Special Functions

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1 \hypertarget{v11-resolution_full-a-visual-atlas-of-finite-resolution}{%
2 \chapter[Finite-resolution certificates]{A visual atlas of finite
3 resolution}\label{app:resolution}\label{v11-resolution_full-a-visual-atlas-of-finite-resolution}}
4 
5 A structural change need not produce a large change in the measurements
6 we can make. A tiny feedback tap can change the order of an entire
7 function. An excluded prime can change the exact arithmetic inverse. A
8 negative atom can destroy positivity of a measure. Each change becomes
9 visible at a different resolution.
10 
11 This atlas makes those statements quantitative. Its three controls are
12 observation radius, absolute precision and polynomial degree. Each
13 example includes a bound that can be evaluated without trusting the
14 appearance of a curve. The large radii below are sufficient guarantees
15 from conservative estimates. The degree bounds bracket a detection
16 problem. Neither is presented as an optimal threshold.
17 
18 \begin{longtable}[]{@{}
19   >{\raggedright\arraybackslash}P{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
20   >{\raggedright\arraybackslash}P{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
21   >{\raggedright\arraybackslash}P{(\columnwidth - 6\tabcolsep) * \real{0.2500}}
22   >{\raggedright\arraybackslash}P{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}}
23 \toprule\noalign{}
24 \begin{minipage}[b]{\linewidth}\raggedright
25 Family
26 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
27 Structural difference
28 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
29 Finite observation
30 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
31 Certificate that resolves it
32 \end{minipage} \\
33 \midrule\noalign{}
34 \endhead
35 \bottomrule\noalign{}
36 \endlastfoot
37 Positive feedback & A nonzero fast tap changes entire order & Values on
38 a fixed complex disk & A sufficient radius for a factor of two
39 separation \\
40 Arithmetic depth & The excluded prime set changes & Values or finitely
41 many derivatives on a disk & A bracket for the negative depth needed to
42 reach an absolute tolerance \\
43 Signed sieve & One prime shelf has negative mass & All polynomial
44 quadratic forms up to a given degree & An explicit rational polynomial
45 with negative quadratic form \\
46 \end{longtable}
47 
48 \Needspace{9\baselineskip}
49 \hypertarget{v11-resolution_full-a-weak-tap-that-changes-global-growth}{%
50 \section{A weak tap that changes global
51 growth}\label{v11-resolution_full-a-weak-tap-that-changes-global-growth}}
52 
53 Let the normalized positive feedback function satisfy
54 
55 \[
56 F_\varepsilon'(z)=1+(1-\varepsilon)F_\varepsilon(z/4)^2
57  +\varepsilon F_\varepsilon(3z/4)^2,
58 \qquad F_\varepsilon(0)=0,
59 \qquad 0\le\varepsilon\le1.
60 \]
61 
62 The base theorem gives a unique entire function with nonnegative odd
63 coefficients. At zero weight, the order is \(1/2\). At every positive
64 weight it is
65 
66 \[
67 \rho=\frac{\log2}{\log(4/3)}\approx2.40942084.
68 \]
69 
70 Thus the order jumps at a parameter value where the entire functions
71 themselves converge uniformly on every compact set.
72 
73 \hypertarget{v11-resolution_full-the-disk-has-a-numerical-error-budget}{%
74 \subsection{The disk has a numerical error
75 budget}\label{v11-resolution_full-the-disk-has-a-numerical-error-budget}}
76 
77 Write \(f=f_{3/4}\) for the normalized diagonal function, and
78 \(f(z)=\sum_{n\ge0}d_nz^{2n+1}\). The inherited tap continuity theorem
79 gives
80 
81 \[
82 \sup_{|z|\le R}|F_\varepsilon(z)-F_0(z)|
83 \le \varepsilon L(R),\qquad
84 L(R)=\sum_{n\ge1}n d_nR^{2n+1}
85 =\frac{Rf'(R)-f(R)}2.
