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langle","display":"\\langle","kind":"use","occurrences":[503,819,822,833]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","occurrences":[503,819,822,833]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[100,388,407,484,539]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[318,601]}],"contexts":["$$\nD=\\bigoplus_p C_p\\subseteq G_A\\subseteq P=\\prod_p C_p.\n$$","$$\nG_A\\cong G_B\\iff A\\triangle B\\text{ is finite},\\qquad\nG_A\\text{ has an }X\\text{-computable copy}\\iff A\\le_T X.\n$$","The central construction is more stringent. For every set $A\\subseteq\\N$, a countable group $G_A$ is placed between that direct sum $D$ and its product completion $P$. All groups have the same torsion subgroup, the same rational quotient, and the same full unmarked finite-quotient system. Nevertheless, their abstract isomorphism types are classified by $A$ modulo finite changes, and an oracle computes a copy precisely when it computes $A$.","nt. For every set $A\\subseteq\\N$, a countable group $G_A$ is placed between that direct sum $D$ and its product completion $P$. All groups have the same torsion subgroup, the same rational quotient, and the same full unmarked finite-quotient system. Nevertheless, their abstract isomorphism types are classified by $A$ modulo finite changes, and an oracle computes a copy precisely when it computes $A$.","\\section*{How to read the book}\nChapters 1--3 separate the realization contracts and develop the first obstruction examples. Chapters 4--8 give the main construction and its proofs. Chapter 4 also describes the general ambient mechanism: rational subspaces of $P/D$ correspond to pure intermediate groups. Chapters 9--10 extract the reconstruction and grammar consequences. Chapter 11 provides calculations that can be followed by hand and reproduced in Python. Chapter 12 returns to the interpretation in Realization Calculus. The appendices contain implementation guidance, a theorem map, a formal proof guide, and a glossary.","\\chapter*{Notation and principal results}\n\\addcontentsline{toc}{chapter}{Notation and principal results}\n\\markboth{Notation and principal results}{}\nAll groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by\n\\[\np_0=2,\\quad p_1=3,\\quad p_2=5,\\quad p_3=7,\\quad\\ldots.\n\\]\n\\begin{longtable}{@{}p{0.23\\textwidth}p{0.71\\textwidth}@{}}\n\\toprule\nSymbol & Meaning","midrule\n$C_p=\\F_p$ & The additive cyclic group of prime order $p$.","$D=\\bigoplus_p C_p$ & Finite-support prime-coordinate vectors.","$P=\\prod_p C_p$ & All prime-coordinate vectors, with product topology.","$q:P\\to V$ & The quotient map to $V=P/D$.","$D_B$ & The subgroup supported on prime indices in $B\\subseteq\\N$.","$H_W=q^{-1}(W)$ & The intermediate group associated to a rational subspace $W\\le V$.","$S_A$ & The prime set $\\{p_{2n+\\one_A(n)}:n\\in\\N\\}$.","$u_A$ & The coordinate indicator of $S_A$ in $P$.","$G_A$ & The group $q^{-1}(\\Q q(u_A))$.","$s_A(r)$ & The specified rational lift of $r q(u_A)$ to $P$.","$c_A(r,t)$ & The finite cocycle $s_A(r)+s_A(t)-s_A(r+t)$.","$e_p$ & The vector supported at $p$ with value one.","$p\\mid_G g$ & There exists $y\\in G$ with $py=g$.","$A\\eqfin B$ & The symmetric difference $A\\triangle B$ is finite.","$\\rad(m)$ & The product of the distinct prime divisors of $m$, with $\\rad(1)=1$.","$\\widehat G$ & The profinite completion, not the Pontryagin dual.","$A\\le_T X$ & Membership in $A$ is computable using oracle $X$.","$K_{\\mathfrak G}(G)$ & Least admissible finite description length, or $+\\infty$.","\\noindent The main statements have the following form:\n\\begin{align*}\n\\Tor(G_A)&=D,&G_A/D&\\cong\\Q,\\\\\nG_A/mG_A&\\cong C_{\\rad(m)},&\\widehat{G_A}&\\cong P,\\\\\nG_A\\cong G_B&\\iff A\\eqfin B,&&\\\\\nG_A\\text{ has an }X\\text{-computable copy}&\\iff A\\le_T X.&&\n\\end{align*}\nThe extension by $D$ does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.","G_A\\text{ has an }X\\text{-computable copy}&\\iff A\\le_T X.&&\n\\end{align*}\nThe extension by $D$ does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.","The integer group $\\Z$ illustrates only the first distinction. One generator and its integer multiples produce infinitely many elements. The groups developed below separate the other distinctions as well. Their purpose is not to make infinite objects mysterious. It is to account for the information that a construction actually consumes.","\\begin{definition}[Operator-seed realization]\nLet $M$ be an abelian group and let $\\mathcal O\\subseteq\\End(M)$ be a declared collection of endomorphisms. For $S\\subseteq M$, its operator hull is the subgroup generated by all $T(s)$, where $s\\in S$ and $T$ is a finite composition of members of $\\mathcal O$, including the identity. A finite operator-seed realization requires a finite $S$ whose hull is $M$.