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1 \clearpage
2 \section{Valuation Words, Polyphase Refinement, and Local Lift Trees}
3 
4 \subsection{Replacing one exponent by a valuation word}
5 The coefficient $v_p(n)$ compresses a run of valuation shells into one scalar. For a prime power $p^e$, define the valuation word
6 \[
7  \boxed{
8  V_e(T)=1+T+\cdots+T^{e-1}.
9  }
10 \]
11 Set $V_0(T)=0$ for an absent prime. It obeys the concatenation identity
12 \[
13  V_{e+f}(T)=V_e(T)+T^eV_f(T).
14 \]
15 The refined partial derivative
16 \[
17  \partial_{p,T}(n)
18  =
19  \frac npV_{v_p(n)}(T)
20 \]
21 therefore satisfies the twisted product rule
22 \[
23  \boxed{
24  \partial_{p,T}(mn)
25  =
26  n\partial_{p,T}(m)
27  +
28  mT^{v_p(m)}\partial_{p,T}(n).
29  }
30 \]
31 At $T=1$, the ordinary arithmetic partial derivative is recovered.
32 
33 \subsection{Finite polyphase rings}
34 Fix an integer $N\ge2$ and work in
35 \[
36  A_N=\Z[T]/(T^N-1).
37 \]
38 The residue of $V_e(T)$ records the number of valuation shells in each congruence class modulo $N$. Multiplication by $V_e$ is represented by a circulant matrix $C_{e,N}$.
39 
40 \begin{theorem}[Polyphase Smith normal form]
41 If $\gcd(e,N)=1$, then
42 \[
43  \boxed{
44  \SNF(C_{e,N})
45  =
46  \operatorname{diag}(1,\ldots,1,e).
47  }
48 \]
49 \end{theorem}
50 
51 \begin{proof}
52 Let
53 \[
54  I_N=\ker\bigl(A_N\xrightarrow{\,T\mapsto1\,}\Z\bigr)
55 \]
56 be the augmentation ideal. It is a free abelian group of rank $N-1$, and multiplication by $V_e$ preserves it. After extension to $\C$, the eigenvalues on $I_N\otimes\C$ are
57 \[
58  V_e(\zeta)
59  =
60  \frac{1-\zeta^e}{1-\zeta},
61  \qquad
62  \zeta^N=1,
63  \quad
64  \zeta\neq1.
65 \]
66 Because $\gcd(e,N)=1$, the map $\zeta\mapsto\zeta^e$ permutes the nontrivial $N$th roots. Therefore
67 \[
68  \det\bigl(V_e\mid I_N\bigr)
69  =
70  \prod_{\substack{\zeta^N=1\\\zeta\neq1}}
71  \frac{1-\zeta^e}{1-\zeta}
72  =1.
73 \]
74 Thus multiplication by $V_e$ is an automorphism of the integral lattice $I_N$. On the quotient
75 \[
76  A_N/I_N\cong\Z,
77 \]
78 it acts as multiplication by $V_e(1)=e$. Applying the snake lemma to
79 \[
80  0\longrightarrow I_N\longrightarrow A_N\longrightarrow\Z\longrightarrow0
81 \]
82 shows that the full map is injective and has cokernel $\Z/e\Z$. Hence its Smith normal form is
83 \[
84  \operatorname{diag}(1,\ldots,1,e).
85 \]
86 \end{proof}
87 
88 The refined block has $N-1$ integrally invertible phase channels and one aggregate multiplicity channel. Every local denominator lies in the aggregate channel.
89 
90 \subsection{Dyadic and odd-adic primary lifts}
91 Factor
92 \[
93  e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
94 \]
95 The cokernel of the valuation block is
96 \[
97  \Z/e\Z
98  \cong
99  \Z/2^\alpha\Z
100  \times
101  \prod_{\ell\text{ odd}}
102  \Z/\ell^{\alpha_\ell}\Z.
103 \]
104 The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its $2$-primary component must be treated $2$-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
105 
106 This Smith decomposition identifies where separate $2$-primary bookkeeping first enters. The additional one-level shift in multiplicative principal-unit coordinates at the prime $2$ comes from the local $2$-adic logarithm and is established later. It is not a consequence of the Smith theorem alone.
107 
108 \subsection{Coarse and phase derivatives}
109 Choose a polyphase order $N$ coprime to every active valuation exponent. After the Smith decomposition, each refined prime packet has one coarse coordinate and integrally invertible phase coordinates in the common augmentation lattice. The additive equation separates accordingly.
110 
111 The coarse equation recovers the original exponent-weighted row
112 \[
113  \sum_{p\mid a}\frac{av_p(a)}p t_p
114  +
115  \sum_{p\mid b}\frac{bv_p(b)}p t_p
116  -
117  \sum_{p\mid c}\frac{cv_p(c)}p t_p
118  =0.
119 \]
120 The phase equation is exponent free:
121 \[
122  \boxed{
123  \sum_{p\mid a}\frac ap y_p
124  +
125  \sum_{p\mid b}\frac bp y_p
126  -
127  \sum_{p\mid c}\frac cp y_p
128  =0.