86 \tag{RA-1}
87 \]
88 
89 The following elementary certificate makes the unit disk example fully
90 computable. Positive coefficient domination by tangent, followed by
91 \(\sin x\le x\) and \(\cos x\ge1-x^2/2\), gives
92 
93 \[
94 f(4/3)\le\tan(4/3)\le12.
95 \]
96 
97 For \(N\ge1\) and \(0\le R\le1\), put \(t=3R/4\). If
98 \((N+2)t^2\le N+1\), then \(n t^{2n+1}\) is nonincreasing for
99 \(n\ge N+1\), and hence
100 
101 \[
102 0\le L(R)-\sum_{n=1}^{N}n d_nR^{2n+1}
103 \le12(N+1)t^{2N+3}.
104 \tag{RA-2}
105 \]
106 
107 Indeed the left side is a sum of \(d_n(4/3)^{2n+1}\) multiplied by
108 \(n t^{2n+1}\). Bound the multiplier by its first value, then use the
109 positive total mass \(f(4/3)\le12\). The coefficients \(d_n\) are
110 rational. With \(N=40\), the exact rational upper bound is less than
111 
112 \[
113 L(1)<0.241890267.
114 \]
115 
116 The decimal shown here is rounded upward from the rational certificate.
117 Consequently
118 
119 \[
120 \boxed{\sup_{|z|\le1}|F_{10^{-12}}(z)-F_0(z)|<2.41890267\times10^{-13},}
121 \]
122 
123 although their entire orders are different. If observations have an
124 absolute error allowance at least half this bound, midpoint data can be
125 consistent with both functions throughout the disk. This is an ambiguity
126 under a stated noise allowance, not equality of their exact values.
127 
128 See Figure~\ref{fig:resolution_01_weak_tap} in Chapter~\ref{ch:resolution}.
129 
130 \textbf{How to read the figure.} The left curves divide the difference
131 by the tap weight, so the relevant small quantity is the plotted height
132 multiplied by \(\varepsilon\). The right panel records the proved order,
133 which is determined by arbitrarily large radii. The curve computation
134 uses powers through \(z^{201}\), with a normalized analytic omitted-tail
135 bound below \(5.26\times10^{-23}\) on the unit interval. Ordinary
136 floating rounding is separate from that analytic tail bound.
137 
138 \hypertarget{v11-resolution_full-a-radius-at-which-a-difference-is-guaranteed}{%
139 \subsection{A radius at which a difference is
140 guaranteed}\label{v11-resolution_full-a-radius-at-which-a-difference-is-guaranteed}}
141 
142 The base lower-type theorem supplies, for every \(0<\varepsilon\le1\),
143 
144 \[
145 \log F_\varepsilon(R)\ge c_\varepsilon(R+3/\varepsilon)^\rho
146 \ge c_\varepsilon R^\rho\qquad(R\ge2),
147 \] \[
148 c_\varepsilon=\frac{\log2}{(4/\varepsilon+8/3)^\rho}.
149 \]
150 
151 For the slow function, the base upper-type construction with
152 \(q=r=1/4\), \(N=2\), yields
153 
154 \[
155 \log F_0(R)\le C_0\sqrt R\qquad(R\ge e),
156 \qquad C_0=e^{-1/2}(1+22\log2)\approx9.85566103.
157 \]
158 
159 The value of \(C_0\) comes directly from the earlier explicit constants:
160 \(V_2=15\log2\), \(\Delta=2\log2\), and \(B_+=5\log2\). Thus
161 \(V_2+1+\Delta+B_+=1+22\log2\).
162 
163 Define
164 
165 \[
166 R_2(\varepsilon)=\max\left\{
167  e,\left(\frac{2C_0}{c_\varepsilon}\right)^{1/(\rho-1/2)},
168  \left(\frac{2\log2}{c_\varepsilon}\right)^{1/\rho}\right\}.