\n\\end{definition}\nThe allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In particular, one cannot evade a generation obstruction by introducing a map that simply returns the desired answer.","in{definition}[Operator-seed realization]\nLet $M$ be an abelian group and let $\\mathcal O\\subseteq\\End(M)$ be a declared collection of endomorphisms. For $S\\subseteq M$, its operator hull is the subgroup generated by all $T(s)$, where $s\\in S$ and $T$ is a finite composition of members of $\\mathcal O$, including the identity. A finite operator-seed realization requires a finite $S$ whose hull is $M$.\n\\end{definition}\nThe allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In particular, one cannot evade a generation obstruction by introducing a map that simply returns the desired answer.","If all endomorphisms are allowed, this is module generation over the unital ring $\\End(M)$. We write $\\srank_{\\End(M)}(M)$ for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.","ription grammar consists of a countable alphabet, a class of valid finite strings, and a semantics assigning each valid string one group isomorphism type. The semantics need not be computable. Any permitted parameters must have a declared coding convention.\n\\end{definition}\nThis definition allows a finite sentence to specify a noncomputable group. It does not count an arbitrary real or subset of $\\N$ as a free symbol. If such a parameter is admitted, its information must be recorded separately.","\\begin{definition}[Computable copy]\nA computable copy of a countable group is an isomorphic group on a computably named domain, with decidable equality and computable group operations. An $X$-computable copy permits an oracle $X$ in those computations. Infinite groups may equivalently be presented on $\\N$.\n\\end{definition}\nA presentation by generators and relators is a different object. Enumerating the relators need not decide whether two words represent the same element. The effective presentations in this book always include decidable equality in the named copy.","\\begin{definition}[Topological generation]\nFor a topological group $M$, a subset $S$ topologically generates $M$ when the closure of the subgroup generated by $S$ is all of $M$. Its least possible size is denoted $\\trank(M)$. An operator version takes the closure of the operator hull instead.\n\\end{definition}\nThe closure is taken in the stated topology. A change of topology is a change of contract, not an innocuous notational adjustment.","\\begin{proposition}[Countable-code bound]\n\\label{thm:codecount}\nLet $\\mathfrak G$ be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most $\\aleph_0$.\n\\end{proposition}\n\\begin{proof}\nFor each length $n$, the set of strings of length $n$ is countable. The union over $n\\in\\N$ is countable. The valid strings form a subset, and a single-valued image of a countable set is countable.\n\\end{proof}\nThis is a boundary of description, not of existence. It also says nothing about whether a particular named group lies outside a chosen grammar. The later constructions supply structural examples and an explicit relative diagonal argument.","\\begin{definition}[Grammar-relative description complexity]\nFor a fixed binary coding of $\\mathfrak G$, let $\\lvert d\\rvert$ be its bit length and set\n\\[\nK_{\\mathfrak G}(G)=\\inf\\{\\lvert d\\rvert:\\ d\\text{ is valid and }\\operatorname{Realize}_{\\mathfrak G}(d)\\cong G\\},\n\\]\nwith value $+\\infty$ when there is no such description.\n\\end{definition}\nThe value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with $K_{\\mathfrak G}(G)=+\\infty$ may have a short description in an expanded language.","g of $\\mathfrak G$, let $\\lvert d\\rvert$ be its bit length and set\n\\[\nK_{\\mathfrak G}(G)=\\inf\\{\\lvert d\\rvert:\\ d\\text{ is valid and }\\operatorname{Realize}_{\\mathfrak G}(d)\\cong G\\},\n\\]\nwith value $+\\infty$ when there is no such description.