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1 \clearpage
2 \section{Integral Projectors, Cotangent Multiplicity, and Wiles Defect}
3 
4 \subsection{Projector denominator as a congruence exponent}
5 Let $V=U\oplus W$ be a rational decomposition and let $\Lambda\subset V$ be an integral lattice. Let $e_U$ be the rational projector onto $U$. Define
6 \[
7  \Lambda_U=\Lambda\cap U,
8  \qquad
9  \Lambda_W=\Lambda\cap W,
10 \]
11 and
12 \[
13  C_U=\Lambda/(\Lambda_U+\Lambda_W).
14 \]
15 
16 \begin{theorem}[Integral projector theorem]
17 The least positive integer $\eta_U$ satisfying
18 \[
19  \eta_U e_U(\Lambda)\subseteq\Lambda
20 \]
21 is the exponent of the finite quotient $C_U$.
22 \end{theorem}
23 
24 \begin{proof}
25 For $x=u+w\in\Lambda$, the condition $m e_U(x)=mu\in\Lambda_U$ is equivalent to
26 \[
27  mx=mu+mw\in\Lambda_U+\Lambda_W.
28 \]
29 Thus $m$ clears every projector denominator exactly when it annihilates every class in $C_U$.
30 \end{proof}
31 
32 For a primitive rank-one vector $v\in\Z^n$, the orthogonal projector denominator is
33 \[
34  \langle v,v\rangle,
35 \]
36 and the congruence quotient is cyclic of that order.
37 
38 \subsection{Why one cyclic quotient is not enough for split persistence}
39 The split persistence module
40 \[
41  \mathcal P_c
42  \cong
43  \bigoplus_{p^e\parallel c}
44  (\Z/p\Z)^{e-1}
45 \]
46 has large $p$-rank. The $p$-torsion of a cyclic target has rank at most one. Therefore every map from $\mathcal P_c$ to a cyclic group has a forced kernel of order at least
47 \[
48  \prod_{p^e\parallel c}p^{\max(e-2,0)}.
49 \]
50 This is a true obstruction to the split source. It motivates the Hecke cotangent module
51 \[
52  \mathfrak p/\mathfrak p^2,
53 \]
54 which can retain multiple independent congruence directions before cyclic compression.
55 
56 \subsection{A toy Hecke order}
57 Consider
58 \[
59  \mathbb T(\mathbf m)
60  =
61  \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
62 \]
63 For the target coordinate $x_0$, the cotangent module is
64 \[
65  \boxed{
66  \mathfrak p/\mathfrak p^2
67  \cong
68  \bigoplus_{i=1}^{r}\Z/m_i\Z.
69  }
70 \]
71 The cyclic congruence quotient is
72 \[
73  \boxed{
74  \Z/\lcm(m_1,\ldots,m_r)\Z.
75  }
76 \]
77 The quotient of cotangent order by congruence order is
78 \[
79  \frac{\prod_i m_i}{\lcm(m_i)}.
80 \]
81 This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
82 
83 The comparison between cotangent and congruence data, including non-complete-intersection defects and local--global formulas, is developed in the Wiles-defect literature~\cite{BKM1,BKM2}.
84 
85 \subsection{Complete-intersection control case}
86 Let $F\in\Z[X]$ be monic and squarefree over $\Q$, with an integer simple root $a_0$. Put
87 \[
88  \mathbb T=\Z[X]/(F)
89 \]
90 and let $\lambda:\mathbb T\to\Z$ evaluate at $a_0$. The ring is finite free over $\Z$ and reduced over $\Q$. Then the congruence quotient $\Z/\lambda(\operatorname{Ann}I)$ and cotangent module both have order
91 \[
92  |F'(a_0)|.
93 \]
94 To see this directly, write $F=(X-a_0)G$. The kernel $I$ is generated by $X-a_0$. Modulo $I^2$, the relation is $G(a_0)(X-a_0)=0$, so $I/I^2\cong\Z/F'(a_0)\Z$. The annihilator of $I$ is generated by $G$ modulo $F$, whose evaluation is $G(a_0)=F'(a_0)$. Thus the congruence ideal is $F'(a_0)\Z$ and the defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.
95 
96 \subsection{Two prime indices}
97 The persistence is indexed by conductor primes $q\mid abc$. Congruence depth is indexed by coefficient primes $\ell$. The natural local signal is a matrix
98 \[
99  D_f(\ell,q).
100 \]
101 Let $E/\Q_q$ have split multiplicative reduction with Tate parameter $q_E$, and let $\ell\ne q$. Scalarity of the full local Galois representation on $E[\ell^n]$ forces
102 \[
103  n\le\min\{v_\ell(q-1),v_\ell(v_q(q_E))\}.
104 \]
105 In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force $\ell^n\mid(q-1)$ and $\ell^n\mid v_q(q_E)$~\cite{SerreAbelian1989}. This is a necessary condition, rather than a converse or a condition on Frobenius alone. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-$q$ factor $q^{e-1}$. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
106 

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