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1 \hypertarget{finite-prime-sieve-families-and-explicit-detection-of-signed-atoms}{%
2 \chapter[Prime sieves and signed-atom detection]{Finite-prime sieve families and explicit detection of signed
3 atoms}\label{app:sieve}\label{finite-prime-sieve-families-and-explicit-detection-of-signed-atoms}}
4 
5 The construction extends the one-prime setting to a finite-prime parameter
6 family. The arguments below identify where classical orthogonal-polynomial
7 mechanisms enter the arithmetic construction.
8 
9 \hypertarget{definition-and-exact-coordinates}{%
10 \section{Definition and exact
11 coordinates}\label{definition-and-exact-coordinates}}
12 
13 Let \(F\) be a finite set of distinct primes, let
14 \(\boldsymbol\lambda=(\lambda_p)_{p\in F}\), and put
15 
16 \[
17 x_n=(1-n^{-2})^2,\qquad
18 \kappa_F(n;\boldsymbol\lambda)=\prod_{p\in F}(1+\lambda_p\mathbf1_{p\mid n}),
19 \] \[
20 \nu_{F,\boldsymbol\lambda}
21 =\sum_{n\ge1}\mu(n)^2n^{-4}\kappa_F(n;\boldsymbol\lambda)\delta_{x_n}.
22 \]
23 
24 The mass at \(x_1=0\) is exactly 1. A strength \(\lambda_p=-1\) deletes
25 every squarefree atom divisible by \(p\). Several deletions commute. The
26 depth-\(k\) family uses \(x^k\,d\nu\), for integers \(k\ge0\).
27 
28 Write \(D(s)=\zeta(s)/\zeta(2s)\). For arbitrary finite complex
29 strengths and \(\Re s>1\), absolute convergence permits expansion of the
30 finite prime product. The local squarefree Euler factors give
31 
32 \[
33 Z_F(s;\boldsymbol\lambda)
34 =\sum_{n\ge1}\mu(n)^2n^{-s}\kappa_F(n;\boldsymbol\lambda)
35 =D(s)\prod_{p\in F}\left(1+\frac{\lambda_p}{p^s+1}\right).
36 \]
37 
38 Equivalently, expansion in subsets \(S\subseteq F\) gives the
39 coefficient \(D(s)/\prod_{p\in S}(p^s+1)\) for
40 \(\prod_{p\in S}\lambda_p\). This is an independently useful coordinate
41 formula for prime overlap.
42 
43 Consequently
44 
45 \[
46 M_j(\boldsymbol\lambda)=\int x^j\,d\nu_{F,\boldsymbol\lambda}
47 =\sum_{r=0}^{2j}(-1)^r\binom{2j}{r}
48  D(4+2r)\prod_{p\in F}
49  \left(1+\frac{\lambda_p}{p^{4+2r}+1}\right).
50 \tag{1}
51 \]
52 
53 Depth-\(k\) moments are \(M_{k+j}\). With \(v=\pi^{-2}\), the even-zeta
54 formula makes every \(D(4+2r)\) a rational multiple of \(v^{r+2}\). Thus
55 the moments belong to \(\mathbb Q[v,\lambda_p:p\in F]\), and monic
56 orthogonal-polynomial coefficients belong to
57 \(\mathbb Q(v,\lambda_p:p\in F)\). This is an exact field statement,
58 with no decimal recognition step.
59 
60 For two primes the interaction term is explicit:
61 
62 \[
63 M_j(\lambda_p,\lambda_q)=M_j(0,0)+\lambda_p U_{p,j}
64 +\lambda_q U_{q,j}+\lambda_p\lambda_q U_{pq,j},
65 \]
66 
67 where \(U_{pq,j}=\sum_{pq\mid n}\mu(n)^2n^{-4}x_n^j>0\). More generally,
68 with all other strengths at least \(-1\), every mixed moment derivative
69 in a set of distinct primes is strictly positive. It is the integral
70 over atoms divisible by those primes, with the remaining nonnegative
71 prime factors. Infinitely many such atoms remain after any finite
72 collection of deletions. Pure second derivatives in any one strength
73 vanish because the moments are separately affine.
