Realization Calculus

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1 \clearpage
2 \section{The explicit dyadic arithmetic tower}\label{app:dyadic_tower}
3 
4 No character, quotient-dimension,
5 source attachment, pair-factor, or universal Chapter XI closure is
6 asserted here.
7 
8 \hypertarget{dyadic_tower-statement}{%
9 \subsection{1. Statement}\label{dyadic_tower-statement}}
10 
11 Let F=F\_2((t)), let 0\textless h\_1\textless\ldots\textless h\_d be odd
12 positive integers, and set
13 
14 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
15 q_i=3h_i,  N=2^d.
16 \end{Verbatim}
17 
18 The explicit recursion below constructs B\_d in M\_N(F) such that
19 
20 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
21 F[B_d] is a totally ramified maximal field normalizing
22    the complete period-N lattice chain,
23 v_A(B_d)=-2^(d-1)q_d,
24 k_0(B_d,A)=-q_1,
25 tr_F(B_d)=1.
26 \end{Verbatim}
27 
28 There are explicit native constructors of ranks 2\^{}j for the largest j
29 exponents. Their identity-amplifications give the actual defining
30 sequence with field degrees 2,4,\ldots,2\^{}d and critical levels
31 
32 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
33 n_j=2^(d-j)q_(d-j+1),  1<=j<=d.
34 \end{Verbatim}
35 
36 Every parameter in this ambient defining sequence has the same leading
37 level n\_1. The quantities n\_2,\ldots,n\_d are its successive critical
38 approximation levels, not new leading levels of the ambient matrices.
39 
40 Every native constructor has trace 1. Every correction from an
41 identity-2 amplification of the preceding native constructor to the next
42 native constructor also has trace 1. In full ambient rank 2\^{}d, the
43 earlier amplified approximations instead have trace 0, because their
44 multiplicities are even. This distinction is essential.
45 
46 \hypertarget{dyadic_tower-the-minimal-degree-two-building-block}{%
47 \subsection{2. The minimal degree-two building
48 block}\label{dyadic_tower-the-minimal-degree-two-building-block}}
49 
50 Let L be a complete discretely valued field of characteristic 2 with
51 residue field F\_2, and choose a uniformizer t\_L. For odd
52 q\textgreater0 put a=(q+1)/2 and define
53 
54 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
55 E=L(alpha),  alpha^2-alpha=t_L^(-q),
56 z=t_L^a alpha.
57 \end{Verbatim}
58 
59 The Artin--Schreier polynomial is irreducible: an element x\^{}2-x of
60 negative L-valuation has even negative valuation. Its derivative is 1,
61 and the valuation equation forces ramification index 2. Thus E/L is
62 separable and totally ramified, v\_E(alpha)=-q, and alpha is minimal.
63 
64 The element z has E-valuation 2a-q=1 and satisfies
65 
66 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
67 f(z)=z^2+t_L^a z+t_L=0.
68 \end{Verbatim}
69 
70 Thus 1,z is a uniformizer power basis, and
71 
72 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
73 alpha=z/t_L^a,  Tr_(E/L)(alpha)=1.
74 \end{Verbatim}
75 
76 \hypertarget{dyadic_tower-an-explicit-normalized-corestriction-and-lift}{%
77 \subsection{3. An explicit normalized corestriction and
78 lift}\label{dyadic_tower-an-explicit-normalized-corestriction-and-lift}}
79 
80 On E use the coefficient basis 1,z. Its trace-dual elements are
81 
82 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
83 z_0^*=(z+t_L^a)/t_L^a,
84 z_1^*=1/t_L^a.
85 \end{Verbatim}
86 
87 Let pr(T\_ij)=z\^{}i z\_j\^{}* be the trace corestriction. Put
88 
89 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
90 kappa=t_L^a/z=alpha^(-1),
91 s=kappa pr,
92 u_0=s(P0)=t_L/z^2.
93 \end{Verbatim}
94 
95 Here P0 is constant-coefficient projection, applied separately to each
96 coordinate when acting on E\^{}m. The identity for u\_0 follows from
97 f(z)=0. It is a unit: v\_E(t\_L)=v\_E(z\^{}2)=2. More explicitly,
98 
99 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
100 s(T_i0)=z^i(1+t_L^a/z),
101 s(T_i1)=z^(i-1).
