Realization Calculus: twin edition

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1 \chapter{The paired-prime groups}
2 \label{ch:construction}
3 \chapterintro{A binary selector chooses one prime from each pair. Its indicator determines a rational line in the ambient quotient, and an explicit finite cocycle turns that line into a countable group.}
4 
5 \section{Choosing a rational line}
6 Let $A\subseteq\N$, write $a_n=\one_A(n)$, and define
7 \[
8 S_A=\{p_{2n+a_n}:n\in\N\}.
9 \]
10 Exactly one prime from each pair is selected. Both $S_A$ and its complement are infinite. Let $u_A\in P$ be its indicator:
11 \[
12 (u_A)_p=\begin{cases}1,&p\in S_A,\\0,&p\notin S_A.\end{cases}
13 \]
14 Because $u_A$ has infinite support, $q(u_A)\ne0$. We can therefore define a rational line and its preimage by
15 \[
16 W_A=\Q q(u_A),\qquad G_A=q^{-1}(W_A).
17 \]
18 This definition produces an actual group for every set $A$. It does not claim an effective method for obtaining the selector bits of an arbitrary $A$.
19 
20 \begin{figure}[tbp]
21 \centering\includegraphics[width=\textwidth]{\figfile{02_prime_pairs.pdf}}
22 \caption{Each bit selects exactly one prime from its pair. A filled point is selected. The torsion factor $C_p$ is present in $G_A$ whether or not $p$ is selected. Selection controls the infinite-order lift and its divisibility, not the presence of the torsion factor.}
23 \label{fig:primepairs}
24 \end{figure}
25 
26 \section{A section with finite arithmetic defects}
27 For a reduced rational $r=a/b$, with $b>0$, define
28 \begin{equation}
29 \label{eq:section}
30 (s_A(a/b))_p=
31 \begin{cases}
32 a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\
33 0,&p\notin S_A\text{ or }p\mid b.
34 \end{cases}
35 \end{equation}
36 The inverse is used only when the denominator is a unit in $\F_p$. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by $p$ is defined in $C_p$.
37 
38 The difference $b s_A(a/b)-a u_A$ is supported on the prime divisors of $b$. It follows that
39 \[
40 q(s_A(r))=r q(u_A).
41 \]
42 Thus every element of $G_A$ is $s_A(r)+d$ for some $r\in\Q$ and $d\in D$.
43 
44 \begin{theorem}[Unique normal form]
45 \label{thm:normalform}
46 Every element of $G_A$ has a unique expression
47 \[
48 g=s_A(r)+d,\qquad (r,d)\in\Q\times D.
49 \]
50 Consequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.
51 \end{theorem}
52 \begin{proof}
53 Existence follows from the preimage definition and the displayed quotient identity. If two forms are equal, applying $q$ gives $(r-r')q(u_A)=0$. The vector $q(u_A)$ is nonzero in a rational vector space, so $r=r'$. The finite-support terms are then equal. Both $\Q$ and $D$ are countable, while the quotient is the infinite group $\Q$.
54 \end{proof}
55 The domain can be named without knowing $A$. The group operation on those names will contain the selector information.
56 
57 \section{The cocycle law}
58 Define
59 \[
60 c_A(r,t)=s_A(r)+s_A(t)-s_A(r+t).
61 \]
62 The sum is taken in $P$, but its value belongs to $D$.
63 
64 \begin{lemma}[A finite support bound]
65 \label{thm:cocyclesupport}
66 For all $r,t\in\Q$,
67 \[
68 \supp c_A(r,t)\subseteq
69 \{p:p\mid\den(r)\den(t)\}.
70 \]
71 \end{lemma}
72 \begin{proof}
73 At a prime not dividing either denominator, all three rationals have well-defined reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On an unselected coordinate, all section values are zero. The reduced denominator of $r+t$ has no prime divisor absent from the two original denominators.
74 \end{proof}
75 
76 \begin{theorem}[Explicit group law]
77 \label{thm:cocyclelaw}
78 In normal-form coordinates, addition is
79 \begin{equation}
80 \label{eq:grouplaw}
81 (r,d)+(t,e)=(r+t,d+e+c_A(r,t)).
