ABC: source and evidence reader

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \clearpage
2 \section{Prime-Packet Differentials and Radical Divisibility}
3 
4 \subsection{The prime-coordinate module}
5 Let
6 \[
7  a+b=c,
8  \qquad
9  \gcd(a,b)=1.
10 \]
11 The three integers are pairwise coprime. Write
12 \[
13  S=S_a\sqcup S_b\sqcup S_c
14  =\{p:p\mid abc\}.
15 \]
16 The first object of the calculus is the free prime-packet module
17 \[
18  \Omega_S=\bigoplus_{p\in S}\Z\xi_p.
19 \]
20 First define the global free module over all primes, and restrict it to $\Omega_S$ for $S$-supported integers. For such a positive integer $n$, define the universal multiplicative differential
21 \[
22  \boxed{
23  \mathrm dn
24  =
25  n\sum_{p\mid n}\frac{v_p(n)}p\,\xi_p.
26  }
27 \]
28 Every coefficient is an integer because $p\mid n$. The map obeys the exact Leibniz law
29 \[
30  \mathrm d(mn)=m\,\mathrm dn+n\,\mathrm dm.
31 \]
32 The verification is local. For each prime $p$,
33 \[
34  v_p(mn)=v_p(m)+v_p(n),
35 \]
36 so the $p$-coordinate of the right side is
37 \[
38  mn\frac{v_p(m)+v_p(n)}p.
39 \]
40 
41 A scalar packet is an integer vector
42 \[
43  x=(x_p)_{p\in S}\in\Z^S.
44 \]
45 Pairing with $x$ gives the arithmetic derivative
46 \[
47  \boxed{
48  \mathrm d_x(n)
49  =
50  n\sum_{p\mid n}\frac{v_p(n)}p x_p.
51  }
52 \]
53 The Leibniz rule survives scalarization.
54 
55 Arithmetic derivatives and their connection with $abc$ have an established literature. The present construction is organized here in prime-packet coordinates and later refined into filtered shell objects~\cite{PastenDerivatives,VojtaABC}.
56 
57 \subsection{The additive row}
58 The condition
59 \[
60  \mathrm d_x(a)+\mathrm d_x(b)=\mathrm d_x(c)
61 \]
62 is one integer equation. Define
63 \[
64  \lambda_p=
65  \begin{cases}
66  \dfrac{a v_p(a)}p,&p\mid a,\\[6pt]
67  \dfrac{b v_p(b)}p,&p\mid b,\\[6pt]
68  -\dfrac{c v_p(c)}p,&p\mid c.
69  \end{cases}
70 \]
71 Then
72 \[
73  \boxed{\lambda\cdot x=0.}
74 \]
75 The additive derivative lattice is
76 \[
77  \boxed{
78  \mathcal T_{a,b,c}
79  =
80  \ker(\lambda)\cap\Z^S.
81  }
82 \]
83 Since the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.
84 
85 \subsection{The arithmetic Wronskian}
86 Define
87 \[
88  \boxed{
89  W_x(a,b)
90  =a\,\mathrm d_x(b)-b\,\mathrm d_x(a).
91  }
92 \]
93 Writing
94 \[
95  \mu_p=
96  \begin{cases}
97  -\dfrac{v_p(a)}p,&p\mid a,\\[6pt]
98  \dfrac{v_p(b)}p,&p\mid b,\\[6pt]
99  0,&p\mid c,
100  \end{cases}
101 \]
102 we obtain
103 \[
104  W_x(a,b)=ab\,\mu\cdot x.
105 \]
106 An additive packet is \emph{transverse} when $W_x(a,b)\neq0$. It is degenerate when the two input terms remain derivative-proportional.
107 
108 Over $\R$, the additive projection is
109 \[
110  P_{\mathrm{add}}
111  =
112  I-
113  \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
114 \]
115 The obstruction direction inside the additive hyperplane is
116 \[
117  \mu_\perp=P_{\mathrm{add}}\mu.
