Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

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1 \clearpage
2 \section{Whole finite extensions and dyadic marking criteria}\label{app:dyadic_markings}
3 
4 Evidence: direct reading of the original dyadic contract and ordinary
5 finite group proofs. This note does not prove a dyadic arithmetic field
6 construction, its simple-character existence, nontrivial stabilizer
7 attachment, H5, or a Lean closure. It identifies exactly what a supplied
8 actual host and sector must preserve, and proves the marked finite
9 attachment step for C=\{1\}.
10 
11 \hypertarget{dyadic_markings-the-original-datum-includes-more-than-the-alternating-form}{%
12 \subsection{1. The original datum includes more than the alternating
13 form}\label{dyadic_markings-the-original-datum-includes-more-than-the-alternating-form}}
14 
15 The original text
16 \path{closure_attempt162/inputs/original_reading.txt} states in
17 Definition11.1, starting line4911, that at p=2 a restricted bridge
18 additionally supplies compatible quadratic refinements and an
19 isomorphism of the resulting Heisenberg central extensions lifting the
20 given symplectic isometry. Sections13.1--13.9 define the refinements and
21 their Arf/Gauss signs.
22 
23 The retained visual appendix is particularly explicit, at
24 lines6892--6947: with center \{I,-I\} fixed, squares of lifts carry the
25 quadratic bit. Multiplying a lift by i changes the particular finite
26 central-extension realization. The appendix says that allowing complex
27 scalar rescaling \textbf{without recording this distinction} erases data
28 that the bridge is required to retain.
29 
30 The finite record must therefore retain at least:
31 
32 \begin{itemize}
33 \tightlist
34 \item
35   the labelled symplectic quotient and its pairing;
36 \item
37   the supplied quadratic refinement q;
38 \item
39   the specified finite central extension, its central
40   embedding/character, and any marked lifts or specified extension
41   isomorphism;
42 \item
43   the resulting normalized Gauss sign, and all declared phase operators;
44 \item
45   the complete original character values retained by the scalar pushout.
46 \end{itemize}
47 
48 The last item alone does not retain the preceding finite-subgroup/lift
49 marking unless that marking is explicitly carried along. The raw text
50 does not require literal embeddings of the old arithmetic matrix groups
51 into a new type. Nor does it define a unique quadratic refinement
52 extracted automatically from an unmarked C× scalar extension of every
53 arithmetic type.
54 
55 \hypertarget{dyadic_markings-general-finite-central-extension-criterion}{%
56 \subsection{2. General finite central-extension
57 criterion}\label{dyadic_markings-general-finite-central-extension-criterion}}
58 
59 Let V be an elementary abelian 2-group with ordered basis
60 v\_1,\ldots,v\_m. Let E\_s and E\_t be central extensions of V by the
61 same abelian group Z. Fix the identity on Z and a quotient isomorphism
62 f.~Choose lifts a\_i of v\_i in E\_s and b\_i of f(v\_i) in E\_t. Write
63 
64 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
65 a_i²=s_i,  b_i²=t_i,  [a_i,a_j]=c_ij, [b_i,b_j]=d_ij.
66 \end{Verbatim}
67 
68 Every element has a unique normal form
69 
70 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
71 z a_1^(epsilon_1) ... a_m^(epsilon_m), epsilon_i in {0,1},
72 \end{Verbatim}
73 
74 and analogously on the target. The multiplication is completely
75 determined by the displayed squares and commutators.
76 
77 \textbf{Criterion.} A center-fixing isomorphism lifting f exists
78 precisely when
79 
80 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
81 c_ij=d_ij for every i,j,
82 s_i/t_i belongs to Z² for every i.                     (1)
83 \end{Verbatim}
84 
85 Indeed, every such isomorphism must take a\_i to lambda\_i b\_i for some
86 lambda\_i in Z, forcing lambda\_i²=s\_i/t\_i. Conversely these equations
87 make the assignment a\_i -\textgreater{} lambda\_i b\_i preserve all
88 relations, and the normal forms prove it is bijective.
89 
90 If the \textbf{particular lifts} a\_i and b\_i must correspond exactly,
91 the condition is s\_i=t\_i, with lambda\_i=1. If a full chosen section
92 is marked, rather than only generating lifts, its multiplication
93 coefficients must also correspond:
94 
95 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
96 a(v)a(w)=c_s(v,w)a(v+w),
97 b(fv)b(fw)=c_t(fv,fw)b(f(v+w)),
98 \end{Verbatim}
99 
100 and exact section preservation requires c\_s(v,w)=c\_t(fv,fw) for every
101 pair. Transporting the complete section under an existing group
102 isomorphism retains these identities automatically. Preserving just its
103 squares need not retain an additional independently specified section.