129  }
130 \]
131 Define the phase derivative
132 \[
133  D_y(n)=n\sum_{p\mid n}\frac{y_p}{p}.
134 \]
135 The same radical divisibility argument gives
136 \[
137  \frac{abc}{R}\mid aD_y(b)-bD_y(a).
138 \]
139 
140 Let
141 \[
142  \sigma(a,b)
143  =
144  \sum_{p\mid a}\frac1p
145  +
146  \sum_{p\mid b}\frac1p.
147 \]
148 For primitive triples other than $(1,1,2)$, let $\Phi(a,b,c)$ be the shortest transverse phase packet. Then
149 \[
150  \boxed{
151  c
152  \le
153  R\,\sigma(a,b)\,\Phi(a,b,c).
154  }
155 \]
156 This can be sharper than the exponent-weighted inequality on individual triples, but finite evidence still shows positive power growth rather than a subpolynomial law.
157 
158 \subsection{First-order no-free-refinement principle}
159 The shell-refined factor
160 \[
161  \prod_{r=0}^{e-1}
162  \left(1+\frac{T_r}{p}\right)
163 \]
164 has first-order expansion
165 \[
166  1+\frac1p\sum_rT_r+O(T^2).
167 \]
168 Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
169 
170 \begin{theorem}[First-order no-free-polyphase principle]
171 Let
172 \[
173  S:(T_{p,r})_{p,r}\longmapsto
174  \left(\sum_rT_{p,r}\right)_p
175 \]
176 be shell synthesis. The first-order additive map and first-order Wronskian factor through $S$. Hence
177 \[
178  \Ker S
179 \]
180 is contained in the kernels of both first-order maps.
181 \end{theorem}
182 
183 Polyphase splitting may lower displayed child coordinates. Exact synthesis multiplies by the corresponding reconstruction norm. At first order, refinement cannot create a new obstruction direction.
184 
185 \subsection{Second-order phase energy}
186 At second order,
187 \[
188  \prod_{r=0}^{e-1}
189  \left(1+\frac{T_r}{p}\right)
190  =
191  1+
192  \frac1p\sum_rT_r+
193  \frac1{p^2}\sum_{r<s}T_rT_s+
194  O(T^3).
195 \]
196 Let $\widehat T_j$ be the discrete Fourier transform of the shell coordinates. Parseval gives
197 \[
198  \sum_{r<s}T_rT_s
199  =
200  \frac12\left[
201  \widehat T_0^2
202  -
203  \frac1e\sum_j
204  \widehat T_j\widehat T_{-j}
205  \right].
206 \]
207 For real coordinates,
208 \[
209  \sum_{r<s}T_rT_s
210  =
211  \frac12\left[
212  \left(1-\frac1e\right)|\widehat T_0|^2
213  -
214  \frac1e\sum_{j\neq0}|\widehat T_j|^2
215  \right].
216 \]
217 Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this curvature is not itself a lifting obstruction.
218 
219 \subsection{Exact local lift gauges}
220 Let
221 \[
222  r_u=\rad(u),
223  \qquad
224  P_u=\frac{u}{r_u}.
225 \]
226 A shell packet $y=(y_{p,r})$ produces
227 \[
228  D_y(u)
229  =
230  u\sum_{p\mid u}\frac1p
231  \sum_{r=0}^{v_p(u)-1}y_{p,r}
232  =P_ut_u,
233 \]
234 where
235 \[
236  t_u=
237  \sum_{p\mid u}
238  \frac{r_u}{p}
239  \sum_r y_{p,r}.
240 \]
241 Because
242 \[
243  \gcd_{p\mid u}\frac{r_u}{p}=1,
244 \]
245 every integer $t_u$ is attainable when $u>1$.
246 
247 The shell lift gauge is
248 \[
249  \rho_u^{\mathrm{sh}}(t)
250  =
251  \min\left\{
252  \norm{y}_\infty:
253  \sum_{p\mid u}
254  \frac{r_u}{p}
255  \sum_r y_{p,r}=t
256  \right\}.
257 \]
258 At the aggregate level,
259 \[
260  P_at_a+P_bt_b=P_ct_c,
261 \]
262 and
263 \[
264  W_y(a,b)
265  =
266  P_aP_b(r_at_b-r_bt_a).
267 \]
268 The radical divisor is attained sharply when
269 \[
270  r_at_b-r_bt_a=P_c.
271 \]
272 When $a,b>1$, all sharp aggregate solutions form a one-parameter family
273 \[
274  \begin{aligned}
275  t_a(n)&=t_a^{(0)}+nr_a,\\
276  t_b(n)&=t_b^{(0)}+nr_b,\\
277  t_c(n)&=t_c^{(0)}+nr_c.
278  \end{aligned}
279 \]
280 The unit-sector cases have no moving parameter. If $a=1$, then
281 \[
282  (t_a,t_b,t_c)=(0,P_c,P_b),
283 \]
284 and if $b=1$, then
285 \[
286  (t_a,t_b,t_c)=(-P_c,0,-P_a).
287 \]
288 For obstruction multiple $k$, these fixed triples are multiplied by $k$. Thus the high-dimensional sharp search reduces either to one integer parameter plus three local lift gauges, or to one fixed aggregate triple in a unit-sector case.
289 

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