\n\\end{definition}\nThe value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with $K_{\\mathfrak G}(G)=+\\infty$ may have a short description in an expanded language.","\\begin{boundary}[No uncharged infinite parameters]\nThe expression $G=q^{-1}(\\Q q(u))$ is short. If $u$ is an arbitrary infinite coordinate vector, that expression is not a finite specification of $G$ without a specification of $u$. Counting its surrounding symbols does not count the information in the parameter.\n\\end{boundary}\nThis boundary will become exact: the computational information in a selected $u_A$ is precisely what is needed to compute the corresponding group.","boundary}[No uncharged infinite parameters]\nThe expression $G=q^{-1}(\\Q q(u))$ is short. If $u$ is an arbitrary infinite coordinate vector, that expression is not a finite specification of $G$ without a specification of $u$. Counting its surrounding symbols does not count the information in the parameter.\n\\end{boundary}\nThis boundary will become exact: the computational information in a selected $u_A$ is precisely what is needed to compute the corresponding group.","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachab","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coo","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final as","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omi","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not suff","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient.","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the part","\\begin{theorem}[Reachability obstruction]\n\\label{thm:reach}\nLet $M=\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the particular seed, or its image ma…","\\bigoplus_{i\\in I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the particular seed, or its image may be a proper subgroup of the target packet. The theorem therefore …","n I}M_i$ be a direct sum of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the particular seed, or its image may be a proper subgroup of the target packet. The theorem therefore gives a non-ge…","um of abelian groups with coordinate projections $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the particular seed, or its image may be a proper subgroup of the target packet. The theorem therefore gives a non-generation certificate b…","ctions $P_i$. Suppose every primitive operator $T$ is an endomorphism and its transfer block $P_jTP_i$ vanishes unless $i\\to j$ is an allowed edge of a directed graph $E$. If $F$ is the union of the supports of a seed family $S$, then\n\\[\n\\Hull_{\\mathcal O}(S)\\subseteq\n\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}M_j.\n\\]\nIf $M_j\\ne0$ and $j$ is not reachable, $P_j$ annihilates the entire hull but not $M$.\n\\end{theorem}\n\\begin{proof}\nAn element of the direct sum has finite support. Applying one primitive operator can move its support only along allowed edges. Induction on word length confines every operator image of a seed to packets reachable by a finite path. Finite sums and inverses cannot introduce a coordinate outside that set. The final assertion follows by taking the projection onto an omitted nonzero packet.\n\\end{proof}\nReachability is necessary, not sufficient. A permitted path may carry zero action on the particular seed, or its image may be a proper subgroup of the target packet. The theorem therefore gives a non-generation certificate but does not assert generation whenever the","The distinction is easiest to express in terms of an embedding\n\\[\nG\\longrightarrow \\varprojlim G/N.\n\\]\nResidual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original $G$ can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish every element once the object is supplied.","\\section{Finite stages and two connecting systems}\nFor $N\\ge1$, define\n\\[\nA_N=\\prod_{i<N}C_{p_i},\\qquad m_N=\\prod_{i<N}p_i.\n\\]\nThe Chinese remainder map sends $k\\bmod m_N$ to $(k\\bmod p_i)_{i<N}$. It is an isomorphism, so $A_N$ is cyclic and its all-one vector is a generator.","\\begin{center}\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\n$N$&Prime factors&$m_N$&Finite group","midrule\n1&$2$&2&$C_2$","2&$2,3$&6&$C_6$","3&$2,3,5$&30&$C_{30}$","4&$2,3,5,7$&210&$C_{210}$","5&$2,3,5,7,11$&2310&$C_{2310}$","Zero extension gives injections $A_N\\to A_{N+1}$. Forgetting the last coordinate gives surjections $A_{N+1}\\to A_N$. These are different diagrams. Their limits are\n\\[\nD=\\varinjlim_N A_N=\\bigoplus_p C_p,\n\\qquad\nP=\\varprojlim_N A_N=\\prod_p C_p.