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1 \clearpage
2 \section{The Exact Obstruction Complex and Formal Shell Geometry}
3 
4 \subsection{A joint exponent-free two-row complex}
5 The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime $p\mid abc$, define
6 \[
7  \Lambda_p=
8  \begin{cases}
9  a/p,&p\mid a,\\
10  b/p,&p\mid b,\\
11  -c/p,&p\mid c,
12  \end{cases}
13  \qquad
14  \mathsf w_p=
15  \begin{cases}
16  -ab/p,&p\mid a,\\
17  ab/p,&p\mid b,\\
18  0,&p\mid c.
19  \end{cases}
20 \]
21 Consider
22 \[
23  \mathsf M_{a,b,c}
24  =
25  \begin{pmatrix}
26  \Lambda\\
27  \mathsf w
28  \end{pmatrix}:
29  \Z^S\longrightarrow\Z^2.
30 \]
31 The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set
32 \[
33  Q=\frac{abc}{R}.
34 \]
35 
36 \begin{theorem}[Smith obstruction theorem]\label{thm:phase-smith}
37 Let $a+b=c$ be primitive, with at least two of $a,b,c$ greater than one. Then
38 \[
39  \boxed{
40  \SNF(\mathsf M_{a,b,c})
41  =
42  \operatorname{diag}(1,Q).
43  }
44 \]
45 Consequently,
46 \[
47  \boxed{
48  \operatorname{coker}(\mathsf M_{a,b,c})_{\mathrm{tors}}
49  \cong
50  \Z/Q\Z.
51  }
52 \]
53 The exceptional triple $(1,1,2)$ has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.
54 \end{theorem}
55 
56 \begin{proof}
57 The gcd of the entries of $\Lambda$ is one. Indeed, every prime dividing one sector is absent from every other nonempty sector, and an entry from another sector is not divisible by it. Hence the first Smith factor is one.
58 
59 Rows restricted to one sector are proportional, so same-sector $2\times2$ minors vanish. If $p$ and $q$ lie in two different nonempty sectors, the corresponding minor is
60 \[
61  \pm\frac{abc}{pq}
62  =
63  \pm Q\frac{R}{pq}.
64 \]
65 Every nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\ell\mid R$, choose $p=\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.
66 \end{proof}
67 
68 \subsection{The exact lattice extension}
69 Throughout the remaining height and exact-sequence constructions, exclude $(1,1,2)$. Define
70 \[
71  L=\ker(\Lambda)\cap\Z^S
72 \]
73 and
74 \[
75  L_0=L\cap\ker(\mathsf w).
76 \]
77 Because $\Lambda$ is primitive, choose $t\in\Z^S$ with $\Lambda(t)=1$. Then
78 \[
79  \Z^S=L\oplus\Z t.
80 \]
81 The image of $\mathsf M_{a,b,c}$ is generated by $(1,\mathsf w(t))$ together with $(0,\mathsf w(L))$. Its index in $\Z^2$ is therefore the index of $\mathsf w(L)$ in $\Z$. Theorem~\ref{thm:phase-smith} shows that this index is $Q$, so
82 \[
83  \mathsf w(L)=Q\Z.
84 \]
85 Hence
86 \[
87  \omega(x)=\frac{\mathsf w\cdot x}{Q}
88 \]
89 is an integer-valued surjection and gives the exact sequence
90 \[
91  \boxed{
92  0\longrightarrow L_0
93  \longrightarrow L
94  \xrightarrow{\omega}
95  \Z
96  \longrightarrow0.
97  }
98 \]
99 The extension splits algebraically. The difficult quantity is the norm of a section, not its existence.
100 
101 For $k\in\Z$, define
102 \[
103  H(k)
104  =
105  \min\left\{
106  \norm{x}_\infty:
107  x\in L,
108  \omega(x)=k
109  \right\}.
110 \]
111 Three heights must be distinguished:
112 \[
113  H(1),
114  \qquad
115  \Phi=\min_{k\neq0}H(k),
116  \qquad
117  \overline H=
118  \inf_{k\ge1}\frac{H(k)}k.
