Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{The trace-zero Artin--Schreier tower}\label{app:odd_tower}
3 
4 Date: 2026-10-05. Ordinary proof, submitted for independent audit.
5 
6 \hypertarget{odd_tower-exact-statement}{%
7 \subsection{Exact statement}\label{odd_tower-exact-statement}}
8 
9 Let p be odd, F=F\_p((t)), and
10 0\textless h\_1\textless\ldots\textless h\_d with p not dividing any
11 h\_i. Put q\_i=(p+1)h\_i. The constructor below gives an actual matrix
12 B\_d over F of size p\^{}d with exactly the field, simplicity,
13 approximation-level, odd-prime quotient, and conductor conclusions of
14 \path{RECURSIVE_TOWER166.md}. In addition, its matrix trace is exactly zero.
15 Every native shorter constructor and every amplified arithmetic
16 approximation also has matrix trace zero. Therefore every difference
17 between consecutive arithmetic approximations has matrix trace zero.
18 
19 The trace statement is an identity over the current coefficient field,
20 not merely a congruence, a field trace inferred from characteristic
21 zero, or a division by the matrix size. All coefficient fields have
22 characteristic p, so division by p\^{}j would be invalid.
23 
24 \hypertarget{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter}{%
25 \subsection{1. An explicit separable minimal extension with a traceless
26 parameter}\label{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter}}
27 
28 At an arbitrary recursive step let L have residue field F\_p and
29 uniformizer t\_L. Given q\textgreater0 prime to p, set
30 
31 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
32 E=L(alpha),  alpha^p-alpha=t_L^(-q).
33 \end{Verbatim}
34 
35 The right side is not an Artin--Schreier coboundary in L: the negative
36 valuation of x\^{}p-x, if x has negative valuation, is divisible by p.
37 Thus the polynomial is irreducible of degree p.~The extension is
38 separable, and the valuation equation forces ramification index p and
39 
40 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
41 v_E(alpha)=-q.
42 \end{Verbatim}
43 
44 It is therefore totally ramified and alpha is minimal over L. For odd p
45 its field trace is
46 
47 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
48 Tr_(E/L)(alpha)=sum_(c in F_p)(alpha+c)=0.
49 \end{Verbatim}
50 
51 Choose the unique b in \{1,\ldots,p-1\} with qb congruent to -1 modulo
52 p, put a=(qb+1)/p, and define
53 
54 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
55 z=t_L^a alpha^b.
56 \end{Verbatim}
57 
58 Then v\_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic
59 minimal polynomial \(f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0\)
60 is Eisenstein. It is an explicit polynomial: compute the product over
61 the p conjugates
62 
63 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
64 f(X)=product_(c in F_p) (X-t_L^a(alpha+c)^b)
65 \end{Verbatim}
66 
67 and reduce coefficients using \(\alpha^p-\alpha=t_L^{-q}\).
68 The invariant coefficients belong to L. This is a finite algebraic
69 instruction, with no choice of an unknown extension or matrix.
70 
71 \hypertarget{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse}{%
72 \subsection{2. Normalized corestriction and an explicit filtered right
73 inverse}\label{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse}}
74 
75 Use the basis 1,z,\ldots,z\^{}(p-1) of E over L. If T\_ij is its
76 coefficient matrix unit and pr is the trace corestriction, then
77 
78 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
79 pr(T_ij)=z^i z_j^*,
80 z_j^*=sum_(k=j+1)^p a_k z^(k-j-1) / f'(z),  a_p=1.
81 \end{Verbatim}
82 
83 Set
84 
85 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
86 kappa=-f'(z)/z^(p-1),  s=kappa pr.
87 \end{Verbatim}
88 
89 The displayed dual-basis formula gives
90 
91 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
92 s(T_ij)=-z^(i-j) + terms of strictly larger E-valuation.
93 \end{Verbatim}
94 
95 Indeed v\_E(a\_k)\textgreater=p for 1\textless=k\textless p, so the
96 contribution indexed by a\_k has valuation at least k greater than the
97 first contribution. Applying s entrywise on E\^{}m gives a B-bimodule
98 corestriction from End\_L(E\^{}m) to B=End\_E(E\^{}m). With the complete
99 interleaved period-pm L-chain and period-m E-chain, it preserves grades
100 and satisfies \(s(P^r)=Q^r\).