104 
105 \hypertarget{dyadic_markings-fixed-two-element-center}{%
106 \subsubsection{2.1. Fixed two-element
107 center}\label{dyadic_markings-fixed-two-element-center}}
108 
109 For Z=\{+1,-1\}, Z²=\{1\}. The lift square is independent of the lift
110 within its fiber and defines
111 
112 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
113 a(v)²=(-1)^(q_s(v)).
114 \end{Verbatim}
115 
116 The identity (ab)²=a²b²{[}b,a{]} gives the quadratic refinement law.
117 Criterion(1) becomes the exact requirement
118 
119 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
120 q_t(f(v))=q_s(v) for every v.                          (2)
121 \end{Verbatim}
122 
123 For an unspecified symplectic isomorphism between equal-dimensional
124 nondegenerate spaces, such an f exists exactly when their Arf invariants
125 agree. For a preassigned f, equality of Arf invariants alone is
126 insufficient: (2) must hold for that f.
127 
128 \hypertarget{dyadic_markings-four-element-and-complex-centers}{%
129 \subsubsection{2.2. Four-element and complex
130 centers}\label{dyadic_markings-four-element-and-complex-centers}}
131 
132 For Z=mu\_4, multiplication of a lift by i changes its square by -1. The
133 unmarked extension criterion preserves square classes modulo Z²=mu\_2.
134 In particular, squares +1 and -1 do not distinguish unmarked mu\_4
135 extensions. A separately marked finite mu\_2-central extension subgroup
136 (such as the full D8 or Q8), or prescribed lifts, still distinguishes
137 them.
138 
139 For Z=C× every square ratio has a square root. Thus matching commutators
140 is sufficient for an unmarked center-fixing extension isomorphism. This
141 fact does not delete a marked finite subgroup from the objects being
142 compared.
143 
144 \hypertarget{dyadic_markings-exact-d8q8-check}{%
145 \subsection{3. Exact D8/Q8 check}\label{dyadic_markings-exact-d8q8-check}}
146 
147 In dimension two over C put
148 
149 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
150 X=[[0,1],[1,0]],       Z=[[1,0],[0,-1]].
151 \end{Verbatim}
152 
153 Then X²=Z²=I, XZ=-ZX, and {[}X,Z{]}=-I. The group generated by X and Z
154 has order8 and is D8. Its refinement on V=F2² is
155 
156 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
157 q_D(x,y)=xy,       Arf(q_D)=0,
158 \end{Verbatim}
159 
160 with square table (+1,+1,+1,-1) at (0,e,f,e+f), and Gauss sum +2.
161 
162 The generators iX,iZ also anticommute, but each squares to -I and their
163 product also squares to -I. They generate Q8, with
164 
165 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
166 q_Q(x,y)=xy+x+y,   Arf(q_Q)=1,
167 \end{Verbatim}
168 
169 square table (+1,-1,-1,-1), and Gauss sum -2.
170 
171 Both C× pushouts identify with the same ambient matrix group
172 
173 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
174 P={z X^a Z^b : z in C×, a,b in {0,1}}.
175 \end{Verbatim}
176 
177 Nevertheless the marked subgroups
178 
179 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
180 H_D=<X,Z>,          H_Q=<iX,iZ>
181 \end{Verbatim}
182 
183 are different. No isomorphism fixing the two-element center can map H\_D
184 onto H\_Q: it would preserve q and hence the Arf invariant, which
185 differs. There is also no center-fixing automorphism of P carrying the
186 marked H\_D onto H\_Q. Their unmarked pushouts are isomorphic, but the
187 marked objects (P,H\_D) and (P,H\_Q) are not.
188 
189 Conversely a Q8 source embeds perfectly into the same P by sending its
190 generators to iX and iZ. This preserves every source square and its
191 negative Gauss sign. It does not identify those generators with the
192 independently marked D8 lifts X,Z. This is the distinction needed for
193 the arithmetic sector.
194 
195 \hypertarget{dyadic_markings-marked-scalar-pushout-attachment-with-c1}{%
196 \subsection{4. Marked scalar-pushout attachment with
197 C=\{1\}}\label{dyadic_markings-marked-scalar-pushout-attachment-with-c1}}
198 
199 Assume an actual type, its full character theta, and its Heisenberg
200 quotient W have already been constructed. Let P\_W be its actual C×
201 scalar pushout, acting on its actual Heisenberg representation eta\_W.