74 
75 \hypertarget{the-exact-positivity-domain}{%
76 \section{The exact positivity
77 domain}\label{the-exact-positivity-domain}}
78 
79 \textbf{Theorem C.1.} For real strengths, \(\nu_{F,\boldsymbol\lambda}\)
80 is a positive measure if and only if \(\lambda_p\ge-1\) for every
81 \(p\in F\). On this closed orthant, every finite Hankel matrix of every
82 nonnegative integer depth is strictly positive definite.
83 
84 \textbf{Proof.} The stated inequalities make each atom weight
85 nonnegative. The primes outside \(F\) supply infinitely many positive
86 atoms, even when all strengths are deletion endpoints. A nonzero
87 polynomial cannot vanish on infinitely many distinct nodes; therefore
88 its squared norm is strictly positive. Conversely, if \(\lambda_p<-1\),
89 the squarefree atom \(n=p\) has mass \((1+\lambda_p)p^{-4}<0\): no other
90 strength appears at that atom. The same argument works after every depth
91 tilt because \(x_p>0\). \(\square\)
92 
93 The positive monic families \(P_m^{(k;F,\boldsymbol\lambda)}\)
94 consequently exist at all degrees throughout the closed orthant. Their
95 Hankel determinants never vanish there, including at simultaneous
96 deletion endpoints. Each finite-degree coefficient is analytic in a
97 neighborhood of every point of that orthant. This neighborhood is a
98 finite-degree statement; the next theorem explains why no common
99 enlargement of the positivity domain works at all degrees.
100 
101 Together, Theorems C.1 and C.2 also prove the exact equivalence: every
102 finite Hankel matrix is positive semidefinite if and only if every
103 strength is at least \(-1\). Thus this is a complete positivity-domain
104 result, not only a sufficient condition.
105 
106 \hypertarget{an-explicit-polynomial-detector-for-every-bad-strength}{%
107 \section{An explicit polynomial detector for every bad
108 strength}\label{an-explicit-polynomial-detector-for-every-bad-strength}}
109 
110 For an integer \(a\ge2\), define
111 
112 \[
113 L_a(x)=\prod_{n=1}^{a-1}\frac{x-x_n}{x_a-x_n},\qquad
114 C_a=L_a(1)=\prod_{n=1}^{a-1}\frac{1-x_n}{x_a-x_n}>0,
115 \] \[
116 r_a=\frac{1-x_{a+1}}{1-x_a}
117 =\frac{a^4\bigl(2(a+1)^2-1\bigr)}{(a+1)^4(2a^2-1)}\in(0,1),
118 \] \[
119 f_{a,m}(x)=L_a(x)\left(\frac{1-x}{1-x_a}\right)^m,
120 \qquad m=0,1,2,\ldots.
121 \tag{2}
122 \]
123 
124 The degree is \(a-1+m\); all lower nodes are killed exactly;
125 \(f_{a,m}(x_a)=1\). At every higher node,
126 \(|f_{a,m}(x_n)|\le C_a r_a^m\), because each factor of \(L_a\)
127 increases on \([x_a,1]\) and \(1-x_n\le1-x_{a+1}\).
128 
129 Set \(C_F=\prod_{q\in F}\max(1,|1+\lambda_q|)\). Then
130 \(|\kappa_F(n)|\le C_F\), and the elementary decreasing-series bound
131 gives
132 
133 \[
134 \sum_{n>a}|\mu(n)^2n^{-4}\kappa_F(n)|
135 \le\frac{C_F}{3a^3}.
136 \tag{3}
137 \]
138 
139 \textbf{Theorem C.2.} If \(\lambda_p<-1\), then
140 
141 \[
142 \int f_{p,m}(x)^2\,d\nu_{F,\boldsymbol\lambda}(x)
143 \le\frac{1+\lambda_p}{p^4}
144 +\frac{C_F C_p^2}{3p^3}r_p^{2m}.
