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1 \pageheading{ARITHMETIC DEPTH\; /\; INVERSE}{The function encodes its parameters}
2 The complete exact analytic germ determines
3 \[
4 \boxed{a,\quad\beta,\quad\operatorname{rad}(Q).}
5 \]
6 Precisely, two admissible triples define the same germ if and only if these
7 three quantities agree. Prime powers in $Q$ are invisible because the terms
8 use only coprimality. A finite Taylor polynomial or noisy measurements are
9 different inputs from the complete germ in this theorem.
10 
11 Let the support integers be $n_1<n_2<\cdots$, frequencies
12 $\lambda_j=-2\log(1-n_j^{-\beta})$, and weights $w_j=n_j^{-2a}$.
13 Starting at $R_1=A$, peel the ordered atoms:
14 \[
15 \lambda_j=\lim_{t\to\infty}\frac{\log R_j(-t)}t,\qquad
16 w_j=\lim_{t\to\infty}e^{-\lambda_jt}R_j(-t),
17 \]
18 \[
19 R_{j+1}(s)=R_j(s)-w_je^{-\lambda_js}.
20 \]
21 Every limit follows from the isolated largest remaining frequency and the
22 positive summable tail. Thus the exact function determines its spectral list.
23 
24 The counting law $N_Q(x)=\delta_Qx+O_Q(\sqrt x)$ gives
25 \[
26 \lambda_j=2\delta_Q^\beta j^{-\beta}
27 [1+O_{Q,\beta}(j^{-1/2}+j^{-\beta})],
28 \quad
29 \beta=-\lim_{j\to\infty}\frac{\log\lambda_j}{\log j}.
30 \]
31 Once $\beta$ is recovered,
32 \[
33 n_j=(1-e^{-\lambda_j/2})^{-1/\beta},\qquad
34 a=-\frac{\log w_j}{2\log n_j}.
35 \]
36 Prime membership in this integer support then recovers the excluded prime
37 set. The reconstruction is exact from infinite data; it does not certify
38 that a finite search has found every excluded prime.
39 
40 Classical Laplace uniqueness provides a comparison for the inverse mechanism.
41 The parameter-specific extraction, frequency-rank law, and explicit instability
42 are proved locally in the complete arithmetic route.
43 
44 \pageheading{ARITHMETIC DEPTH\; /\; RESOLUTION}{Separation improves while subtraction worsens}
45 \fig{10_arithmetic_inverse}{The frequency-rank estimates converge toward
46 the depth exponent. Later-atom peeling at large negative depth also has a
47 rapidly growing subtraction condition factor. These are finite examples
48 from the checked arithmetic run, not certified floating-point intervals.}
49 If earlier atoms have been removed exactly, set
50 $\Delta_j=\lambda_j-\lambda_{j+1}>0$, $M_{j+1}=\sum_{k>j}w_k$.
51 The logarithmic secant
52 \[
53 \widehat\lambda_j(t,h)=\frac{\log R_j(-t-h)-\log R_j(-t)}h
54 \]
55 obeys, for $t\ge0$, $h>0$,
56 \[
57 0\le\lambda_j-\widehat\lambda_j(t,h)
58 \le\frac1h\log\left(1+\frac{M_{j+1}}{w_j}e^{-\Delta_jt}\right).
59 \]
60 This improves with negative depth. But relative error in $A(-t)$ can be
61 amplified after subtraction by
62 \[
63 \kappa_j(t)=\frac{A(-t)}{R_j(-t)}
64 \sim\frac{w_1}{w_j}e^{(\lambda_1-\lambda_j)t}.
65 \]
66 For the displayed $Q=1$ example, the third-atom subtraction at $t=500$
67 has a condition factor of roughly $10^{108.8}$. Exact uniqueness therefore
68 does not promise easy recovery from ordinary precision.
69 
70 \pageheading{ARITHMETIC DEPTH\; /\; STABILITY}{A large prime can hide in every finite window}
71 For a prime $p\nmid Q$, any $R\ge0$, and integer $k\ge0$,
72 \[
73 \boxed{\sup_{|s|\le R}|A_{a,\beta,Q}^{(k)}(s)-A_{a,\beta,Qp}^{(k)}(s)|
74 \le\lambda_p^k e^{R\lambda_p}\frac{Z_Q(2a)}{p^{2a}+1}.}
75 \]
76 The difference consists of the $p$-divisible admissible squarefree atoms.
77 Their frequencies are at most $\lambda_p$ and their total weight is the
78 Euler-factor expression on the right. Termwise differentiation proves the
79 bound for every derivative.
80 \fig{11_prime_instability}{The proved compact-window envelope shrinks as
81 the excluded prime increases. At $a=\beta=2$, $Q=1$, $R=5$, excluding
82 $p=10007$ changes the entire window by at most approximately
83 $1.0749\times10^{-16}$. The depth-zero difference is known exactly.}
84 The radical differs for every such prime, while all bounded-window derivatives
85 converge to those of $A_Q$. Without a bound on excluded primes, there is no
86 uniformly stable inverse from these windows to the radical parameter.
87 
88 The same distinction appears in weak taps: exact structural invariants can
89 change under perturbations that are small at finite resolution.
90 
91 \pageheading{ARITHMETIC DEPTH\; /\; REALIZATION}{An infinite feature space with arithmetic labels}
92 For the ordered support, define $\phi_s(j)=\sqrt{w_j}e^{-\lambda_js}$.
93 The kernel $A(s+\overline t)$ is its weighted $\ell^2$ Gram kernel.
94 Summability and bounded frequencies make every feature vector well defined.
95 
96 If a weighted $\ell^2$ combination produces the zero entire function,
97 Cauchy--Schwarz makes its coefficients absolutely summable. Successive
98 dominant-frequency peeling then forces every coefficient to vanish. This
99 gives an injective Hilbert realization, and all finite derivative Hankel
100 matrices have full rank: a nonzero polynomial in $\lambda$ cannot vanish
101 on the infinite frequency support.
102 
103 \claim{Fixed-modulus character resolution}
104 For characters $\chi,\psi$ modulo a fixed $Q$, let
105 $\eta=\chi\overline\psi$. When $\eta$ is nonprincipal, the elementary
106 squarefree character sum is $O_Q(\sqrt x)$, with no density term.
107 The same sector partial-integration argument therefore gives
108 \[
109 A_{\chi,\psi}(s)
110 =O_{a,\beta,Q,\epsilon}(|s|^{-(2a-1/2)/\beta}),
111 \qquad |\arg s|\le\pi/2-\epsilon.
112 \]
113 Compared with the principal $|s|^{-(2a-1)/\beta}$ scale, this is faster
114 by $|s|^{-1/(2\beta)}$. The modulus is fixed; the constants are not asserted uniform as $Q$ grows. Individual off-diagonal functions can be complex and are not asserted
115 completely monotone.
116 

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