202 Let V\_src be a finite nondegenerate symplectic F2-space of dimension2s,
203 with its complete supplied dyadic marked extension data. For several
204 breaks use their labelled orthogonal sum, balanced central product, and
205 supplied product refinement.
206 
207 Assume the following precise finite contract:
208 
209 \begin{enumerate}
210 \def\labelenumi{\arabic{enumi}.}
211 \tightlist
212 \item
213   The source full-character pushout P\_src has quotient V\_src and the
214   same represented scalar center as P\_W.
215 \item
216   The supplied finite central extensions and their marked lifts are
217   retained as subgroups/elements of P\_src, or as already specified maps
218   into P\_src. Any central character kernel already factored out must be
219   recorded; a scalar pushout cannot recover invisible elements of that
220   kernel.
221 \item
222   The host has enough symplectic capacity, dim W\textgreater=2s, and the
223   new attachment into this host has no independently fixed conflicting
224   quadratic marking.
225 \item
226   The declared stabilizer is C=\{1\}; the graph has one vertex.
227 \end{enumerate}
228 
229 Then a complete marked attachment exists in the \textbf{same actual
230 candidate}.
231 
232 \textbf{Proof.} Choose a symplectic embedding f:V\_src -\textgreater{} W
233 and let V be its image. The restriction P\_V of P\_W and P\_src have the
234 same commutator. Criterion(1) with center C× produces an isomorphism
235 
236 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
237 F:P_src -> P_V
238 \end{Verbatim}
239 
240 fixing every scalar and lifting f.~Carry every supplied finite subgroup
241 H\_src to the actual subgroup F(H\_src) of P\_V, and every marked lift a
242 to F(a). This operation does not erase square data:
243 
244 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
245 F(a)²=F(a²),        [F(a),F(b)]=F([a,b]).              (3)
246 \end{Verbatim}
247 
248 Since F fixes the scalar center, a²=(-1)\^{}q(v) implies
249 F(a)²=(-1)\^{}q(v). The transported refinement is therefore exactly q,
250 not a new choice with the same polar form. All marked section products,
251 specified extension maps, and finite group relations are preserved by
252 the same F. Consequently the Arf invariant and normalized Gauss sign are
253 unchanged.
254 
255 Every original h in a full character domain still maps to theta\_i(h),
256 because F fixes the entire scalar center. Thus the dyadic marking is
257 retained together with the full original character values.
258 
259 Write W=V perpendicular-sum U. Finite Stone--von Neumann gives
260 
261 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
262 eta_W isomorphic to eta_src tensor eta_U.
263 \end{Verbatim}
264 
265 Choose a line in eta\_U and its projector e. It lies in the actual
266 Heisenberg image algebra, and the marked source subgroup acts on e
267 eta\_W by precisely its supplied irreducible Heisenberg representation.
268 Its lift squares remain the operators in (3). The existing sector
269 recovery theorem then recovers the unchanged inducing representation
270 Lambda, and the same arithmetic candidate. There is no C-equivariance
271 equation to solve because C=\{1\}. QED.
272 
273 This realizes an actual finite marked subgroup in the scalar extension
274 of the actual type, with actual represented operators that are complex
275 scalars times type operators. It does not assert that every marked
276 source element is the image of an unscaled element of the old or new
277 arithmetic matrix group. That stronger literal-embedding requirement is
278 absent from the present contract.
279 
280 It also does not assert that the newly marked subgroup equals a
281 separately specified canonical native subgroup of the host. The marked
282 subgroup is part of the result and must be retained in the artifact.
283 Omitting it and recording only the unmarked P\_W would reproduce the
284 loss displayed by D8/Q8.
285 
286 \hypertarget{dyadic_markings-if-the-host-refinement-is-independently-fixed}{%
287 \subsection{5. If the host refinement is independently
288 fixed}\label{dyadic_markings-if-the-host-refinement-is-independently-fixed}}
289 
290 Suppose instead that the actual host arrives with a particular
291 nonsingular quadratic refinement q\_W and its actual finite mu\_2
292 central extension H\_W. An attachment into that marked extension
293 requires a quadratic isometric embedding, not merely a symplectic
294 embedding. There is a sharp capacity test.