145 \tag{4}
146 \]
147 
148 Hence the Hankel matrix of order \(p+m\) is not positive semidefinite
149 whenever
150 
151 \[
152 \boxed{\frac{p C_F C_p^2}{3}r_p^{2m}<-(1+\lambda_p).}
153 \tag{5}
154 \]
155 
156 All quantities in (5) are rational when the strengths are rational. It
157 provides an exact finite certificate without calculating Hankel
158 eigenvalues or relying on floating-point root finding. A suitable finite
159 \(m\) exists because \(r_p<1\).
160 
161 \textbf{Proof.} The polynomial kills all atoms below \(p\), equals 1 at
162 the negative atom \(p\), and has the stated uniform bound above \(p\).
163 Replacing all higher signed contributions by their absolute values and
164 applying (3) proves (4). The coefficients of \(f_{p,m}\) then supply a
165 negative quadratic form for \([M_{i+j}]_{i,j<p+m}\) when (5) holds.
166 \(\square\)
167 
168 For depth \(k\), the same polynomial satisfies
169 
170 \[
171 \int x^k f_{p,m}(x)^2\,d\nu
172 \le\frac{(1+\lambda_p)x_p^k}{p^4}
173 +\frac{C_F C_p^2}{3p^3}r_p^{2m}.
174 \]
175 
176 Replace the right side of (5) by \(-(1+\lambda_p)x_p^k\) to obtain an
177 explicit depth-\(k\) detector. This proof extends to any finite signed
178 atomic measure with these nodes and the same summable weight envelope.
179 Its mechanism is an isolating polynomial followed by exponential
180 suppression of the higher nodes.
181 
182 \hypertarget{conditional-prime-fibers-and-all-prime-margin-bounds}{%
183 \section{Conditional prime fibers and all-prime margin
184 bounds}\label{conditional-prime-fibers-and-all-prime-margin-bounds}}
185 
186 Fix \(p\in F\), with all other strengths at least \(-1\), and write
187 \(\kappa(n)=\prod_{q\in F\setminus\{p\}}(1+\lambda_q\mathbf1_{q\mid n})\).
188 Separate the positive measure into its \(p\)-free and \(p\)-divisible
189 parts:
190 
191 \[
192 A_m=[\sum_{p\nmid n}\mu(n)^2n^{-4}\kappa(n)x_n^{i+j}]_{i,j<m},
193 \quad
194 B_m=[\sum_{p\mid n}\mu(n)^2n^{-4}\kappa(n)x_n^{i+j}]_{i,j<m}.
195 \]
196 
197 Both matrices are positive definite. For \(A_m\), use infinitely many
198 primes outside \(F\); for \(B_m\), use infinitely many squarefree
199 integers \(p\ell\), with primes \(\ell\) outside \(F\). This remains
200 true at every deletion endpoint of the other strengths.
201 
202 \textbf{Theorem C.3.} The one-variable fiber
203 
204 \[
205 \tau_m(\lambda_p)=\det\bigl(A_m+(1+\lambda_p)B_m\bigr)
206 \]
207 
208 has exact degree \(m\), and all its roots, with multiplicity, are real
209 and strictly less than \(-1\). The root nearest the positivity boundary
210 is \(-1-\delta_m\), with
211 
212 \[
213 \delta_m=\min_{0\ne f\in\mathbb R[x],\ \deg f<m}
214 \frac{\int f^2\,d\nu_{p\text{-free}}}{\int f^2\,d\nu_{p\text{-divisible}}}>0.
