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1 \hypertarget{arithmetic-depth-an-exact-inverse-its-conditioning-and-character-resolution}{%
2 \chapter[Arithmetic inverse and character resolution]{Arithmetic depth: an exact inverse, its conditioning, and
3 character
4 resolution}\label{app:arithmetic}\label{arithmetic-depth-an-exact-inverse-its-conditioning-and-character-resolution}}
5 
6 
7 
8 The entire squarefree depth function determines its arithmetic
9 parameters exactly, with one precise equivalence: powers of the primes
10 in the modulus are invisible. This is a local mathematical theorem. It
11 does not assert historical priority for Laplace inversion or for this
12 particular arithmetic application.
13 
14 \hypertarget{claim-card-and-conventions}{%
15 \section{Definitions and conventions}\label{claim-card-and-conventions}}
16 
17 For real \(a>1/2\), real \(\beta>0\), and integer \(Q\ge1\), write
18 
19 \[
20  A_{a,\beta,Q}(s)=\sum_{\substack{n\ge2\\ \mu(n)^2=1\\ (n,Q)=1}}
21  n^{-2a}\exp(-\lambda_n s),\qquad
22  \lambda_n=-2\log(1-n^{-\beta}).
23 \]
24 
25 The logarithm is real. The omitted \(n=1\) atom is not restored in any
26 inverse formula. Let \(n_1<n_2<\cdots\) be the support integers, and
27 abbreviate \(\lambda_j=\lambda_{n_j}\), \(w_j=n_j^{-2a}\). Then
28 \(\lambda_1>\lambda_2>\cdots>0\), \(\lambda_j\to0\), and
29 \(\sum_jw_j=M=Z_Q(2a)-1<\infty\), where
30 
31 \[
32  Z_Q(v)=\frac{\zeta(v)}{\zeta(2v)}
33           \prod_{p\mid Q}(1+p^{-v})^{-1},\qquad v>1.
34 \]
35 
36 Here a germ means the complete exact analytic germ, equivalently all
37 exact derivatives at a point, rather than a finite Taylor polynomial or
38 noisy measurements. The base route proves that this germ has a unique
39 entire continuation, with
40 \(|A(s)|\le M\exp(\lambda_1\max\{-\Re s,0\})\).
41 
42 \hypertarget{exact-inverse-theorem}{%
43 \section{Exact inverse theorem}\label{exact-inverse-theorem}}
44 
45 \textbf{Theorem B.1 (AI-1).} Two admissible parameter triples define the same
46 analytic germ if and only if
47 
48 \[
49  a=a',\qquad \beta=\beta',\qquad
50  \operatorname{rad}(Q)=\operatorname{rad}(Q').
51 \]
52 
53 The function itself therefore encodes the weight exponent, the depth
54 exponent, and the finite excluded prime set. It does not encode
55 valuations \(v_p(Q)\).
56 
57 \hypertarget{step-1-recover-the-ordered-spectral-atoms}{%
58 \subsection{Step 1: recover the ordered spectral
59 atoms}\label{step-1-recover-the-ordered-spectral-atoms}}
60 
61 Starting with \(R_1=A\), define successively
62 
63 \[
64  \lambda_j=\lim_{t\to\infty}\frac{\log R_j(-t)}t,
65  \qquad
66  w_j=\lim_{t\to\infty}e^{-\lambda_jt}R_j(-t),
67  \qquad
68  R_{j+1}(s)=R_j(s)-w_je^{-\lambda_js}.
69 \]
70 
71 Every limit exists and recovers exactly the \(j\)-th support frequency
72 and weight. Indeed, if \(M_{j+1}=\sum_{k>j}w_k\), then for every
73 \(t\ge0\),
74 
75 \[
76  R_j(-t)=w_je^{\lambda_jt}(1+r_j(t)),\qquad
77  0\le r_j(t)\le\frac{M_{j+1}}{w_j}
78                     e^{-(\lambda_j-\lambda_{j+1})t}.