102 \end{Verbatim}
103 
104 Since v\_E(t\_L\^{}a/z)=2a-1=q\textgreater0, these preserve the expected
105 leading grades. On End\_L(E\^{}m), take the complete interleaved chain
106 in the basis w\_(mi+b)=z\^{}i v\_b. Its centralizer is M\_m(E) with its
107 complete E-chain. The formulas give s(P\^{}r) contained in Q\^{}r for
108 all integers r.
109 
110 For C in M\_m(E), define the explicit L-linear operator
111 
112 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
113 Lift_E(C)=u_0^(-1) C P0=(z^2/t_L) C P0.
114 \end{Verbatim}
115 
116 B-bimodularity gives s(Lift\_E(C))=C. The lift preserves exact
117 valuation: u\_0\^{}(-1) is an E-unit, and expansion in 1,z gives
118 coefficient valuations distinct modulo 2. Each coefficient entry has the
119 same grade in the interleaved L-chain and the E-chain. In particular
120 \(s(P^r)=Q^r\).
121 
122 \hypertarget{dyadic_tower-exact-trace-transport}{%
123 \subsection{4. Exact trace transport}\label{dyadic_tower-exact-trace-transport}}
124 
125 For every C in M\_m(E), ordinary matrix traces satisfy
126 
127 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
128 tr_L(Lift_E(C))
129    =Tr_(E/L)(z_0^* u_0^(-1) tr_E(C))
130    =Tr_(E/L)(kappa^(-1) tr_E(C))
131    =Tr_(E/L)(alpha tr_E(C)).                 (T2)
132 \end{Verbatim}
133 
134 The first equality follows by taking the L-trace of each rank-one
135 operator yP0, which is the constant coefficient of y. This coefficient
136 is Tr\_(E/L)(z\_0\^{}* y).
137 
138 Thus the trace of a lift is 1 whenever the inner ordinary E-matrix trace
139 is 1. It is not asserted that arbitrary trace values are unchanged.
140 
141 \hypertarget{dyadic_tower-recursive-actual-matrices-and-fields}{%
142 \subsection{5. Recursive actual matrices and
143 fields}\label{dyadic_tower-recursive-actual-matrices-and-fields}}
144 
145 Define B(L,t\_L;h\_1,\ldots,h\_d) as follows. For d=1 use the
146 multiplication matrix of alpha from Section~2 with q=q\_1. For
147 d\textgreater=2 apply Section~2 with q=q\_d, set m=2\^{}(d-1),
148 recursively construct
149 
150 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
151 C=B(E,z;h_1,...,h_(d-1)),
152 \end{Verbatim}
153 
154 and put
155 
156 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
157 B(L,t_L;h_1,...,h_d)=alpha I_m+(z^2/t_L) C P0.       (D2)
158 \end{Verbatim}
159 
160 This is a finite algebraic recipe. It uses specified separable quadratic
161 extensions and no unknown support, genericity choice, or correspondence
162 inverse.
163 
164 The degree and simplicity proof is by induction. The base alpha is
165 minimal and generates E. At a later step let gamma=alpha I\_m and
166 b=Lift\_E(C). The outer minimal stratum has
167 
168 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
169 n=2^(d-1)q_d,  k_0(gamma,A)=-n.
170 \end{Verbatim}
171 
172 The inner induction hypothesis gives an actual maximal field E{[}C{]}
173 normalizing the complete E-chain with
174 
175 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
176 R=2^(d-2)q_(d-1),  v(C)=-R,  k_0(C)=-q_1.
177 \end{Verbatim}
178 
179 Because h\_d\textgreater h\_(d-1), we have n\textgreater2R. Therefore
180 
181 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
182 -2R > -n-q_1 = k_0(gamma,A)+k_0(C).
183 \end{Verbatim}
184 
185 The derived stratum with parameter s(b)=C is pure, and its field is
186 maximal. Bushnell--Kutzko Remark 2.2.7 consequently gives the field
187 generated by the actual gamma+b, not merely an equivalent replacement.
188 It has ramification and degree 2\^{}d, residue degree one, normalizes
189 the complete chain, and has k\_0=max(-n,-q\_1)=-q\_1. Since
190 v(b)\textgreater v(gamma), its level is n and {[}A,n,0,gamma+b{]} is
191 simple.
192 
193 The derived model E{[}C{]} is not identified with L{[}gamma+b{]}. In
194 particular no literal embedding of E into the latter field is claimed.