215 \tag{6}
216 \]
217 
218 The sequence \(\delta_m\) is nonincreasing, and, with
219 \(C_\kappa=\prod_{q\ne p}\max(1,1+\lambda_q)\),
220 
221 \[
222 \boxed{\delta_{p+m}\le\frac{p C_\kappa C_p^2}{3}r_p^{2m}.}
223 \tag{7}
224 \]
225 
226 \textbf{Proof.} The definite pencil is congruent to
227 \(I+(1+\lambda_p)A_m^{-1/2}B_mA_m^{-1/2}\), whose eigenvalues have
228 positive slopes. Its roots are \(-1-1/\eta_i\), with \(\eta_i>0\). This
229 proves the degree and root statements, and the reciprocal largest
230 eigenvalue is exactly (6). Polynomial-space inclusion proves
231 monotonicity. Test (6) at order \(p+m\) with (2). The denominator
232 includes the atom \(p\), of mass \(p^{-4}\), because \(\kappa(p)=1\).
233 The numerator kills every lower atom, has no atom at \(p\), and its
234 higher tail is bounded by \(C_\kappa C_p^2r_p^{2m}/(3p^3)\). This proves
235 (7) and therefore \(\delta_m\to0\). \(\square\)
236 
237 A sharper arithmetic bound uses the exact remaining tail mass
238 
239 \[
240 A_{\rm tail}=D(4)\frac{p^4}{p^4+1}
241 \prod_{q\ne p}\left(1+\frac{\lambda_q}{q^4+1}\right)
242 -\sum_{1\le n<p}\mu(n)^2n^{-4}\kappa(n)>0,
243 \]
244 
245 and replaces (7) by \(\delta_{p+m}\le p^4 C_p^2 A_{\rm tail}r_p^{2m}\).
246 Every finite-degree coefficient can have poles only at the fiber's
247 Hankel roots; numerator cancellations can remove a possible pole. The
248 theorem does not assert simple roots or global multivariate real
249 stability.
250 
251 At depth zero, the initial margin is exactly \(\delta_1=p^4\),
252 independently of all other safe strengths: the total \(p\)-free and
253 \(p\)-divisible masses have ratio \(p^4\), since all the other Euler
254 factors cancel. The higher-order margins contain the arithmetic overlap
255 information.
256 
257 The same argument applies to every nonnegative depth, replacing each
258 integral by its \(x^k\) tilt. The denominator's explicit atom
259 contribution becomes \(p^{-4}x_p^k\), and (7) acquires the factor
260 \(x_p^{-k}\).
261 
262 \hypertarget{resolution-nonconvex-finite-tests-and-a-different-additive-family}{%
263 \section{Resolution, nonconvex finite tests, and a different additive
264 family}\label{resolution-nonconvex-finite-tests-and-a-different-additive-family}}
265 
266 For \(F=\{2,3\}\), the order-one Hankel matrix is the scalar
267 
268 \[
269 M_0(\lambda_2,\lambda_3)=105\pi^{-4}
270 \left(1+\frac{\lambda_2}{17}\right)
271 \left(1+\frac{\lambda_3}{82}\right).
272 \]
273 
274 Thus \((0,0)\) and \((-34,-164)\) have the same positive order-one
275 Hankel matrix. Their midpoint \((-17,-82)\) has zero total mass. The
276 unrestricted real parameter set passing the order-one
277 positive-definiteness test is therefore nonconvex. This is a concrete
278 warning about a finite test, while the true positivity domain in Theorem
279 C.1 is a convex closed orthant.
280 
281 The positive-semidefinite feasible set is nonconvex as well: the points
282 \((0,0)\) and \((-34,-246)\) have positive mass factors 1 and 2, while
283 their convex combination with weights \(3/5\) and \(2/5\), namely
284 \((-68/5,-492/5)\), has mass factor \(-1/25\).
285 
286 An additive prime family is a different deformation:
287 
288 \[
289 \widetilde\nu_{F,\boldsymbol\gamma}
290 =\sum_{n\ge1}\mu(n)^2n^{-4}
291 \left(1+\sum_{p\in F}\gamma_p\mathbf1_{p\mid n}\right)\delta_{x_n}.