79 \]
80 
81 The bound follows by replacing every tail frequency by its largest value
82 \(\lambda_{j+1}\). Taking logarithms and dividing by \(t\) proves the
83 first limit. Multiplying by \(e^{-\lambda_jt}\) proves the second.
84 Removing this exact entire atom leaves the same hypotheses for the next
85 step. Thus the complete ordered atom list is intrinsic to the entire
86 function.
87 
88 No approximation to a large negative-depth value is used in this
89 uniqueness proof. It is an exact limiting argument.
90 
91 \hypertarget{step-2-recover-beta-without-knowing-the-integer-support}{%
92 \subsection{\texorpdfstring{Step 2: recover \(\beta\) without knowing
93 the integer
94 support}{Step 2: recover \textbackslash beta without knowing the integer support}}\label{step-2-recover-beta-without-knowing-the-integer-support}}
95 
96 Put
97 
98 \[
99  \delta_Q=\frac1{\zeta(2)}\prod_{p\mid Q}\frac{p}{p+1}>0.
100 \]
101 
102 The base route establishes the elementary counting estimate
103 
104 \[
105  N_Q(x):=\#\{2\le n\le x:\mu(n)^2=1,(n,Q)=1\}
106        =\delta_Qx+O_Q(\sqrt{x}).
107 \]
108 
109 Since \(N_Q(n_j)=j\), it follows that
110 
111 \[
112  n_j=\frac{j}{\delta_Q}\left(1+O_Q(j^{-1/2})\right),
113  \quad
114  \lambda_j=2\delta_Q^{\beta}j^{-\beta}
115      \left(1+O_{Q,\beta}(j^{-1/2}+j^{-\beta})\right).
116 \]
117 
118 For completeness, the first estimate follows initially from
119 \(j=\delta_Qn_j+O_Q(\sqrt{n_j})\), which gives \(n_j\asymp_Qj\);
120 substituting this back improves the error to \(O_Q(\sqrt j)\). The
121 second uses \(-\log(1-u)=u(1+O(u))\) as \(u\downarrow0\).
122 
123 Consequently the already recovered frequencies determine
124 
125 \[
126  \boxed{\displaystyle
127  \beta=-\lim_{j\to\infty}\frac{\log\lambda_j}{\log j}.}
128 \]
129 
130 A rank ratio avoids the slowly decaying constant divided by \(\log j\):
131 for every fixed real \(c>1\),
132 
133 \[
134  -\frac{\log(\lambda_{\lfloor cj\rfloor}/\lambda_j)}{\log c}
135  =\beta+O_{Q,\beta,c}(j^{-1/2}+j^{-\beta}).
136 \]
137 
138 The floor contributes an additional \(O(j^{-1})\), which is absorbed by
139 the displayed error. These are fixed-parameter asymptotics. They do not
140 give a constant uniform in arbitrary moduli.
141 
142 \hypertarget{step-3-recover-the-support-integers-and-the-weight-exponent}{%
143 \subsection{Step 3: recover the support integers and the weight
144 exponent}\label{step-3-recover-the-support-integers-and-the-weight-exponent}}
145 
146 Once \(\beta\) is known, every spectral atom gives its exact integer
147 label:
148 
149 \[
150  \boxed{\displaystyle
151  n_j=(1-e^{-\lambda_j/2})^{-1/\beta},\qquad
152  a=-\frac{\log w_j}{2\log n_j}.}
153 \]
154 
155 The recovered real number in the first formula is an integer by the
156 family definition; rounding is not part of the exact theorem. Since
157 \(n_j\ge2\), the denominator in the second formula is nonzero. Every
158 recovered atom yields the same value of \(a\).