169 \tag{RA-3}
170 \]
171 
172 Then, for every \(R\ge R_2(\varepsilon)\),
173 
174 \[
175 \boxed{F_\varepsilon(R)\ge2F_0(R).}
176 \]
177 
178 To prove this, the last two constraints imply that
179 \(c_\varepsilon R^\rho\) is at least both \(2C_0\sqrt R\) and
180 \(2\log2\). It is therefore at least their half-sum
181 \(C_0\sqrt R+\log2\). Exponentiating proves the claim.
182 
183 \begin{longtable}[]{@{}
184   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.2727}}
185   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}
186   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}@{}}
187 \toprule\noalign{}
188 \begin{minipage}[b]{\linewidth}\raggedright
189 Tap weight
190 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
191 Difference on the unit disk is at most
192 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
193 Sufficient radius \(R_2\), rounded
194 \end{minipage} \\
195 \midrule\noalign{}
196 \endhead
197 \bottomrule\noalign{}
198 \endlastfoot
199 \(10^{-3}\) & \(2.419\times10^{-4}\) & \(2.029\times10^5\) \\
200 \(10^{-6}\) & \(2.419\times10^{-7}\) & \(1.237\times10^9\) \\
201 \(10^{-9}\) & \(2.419\times10^{-10}\) & \(7.549\times10^{12}\) \\
202 \(10^{-12}\) & \(2.419\times10^{-13}\) & \(4.608\times10^{16}\) \\
203 \end{longtable}
204 
205 The radius column displays the analytic formula approximately. For a
206 certificate use the formula itself or a rigorously rounded upper value.
207 These large numbers reflect the conservatism of the global comparison.
208 They do not mean that observations at all smaller radii fail. The
209 formula grows proportionally to \(\varepsilon^{-\rho/(\rho-1/2)}\) as
210 the tap vanishes.
211 
212 See Figure~\ref{fig:resolution_02_growth_resolution} in Chapter~\ref{ch:resolution}.
213 
214 \hypertarget{v11-resolution_full-an-excluded-prime-and-the-precision-needed-to-see-it}{%
215 \section{An excluded prime and the precision needed to see
216 it}\label{v11-resolution_full-an-excluded-prime-and-the-precision-needed-to-see-it}}
217 
218 Fix \(a=\beta=2\), and use the arithmetic depth family
219 
220 \[
221 A_Q(s)=\sum_{\substack{n\ge2\\\mu(n)^2=1\\(n,Q)=1}}
222  n^{-4}(1-n^{-2})^{2s}.
223 \]
224 
225 For a prime \(p\), let \(D_p=A_1-A_p\). This positive difference
226 consists exactly of the squarefree atoms divisible by \(p\). Its largest
227 frequency is
228 
229 \[
230 \lambda_p=-2\log(1-p^{-2}),
231 \]
232 
233 and its total mass is \(D(4)/(p^4+1)\), where
234 \(D(4)=\zeta(4)/\zeta(8)=105/\pi^4\).
235 
236 \hypertarget{v11-resolution_full-exact-disk-bounds-include-derivative-measurements}{%
237 \subsection{Exact disk bounds include derivative
238 measurements}\label{v11-resolution_full-exact-disk-bounds-include-derivative-measurements}}
239 
240 For every integer \(k\ge0\) and \(R\ge0\),
241 
242 \[
243 \sup_{|s|\le R}|D_p^{(k)}(s)|
244 \le \lambda_p^k e^{R\lambda_p}\frac{D(4)}{p^4+1}.
245 \tag{RA-4}
246 \]
247 
248 This is the inherited arithmetic inverse theorem, now read as a
249 precision requirement. For fixed \(R\), the envelope is of order
250 \(p^{-4-2k}\). Each additional derivative suppresses the large-prime
251 difference even further in absolute units. This does not rule out
252 methods that rescale derivatives and account for their corresponding
253 noise amplification.
