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1 \clearpage
2 \section{Divisor-Polyphase Persistence and the Hecke Filter Bank}
3 
4 \subsection{Exact-order divisor packets}
5 Let
6 \[
7  n=\prod_p p^{e_p}.
8 \]
9 For every $d\ge1$, define
10 \[
11  \boxed{
12  B_d(n)
13  =
14  \prod_{\substack{p\mid n\\d\mid e_p}}p.
15  }
16 \]
17 Since
18 \[
19  \sum_{d\mid e}\varphi(d)=e,
20 \]
21 we have the perfect reconstruction identity
22 \[
23  \boxed{
24  n
25  =
26  \prod_{d\ge1}
27  B_d(n)^{\varphi(d)}.
28  }
29 \]
30 The first packet is
31 \[
32  B_1(n)=\rad(n).
33 \]
34 The powerful part is
35 \[
36  \boxed{
37  \frac{n}{\rad(n)}
38  =
39  \prod_{d\ge2}
40  B_d(n)^{\varphi(d)}.
41  }
42 \]
43 This is a multiplicative polyphase decomposition indexed by exact character orders.
44 
45 \subsection{Quality as input-output persistence imbalance}
46 For a primitive triple,
47 \[
48  q-1
49  =
50  \frac{
51  \log(c/R_c)-\log(R_aR_b)
52  }{
53  \log R
54  }.
55 \]
56 Using the divisor packets,
57 \[
58  \boxed{
59  q-1
60  =
61  \frac{
62  \sum_{d\ge2}\varphi(d)\log B_d(c)
63  -
64  \log(R_aR_b)
65  }{\log R}.
66  }
67 \]
68 A high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.
69 
70 Strong $abc$ is equivalent to
71 \[
72  \prod_{d\ge2}
73  B_d(c)^{\varphi(d)}
74  \le
75  C_\eps R_aR_bR^\eps.
76 \]
77 The decomposition makes the target transparent, but not easier by itself. A global channel must explain why the exact-order packets cannot all be large simultaneously.
78 
79 \subsection{Local Tate-depth interpretation}
80 At an odd prime $p^e\parallel c$, the Frey discriminant has local depth $2e$. The divisors of $e$ organize the levels at which prime-power coefficient systems can become insensitive to the local extension. The exact-order divisor packet therefore has a local representation-theoretic interpretation, although local disappearance does not automatically produce a global low-level eigenform.
81 
82 \subsection{Atkin--Lehner Walsh packets}
83 Let
84 \[
85  W_1,\ldots,W_r
86 \]
87 be commuting Atkin--Lehner involutions. For a sign vector
88 \[
89  \varepsilon\in\{\pm1\}^r,
90 \]
91 define
92 \[
93  A_\varepsilon
94  =
95  \prod_{j=1}^{r}(I+\varepsilon_jW_j).
96 \]
97 Then
98 \[
99  \boxed{
100  A_\varepsilon^2=2^rA_\varepsilon,
101  }
102 \]
103 \[
104  A_\varepsilon A_{\varepsilon'}=0
105  \quad(\varepsilon\neq\varepsilon'),
106 \]
107 and
108 \[
109  \boxed{
110  \sum_\varepsilon A_\varepsilon=2^rI.
111  }
112 \]
113 The unnormalized analysis packets are integral. The common synthesis denominator is $2^r$, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
114 
115 The automorphic interpretation uses the standard Atkin--Lehner and modular-newform framework. The integral separation problem is related to congruence modules and modular degrees~\cite{Wiles,TaylorWiles,BCDT,Agashe,CesnaviciusManin}.
116 
117 \subsection{The same-sign separation problem}
118 After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let $\Lambda$ be an integral Hecke lattice and let $e_f$ be the rational projector onto the Frey eigenline. Define
119 \[
120  \eta_f
121  =
122  \min\{m\ge1:m e_f(\Lambda)\subseteq\Lambda\}.
123 \]
124 This is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.
125 

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