118 \]
119 When $\mu_\perp\neq0$, the rank-one obstruction projector is
120 \[
121  P_{\mathrm{obs}}
122  =
123  \frac{\mu_\perp\mu_\perp^{\mathsf T}}
124  {\norm{\mu_\perp}_2^2}.
125 \]
126 This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
127 
128 \subsection{Radical divisibility}
129 Set
130 \[
131  R=\rad(abc).
132 \]
133 The central first-order theorem is an exact divisibility law.
134 
135 \begin{theorem}[Wronskian divisibility]
136 For every additive packet $x\in\mathcal T_{a,b,c}$,
137 \[
138  \boxed{
139  \frac{abc}{R}\mid W_x(a,b).
140  }
141 \]
142 \end{theorem}
143 
144 \begin{proof}
145 Let $p^e\parallel a$. Every term of $\mathrm d_x(a)$ is divisible by $p^{e-1}$. Since $p\nmid b$, the term $b\,\mathrm d_x(a)$ is divisible by $p^{e-1}$. Also $p^e\mid a$, so $a\,\mathrm d_x(b)$ is divisible by $p^e$, hence by $p^{e-1}$. Therefore
146 \[
147  p^{e-1}\mid W_x(a,b).
148 \]
149 The same argument applies to primes dividing $b$.
150 
151 Now let $p^e\parallel c$. Additivity gives
152 \[
153  \mathrm d_x(c)=\mathrm d_x(a)+\mathrm d_x(b).
154 \]
155 Using $c=a+b$,
156 \[
157  \begin{aligned}
158  W_x(a,b)
159  &=a\,\mathrm d_x(b)-b\,\mathrm d_x(a)\\
160  &=a\bigl(\mathrm d_x(c)-\mathrm d_x(a)\bigr)-b\,\mathrm d_x(a)\\
161  &=a\,\mathrm d_x(c)-c\,\mathrm d_x(a).
162  \end{aligned}
163 \]
164 The first term is divisible by $p^{e-1}$ because $\mathrm d_x(c)$ is. The second is divisible by $p^e$. Hence $p^{e-1}\mid W_x(a,b)$.
165 
166 The prime supports of $a$, $b$, and $c$ are disjoint. Multiplying all local divisibilities gives
167 \[
168  \prod_{p^e\parallel abc}p^{e-1}
169  =
170  \frac{abc}{R}
171  \mid W_x(a,b).
172 \]
173 \end{proof}
174 
175 For a transverse packet,
176 \[
177  \boxed{
178  |W_x(a,b)|\ge\frac{abc}{R}.
179  }
180 \]
181 This is the integer analogue of the repeated-root divisor appearing in the polynomial Mason--Stothers theorem.
182 
183 \subsection{The master height inequality}
184 For every positive $n$,
185 \[
186  \begin{aligned}
187  |\mathrm d_x(n)|
188  &\le
189  n\norm{x}_\infty
190  \sum_{p\mid n}\frac{v_p(n)}p.
191  \end{aligned}
192 \]
193 Since $p\ge2$,
194 \[
195  \frac1p\le\frac1{2\log2}\log p.
196 \]
197 Therefore
198 \[
199  \sum_{p\mid n}\frac{v_p(n)}p
200  \le
201  \frac{\log n}{2\log2},
202 \]
203 and
204 \[
205  |\mathrm d_x(n)|
206  \le
207  \frac{n\log n}{2\log2}\norm{x}_\infty.
208 \]
209 Consequently,
210 \[
211  \begin{aligned}
212  |W_x(a,b)|
213  &\le
214  a|\mathrm d_x(b)|+b|\mathrm d_x(a)|\\
215  &\le
216  \frac{ab}{\log2}\norm{x}_\infty\log c.
217  \end{aligned}
218 \]
219 Combining this with radical divisibility gives
220 \[
221  \boxed{
222  c
223  \le
224  \frac{R}{\log2}\norm{x}_\infty\log c.