254 
255 At integer radius \(R=5\), an especially simple rational certificate
256 follows from \(D(4)<\zeta(4)<4/3\):
257 
258 \[
259 \boxed{\sup_{|s|\le5}|D_p(s)|
260  <\frac{4}{3(p^4+1)}(1-p^{-2})^{-10}.}
261 \tag{RA-5}
262 \]
263 
264 For derivatives, multiply this rational bound by \([2/(p^2-1)]^k\),
265 using \(-\log(1-u)\le u/(1-u)\). Thus this part of the certificate needs
266 no transcendental evaluation at all.
267 
268 \begin{longtable}[]{@{}lr@{}}
269 \toprule\noalign{}
270 Excluded prime & Rational upper bound on \(|s|\le5\), rounded upward \\
271 \midrule\noalign{}
272 \endhead
273 \bottomrule\noalign{}
274 \endlastfoot
275 101 & \(1.283\times10^{-8}\) \\
276 1009 & \(1.287\times10^{-12}\) \\
277 10007 & \(1.330\times10^{-16}\) \\
278 \end{longtable}
279 
280 See Figure~\ref{fig:resolution_03_prime_precision} in Chapter~\ref{ch:resolution}.
281 
282 \hypertarget{v11-resolution_full-increasing-negative-depth-gives-a-two-sided-threshold-bracket}{%
283 \subsection{Increasing negative depth gives a two-sided threshold
284 bracket}\label{v11-resolution_full-increasing-negative-depth-gives-a-two-sided-threshold-bracket}}
285 
286 There is also an elementary lower bound, because the removed prime atom
287 is present:
288 
289 \[
290 \boxed{p^{-4}e^{\lambda_pT}\le D_p(-T)
291 \le\frac{D(4)}{p^4+1}e^{\lambda_pT}\qquad(T\ge0).}
292 \tag{RA-6}
293 \]
294 
295 Choose an absolute tolerance \(\eta>D(4)/(p^4+1)\). The strict
296 inequality makes the initial difference smaller than the tolerance.
297 Define
298 
299 \[
300 T_-=\frac{\log\bigl(\eta(p^4+1)/D(4)\bigr)}{\lambda_p},
301 \qquad T_+=\frac{\log(\eta p^4)}{\lambda_p}.
302 \tag{RA-7}
303 \]
304 
305 Then \(D_p(-T)\le\eta\) for \(0\le T\le T_-\), while \(D_p(-T)\ge\eta\)
306 for \(T\ge T_+\). The true crossing exists and is unique because
307 \(D_p(-T)\) is continuous and strictly increasing. Therefore it belongs
308 to \([T_-,T_+]\). If the tolerance hypothesis fails, the interval
309 starting at zero is not certified by this formula. The executable record
310 represents an absent lower interval by \texttt{null}.
311 
312 The bracket width and its large-prime scale are explicit:
313 
314 \[
315 T_+-T_-=\frac{\log\bigl(D(4)p^4/(p^4+1)\bigr)}{\lambda_p},
316 \qquad
317 T_+=\frac{p^2}{2}\bigl(4\log p+\log\eta\bigr)(1+O(p^{-2})).
318 \]
319 
320 For \(\eta=10^{-6}\), rounded values are:
321 
322 \begin{longtable}[]{@{}
323   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.2727}}
324   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}
325   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}@{}}
326 \toprule\noalign{}
327 \begin{minipage}[b]{\linewidth}\raggedright
328 Prime
329 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
330 No larger than tolerance through \(T_-\)
331 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
332 At least tolerance by \(T_+\)
333 \end{minipage} \\
334 \midrule\noalign{}
335 \endhead
336 \bottomrule\noalign{}
337 \endlastfoot
338 101 & 23,307.789 & 23,690.516 \\
339 1009 & 7,012,695.633 & 7,050,894.424 \\
340 10007 & 1,149,287,766.447 & 1,153,045,061.461 \\
341 \end{longtable}
342 
343 These are approximate evaluations of exact analytic endpoints, not
344 directed-rounding intervals.
