\frac{M_0M_3-M_1M_2}{\Delta}x+
382 \frac{M_1M_3-M_2^2}{\Delta}.
383 \]
384 The infinitely many positive nodes make every Hankel matrix strictly positive
385 definite throughout $\lambda\ge-1$. Hence all degrees exist, with positive
386 norms, real simple zeros in $(0,1)$, and interlacing.
387 
388 \pageheading{FAMILY 3\; /\; BOUNDARY}{Finite positivity can conceal a signed defect}
389 Let $\nu_{\mathrm{non}p}$ and $\nu_p$ be the restrictions of the
390 undeformed measure $\nu_{p,0}$ to indices not divisible by $p$ and divisible
391 by $p$, respectively. Then
392 $\nu_{p,\lambda}=\nu_{\mathrm{non}p}+(1+\lambda)\nu_p$.
393 Let $A_m,B_m$ be the order-$m$ Hankel matrices of these two unscaled
394 positive measures. Both are positive definite.
395 Thus
396 \[
397 \tau_m(\lambda)=\det(A_m+(1+\lambda)B_m)
398 =\det A_m\prod_{i=1}^{m}[1+(1+\lambda)\eta_i],
399 \]
400 where $\eta_i>0$ are the eigenvalues of $A_m^{-1/2}B_mA_m^{-1/2}$.
401 Every determinant root is real and below $-1$; possible coefficient poles
402 occur there, although numerator cancellation is possible.
403 \fig{05_sieve_boundary}{The line is a proved bound for $p=2$; dots are
404 ordinary high-precision pencil roots. A defect $\lambda=-1.001$ eventually
405 violates positivity even though it can pass low-order checks.}
406 The nearest root is $-1-\delta_m$, with
407 \[
408 \delta_m=\min_{\substack{f\in\mathbb R[x]\setminus\{0\}\\\deg f<m}}
409 \frac{\int f^2\dd\nu_{\mathrm{non}p}}{\int f^2\dd\nu_p}>0,
410 \qquad\delta_m\downarrow0.
411 \]
412 To prove the limit, approximate a continuous bump supported at the isolated
413 node $x_p$ by polynomials on $[0,1]$. The numerator tends to zero while the
414 denominator retains the $p^{-4}$ atom. Hence every $\lambda<-1$ eventually
415 produces a negative finite-degree quadratic form.
416 
417 For $p=2$, test with $f(x)=x(1-x)^{m-2}$, $m\ge2$:
418 \[
419 \delta_m\le\frac{4096}{81}\left(\frac{16M_0}{17}-1\right)
420 \left(\frac{272}{567}\right)^{2(m-2)},\qquad M_0=105\pi^{-4}.
421 \]
422 This follows from $x_2=9/16$ and $x_n\ge64/81$ at nonzero odd nodes.
423 Exactly $\delta_1=p^4$; the limit is an infinite-resolution phenomenon.
424 
425 \pageheading{FAMILY 3\; /\; TRANSPORT}{Classical operations with arithmetic coordinates}
426 The depth tilt $x^k$ is a Christoffel transformation. Suppressing $p,\lambda$,
427 \[
428 P_m^{(k+1)}(x)=\frac{P_{m+1}^{(k)}(x)
429 -[P_{m+1}^{(k)}(0)/P_m^{(k)}(0)]P_m^{(k)}(x)}{x}.
430 \]
431 The numerator vanishes at zero. All roots are positive, so the displayed
432 denominator coefficient is nonzero. For shifted Hankel determinants $\tau_m^{(k)}=\det[M_{k+i+j}]_{i,j<m}$,
433 with $\tau_0^{(k)}=1$,
434 \[
435 \tau_m^{(k)}\tau_m^{(k+2)}-(\tau_m^{(k+1)})^2
436 =\tau_{m+1}^{(k)}\tau_{m-1}^{(k+2)}>0,\qquad m\ge1.
437 \]
438 This is the Desnanot--Jacobi identity applied to exact arithmetic moments.
439 The construction mechanism is classical.
440 
441 \claim{An endpoint mass is numerically delicate}
442 For the undeformed depth-zero measure, set $\nu_\theta=\nu+\theta\delta_0$,
443 $\theta\ge-1$. Write $h_j=\int P_j(x)^2\dd\nu(x)$ and
444 $K_m(x,y)=\sum_{j=0}^{m}P_j(x)P_j(y)/h_j$.