159 
160 \hypertarget{step-4-recover-the-excluded-primes}{%
161 \subsection{Step 4: recover the excluded
162 primes}\label{step-4-recover-the-excluded-primes}}
163 
164 A prime \(p\) lies in the recovered integer support exactly when
165 \(p\nmid Q\): primes are squarefree, and no other restriction is
166 imposed. Hence
167 
168 \[
169  \boxed{\displaystyle
170  \operatorname{rad}(Q)=\prod_{\substack{p\ \mathrm{prime}\\
171                                       p\notin\{n_j:j\ge1\}}}p.}
172 \]
173 
174 The product is finite because the parameters are assumed to come from
175 this family. This formula is an exact reconstruction from infinite data,
176 not an algorithm that can certify completion of the prime list after
177 finitely many observations. Conversely, equal \(a,\beta\) and equal
178 radicals produce exactly the same terms. This proves both directions of
179 AI-1.
180 
181 \hypertarget{a-constructive-counting-envelope}{%
182 \subsection{A constructive counting
183 envelope}\label{a-constructive-counting-envelope}}
184 
185 The rank asymptotic can be made effective without an RH input. Write
186 \(\omega(Q)=\#\{p:p\mid Q\}\). For real \(x\ge4\), inclusion and
187 exclusion gives
188 
189 \[
190  |N_Q(x)-\delta_Qx|\le C_Q\sqrt x+1,
191  \qquad C_Q=2^{\omega(Q)}+2\frac{\varphi(Q)}Q.
192 \]
193 
194 To verify the constants, expand \(\mu(n)^2=\sum_{d^2\mid n}\mu(d)\). For
195 \((d,Q)=1\), the number of \(m\le x/d^2\) coprime to \(Q\) differs from
196 \((\varphi(Q)/Q)x/d^2\) by at most \(2^{\omega(Q)}\). Summing over at
197 most \(\sqrt x\) values of \(d\) gives the first part of \(C_Q\). The
198 omitted main-term tail is at most
199 \((\varphi(Q)/Q)x/\lfloor\sqrt x\rfloor\le2(\varphi(Q)/Q)\sqrt x\).
200 Removing \(n=1\) adds one. The main-term density is
201 \((\varphi(Q)/Q)\sum_{(d,Q)=1}\mu(d)d^{-2}=\delta_Q\) by its absolutely
202 convergent Euler product.
203 
204 For \(j\ge3\), therefore,
205 
206 \[
207  \left(\frac{\sqrt{C_Q^2+4\delta_Q(j-1)}-C_Q}{2\delta_Q}\right)^2
208  \le n_j\le
209  \left(\frac{\sqrt{C_Q^2+4\delta_Q(j+1)}+C_Q}{2\delta_Q}\right)^2.
210 \]
211 
212 The lower and upper bounds are the positive solutions of the
213 corresponding quadratic inequalities in \(\sqrt{n_j}\). At \(j\ge3\),
214 all possible supports have \(n_j\ge5\), so the \(x\ge4\) counting
215 envelope applies.
216 
217 \hypertarget{what-finite-negative-depth-measurements-can-recover}{%
218 \section{What finite negative-depth measurements can
219 recover}\label{what-finite-negative-depth-measurements-can-recover}}
220 
221 \textbf{Theorem B.2 (AI-2), gap bound.} Suppose the first \(j-1\) atoms have
222 already been removed exactly, put
223 \(\Delta_j=\lambda_j-\lambda_{j+1}>0\), and fix \(h>0\). Define
224 
225 \[
226  \widehat\lambda_j(t,h)
227  =\frac{\log R_j(-t-h)-\log R_j(-t)}h.
228 \]
229 
230 Then for every \(t\ge0\),
231 
232 \[
233  0\le\lambda_j-\widehat\lambda_j(t,h)
234  \le\frac1h\log\!\left(1+\frac{M_{j+1}}{w_j}e^{-\Delta_jt}\right),
235 \]
236 
237 and
238 
239 \[
240  0\le e^{-\lambda_jt}R_j(-t)-w_j
241      \le M_{j+1}e^{-\Delta_jt}.
242 \]
243 
244 \textbf{Proof.} The preceding positive remainder formula gives
245 \(\widehat\lambda_j=\lambda_j+h^{-1}[\log(1+r_j(t+h))-\log(1+r_j(t))]\).