345 
346 \hypertarget{v11-resolution_full-the-direct-difference-and-subtraction-of-full-functions-have-different-conditioning}{%
347 \subsection{The direct difference and subtraction of full functions
348 have different
349 conditioning}\label{v11-resolution_full-the-direct-difference-and-subtraction-of-full-functions-have-different-conditioning}}
350 
351 The positive series for \(D_p\) can be evaluated directly if the
352 excluded prime is already specified. Recovering it by subtracting
353 independently evaluated full functions can be much harder. The atom
354 \(n=2\) in \(A_1\) gives
355 
356 \[
357 \boxed{\frac{A_1(-T)}{D_p(-T)}
358 \ge\frac{p^4+1}{16D(4)}e^{(\lambda_2-\lambda_p)T}.}
359 \tag{RA-8}
360 \]
361 
362 At the sufficient endpoint \(T_+\), the base-ten logarithm of this lower
363 bound is about 5,924 for \(p=101\), 1,761,864 for \(p=1009\), and
364 288,119,991 for \(p=10007\). These are conditioning scales for
365 subtracting full signals. They are not running times, required precision
366 for every possible algorithm, or claimed numerical evaluations of the
367 full functions at those enormous depths. The logarithmic bound is
368 computed directly without forming those functions.
369 
370 See Figure~\ref{fig:resolution_04_prime_depth} in Chapter~\ref{ch:resolution}.
371 
372 \hypertarget{v11-resolution_full-a-negative-atom-that-every-low-degree-test-misses}{%
373 \section{A negative atom that every low-degree test
374 misses}\label{v11-resolution_full-a-negative-atom-that-every-low-degree-test-misses}}
375 
376 Consider the one-prime signed sieve at \(p=2\), with
377 \(\lambda_2=-1-\delta\), \(\delta>0\). Its nodes are
378 \(x_n=(1-n^{-2})^2\), and its quadratic form is
379 
380 \[
381 Q_\delta(f)=\sum_{\substack{n\ge1\\\mu(n)^2=1\\n\text{ odd}}}n^{-4}f(x_n)^2
382 -\delta\sum_{\substack{n\ge1\\\mu(n)^2=1\\n\text{ even}}}n^{-4}f(x_n)^2.
383 \]
384 
385 Every even squarefree atom has negative mass. We will certify that all nonzero real
386 polynomials through degree four nevertheless have positive quadratic form for
387 \(\delta=10^{-12}\), and then construct a degree-seven polynomial of
388 negative quadratic form. This is an assertion about every polynomial in a
389 finite-dimensional space, not a plot of a few test vectors.
390 
391 \hypertarget{v11-resolution_full-a-rational-certificate-for-every-polynomial-through-a-chosen-degree}{%
392 \subsection{A rational certificate for every polynomial through a
393 chosen
394 degree}\label{v11-resolution_full-a-rational-certificate-for-every-polynomial-through-a-chosen-degree}}
395 
396 Choose \(d+1\) distinct odd squarefree labels \(n_0,\ldots,n_d\),
397 including 1, and set \(y_i=x_{n_i}\). Their Lagrange polynomials are
398 
399 \[
400 \ell_i(x)=\prod_{j\ne i}\frac{x-y_j}{y_i-y_j},
401 \quad
402 b_i=\prod_{j\ne i}\frac{\max(y_j,1-y_j)}{|y_i-y_j|},
403 \quad K_d=\sum_{i=0}^{d}n_i^4b_i^2.
