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1 \section{Cross-Prime Logarithmic Coherence}
2 
3 \subsection{The signed input exponent signal}
4 Let
5 \[
6  S_{ab}=\{q:q\mid ab\}.
7 \]
8 Define
9 \[
10  \nu_q=v_q(b)-v_q(a).
11 \]
12 Then
13 \[
14  -\frac ba=-\prod_{q\mid ab}q^{\nu_q}.
15 \]
16 For every output prime $p\mid c$, the same input exponent packet produces the deep principal unit
17 \[
18  1-\frac ca.
19 \]
20 The output persistence is therefore a simultaneous cross-prime cancellation phenomenon.
21 
22 \subsection{Odd logarithmic rows}
23 For odd $p^e\parallel c$, define the finite-depth row coefficient
24 \[
25  \ell_{p,e}(q)
26  =
27  \frac1p\log_p\langle q\rangle_p
28  \pmod{p^{e-1}}.
29 \]
30 The balanced relation gives
31 \[
32  \boxed{
33  \sum_{q\mid ab}
34  \nu_q\ell_{p,e}(q)
35  \equiv0
36  \pmod{p^{e-1}}.
37  }
38 \]
39 Define the truncated valuation on $\Z/p^{e-1}\Z$ by
40 \[
41  \bar v_p(0)=e-1,
42  \qquad
43  \bar v_p(x)=\min\{e-1,v_p(\widetilde x)\}
44 \]
45 for any integer lift $\widetilde x$ of a nonzero residue. Then
46 \[
47  \boxed{
48  \bar v_p(\ell_{p,e}(q))
49  =
50  \min\{e-1,v_p(q^{p-1}-1)-1\}.
51  }
52 \]
53 Exceptional entrywise depth is a generalized Wieferich event.
54 
55 \subsection{Weighted dyadic row}
56 For $2^e\parallel c$ with $e\ge2$ and an odd input prime $q$, define
57 \[
58  \ell_{2,e}(q)
59  =
60  \frac14
61  \log_2\left((-1)^{(q-1)/2}q\right)
62  \pmod{2^{e-2}}.
63 \]
64 With the analogous truncated valuation on $\Z/2^{e-2}\Z$,
65 \[
66  \bar v_2(\ell_{2,e}(q))
67  =
68  \min\{e-2,v_2(q^2-1)-3\}.
69 \]
70 At $e=1$, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.
71 
72 \subsection{Sequential factorization of the powerful part}
73 The factorization is defined prime by prime. For an odd repeated output prime $p^e\parallel c$, set
74 \[
75  d_p=e-1.
76 \]
77 For a repeated dyadic prime $2^e\parallel c$, remove one sign layer and set
78 \[
79  s_2=1,
80  \qquad
81  d_2=e-2.
82 \]
83 If $2\nmid M$, take $s_2=0$. Put $\epsilon_2=2^{s_2}$. Let $K_{\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put
84 \[
85  r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
86 \]
87 The balanced order formula implies
88 \[
89  K_{\mathrm{Kum}}\mid M/\epsilon_2,
90 \]
91 so $K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}$. Write $K=K_{\mathrm{Kum}}$. Divide the input exponent vector by $K$. Define $a_{p,q}$ as the original odd or weighted dyadic coefficient, now reduced modulo $p^{d_p-r_p}$. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by $K$. Define
92 \[
93  w_p
94  =
95  \min\left
96  \{
97  d_p-r_p,
98  \min_{q\mid ab}\bar v_p(a_{p,q})
99  \right\},
100  \qquad
101  \chi_p=d_p-r_p-w_p.
102 \]
103 The normalized global factors are
104 \[
105  K=\prod_{p\mid c}p^{r_p},
106  \qquad
107  W=\prod_{p\mid c}p^{w_p},
108  \qquad
109  C=\prod_{p\mid c}p^{\chi_p}.
110 \]
111 The balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence
112 \[
113  \boxed{
114  M=\epsilon_2KWC.
115  }
116 \]
117 The order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\chi_p}$.
118 
119 \subsection{The coherence character}
120 Set $\chi_p=0$ for absent or squarefree output channels. When the residual modulus is one, use the trivial row and character. After normalization, every output prime with nontrivial residual depth produces a primitive row
121 \[
122  g_p
123  \in
124  (\Z/p^{\chi_p}\Z)^{S_{ab}}
125 \]
126 satisfying
127 \[
128  g_p\cdot\nu_0
129  \equiv0
130  \pmod{p^{\chi_p}},
131 \]
132 where $\nu_0$ is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row
133 \[
134  g\in(\Z/C\Z)^{S_{ab}}.
135 \]
136 Define the coherence character
137 \[
138  \boxed{
139  \chi_{\mathrm{coh}}(x)
140  =
141  \exp\left(\frac{2\pi i}{C}g\cdot x\right).
142  }
143 \]
144 Its kernel is the adelic coherence lattice
145 \[
146  \boxed{
147  \Lambda_{\mathrm{coh}}
148  =
149  \{x\in\Z^{S_{ab}}:g\cdot x\equiv0\pmod C\}.