246 Each tail factor decreases, so \(0\le r_j(t+h)\le r_j(t)\). The two
247 inequalities now follow from the bound on \(r_j(t)\). For the amplitude,
248 multiply the positive tail bound by \(e^{-\lambda_jt}\).
249 
250 In particular, to guarantee frequency error at most \(\varepsilon>0\)
251 from exact residual values, it suffices to take
252 
253 \[
254  t\ge\max\left\{0,\frac1{\Delta_j}
255       \log\frac{M_{j+1}}{w_j(e^{h\varepsilon}-1)}\right\}.
256 \]
257 
258 The parameters in this certificate are properties of the true tail. If
259 they are only estimated from data, this is a conditional recovery bound
260 rather than an independent numerical certificate.
261 
262 \hypertarget{positive-tail-envelopes-for-reproducible-examples}{%
263 \subsection{Positive tail envelopes for reproducible
264 examples}\label{positive-tail-envelopes-for-reproducible-examples}}
265 
266 For integer \(L\ge n_j\), a direct sum through \(n\le L\) at depth
267 \(-t\), \(t\ge0\), omits at most
268 
269 \[
270  E_L(t)=e^{t\lambda_{L+1}}\frac{L^{1-2a}}{2a-1}.
271 \]
272 
273 Indeed all omitted frequencies are at most \(\lambda_{L+1}\), and
274 \(\sum_{n>L}n^{-2a}\le\int_L^\infty x^{-2a}\,dx\). Ignoring
275 squarefreeness and coprimality makes this a valid upper bound. If
276 \(r_0,r_1\) are the two positive truncated residual sums at \(t,t+h\),
277 the true secant belongs to
278 
279 \[
280  \left[
281  \frac{\log r_1-\log(r_0+E_L(t))}h,
282  \frac{\log(r_1+E_L(t+h))-\log r_0}h
283  \right].
284 \]
285 
286 This interval treats truncation exactly at the mathematical level. Its
287 displayed floating-point endpoints are not directed-rounding
288 certificates.
289 
290 \hypertarget{cancellation-is-a-real-limitation}{%
291 \subsection{Cancellation is a real
292 limitation}\label{cancellation-is-a-real-limitation}}
293 
294 Evaluating a residual by subtracting previous atoms is different from
295 evaluating its known positive series. If an observation of \(A(-t)\) has
296 relative error at most \(\eta\) and the previous atoms are exact, then
297 the residual error can have relative size
298 
299 \[
300  \eta\,\kappa_j(t),\qquad
301  \kappa_j(t)=\frac{A(-t)}{R_j(-t)}
302  \sim\frac{w_1}{w_j}e^{(\lambda_1-\lambda_j)t}.
303 \]
304 
305 Thus the large negative depth that separates nearby atoms also worsens
306 cancellation for later atoms. Uncertain previously subtracted
307 frequencies introduce further error. The computation evaluates residuals
308 as positive sums and reports \(\log_{10}\kappa_j\) so this conditioning
309 cost remains visible.
310 
311 \hypertarget{exact-uniqueness-does-not-imply-uniform-prime-recovery}{%
312 \section{Exact uniqueness does not imply uniform prime
313 recovery}\label{exact-uniqueness-does-not-imply-uniform-prime-recovery}}
314 
315 \textbf{Theorem B.3 (AI-3), bounded-window instability.} Let \(p\) be a prime
316 not dividing \(Q\), and let \(R\ge0\). For every integer \(k\ge0\),
317 
318 \[
319  \boxed{\displaystyle
320  \sup_{|s|\le R}|A_{a,\beta,Q}^{(k)}(s)-A_{a,\beta,Qp}^{(k)}(s)|
321  \le \lambda_p^k e^{R\lambda_p}
322       \frac{Z_Q(2a)}{p^{2a}+1}.}
323 \]
324 
325 Consequently adding larger and larger excluded primes gives convergence
326 in every derivative on every bounded complex-depth window, even though
327 the recovered radical changes each time.