445 The Uvarov formula~\cite{Uvarov1969} is
446 \[
447 P_m^{[\theta]}(x)=P_m(x)-
448 \frac{\theta P_m(0)}{1+\theta K_{m-1}(0,0)}K_{m-1}(x,0),
449 \]
450 with $K_{-1}=0$ for degree zero. The mass at zero is recovered by
451 \[
452 1\le\frac1{K_m(0,0)}\le1+(M_0-1)(7/16)^{2m}.
453 \]
454 The lower bound comes from the zero atom, whose mass is one. The test polynomial
455 $(1-x)^m$ gives the upper bound since every other node is at least $9/16$.
456 At finite degree $K_m(0,0)<1$, because no nonzero polynomial can vanish at
457 all the other infinitely many nodes. Complete mass removal is valid, while
458 $1-K_m(0,0)$ becomes very small. This explains the conditioning problem.
459 
460 \claim{The resolvent is non-D-finite}
461 For $\lambda\ge-1$, the Stieltjes function $S_{p,\lambda}(z)=\int(z-x)^{-1}\dd\nu_{p,\lambda}(x)$
462 has infinitely many isolated poles at the positive-weight nodes $x_n\to1$.
463 A solution of a finite rational-coefficient linear ODE has only finitely many
464 possible finite singular points. The germ at $z=2$ therefore is not D-finite.
465 This is a general infinite-atomic-resolvent mechanism, specialized here to
466 arithmetic data.
467 
468 The full packet sequence also cannot be an affine copy of a continuous Jacobi
469 sequence: compact moment determinacy would equate a purely atomic measure with
470 a continuous one. An inherited Laguerre scaling limit remains compatible with
471 this separation of the complete families.
472 
473 \input{sections/multi_prime.tex}
474 
475 
476 \MonographPart{III}{Orbit spectra and exact coordinates}{Spectral continuation and the coefficient language of polyphase transforms.}
477 \chapter{Spectral functions of zero orbits}\label{ch:spectra}
478 \pageheading{FAMILY 4}{One spectral address for each rotation orbit}
479 For integer $N\ge2$, define the classical Mittag--Leffler seed~\cite[\S10.46]{DLMF}
480 \[
481 d_N(w)=\sum_{n\ge0}\frac{(-1)^nw^n}{(Nn)!}=E_{N,1}(-w),
482 \qquad D_N(z)=d_N(z^N).
483 \]
484 Let $w_j$ be the nonzero zeros of $d_N$, counted with multiplicity. Each
485 corresponds to one rotation orbit of $N$ zeros of $D_N$. Using
486 $\operatorname{Arg}w\in(-\pi,\pi]$, define
487 \[
488 \boxed{\mathcal Z_N(s)=\sum_jw_j^{-s},\qquad\Re s>1/N.}
489 \]
490 \fig{06_orbit_zeros}{For $N=4$, the four real/imaginary zeros in each orbit
491 share one positive coordinate $w=z^4$. The plot shows the first six orbits.
492 Multiplicity is retained by the definition; simplicity is not assumed in general.}
493 The bound $|d_N(w)|\le e^{|w|^{1/N}}$ and the finite-order zero-counting theorem
494 give absolute local uniform convergence in the stated half-plane. Since the
495 order is below one and $d_N(0)=1$, the Hadamard factorization has genus zero:
496 \[
497 d_N(w)=\prod_j(1-w/w_j).
498 \]
499 At positive integers, a log derivative gives
500 \[
501 -w\frac{d_N'(w)}{d_N(w)}=\sum_{k\ge1}\mathcal Z_N(k)w^k.
502 \]
503 If $a_n=(-1)^n/(Nn)!$ and $b_k=\mathcal Z_N(k)$, then
504 \[
505 b_k=-ka_k-\sum_{j=1}^{k-1}a_jb_{k-j}.
506 \]
507 Thus every positive integer value is rational without numerical zero finding.
508 
509 \pageheading{FAMILY 4\; /\; REDUCTIONS}{Classical anchors and the open spectral frontier}
510 The first values are
511 \[
512 \begin{aligned}
513 \mathcal Z_N(1)&=\frac1{N!},\\
514 \mathcal Z_N(2)&=\frac1{(N!)^2}-\frac2{(2N)!},\\
515 \mathcal Z_N(3)&=\frac1{(N!)^3}-\frac3{N!(2N)!}+\frac3{(3N)!}.