101 
102 Let P0 extract the constant coefficient on every E-coordinate. The
103 constant dual coefficient satisfies
104 
105 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
106 z_0^*=-a_0/(z f'(z)),
107 u_0=s(P0)=kappa z_0^*=a_0/z^p in O_E^times.
108 \end{Verbatim}
109 
110 For any C in M\_m(E), define the actual L-linear matrix
111 
112 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
113 Lift_E(C)=u_0^(-1) C P0.
114 \end{Verbatim}
115 
116 Then
117 
118 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
119 s(Lift_E(C))=C.
120 \end{Verbatim}
121 
122 This lift preserves the exact filtration valuation. To see this, first
123 multiply C by the unit u\_0\^{}(-1), which does not change its valuation
124 in the E-chain. Expand each entry in the basis 1,z,\ldots,z\^{}(p-1).
125 The only nonzero columns of Lift are its constant-coefficient columns.
126 The L-grade of a coefficient c\_i z\^{}i E\_uv is
127 
128 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
129 pm*v_L(c_i)+mi+u-v,
130 \end{Verbatim}
131 
132 equal to its E-grade, and distinct i have valuations distinct modulo p.
133 Thus the minimum cannot cancel. This also proves the stated surjectivity
134 \(s(P^r)=Q^r\) without a compactness or lifting-existence
135 assumption.
136 
137 \hypertarget{odd_tower-exact-trace-identity}{%
138 \subsection{3. Exact trace identity}\label{odd_tower-exact-trace-identity}}
139 
140 For y in E, extraction of its constant coefficient is
141 
142 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
143 coeff_0(y)=Tr_(E/L)(z_0^* y).
144 \end{Verbatim}
145 
146 The ordinary L-matrix trace of the rank-one operator yP0 is coeff\_0(y).
147 Consequently, for every C in M\_m(E),
148 
149 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
150 tr_L(Lift_E(C))
151    =Tr_(E/L)(z_0^* u_0^(-1) tr_E(C))
152    =Tr_(E/L)(kappa^(-1) tr_E(C)).             (T)
153 \end{Verbatim}
154 
155 Here tr\_E(C) is the ordinary m-by-m matrix trace over E. In particular,
156 if tr\_E(C)=0, then the lift has ordinary L-matrix trace zero exactly.
157 
158 \hypertarget{odd_tower-the-recursive-actual-matrix}{%
159 \subsection{4. The recursive actual
160 matrix}\label{odd_tower-the-recursive-actual-matrix}}
161 
162 Define B(L,t\_L;h\_1,\ldots,h\_d) by induction. For d=1 use the
163 multiplication matrix of the alpha from Section1 with q=q\_1. For
164 d\textgreater=2 use Section1 with q=q\_d, put m=p\^{}(d-1), recursively
165 construct
166 
167 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
168 C=B(E,z;h_1,...,h_(d-1)),
169 \end{Verbatim}
170 
171 and set
172 
173 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
174 B(L,t_L;h_1,...,h_d)=alpha I_m+Lift_E(C).       (ASR)
175 \end{Verbatim}
176 
177 Everything in (ASR) is explicit in finite field operations and the
178 displayed degree-p polynomial extensions. It differs from the old
179 conjugacy-preserving two-summand construction. Preserving old lower
180 summand conjugacy classes is not claimed.
181 
182 Inductively tr\_E(C)=0. Equation (T) and the Artin--Schreier trace give
183 
184 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
185 tr_L(B)=m Tr_(E/L)(alpha)+tr_L(Lift_E(C))=0.
186 \end{Verbatim}
187 
188 For the induction base the trace is Tr\_(E/L)(alpha)=0. Thus this
189 argument proves tracelessness in the current base field for every native
190 rank, not merely tracelessness after an amplification whose size is
191 divisible by p.
192 
193 \hypertarget{odd_tower-actual-fields-and-profiles}{%
194 \subsection{5. Actual fields and
195 profiles}\label{odd_tower-actual-fields-and-profiles}}
196 
197 Set gamma=alpha I\_m and b=Lift\_E(C). The minimal outer stratum has
198 
199 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
200 n=p^(d-1)q_d,  k_0(gamma)=-n.
201 \end{Verbatim}
202 
203 The inner actual maximal field supplied by induction has
204 
205 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
206 R=p^(d-2)q_(d-1),  v(C)=-R,  k_0(C)=-q_1.
207 \end{Verbatim}
208 
209 We have s(b)=C, v(b)=-R, and n\textgreater pR\textgreater2R because p is
210 odd and the h\_i increase. Hence
211 
212 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
213 -2R > -n-q_1 = k_0(gamma)+k_0(C).