328 
329 \textbf{Proof.} The difference consists exactly of the squarefree
330 integers divisible by \(p\) and coprime to \(Q\), all of which satisfy
331 \(n\ge p\). Their frequencies are at most \(\lambda_p\). The Euler
332 factor at \(p\) shows that their total weight is
333 
334 \[
335  \sum_{\substack{n\ge1\\\mu(n)^2=1,(n,Q)=1\\p\mid n}}n^{-2a}
336  =\frac{Z_Q(2a)}{p^{2a}+1}.
337 \]
338 
339 There is no \(n=1\) term in this difference. Bound the \(k\)-th
340 derivative of every exponential by \(\lambda_p^ke^{R\lambda_p}\) and
341 sum. Since \(\lambda_p\to0\) and \(p^{-2a}\to0\), the bound tends to
342 zero. This proves the derivative convergence and the claimed
343 instability.
344 
345 More explicitly, for any bounded window and positive measurement
346 tolerance, two distinct radicals in this family can produce functions
347 closer than that tolerance throughout the window. This defeats a
348 recovery guarantee uniform over unbounded excluded primes. It does not
349 refute recovery under an a priori modulus bound or from complete exact
350 analytic data.
351 
352 For \(a=\beta=2,Q=1\), the mass removed by \(p\) is exactly
353 \(105/[\pi^4(p^4+1)]\). The examples \(p=101,1009,10007\) quantify how
354 quickly large-prime distinctions disappear on \(|s|\le5\).
355 
356 \hypertarget{the-function-as-an-arithmetic-hilbert-feature-space}{%
357 \section{The function as an arithmetic Hilbert feature
358 space}\label{the-function-as-an-arithmetic-hilbert-feature-space}}
359 
360 This section spells out the classical feature-space construction for
361 this arithmetic spectrum. It supplies a precise space in which the
362 complex-depth packets live.
363 
364 For \(c=(c_j)\in\ell^2\), define
365 
366 \[
367  (Uc)(s)=\sum_{j\ge1}c_j\sqrt{w_j}\,e^{-\lambda_js}.
368 \]
369 
370 The Cauchy--Schwarz inequality gives uniform convergence on bounded complex sets and
371 
372 \[
373  |(Uc)(s)|\le\|c\|_{\ell^2}\sqrt{A(2\Re s)}.
374 \]
375 
376 The map is injective: \(\sum|c_j|\sqrt{w_j}\le\|c\|\sqrt M<\infty\), so
377 the same spectral peeling argument, now for complex coefficients, forces
378 the first coefficient and then every coefficient to vanish whenever
379 \(Uc=0\). Positivity was needed to recover unknown frequencies in AI-1;
380 injectivity here only needs absolute summability and the known ordered
381 support. Specifically, after dividing by the largest remaining
382 exponential, the tail is bounded by
383 \(e^{-\Delta_jt}\sum_{k>j}|c_k|\sqrt{w_k}\), which tends to zero.
384 
385 The image \(\mathcal H_{a,\beta,Q}=U\ell^2\), with
386 \(\|Uc\|_{\mathcal H}=\|c\|_{\ell^2}\), is therefore a complete Hilbert
387 space of entire functions. Its reproducing kernel is
388 
389 \[
390  K(s,t)=A(s+\overline t).
391 \]
392 
393 Using an inner product linear in the first argument, the coefficient
394 vector of \(K(\cdot,t)\) is \((\sqrt{w_j}e^{-\lambda_j\overline t})_j\),
395 so \(\langle Uc,K(\cdot,t)\rangle=(Uc)(t)\). This also verifies the sign
396 and conjugation convention directly. Arbitrarily large finite Gram
397 matrices at distinct complex depths are strictly positive, as proved in
398 the base route. Thus the feature space has infinite rank, not a
399 finite-dimensional Prony model.