516 \end{aligned}
517 \]
518 \begin{tabular}{@{}llll@{}}\toprule
519 $N$ & $\mathcal Z_N(1)$ & $\mathcal Z_N(2)$ & $\mathcal Z_N(3)$\\\midrule
520 2 & $1/2$ & $1/6$ & $1/15$\\
521 3 & $1/6$ & $1/40$ & $53/13440$\\
522 4 & $1/24$ & $17/10080$ & $691/9979200$\\
523 5 & $1/120$ & $5/72576$ & $997/1743565824$\\
524 \bottomrule\end{tabular}
525 
526 \claim{Two exact Hurwitz-zeta reductions}
527 For $N=2$, $d_2(w)=\cos\sqrt w$, whose zeros are
528 $w_j=\pi^2(j+1/2)^2$, $j\ge0$. Therefore
529 \[
530 \mathcal Z_2(s)=\pi^{-2s}\zeta(2s,1/2).
531 \]
532 For $N=4$,
533 \[
534 D_4(z)=\cos(z/\sqrt2)\cosh(z/\sqrt2),
535 \]
536 and the rotation quotient has one zero
537 $w_j=(\sqrt2\pi)^4(j+1/2)^4$ per orbit. Hence
538 \[
539 \mathcal Z_4(s)=(\sqrt2\pi)^{-4s}\zeta(4s,1/2).
540 \]
541 These anchors are classical reductions. The general spectral-zeta mechanism
542 also has substantial antecedents, including hypergeometric zero zeta functions.
543 
544 \claim{Global meromorphic continuation}
545 The half-plane definition has a meromorphic extension to the
546 whole complex plane. The following sections prove its sole pole, special values,
547 and regularized product using a branch-safe Mellin representation.
548 These are worked spectral results from established analytic mechanisms.
549 Residue-weighted sums involving derivatives at zeros remain a separate
550 question, especially when multiplicities occur.
551 
552 \input{sections/spectral_continuation.tex}
553 \chapter{Sequences, trees and polyphase}\label{ch:sequences}
554 \input{sections/sequence_polyphase.tex}
555 
556 
557 \MonographPart{IV}{Coupled systems and finite resolution}{Interacting channels, global growth and finite observation.}
558 \chapter{Coupled contraction functions}\label{ch:coupled}
559 \input{sections_v11/extension_overview.tex}
560 \input{sections_v11/coupled_visual.tex}
561 \chapter{Limits of finite observation}\label{ch:resolution}
562 \input{sections_v11/resolution_visual.tex}
563 \addtocontents{toc}{\protect\newpage}
564 \MonographPart{V}{Proofs and mathematical evidence}{Complete arguments, reproducible calculations and their precise limits.}
565 \input{sections/proof_map.tex}
566 \clearpage\input{sections_v11/coupled_growth_full.tex}
567 \clearpage\input{sections_v11/coupled_inverse_full.tex}
568 \clearpage\input{sections_v11/coupled_structure_full.tex}
569 \clearpage\input{sections_v11/resolution_full.tex}
570 \clearpage\input{sections_v11/coupled_lab_full.tex}
571 
572 \backmatter
573 \FrontChapter{Evidence and limitations}\label{evidence}
574 The main chapters develop entire existence, nonrescaling, arithmetic
575 non-D-finiteness, one-prime transport and base orbit convergence.
576 Appendices A--J supply the extended arguments. The preserved foundational
577 review records cover Appendices A--E and their exact API. The coupled-family
578 records cover the proofs, resolution certificates and exact APIs underlying
579 Appendices F--J. These are project review records, not claims of external
580 peer review or kernel certification.
581 
582 Classical external leaves are named explicitly. In particular, ordinary
583 orthogonal-polynomial theory, Hadamard factorization, Laplace uniqueness,
584 and spectral Mellin regularization are established machinery. The local
585 proofs verify their application and do not presume historical novelty.
586 
587 \input{proofread_verification.tex}
588 \input{sections_v11/release_status.tex}
589 \input{sections/release_status.tex}
590 \input{evidence_notes.tex}
591 \input{notation.tex}
592 \input{references.tex}
593 \end{document}
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