82 \end{equation}
83 The identity is $(0,0)$ and the inverse of $(r,d)$ is $(-r,-d)$. The law is associative and commutative, and the map $(r,d)\mapsto s_A(r)+d$ is an isomorphism onto the subgroup $G_A\le P$.
84 \end{theorem}
85 \begin{proof}
86 Substitution into addition in $P$ gives the formula. The section satisfies $s_A(0)=0$ and $s_A(-r)=-s_A(r)$, so $c_A(r,0)=0$ and $c_A(r,-r)=0$. These identities give the stated identity and inverse.
87 
88 Symmetry is immediate. The cocycle equation
89 \[
90 c_A(r,t)+c_A(r+t,v)=c_A(t,v)+c_A(r,t+v)
91 \]
92 follows by expanding and cancelling the section terms. Expanding the two associations in \eqref{eq:grouplaw} now proves associativity. Uniqueness of normal forms makes the ambient map bijective, and the addition formula makes it a homomorphism.
93 \end{proof}
94 This is the standard symmetric cocycle construction of an abelian extension \cite[Section 3.2]{Lupini}. Here the particular section makes each individual correction finitely supported and exactly computable from finitely many selector queries.
95 
96 \begin{proposition}[One cocycle value reads one coordinate]
97 \label{thm:cocyclebit}
98 For every prime $p$,
99 \[
100 c_A(1/p,(p-1)/p)=-\one_{S_A}(p)e_p.
101 \]
102 \end{proposition}
103 \begin{proof}
104 The support bound leaves only the coordinate $p$. At that coordinate both fractional lifts are zero, and the lift of their sum, which is one, is $\one_{S_A}(p)$.
105 \end{proof}
106 Dropping the cocycle is therefore not a harmless simplification. The direct product operation on $\Q\times D$ removes exactly these defects and gives a different, split extension.
107 
108 \section{Structure of the extension}
109 By Chapter~\ref{ch:purity}, the group is pure in $P$, reduced, residually finite, and dense. Its torsion is $D$, and the rational coordinate gives the exact sequence
110 \begin{equation}
111 \label{eq:extension}
112 0\longrightarrow D\longrightarrow G_A\xrightarrow{\rho_A}\Q\longrightarrow0,
113 \qquad \rho_A(s_A(r)+d)=r.
114 \end{equation}
115 It does not split.
116 
117 There is an equivalent cocycle obstruction. Suppose a function $b:\Q\to D$ modified the section to an additive map $r\mapsto s_A(r)+b(r)$. Then
118 \[
119 c_A(r,t)+b(r)+b(t)-b(r+t)=0
120 \]
121 for every $r,t$. The modified section would split \eqref{eq:extension}. Hence no such function exists. The cocycle is the defect of a section into $P$, but it is not removable by a correction valued in $D$.
122 
123 \begin{proposition}[Finitely generated subgroups]
124 \label{thm:fgsubgroups}
125 Every finitely generated subgroup of $G_A$ is either finite cyclic or isomorphic to $\Z\oplus C_m$ for a squarefree positive integer $m$, with $C_1$ trivial.
126 \end{proposition}
127 \begin{proof}
128 Let $L\le G_A$ be generated by finitely many elements. Its image in $\Q$ is a finitely generated subgroup of $\Q$, hence either zero or infinite cyclic. If it is zero, then $L\le D$ and its finite support places it inside a finite cyclic group.
129 
130 Otherwise choose a lift $x\in L$ of a generator of its rational image. Subtracting suitable integer multiples of $x$ from the given generators produces finitely many elements of $D$. They generate the kernel of $L\to\rho_A(L)$, which is finite cyclic of squarefree order. The subgroup $\langle x\rangle$ intersects that torsion kernel trivially, giving the direct sum.
131 \end{proof}
132 The extensions over finitely generated rational images split, but the extension over all of $\Q$ does not. This is another exact distinction between finite algebraic stages and the global realization.

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