400 
401 The monomial moment matrices also have infinite rank:
402 \(m_k=(-1)^kA^{(k)}(0)=\sum_jw_j\lambda_j^k\), and for every nonzero
403 polynomial \(P\), \(\sum_jw_j|P(\lambda_j)|^2>0\), because a nonzero
404 polynomial cannot vanish at infinitely many distinct frequencies. All
405 finite derivative Hankel matrices are strictly positive definite.
406 
407 \hypertarget{complete-fixed-modulus-character-resolution}{%
408 \section{Complete fixed-modulus character
409 resolution}\label{complete-fixed-modulus-character-resolution}}
410 
411 Let \(\chi,\psi\) be Dirichlet characters modulo the same fixed integer
412 \(Q\), extended by zero outside the units, and put
413 \(\eta=\chi\overline\psi\). The base extension is
414 
415 \[
416  A_{\chi,\psi}(s)=\sum_{n\ge2}\mu(n)^2
417             \eta(n)n^{-2a}(1-n^{-\beta})^{2s}.
418 \]
419 
420 The endpoint subtraction in its Newton expansion is still exactly one:
421 
422 \[
423  A_{\chi,\psi}(s)
424  =\sum_{j\ge0}(-1)^j\binom{2s}{j}
425  \left[\frac{L(2a+j\beta,\eta)}
426               {L(4a+2j\beta,\eta^2)}-1\right].
427 \]
428 
429 Every character takes the value one at \(n=1\); at primes dividing
430 \(Q\), it is zero. The Euler product identity is valid even when
431 \(\eta\) is imprimitive, because the actual Dirichlet character and its
432 actual modulus are retained. All arguments of \(L\) lie in \(\Re v>1\),
433 where its Euler product converges absolutely and is nonzero. No unproved
434 zero-free-region hypothesis is used.
435 
436 \textbf{Theorem B.4 (AI-4), character sector decay.} Fix \(a>1/2,\beta>0,Q\),
437 and a closed sector \(|\arg s|\le\pi/2-\varepsilon\),
438 \(0<\varepsilon<\pi/2\). If \(\chi\ne\psi\), then uniformly in that
439 sector,
440 
441 \[
442  A_{\chi,\psi}(s)
443  =O_{a,\beta,Q,\varepsilon}\bigl(|s|^{-(2a-1/2)/\beta}\bigr),
444  \qquad
445  \frac{A_{\chi,\psi}(s)}{A_{a,\beta,Q}(s)}
446  =O_{a,\beta,Q,\varepsilon}\bigl(|s|^{-1/(2\beta)}\bigr).
447 \]
448 
449 These statements are uniform over the finite character set of this fixed
450 modulus. They are not uniform as \(Q\) grows.
451 
452 \textbf{Proof.} Distinct characters give a nonprincipal \(\eta\). Its
453 sum over a complete period is zero, so \(|\sum_{m\le y}\eta(m)|\le Q\).
454 Hence
455 
456 \[
457  \sum_{n\le x}\mu(n)^2\eta(n)
458  =\sum_{d\le\sqrt x}\mu(d)\eta(d)^2
459      \sum_{m\le x/d^2}\eta(m)=O_Q(\sqrt x).
460 \]
461 
462 Use partial integration against \(g_s(x)=x^{-2a}(1-x^{-\beta})^{2s}\),
463 whose endpoint values vanish when \(\Re s>0\). The counting bound
464 reduces the magnitude to a constant times
465 \(\int_1^\infty\sqrt x\,|g_s'(x)|\,dx\). Writing \(u=\Re s\) and
466 \(\gamma=(2a-1/2)/\beta>0\), direct substitution \(y=x^{-\beta}\) bounds
467 its two derivative terms by
468 
469 \[
470  \frac{2a}{\beta}B(\gamma,2u+1)
471  +2|s|B(\gamma+1,2u),
472 \]
473 
474 which is \(O_{a,\beta,\varepsilon}(|s|^{-\gamma})\) because
475 \(u\ge |s|\sin\varepsilon\). The principal diagonal has the nonzero
476 leading term \(\delta_Q\Gamma(\alpha)\beta^{-1}(2s)^{-\alpha}\),
477 \(\alpha=(2a-1)/\beta\), by the base sector theorem. Its modulus is
478 bounded below by a positive fixed-sector multiple of \(|s|^{-\alpha}\)
479 for sufficiently large \(|s|\). Division gives the second bound, since
480 \(\gamma-\alpha=1/(2\beta)\).