404 \]
405 
406 For \(x\in[0,1]\), we have \(|\ell_i(x)|\le b_i\). Interpolation and
407 weighted Cauchy--Schwarz give, for every real polynomial of degree at
408 most \(d\),
409 
410 \[
411 |f(x)|^2\le
412 \left(\sum_{i=0}^{d}n_i^{-4}f(y_i)^2\right)
413 \left(\sum_{i=0}^{d}n_i^4\ell_i(x)^2\right)
414 \le K_d\sum_{i=0}^{d}n_i^{-4}f(y_i)^2.
415 \]
416 
417 The total even mass satisfies
418 \(\sum_{n\text{ even}}\mu(n)^2n^{-4}<\frac1{16}(1+\frac13)=1/12\).
419 Keeping the selected nodes in the positive part yields the explicit
420 bound
421 
422 \[
423 \boxed{Q_\delta(f)\ge
424 \left(1-\frac{\delta K_d}{12}\right)
425 \sum_{i=0}^{d}n_i^{-4}f(y_i)^2.}
426 \tag{RA-9}
427 \]
428 
429 A nonzero polynomial of degree at most \(d\) cannot vanish at all
430 \(d+1\) nodes. Thus \(\delta<12/K_d\) proves strict positive
431 definiteness of the Hankel matrix of order \(d+1\).
432 
433 For \(d=4\), use labels \(1,3,5,7,11\). Exact rational arithmetic gives
434 
435 \[
436 K_4\approx1.50393441996\times10^{11},\qquad
437 12/K_4\approx7.9790713217\times10^{-11}>10^{-12}.
438 \]
439 
440 The inequality is checked as an exact rational comparison. Therefore
441 every nonzero polynomial of degree at most four has positive quadratic form in the
442 chosen signed measure.
443 
444 \hypertarget{v11-resolution_full-a-shorter-detector-by-killing-the-first-positive-shelves}{%
445 \subsection{A shorter detector by killing the first positive
446 shelves}\label{v11-resolution_full-a-shorter-detector-by-killing-the-first-positive-shelves}}
447 
448 Let \(S\) be exactly the set of all odd squarefree labels \(n\le K\),
449 where \(K\) is an odd squarefree integer, and let \(n_+\) be the next
450 such label. Assume \(n_+>2\), and define
451 
452 \[
453 P_{S,m}(x)=\prod_{n\in S}\frac{x-x_n}{x_2-x_n}
454 \left(\frac{1-x}{1-x_2}\right)^m,
455 \] \[
456 C_S=\prod_{n\in S}\frac{1-x_n}{|x_2-x_n|},
457 \qquad r_S=\frac{1-x_{n_+}}{1-x_2}<1.
458 \]
459 
460 This polynomial has degree \(|S|+m\), equals one at \(x_2\), and kills
461 every positive atom through the cutoff. At each remaining positive node,
462 the product of the absolute root factors is at most \(C_S\), and the
463 last factor is at most \(r_S^m\). Dropping all other negative atoms and
464 bounding the positive tail by the integral
465 \(\sum_{n>K}n^{-4}\le1/(3K^3)\) gives
466 
467 \[
468 \boxed{Q_\delta(P_{S,m})\le-\frac\delta{16}
469  +\frac{C_S^2}{3K^3}r_S^{2m}.}
470 \tag{RA-10}
471 \]
472 
473 For any chosen \(S\), a computable sufficient exponent is the first
474 nonnegative integer \(m\) for which
475 
476 \[
477  r_S^{2m}<\frac{3K^3\delta}{16C_S^2}.
478 \]
479 
480 It can be found by multiplying rational numbers, without logarithms or
481 eigenvalues. Searching finitely many choices of \(S\) improves the
482 sufficient degree but does not prove its optimality.