195 
196 \hypertarget{dyadic_tower-trace-1-at-every-native-stage}{%
197 \subsection{6. Trace 1 at every native
198 stage}\label{dyadic_tower-trace-1-at-every-native-stage}}
199 
200 The base matrix has trace Tr\_(E/L)(alpha)=1. Inductively tr\_E(C)=1, so
201 (T2) gives tr\_L(Lift\_E(C))=1. For d\textgreater=2 the L-trace of alpha
202 I\_m is
203 
204 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
205 m Tr_(E/L)(alpha)=0,
206 \end{Verbatim}
207 
208 because m is even. Hence tr\_L(B)=1 at every native rank.
209 
210 For any j\textgreater=1, let A\_j denote the native rank-2\^{}j
211 constructor using a fixed list of the largest j exponents. The next
212 native rank-2\^{}(j+1) constructor A\_(j+1) uses the same largest
213 exponents and adds the next smaller one. With their compatible
214 interleaved bases, put
215 
216 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
217 Delta_(j+1)=A_(j+1)-A_j tensor I_2.
218 \end{Verbatim}
219 
220 Then
221 
222 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
223 tr(Delta_(j+1))=1-2*1=1                    in characteristic 2.
224 \end{Verbatim}
225 
226 This native trace identity is stronger and more relevant than a claim
227 obtained merely by amplifying to a rank divisible by two.
228 
229 \hypertarget{dyadic_tower-full-arithmetic-defining-sequence}{%
230 \subsection{7. Full arithmetic defining
231 sequence}\label{dyadic_tower-full-arithmetic-defining-sequence}}
232 
233 For the final rank 2\^{}d define
234 
235 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
236 B^[j]=B(F,t;h_(d-j+1),...,h_d) tensor I_(2^(d-j)).
237 \end{Verbatim}
238 
239 All truncations use the same top q\_d to select the first extension and
240 uniformizer, the same next q\_(d-1) to select the next extension, and so
241 on. The coefficient projection and lift commute with amplification. Thus
242 these are literal truncations of (D2).
243 
244 Their parameter fields have degree 2\^{}j, and their critical levels in
245 the ambient complete chain are n\_j=2\^{}(d-j)q\_(d-j+1).
246 Valuation-preserving lifts give
247 
248 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
249 v_A(B^[j+1]-B^[j])=-n_(j+1).
250 \end{Verbatim}
251 
252 Later differences have strictly greater valuation, since
253 n\_j\textgreater2n\_(j+1). This proves that the displayed parameters
254 form an actual defining sequence of simple approximations.
255 
256 Their ambient traces are
257 
258 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
259 tr_F(B^[j])=0 for j<d,  tr_F(B^[d])=1.
260 \end{Verbatim}
261 
262 Consequently ambient corrections before the last step have trace 0, and
263 the last correction has trace 1. Every one of these trace values is
264 integral. On a scalar principal unit u, the additive correction
265 character has value psi\_F((u-1)tr(delta)). This is 1 for an additive
266 character trivial on the maximal ideal. No assertion about full simple
267 character extension follows here without its separate intersection and
268 transfer argument.
269 
270 \hypertarget{dyadic_tower-concrete-rank-eight-instance-and-limits}{%
271 \subsection{8. Concrete rank-eight instance and
272 limits}\label{dyadic_tower-concrete-rank-eight-instance-and-limits}}
273 
274 For h=(1,3,5), the construction gives actual approximation-field degrees
275 2,4,8 and critical levels 60,18,3. The supplementary exact program
276 constructs its Laurent polynomial matrices, verifies the normalized lift
277 identities, and checks the native/ambient trace distinctions and matrix
278 grades. It does not prove field existence or simplicity by bounded
279 computation: those are consequences of Section~5.
280 
281 The odd-prime Heisenberg quotient and conductor counting from Block166
282 cannot simply be copied: q\_1 is odd here. This note makes no dyadic
283 quotient, full-character, stabilizer, phase, source-attachment, or H5
284 claim. It also changes lower parameter conjugacy classes, as in the
285 corresponding odd-prime Artin--Schreier construction.
286 
287 \hypertarget{dyadic_tower-dependencies-and-provenance}{%
288 \subsection{Dependencies and
289 provenance}\label{dyadic_tower-dependencies-and-provenance}}
290 
291 Primary theorem: Bushnell--Kutzko, The admissible dual of GL(N) via
292 compact open subgroups, Remark 2.2.7 (printed p. 55), together with the
293 amplification normalization and defining-sequence conventions already
294 checked in Block166.
295 
296 Retained primary source text SHA256:
297 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8}.
298 
299 Block166 inputs are pinned separately in \path{INPUT_PINS167.json} and are not
300 modified by this continuation. The Artin--Schreier, trace-dual, and
301 trace calculations needed specifically for p=2 are proved above.

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