481 
482 At real positive depth, the matrix \((A_{\chi,\psi}(s))\) on any \(d\)
483 distinct characters is positive semidefinite and has identical diagonal
484 \(A_{a,\beta,Q}(s)\). Its normalized matrix tends to \(I_d\), with
485 operator norm error \(O_{a,\beta,Q,d}(s^{-1/(2\beta)})\). Indeed, the
486 off-diagonal entry bound controls the maximum row and column sums, and
487 thus the operator norm. This gives quantitative separation of a fixed
488 character bank. It is consistent with a complex or signed off-diagonal
489 entry and makes no complete-monotonicity assertion for that entry.
490 
491 \hypertarget{classical-comparison-and-limits-of-novelty}{%
492 \section{Classical comparison and limits of
493 novelty}\label{classical-comparison-and-limits-of-novelty}}
494 
495 Uniqueness of a positive measure from its Laplace transform is classical.
496 Schilling, Song and Vondraček~\cite[Proposition~1.2 and Theorem~1.4]{Schilling2010}
497 give the uniqueness and Bernstein representation results. Here the measure is
498 $\sum_jw_j\delta_{\lambda_j}$, the transform variable is real depth $s>0$,
499 and finiteness follows from $\sum_jw_j<\infty$.
500 The exact recovery of $a$, $\beta$ and $\operatorname{rad}(Q)$, the rank
501 asymptotic and the large-prime instability estimates are proved above.
502 They are not attributed to that reference.
503 
504 The Hilbert feature-space construction and finite Prony reconstruction are
505 also classical. This chapter supplies their realization for the arithmetic
506 spectrum. It does not claim historical priority for these mechanisms or for
507 the parameter-specific inverse and character results. A bounded literature
508 comparison cannot establish the absence of earlier work.
509 
510 \hypertarget{computation-contract-and-handoff}{%
511 \section{Computational verification}\label{computation-contract-and-handoff}}
512 
513 \path{checks/arithmetic_inverse.py} is deterministic. Its exact
514 checks use rational finite atomic surrogates with \(2a=4\), integer
515 \(\beta=1,2,3\), \(Q=1,2,6,30\), and one through five atoms. They
516 independently construct the annihilating polynomial from moments and
517 from its known rational nodes, verify the positive Vandermonde formula
518 for Hankel determinants, reconstruct weights, recover integer labels at
519 known \(\beta\), and test further exact recurrence moments. These
520 surrogates do not turn the infinite family's Hankel matrices into
521 finite-rank matrices.
522 
523 The high-precision examples use 180 decimal digits. They show two-rank
524 \(\beta\) estimates at ranks 100, 1000, and 10000, conditional
525 negative-depth peeling for the first three atoms at \(t=20,100,500\),
526 and large-prime bounded-window envelopes. Positive direct sums use
527 \(n\le5000\), with the analytic omitted-tail envelope above. Rank
528 enumeration uses \(n\le100000\). Actual cancellation factors are
529 reported. Numerical values and floating-point endpoints are
530 illustrations, not ball-arithmetic certificates.
531 
532 From the package's \texttt{research/} directory, run
533 \texttt{\small python checks/arithmetic\_inverse.py} in a Python environment
534 with \texttt{mpmath} installed. Results are written to
535 \path{evidence/arithmetic_inverse_checks.json}, including
536 input/source hashes, arithmetic conventions, tested ranges, software
537 versions and elapsed time. Earlier retrieval failures remain in the archived
538 run history. Theorems B.1--B.4 rest on the local proofs above. Historical
539 priority remains unresolved.

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