292 \]
293 
294 Its exact positivity domain is the polyhedron
295 
296 \[
297 1+\sum_{p\in S}\gamma_p\ge0\quad(S\subseteq F),
298 \quad\text{equivalently}\quad
299 1+\sum_{\gamma_p<0}\gamma_p\ge0.
300 \]
301 
302 Necessity uses the squarefree integer \(n=\prod_{p\in S}p\); sufficiency
303 checks every atom's prime subset. Its finite Hankel matrices are affine
304 functions of the strengths, so their positive-semidefinite feasible sets
305 are convex spectrahedra. In contrast, the multiplicative family includes
306 the arithmetic prime-overlap terms and has polynomial parameter
307 dependence. For example, additive strengths \(\gamma_2=\gamma_3=-3/4\)
308 leave both individual prime weights positive but make the \(n=6\) weight
309 negative; multiplicative strengths \(\lambda_2=\lambda_3=-3/4\) remain
310 safely positive.
311 
312 \hypertarget{a-limited-nonreduction-statement}{%
313 \section{A limited nonreduction
314 statement}\label{a-limited-nonreduction-statement}}
315 
316 If at least two strengths in a positive multiplicative family are
317 nonzero, its entire depth-zero monic polynomial sequence cannot equal
318 any one-prime sequence on the same variable and node system. A direct
319 proof avoids importing a determinacy theorem. Write
320 \(P_m=x^m+\sum_{j<m}c_{mj}x^j\). The equations \(\int P_m\,d\nu=0\)
321 recursively determine every normalized moment from the monic sequence.
322 Equal sequences therefore give equal normalized Stieltjes series for
323 \(|z|>1\). Analytic continuation on the connected complement of the
324 common closed node set makes their Stieltjes functions equal, and
325 residues at each isolated node give equal normalized atom weights. The
326 mass 1 at zero fixes the normalization constant. Finally, the atom at
327 each prime \(p\) reads off \(1+\lambda_p\), whereas a one-prime
328 deformation changes at most one such atom. This is a precise separator
329 from the one-prime family. It
330 does not address arbitrary affine variable changes.
331 
332 The multi-prime Stieltjes kernel also retains infinitely many isolated
333 poles at primes outside \(F\), so the preceding one-prime proof of
334 non-D-finiteness applies unchanged. Orthogonal-polynomial existence,
335 definite-pencil spectral theory, and compact moment determinacy are
336 classical mechanisms. The specific arithmetic coordinates, parameter
337 geometry, and explicit finite-resolution detector are the objects
338 developed here.
339 
340 \hypertarget{worked-arithmetic-examples}{%
341 \section{Worked arithmetic
342 examples}\label{worked-arithmetic-examples}}
343 
344 For \(F=\{2,3\}\), the following coefficients are rounded high-precision
345 evaluations of (1) and the Hankel formulas. They are not exact decimal
346 constants.
347 
348 \begin{longtable}[]{@{}
349   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
350   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
351   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}@{}}
352 \toprule\noalign{}
353 \begin{minipage}[b]{\linewidth}\raggedright
354 \((\lambda_2,\lambda_3)\)
355 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
356 \(P_1(x)\)
357 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
358 \(P_2(x)\)
359 \end{minipage} \\
360 \midrule\noalign{}
361 \endhead
362 \bottomrule\noalign{}
363 \endlastfoot
364 \((-1,-1)\) & \(x-0.00200014863265043361\) &
365 \(x^2-0.933902261393426053x+0.000000921271332319859170\) \\
366 \((-1/2,1)\) & \(x-0.0377167661732468061\) &
367 \(x^2-0.725293001442481628x+0.000932648146085806112\) \\
368 \((1,1)\) & \(x-0.0822907496962840844\) &
369 \(x^2-0.655873200757636623x+0.00211153000026086129\) \\
370 \end{longtable}
371 
372 Simultaneously deleting \(2,3,5\) gives
373 \(P_1(x)=x-0.000528721422047782375\) and
374 \(P_2(x)=x^2-0.966491641778821472x+0.0000000795531413796910514\), again
375 rounded. This removes the low prime shelves while retaining the fixed
376 mass at zero and an infinite tail.