225  }
226 \]
227 
228 Except for $(1,1,2)$, define the shortest transverse packet
229 \[
230  \boxed{
231  \Theta(a,b,c)
232  =
233  \min\left\{
234  \norm{x}_\infty:
235  x\in\mathcal T_{a,b,c},
236  W_x(a,b)\neq0
237  \right\}.
238  }
239 \]
240 Then
241 \[
242  c
243  \le
244  \frac{R}{\log2}\Theta(a,b,c)\log c.
245 \]
246 The exceptional triple $(1,1,2)$ has zero input derivatives and is omitted from $\Theta$. It is absorbed into the constant in the conjecture. A subpolynomial estimate
247 \[
248  \Theta(a,b,c)=c^{o(1)}
249 \]
250 would imply the strong $abc$ inequality. Later sections explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.
251 
252 \subsection{Three-sector compression}
253 For the empty unit sector, set $g_1=0$, $t_1=0$, and $\rho_1(0)=0$. Any nonzero target in that sector is infeasible. Within a nonempty $a$-sector, define the integer row
254 \[
255  \lambda^{(a)}
256  =
257  \left(\frac{a v_p(a)}p\right)_{p\mid a}
258 \]
259 and let
260 \[
261  g_a=\gcd\left\{\frac{a v_p(a)}p:p\mid a\right\}.
262 \]
263 Define $g_b$ and $g_c$ similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes
264 \[
265  \boxed{
266  g_at_a+g_bt_b-g_ct_c=0.
267  }
268 \]
269 The Wronskian becomes
270 \[
271  \boxed{
272  a g_bt_b-b g_at_a\neq0.
273  }
274 \]
275 All remaining sector coordinates are lift gauges. They matter only because they determine the cost of realizing an aggregate channel by an integral prime packet.
276 
277 For each term define
278 \[
279  \rho_a(t)
280  =
281  \min\left\{
282  \norm{x^{(a)}}_\infty:
283  \lambda^{(a)}\cdot x^{(a)}=g_at
284  \right\},
285 \]
286 with analogous functions for $b$ and $c$. Then
287 \[
288  \Theta(a,b,c)
289  =
290  \min
291  \max\{\rho_a(t_a),\rho_b(t_b),\rho_c(t_c)\}
292 \]
293 over aggregate solutions satisfying additivity and transversality.
294 
295 This reduction separates the global two-dimensional arithmetic geometry from the local lifting cost. It is the template used repeatedly later: compress first, then retain the exact synthesis norm of the discarded coordinates.
296 
297 \subsection{Worked example: \texorpdfstring{$3+125=128$}{3+125=128}}
298 The prime packets are $3$, $5$, and $2$. The additive row is
299 \[
300  \lambda=(1,75,-448).
301 \]
302 A shortest transverse packet is
303 \[
304  x=(-2,6,1).
305 \]
306 It gives
307 \[
308  \mathrm d_x(3)=-2,
309  \qquad
310  \mathrm d_x(125)=450,
311  \qquad
312  \mathrm d_x(128)=448.
313 \]
314 Thus additivity holds. The Wronskian is
315 \[
316  W_x(3,125)
317  =3(450)-125(-2)
318  =1600.
319 \]
320 The radical is
321 \[
322  R=2\cdot3\cdot5=30,
323 \]
324 and
325 \[
326  \frac{3\cdot125\cdot128}{30}=1600.
327 \]
328 Hence the radical lower bound is attained exactly.
329 
330 The minimality is elementary. At height at most five, $|x_3+75x_5|\le380<448$, so additivity forces $x_2=0$. Then $x_3=-75x_5$ and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves $\Theta=6$.
331 
332 This example becomes a recurring calibration point. Every later refinement must explain why the sharp integer divisor is visible while the corresponding height bound remains nontrivial.
333 

Back to source controls

Back to ABC: source and evidence reader