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1 \section{Kummer Globalization and the Fermat Character Packet}
2 
3 \subsection{The contact modulus}
4 Let
5 \[
6  c=\prod_{i=1}^{r}p_i^{e_i},
7  \qquad
8  R_c=\prod_i p_i,
9  \qquad
10  M=\frac{c}{R_c}
11  =\prod_i p_i^{e_i-1}.
12 \]
13 The filtered output contact packet is the cyclic group
14 \[
15  \Z/M\Z.
16 \]
17 The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
18 
19 \subsection{The global Kummer map}
20 Kummer theory gives
21 \[
22  H^1(\Q,\mu_M)
23  \cong
24  \Q^\times/(\Q^\times)^M.
25 \]
26 Define
27 \[
28  \boxed{
29  \kappa_c:
30  \Z/M\Z
31  \longrightarrow
32  H^1(\Q,\mu_M),
33  \qquad
34  k\longmapsto[c^k].
35  }
36 \]
37 The valuation vector of $c$ is
38 \[
39  (e_1,\ldots,e_r)
40  \in
41  (\Z/M\Z)^r.
42 \]
43 
44 \begin{theorem}[Exact Kummer kernel]
45 Let
46 \[
47  g_c=\gcd(M,e_1,\ldots,e_r).
48 \]
49 Then
50 \[
51  \boxed{
52  |\ker\kappa_c|=g_c,
53  \qquad
54  |\im\kappa_c|=M/g_c.
55  }
56 \]
57 \end{theorem}
58 
59 \begin{proof}
60 The class $[c]^k$ is trivial exactly when every prime valuation $ke_i$ is divisible by $M$. The least positive such $k$ is
61 \[
62  \operatorname{lcm}_i
63  \frac{M}{\gcd(M,e_i)}
64  =
65  \frac{M}{\gcd(M,e_1,\ldots,e_r)}.
66 \]
67 This is the order of the image. The kernel size follows.
68 \end{proof}
69 
70 Because
71 \[
72  g_c\le\min_i e_i\le\log_2 c,
73 \]
74 we obtain the universal near-injectivity estimate
75 \[
76  \boxed{
77  |\ker\kappa_c|\le\log_2c.
78  }
79 \]
80 This closes the global-occupancy problem for the contact packet up to logarithmic loss.
81 
82 \subsection{Balanced Kummer class}
83 The class of $c$ does not remember the additive relation. A better coordinate is
84 \[
85  \beta=-\frac ba=1-\frac ca.
86 \]
87 Let
88 \[
89  \nu_q=v_q(b)-v_q(a)
90 \]
91 for input primes $q\mid ab$. The order of $[\beta]$ in
92 \[
93  \Q^\times/(\Q^\times)^M
94 \]
95 is determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If
96 \[
97  g_{\mathrm{val}}
98  =
99  \gcd(M,\{\nu_q\}),
100  \qquad
101  m_0=M/g_{\mathrm{val}},
102 \]
103 then the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.
104 
105 The balanced class retains the input-output relation and usually has full contact order. Its kernel remains logarithmically bounded.
106 
107 \subsection{The Kummer--Fermat lift}
108 Choose algebraic numbers $u$ and $v$ satisfying
109 \[
110  u^M=\frac ac,
111  \qquad
112  v^M=\frac bc.
113 \]
114 Then
115 \[
116  \boxed{u^M+v^M=1.}
117 \]
118 Thus
119 \[
120  P_M=[u:v:1]
121 \]
122 lies on the Fermat curve
123 \[
124  F_M:X^M+Y^M=Z^M.
125 \]
126 The construction is canonical up to the action of $\mu_M^2$.
127 
128 \begin{theorem}[Exact height compression]
129 \[
130  \boxed{
131  h_{\mathcal O(1)}(P_M)=\frac{\log c}{M}.
132  }
133 \]
134 \end{theorem}
135 
136 \begin{proof}
137 The coordinatewise $M$th-power map sends $P_M$ to $[a:b:c]$. Absolute logarithmic projective height multiplies by $M$ under this map. Since the primitive positive triple has height $\log c$, the result follows.
138 \end{proof}
139 
140 The contact modulus divides the displayed coordinate height by the full powerful part.
141 
142 \subsection{Fermat character polyphase}
143 The group $\mu_M^2$ acts by
144 \[
145  (\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
146 \]
147 Let $\sigma$ and $\tau$ denote the two coordinate rotations. For
148 \[
149  (r,s)\in(\Z/M\Z)^2,
150 \]
151 define
152 \[
153  P_{r,s}
154  =
155  \frac1{M^2}
156  \sum_{i,j=0}^{M-1}
157  \zeta_M^{-ri-sj}
158  \sigma^i\tau^j.
159 \]
160 Character orthogonality gives
161 \[
162  P_{r,s}P_{r',s'}
163  =
164  \delta_{r,r'}\delta_{s,s'}P_{r,s}
165 \]
166 and
167 \[
168  \boxed{
169  \sum_{r,s}P_{r,s}=I.
170  }
171 \]
172 This is an exact two-dimensional character filter bank.
173 
174 \subsection{Line-bundle cost of the full Fermat model}
175 For $M\ge4$, adjunction gives
176 \[
177  K_{F_M}\cong\mathcal O_{F_M}(M-3).
178 \]
179 For compatible Weil-height normalizations,
180 \[
181  h_{K_{F_M}}(P_M)
182  =
183  (M-3)h_{\mathcal O(1)}(P_M)+O(1)
184  =
185  \frac{M-3}{M}\log c+O(1).
186 \]
187 If the metric on $K_{F_M}$ is defined by the induced tensor-power metric on $\mathcal O(1)^{\otimes(M-3)}$, the displayed equality holds without the $O(1)$ term. Thus the full plane-curve polarization reintroduces a factor of order $M$ into the height accounting.
188 
189 \begin{proposition}[Line-bundle cost of the full plane model]
190 The Kummer lift reduces the $\mathcal O(1)$ coordinate height by $M$, while the canonical line bundle of the complete Fermat curve has degree factor $M-3$. This rules out a height saving obtained merely by retaining the full plane-curve carrier with its induced polarization. It does not rule out a lower-complexity quotient equipped with a different arithmetic metric.
191 \end{proposition}
192 
193 A successful route must therefore select and control a much smaller quotient of the character bank. Fermat character decompositions and Jacobi-sum factors provide the established ambient framework for such quotients~\cite{AokiFermat,OtsuboYamazaki,ColemanMcCallum}.
194 

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