483 
484 For the worked defect \(\delta=10^{-12}\), take
485 
486 \[
487 S=\{1,3,5,7\},\quad K=7,\quad n_+=11,\quad m=3.
488 \]
489 
490 Then
491 
492 \[
493 C_S=\frac{756482048}{20777538825},\qquad
494 r_S=\frac{3856}{102487},
495 \]
496 
497 and \(P_{S,3}\) has degree seven. The exact rational right side of
498 (RA-10) is strictly negative and is approximately
499 
500 \[
501 \boxed{Q_{10^{-12}}(P_{S,3})\le-5.8845708899\times10^{-14}.}
502 \]
503 
504 The negative atom contributes \(-6.25\times10^{-14}\). The full positive
505 tail contributes at most \(3.6542911008\times10^{-15}\), a bound that
506 covers every omitted atom. The exact certificate is stored as numerator
507 and denominator. The original generic isolating detector needed degree
508 21 for this defect. Using the known sign pattern and killing later
509 positive shelves reduces the exhibited degree to seven.
510 
511 See Figure~\ref{fig:resolution_05_signed_detector} in Chapter~\ref{ch:resolution}.
512 
513 The fifth-order Hankel matrix is therefore positive definite, while the
514 eighth-order matrix is not positive semidefinite. The two intervening
515 polynomial degrees, five and six, are not decided by these certificates.
516 A numerical eigenvalue could suggest a sharper boundary, but it would be
517 a different calculation requiring its own conditioning and error
518 analysis.
519 
520 \hypertarget{v11-resolution_full-what-the-three-examples-establish-together}{%
521 \section{What the three examples establish
522 together}\label{v11-resolution_full-what-the-three-examples-establish-together}}
523 
524 The three resolution controls act on different mathematical objects.
525 Radius controls what part of an entire function is sampled. Precision
526 controls whether two different arithmetic parameters are distinguishable
527 within an observation model. Degree controls the richness of a
528 positivity test. The common lesson has a precise form: a property
529 determined by an unbounded collection of tests need not have a uniform
530 margin on each bounded subcollection.
531 
532 Here the positive tap changes a proved global order. The excluded prime
533 changes an exactly recoverable radical. The signed sieve lies outside
534 the exact positivity domain even while a whole low-degree polynomial
535 space passes. Each conclusion follows from explicit formulas and local
536 proofs. There is no claim that the three mechanisms are identical, and
537 no historical novelty claim for interpolation, spectral peeling, or the
538 use of isolating polynomials.
539 
540 \Needspace{11\baselineskip}
541 \hypertarget{v11-resolution_full-reproduction-and-evidence}{%
542 \section{Reproduction and
543 evidence}\label{v11-resolution_full-reproduction-and-evidence}}
544 
545 Run from the package\textquotesingle s \texttt{research/} directory:
546 
547 \begin{verbatim}
548 python checks/resolution_v11.py
549 python build_resolution_figures.py
550 \end{verbatim}
551 
552 The first command uses only the Python standard library. It records
553 \textbf{985 exact rational assertions} and \textbf{16 floating algebra
554 checks}, with source hashes and exact certificate numerators and
555 denominators. It tests interpolation identities in the monomial basis,
556 all selected root values, sample envelope inequalities, exact negative
557 quadratic forms for finite sums, coefficient continuity bounds and each displayed
558 certificate. The universal analytic conclusions rely on the proofs
559 above, rather than on the finite count.
560 
561 The figure command uses NumPy, Matplotlib and mpmath. Six figures are
562 saved in PDF and PNG form. The weak-tap curves use 100 decimal digits
563 and an explicit positive-coefficient tail bound. Every other curve
564 displays a closed analytic bound or rational certificate, so it has no
565 omitted arithmetic series. Displayed floating values and polynomial
566 curves are not interval arithmetic. The script never evaluates the huge
567 full arithmetic functions in the cancellation example.
568 
569 One preliminary presentation issue was corrected during development: a
570 negative formula for \(T_-\) was initially clamped to zero. That would
571 misleadingly label an already visible difference as invisible at zero.
572 The finished formula requires its stated tolerance hypothesis, and the
573 executable record marks an absent interval by \texttt{null}. The exact
574 assertion run passed. This correction did not affect any of the three
575 listed prime examples.

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