150  }
151 \]
152 Since $g$ is primitive modulo $C$,
153 \[
154  [\Z^{S_{ab}}:\Lambda_{\mathrm{coh}}]=C
155 \]
156 and
157 \[
158  \Z^{S_{ab}}/\Lambda_{\mathrm{coh}}\cong\Z/C\Z.
159 \]
160 The normalized exponent vector lies in the lattice.
161 
162 \subsection{Worked lattice for \texorpdfstring{$3+125=128$}{3+125=128}}
163 Here
164 \[
165  M=64
166 \]
167 and
168 \[
169  (\epsilon_2,K,W,C)=(2,1,1,32).
170 \]
171 The input vector is
172 \[
173  \nu_0=(-1,3)
174 \]
175 on the prime coordinates $(3,5)$. The residual row is
176 \[
177  (29,31)\pmod{32}.
178 \]
179 Thus
180 \[
181  \boxed{
182  \Lambda_{\mathrm{coh}}
183  =
184  \{(x_3,x_5)\in\Z^2:29x_3+31x_5\equiv0\pmod{32}\}.
185  }
186 \]
187 Indeed,
188 \[
189  29(-1)+31(3)=64\equiv0\pmod{32}.
190 \]
191 The five residual dyadic contact layers are exactly the index of this lattice.
192 
193 \subsection{Generalized Wieferich versus collective coherence}
194 There are two qualitatively different sources of large output depth.
195 
196 \begin{enumerate}[leftmargin=1.5em]
197 \item An entrywise defect can make every coefficient in one row divisible by a large power of $p$. This is the factor $W$.
198 \item Ordinary coefficients can cancel coherently against the specific exponent vector. This is the factor $C$.
199 \end{enumerate}
200 
201 In the recovered 361-row coherence corpus, the recorded $W$ factor is usually one while $C$ can be large. The corpus is selected, rather than a statistical sample of all primitive triples. Its frequencies do not establish a generic population law. For its $W=1$ rows, the residual depth is represented by collective cancellation rather than an entrywise Wieferich defect.
202 
203 \subsection{Why abstract lattice geometry is insufficient}
204 Let $v\in\Z^s$ be primitive, with $s\ge2$. For every $C\ge2$, there exists a primitive row $g\pmod C$ satisfying
205 \[
206  g\cdot v\equiv0\pmod C.
207 \]
208 Thus a fixed short vector can lie in primitive congruence lattices of arbitrarily large index.
209 
210 \begin{theorem}[Arbitrary-index annihilator]
211 Abstract lattice geometry cannot bound the coherence index from the length of the known relation vector alone.
212 \end{theorem}
213 
214 The proof completes $v$ to a unimodular basis and chooses a primitive row annihilating its first basis coordinate modulo $C$.
215 
216 The missing theorem must use the arithmetic origin of $g$: its entries are simultaneous $p$-adic logarithms of the actual input primes, and the output primes are exactly the divisors of $a+b$.
217 
218 \subsection{Why positive divisibility is circular}
219 If positive integers $A$ and $B$ satisfy $c\mid A+B$, then
220 \[
221  A+B\ge c,
222 \]
223 so
224 \[
225  \max(A,B)\ge c/2.
226 \]
227 A direct search for a small positive $S$-unit relation divisible by $c$ cannot produce an $abc$ saving. Signed cancellation, a determinant, a regulator, or a functional equation is necessary.
228 
229 \subsection{Exact reformulation of strong \texorpdfstring{$abc$}{abc}}
230 Since
231 \[
232  c=R_c\epsilon_2KWC
233 \]
234 and $\epsilon_2\le2$ while $K\le\log_2c$, the all-$\eps$ strong $abc$ statement is equivalent, after the usual absorption of bounded and logarithmic factors into an arbitrarily small power, to the target
235 \[
236  \boxed{
237  WC
238  \le
239  C_\eps R_aR_bR^\eps.
240  }
241 \]
242 The logarithmic absorption is proved explicitly in Section~\ref{sec:bridge}, without assuming an initial relation $c\ll R$. Conversely, $WC\le M_c$ makes the reverse implication immediate. In the recorded examples with $W=1$, this leaves the coherence conductor bound
243 \[
244  C\le C_\eps R_aR_bR^\eps.
245 \]
246 
247 \subsection{Adelic coherence regulator object}
248 Package the surviving data as
249 \[
250  \boxed{
251  \mathfrak R_{a,b,c}
252  =
253  \left(
254  U_{ab},
255  \beta^{1/K},
256  \{\chi_p\}_{p\mid c},
257  \chi_{\mathrm{coh}},
258  \Lambda_{\mathrm{coh}},
259  \norm{\cdot}_\infty
260  \right).
261  }
262 \]
263 Here $U_{ab}$ is the input $S$-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
264 
265 The open theorem is an adelic slope or regulator inequality bounding the finite conductor $\log C$ by the input radical and an arbitrarily small global leakage term. General simultaneous linear-form theory contains the right kinds of local and archimedean quantities, but known general estimates are much too coarse for the $R^\eps$ scale.
266 

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