214 \end{Verbatim}
215 
216 BK Remark2.2.7 applies to this pure derived maximal field. It gives the
217 field generated by the actual displayed gamma+b, with degree and
218 ramification p\^{}d, residue degree one, normalization of the complete
219 chain, and k\_0=-q\_1. Its valuation remains -n.~Thus {[}A,n,0,B{]} is
220 simple.
221 
222 This repeats the precise argument of \path{RECURSIVE_TOWER166.md} with a new
223 outer minimal extension and normalized right inverse. No step of that
224 argument required the old polynomial e\^{}p+t e+t.
225 
226 \hypertarget{odd_tower-native-approximations-amplification-and-traceless-corrections}{%
227 \subsection{6. Native approximations, amplification, and traceless
228 corrections}\label{odd_tower-native-approximations-amplification-and-traceless-corrections}}
229 
230 For the full rank p\^{}d matrix put, for 1\textless=j\textless=d,
231 
232 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
233 B^[j]=B(F,t;h_(d-j+1),...,h_d) tensor I_(p^(d-j)).
234 \end{Verbatim}
235 
236 These are literal truncations of (ASR). At the outer level the same
237 largest q\_d selects the same Artin--Schreier extension and uniformizer
238 in all truncations. At the next level the same q\_(d-1) does the same,
239 and so on. Thus no hidden change of bases or incompatible parameter
240 fields enters the truncations. Lift\_E commutes with amplification by an
241 identity because P0 acts separately on each coordinate.
242 
243 Every native shorter B has trace zero by Section4. Its amplification
244 therefore has trace zero. In particular
245 
246 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
247 tr_F(B^[j])=0,
248 tr_F(B^[j+1]-B^[j])=0.                        (Z)
249 \end{Verbatim}
250 
251 The same is true at native rank p\^{}(j+1): the correction between
252 B(F,t;h\_(d-j),\ldots,h\_d) and the identity-p amplification of
253 B(F,t;h\_(d-j+1),\ldots,h\_d) has trace zero. This native assertion is
254 the one relevant when constructing characters before each transfer.
255 
256 The field degrees and critical levels of \(B^{[j]}\) are exactly
257 
258 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
259 degree p^j,  n_j=p^(d-j)q_(d-j+1).
260 \end{Verbatim}
261 
262 Valuation-preservation of every Lift gives
263 
264 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
265 v_A(B^[j+1]-B^[j])=-n_(j+1).
266 \end{Verbatim}
267 
268 Thus the approximation and defining-sequence proof of the first note
269 applies unchanged. The odd-prime J/H dimensions, grade-index formula,
270 Swan conductor and self-pair conductor depend on these same levels and
271 centralizer degrees, so all their formulas carry over unchanged.
272 
273 \hypertarget{odd_tower-precisely-what-the-trace-supplies}{%
274 \subsection{7. Precisely what the trace
275 supplies}\label{odd_tower-precisely-what-the-trace-supplies}}
276 
277 For any displayed correction delta and any scalar u in L\^{}times for
278 which its additive correction character is defined,
279 
280 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
281 psi_delta(u I)=psi_L((u-1)tr_L(delta))=1.
282 \end{Verbatim}
283 
284 This proves triviality of the actual additive correction on the scalar
285 subgroup of its domain. It does not, by itself, construct a simple
286 character with prescribed values on larger scalar groups. Compatible
287 extension, transfer, and intersection arguments for the full simple
288 character are separate obligations.
289 
290 The note also does not identify the constructed candidates with any Weil
291 source, choose them by inverse LLC, realize arbitrary stabilizer
292 operators, or prove same-candidate H5. Its new output is an explicit
293 arithmetic defining sequence whose trace vanishes at every native and
294 amplified stage, suitable as input to that separate character argument.
295 
296 \hypertarget{odd_tower-primary-dependency}{%
297 \subsection{Primary dependency}\label{odd_tower-primary-dependency}}
298 
299 Bushnell--Kutzko, The admissible dual of GL(N) via compact open
300 subgroups, Remark2.2.7 (printed p55), with the defining-sequence and
301 ring recursion already pinned in \path{RECURSIVE_TOWER166.md}. Retained source
302 SHA256:
303 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8}. The
304 trace-dual and Artin--Schreier calculations above are given in full.

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