119 \]
120 The shortest nonzero detector may live in a nonprimitive obstruction coset.
121 
122 \subsection{The stable real height}
123 Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let
124 \[
125  \sigma_u=\sum_{p\mid u}\frac1p
126 \]
127 and
128 \[
129  \Delta(a,b,c)
130  =
131  \min\left\{
132  \sigma_a+\sigma_b,
133  \frac cb(\sigma_a+\sigma_c),
134  \frac ca(\sigma_b+\sigma_c)
135  \right\}.
136 \]
137 Then
138 \[
139  \boxed{
140  \overline H
141  =
142  \frac{c}{R\Delta(a,b,c)}.
143  }
144 \]
145 The discrete master inequality becomes
146 \[
147  c\le R\Delta\Phi.
148 \]
149 Since
150 \[
151  \Phi=\overline H\,k_*\Gamma
152 \]
153 for the optimal obstruction multiple $k_*$ and an integrality factor $\Gamma\ge1$,
154 \[
155  \boxed{
156  \Delta\Phi
157  =
158  \frac cR k_*\Gamma.
159  }
160 \]
161 This is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.
162 
163 \subsection{The full shell deformation}
164 Introduce one variable for every valuation shell and define
165 \[
166  \mathscr J_n(\mathbf T)
167  =
168  \prod_{p^e\parallel n}
169  \prod_{r=0}^{e-1}(p+T_{p,r}).
170 \]
171 At $\mathbf T=0$,
172 \[
173  \mathscr J_n(0)=n.
174 \]
175 For $a+b=c$, set
176 \[
177  F_{a,b,c}(\mathbf T)
178  =
179  \mathscr J_a(\mathbf T)
180  +
181  \mathscr J_b(\mathbf T)
182  -
183  \mathscr J_c(\mathbf T).
184 \]
185 The shell gradient has entries
186 \[
187  \frac ap,
188  \qquad
189  \frac bp,
190  \qquad
191  -\frac cp
192 \]
193 repeated according to valuation depth. The gcd of all entries is one.
194 
195 \begin{theorem}[Formal affine-space theorem]
196 Let $d$ be the number of shell variables. After a unimodular integral change of shell coordinates, the $(\mathbf T)$-adic completion of
197 \[
198  \Z[\mathbf T]/(F_{a,b,c})
199 \]
200 at the zero section is isomorphic to a formal power-series ring $\Z[[U_2,\ldots,U_d]]$.
201 \end{theorem}
202 
203 \begin{proof}
204 A primitive gradient row can be completed to a unimodular matrix. In the new coordinates, one variable has linear coefficient one, which is a unit in $\Z$. The formal implicit-function recursion therefore solves uniquely for that variable at every degree with integral coefficients, giving the stated isomorphism of completed rings.
205 \end{proof}
206 
207 \begin{corollary}[Integral jet extension]
208 Every integral first-order tangent vector to $F_{a,b,c}=0$ extends to an integral formal curve through all orders.
209 \end{corollary}
210 
211 The consequence is decisive. Higher shell curvature exists, but there is no formal lifting obstruction to exploit. The missing arithmetic must be a global size or conductor estimate.
212 
213 \subsection{Aggregate products and the integral boundary}
214 Over $\Q$, on the locus where all shell factors are nonzero, product-preserving directions within a sector form a torus. For $a,b>1$, the normalized products
215 \[
216  A=\mathscr J_a/a,\qquad B=\mathscr J_b/b,\qquad C=\mathscr J_c/c
217 \]
218 satisfy $aA+bB=cC$, leaving a two-dimensional aggregate plane and internal product gauges. A unit input sector has no shell variables and leaves a one-dimensional aggregate line.
219 
220 This rational description introduces denominators. It is not an assertion that the integral completion is a torus. The preceding primitive-gradient argument is the integral statement: local formal extension exists through all orders. Arithmetic size remains the separate global obligation.

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