295 
296 \textbf{Theorem.} Let dim V\_src=2s and dim W=2R. With embeddings
297 otherwise free, a quadratic isometric embedding
298 
299 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
300 (V_src,q_src) -> (W,q_W)
301 \end{Verbatim}
302 
303 exists if and only if R\textgreater=s and, in the equal-capacity case
304 R=s,
305 
306 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
307 Arf(q_src)=Arf(q_W).                                  (4)
308 \end{Verbatim}
309 
310 If R\textgreater s, either source Arf type embeds.
311 
312 \textbf{Proof.} Necessity of dimension is immediate. An
313 equal-dimensional embedding is an isometry, forcing (4). If
314 R\textgreater s, choose a nondegenerate quadratic space U of
315 dimension2(R-s) with
316 
317 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
318 Arf(q_U)=Arf(q_W)+Arf(q_src).
319 \end{Verbatim}
320 
321 Both Arf types exist in every positive even dimension, by original
322 Proposition13.7. The orthogonal sum V\_src perpendicular-sum U then has
323 the same dimension and Arf invariant as W. Original Standard Fact13.4
324 gives an isometry to W; restriction to V\_src gives the desired
325 embedding. QED.
326 
327 After this embedding is chosen, the preimage of its image in H\_W has
328 exactly q\_src. The fixed-two-element-center criterion of Section2 gives
329 the extension isomorphism. Generating lifts with the specified squares
330 can be matched. Additional fully prescribed section elements still
331 require their exact product relations, as described in Section2.
332 
333 Thus \textbf{one spare symplectic plane absorbs the Arf mismatch},
334 without changing the host's own refinement. This conclusion remains
335 conditional on the host's actual marked refinement/finite extension
336 being supplied or constructed. It does not manufacture a canonical
337 arithmetic quadratic datum from its dimension.
338 
339 For a preassigned embedding f, the missing equation is instead
340 
341 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
342 q_W(f(v))=q_src(v) for all v.                          (5)
343 \end{Verbatim}
344 
345 Its discrepancy q\_W composed with f minus q\_src is a linear
346 functional. Strict capacity allows a different quadratic embedding when
347 it is free; it does not authorize replacing a fixed original map, fixed
348 depth placement, or fixed transport marking.
349 
350 \hypertarget{dyadic_markings-exact-outcome-for-the-dyadic-construction}{%
351 \subsection{6. Exact outcome for the dyadic
352 construction}\label{dyadic_markings-exact-outcome-for-the-dyadic-construction}}
353 
354 Under the retained one-vertex, C=\{1\}, full scalar-pushout realization
355 contract, an actual host with sufficient symplectic capacity does
356 support every supplied finite dyadic marking, provided the transported
357 finite subgroup, lifts, refinement, and phase are explicitly kept. The
358 proof is Section4, rather than an assertion that commutators determine
359 the dyadic sign.
360 
361 If a particular host-native quadratic extension is fixed in advance, use
362 Section5. Strict capacity suffices for its quadratic embedding; equal
363 capacity requires the Arf equation, and a preassigned embedding requires
364 (5). If only the type quotient is known and a specifically canonical
365 host quadratic datum is requested, constructing and comparing that datum
366 remains an arithmetic obligation. The scalar-pushout proof must not
367 silently claim to have done it.
368 
369 No part of this note handles nontrivial prescribed C actions. In that
370 case quadratic embedding, full extension action, lift markings, and
371 chosen honest phase operators must all be equivariant; dimension and Arf
372 equality alone do not settle those equations.
373 
374 \hypertarget{dyadic_markings-evidence-and-author-self-review}{%
375 \subsection{Evidence and author
376 self-review}\label{dyadic_markings-evidence-and-author-self-review}}
377 
378 Read directly: original Definition11.1, Sections13 and15, and visual
379 appendix pagesV--22/V--23 in the retained raw file; final Block166
380 \path{PROOF166.md}, Section2, and \path{LEADING_LAYER_CAPACITY166.md}, Sections2.1
381 and4. The extension criterion and D8/Q8 calculation are complete finite
382 proofs. The quadratic embedding theorem uses the Arf classification
383 already explicitly supplied by the original paper. The sector conclusion
384 reuses the retained exact recovery theorem and does not assert new
385 arithmetic existence.
386 
387 Self-review checked that complex lift rescaling is never treated as an
388 automorphism changing squares within a fixed marked extension, that the
389 marked subgroup remains in the output, that preassigned maps differ from
390 freely chosen embeddings, and that fixed native refinements require
391 their own actual input. No independent review of this note or new Lean
392 execution is claimed here.

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