\n\\]\nThe first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","Give each $C_p$ its discrete topology and give $P$ the product topology. A basic neighborhood fixes only finitely many coordinates. We give every subgroup of $P$ the induced topology when discussing topological generation, unless explicitly stated otherwise.","\\section{Torsion and endomorphisms}\n\\begin{theorem}[The torsion subgroup of the product]\n\\label{thm:torP}\nThe subgroup $D$ is dense and proper in $P$, and\n\\[\n\\Tor(P)=D.\n\\]\n\\end{theorem}\n\\begin{proof}\nA prescribed finite coordinate pattern is realized by a finite-support vector. Thus $D$ meets every basic open set and is dense. A finite-support vector is annihilated by the product of the finitely many primes in its support, so it is torsion.","Conversely, suppose $mx=0$ for a positive integer $m$. Whenever $p\\nmid m$, multiplication by $m$ is invertible in $C_p$, and hence $x_p=0$. The support of $x$ is contained in the finite set of prime divisors of $m$. Thus $x\\in D$. The all-one vector has infinite support, so it belongs to $P\\setminus D$.\n\\end{proof}","\\begin{proposition}[Diagonal endomorphisms of the direct sum]\n\\label{thm:endD}\nEvery endomorphism of $D$ acts by a scalar on each prime coordinate, and\n\\[\n\\End(D)\\cong\\prod_p\\F_p\n\\]\nas rings.\n\\end{proposition}\n\\begin{proof}\nThe subgroup annihilated by a prime $p$ is exactly the $C_p$ coordinate. Every endomorphism preserves it, because $px=0$ implies $pT(x)=T(px)=0$. Its restriction to that cyclic group is multiplication by a unique scalar in $\\F_p$. An arbitrary element of $D$ is a finite sum of such primary elements, so these restrictions determine the map. Conversely, any family of coordinate scalars preserves finite support and defines an endomorphism. Addition and composition correspond to coordinate addition and multiplication.\n\\end{proof}","e direct sum]\n\\label{thm:endD}\nEvery endomorphism of $D$ acts by a scalar on each prime coordinate, and\n\\[\n\\End(D)\\cong\\prod_p\\F_p\n\\]\nas rings.\n\\end{proposition}\n\\begin{proof}\nThe subgroup annihilated by a prime $p$ is exactly the $C_p$ coordinate. Every endomorphism preserves it, because $px=0$ implies $pT(x)=T(px)=0$. Its restriction to that cyclic group is multiplication by a unique scalar in $\\F_p$. An arbitrary element of $D$ is a finite sum of such primary elements, so these restrictions determine the map. Conversely, any family of coordinate scalars preserves finite support and defines an endomorphism. Addition and composition correspond to coordinate addition and multiplication.\n\\end{proof}","very endomorphism of $D$ acts by a scalar on each prime coordinate, and\n\\[\n\\End(D)\\cong\\prod_p\\F_p\n\\]\nas rings.\n\\end{proposition}\n\\begin{proof}\nThe subgroup annihilated by a prime $p$ is exactly the $C_p$ coordinate. Every endomorphism preserves it, because $px=0$ implies $pT(x)=T(px)=0$. Its restriction to that cyclic group is multiplication by a unique scalar in $\\F_p$. An arbitrary element of $D$ is a finite sum of such primary elements, so these restrictions determine the map. Conversely, any family of coordinate scalars preserves finite support and defines an endomorphism. Addition and composition correspond to coordinate addition and multiplication.\n\\end{proof}","\\begin{corollary}[An infinite seed obstruction]\n\\label{thm:rankD}\nThe endomorphism-seed rank of $D$ is $\\aleph_0$. No finite set topologically generates $D$, even after all endomorphisms are admitted.\n\\end{corollary}\n\\begin{proof}\nThe union of the supports of finitely many seeds is finite. Diagonal endomorphisms preserve its complement as zero coordinates. Any omitted coordinate projection is a certificate from Theorem~\\ref{thm:reach}. Its kernel is closed in the induced topology, so taking closure does not help. The countably many vectors $e_p$ generate $D$ algebraically, proving the exact rank.\n\\end{proof}\nNotice that this conclusion does not depend on restricting the number of endomorphisms. Even the full endomorphism ring cannot create a missing prime component from the available see","\\begin{corollary}[An infinite seed obstruction]\n\\label{thm:rankD}\nThe endomorphism-seed rank of $D$ is $\\aleph_0$. No finite set topologically generates $D$, even after all endomorphisms are admitted.\n\\end{corollary}\n\\begin{proof}\nThe union of the supports of finitely many seeds is finite. Diagonal endomorphisms preserve its complement as zero coordinates. Any omitted coordinate projection is a certificate from Theorem~\\ref{thm:reach}. Its kernel is closed in the induced topology, so taking closure does not help. The countably many vectors $e_p$ generate $D$ algebraically, proving the exact rank.