377 
378 For the variable strength at \(p=2\), the nearest conditional margins
379 are:
380 
381 \begin{longtable}[]{@{}
382   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.2727}}
383   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}
384   >{\raggedleft\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3636}}@{}}
385 \toprule\noalign{}
386 \begin{minipage}[b]{\linewidth}\raggedright
387 Hankel order
388 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
389 Other strength \(\lambda_3=-1\)
390 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
391 Other strength \(\lambda_3=3\)
392 \end{minipage} \\
393 \midrule\noalign{}
394 \endhead
395 \bottomrule\noalign{}
396 \endlastfoot
397 1 & \(16\), exact & \(16\), exact \\
398 2 & \(0.0939109919936561\) & \(1.43829172807931\) \\
399 3 & \(0.000302450514004102\) & \(0.0346655628067259\) \\
400 4 & \(0.000000528145808944195\) & \(0.000223749375519984\) \\
401 \end{longtable}
402 
403 The noninteger entries are approximate symmetric-pencil calculations.
404 They show an actual prime-overlap effect beyond the mass test. In these
405 calculations the defect \(\lambda_2=-1.001\) is visible at order 3 when
406 the 3-shelf is deleted, but only at order 4 when that shelf is
407 strengthened to 3. This comparison is a numerical example, not a
408 universal monotonicity theorem.
409 
410 The separate rational certificates (5) do not depend on those
411 approximate roots. The script constructs negative detectors for primes
412 \(2,3,5,7\), at Hankel orders \(8,11,19,40\), respectively. These
413 sufficient orders are not asserted to be minimal. In the \(p=7\)
414 example, the other strengths are
415 \(\lambda_2=1/2,\lambda_3=-1,\lambda_5=2\), and \(\lambda_7=-1.0001\);
416 the exact rational upper bound on the detector quadratic form is approximately
417 \(-9.74271\times10^{-9}\).
418 
419 \hypertarget{computation-contract}{%
420 \section{Computation contract}\label{computation-contract}}
421 
422 \path{checks/multi_prime_sieve.py} uses exact rational arithmetic
423 for Euler-factor versus subset-expansion comparisons, Bernoulli-based
424 even-zeta coordinates, isolating-polynomial values, and the rational
425 detection criterion (5). It separately evaluates the infinite moments at
426 high precision, tests positive orthogonality grids, compares them with a
427 direct finite squarefree sum plus a proved tail envelope, calculates
428 conditional definite-pencil roots, and evaluates signed detector quadratic forms.
429 These decimal checks are not interval certificates. Universal positivity
430 and negativity conclusions follow from the displayed proofs and exact
431 rational bounds.
432 
433 The passing run records 1,019 exact rational assertions, 102 positive
434 parameter/depth cases, 1,395 lower-monomial orthogonality conditions,
435 381 recurrence identities, 40 conditional-pencil cases, and four exact
436 negative-atom certificates. Working precision is 320 decimal digits.
437 Maximum absolute orthogonality residual was approximately
438 \(6.33\times10^{-322}\), maximum recurrence residual
439 \(2.23\times10^{-303}\), and maximum scale-normalized pencil determinant
440 residual \(5.06\times10^{-321}\). These are finite numerical consistency
441 results, not interval certificates. No execution failure occurred in
442 this route's run.
443 
444 The script records actual versions, source hash, parameter bounds,
445 finite counts, precision, residuals, examples, and any failed criteria
446 in \path{evidence/multi_prime_sieve_checks.json}. It accepts an
447 optional output path so a parent bounded runner can place the result
448 inside a fresh evidence directory. No broad priority search was
449 performed for this specific extension; historical novelty remains
450 unresolved.

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