\n\\end{proof}\nNotice that this conclusion does not depend on restricting the number of endomorphisms. Even the full endomorphism ring cannot create a missing prime component from the available seeds.","rates $D$, even after all endomorphisms are admitted.\n\\end{corollary}\n\\begin{proof}\nThe union of the supports of finitely many seeds is finite. Diagonal endomorphisms preserve its complement as zero coordinates. Any omitted coordinate projection is a certificate from Theorem~\\ref{thm:reach}. Its kernel is closed in the induced topology, so taking closure does not help. The countably many vectors $e_p$ generate $D$ algebraically, proving the exact rank.\n\\end{proof}\nNotice that this conclusion does not depend on restricting the number of endomorphisms. Even the full endomorphism ring cannot create a missing prime component from the available seeds.","after all endomorphisms are admitted.\n\\end{corollary}\n\\begin{proof}\nThe union of the supports of finitely many seeds is finite. Diagonal endomorphisms preserve its complement as zero coordinates. Any omitted coordinate projection is a certificate from Theorem~\\ref{thm:reach}. Its kernel is closed in the induced topology, so taking closure does not help. The countably many vectors $e_p$ generate $D$ algebraically, proving the exact rank.\n\\end{proof}\nNotice that this conclusion does not depend on restricting the number of endomorphisms. Even the full endomorphism ring cannot create a missing prime component from the available seeds.","\\section{A single topological generator after completion}\n\\begin{theorem}[Topological cyclicity]\n\\label{thm:procyclic}\nLet $u=(1,1,1,\\ldots)\\in P$. Then\n\\[\n\\overline{\\langle u\\rangle}=P.\n\\]\nIn particular, $\\trank(P)=1$.\n\\end{theorem}\n\\begin{proof}\nA basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer $k$ with the specified residues. The element $ku$ belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice.\n\\end{proof}\nThe generator $u$ is absent from $D$. The topological statement does not exhibit a finite generating family inside the original dense subgroup. This is the precise mechanism by which completion changes the generation contract.","ical generator after completion}\n\\begin{theorem}[Topological cyclicity]\n\\label{thm:procyclic}\nLet $u=(1,1,1,\\ldots)\\in P$. Then\n\\[\n\\overline{\\langle u\\rangle}=P.\n\\]\nIn particular, $\\trank(P)=1$.\n\\end{theorem}\n\\begin{proof}\nA basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer $k$ with the specified residues. The element $ku$ belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice.\n\\end{proof}\nThe generator $u$ is absent from $D$. The topological statement does not exhibit a finite generating family inside the original dense subgroup. This is the precise mechanism by which completion changes the generation contract.","In particular, $\\trank(P)=1$.\n\\end{theorem}\n\\begin{proof}\nA basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer $k$ with the specified residues. The element $ku$ belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice.\n\\end{proof}\nThe generator $u$ is absent from $D$. The topological statement does not exhibit a finite generating family inside the original dense subgroup. This is the precise mechanism by which completion changes the generation contract.","rank(P)=1$.\n\\end{theorem}\n\\begin{proof}\nA basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer $k$ with the specified residues. The element $ku$ belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice.\n\\end{proof}\nThe generator $u$ is absent from $D$. The topological statement does not exhibit a finite generating family inside the original dense subgroup. This is the precise mechanism by which completion changes the generation contract.","\\begin{figure}[tbp]\n\\centering\\includegraphics[width=\\textwidth]{\\figfile{01_two_limits.pdf}}\n\\caption{The finite stages agree, but the arrows do not. Zero extension produces the finite-support group $D$. Coordinate restriction produces the complete product $P$. The labels state algebraic endomorphism-seed rank for $D$ and ordinary topological rank for $P$.}\n\\label{fig:twolimits}\n\\end{figure}","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and gene","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite compl","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed cofinality statements in this book are proved for the specific groups under c","\\section{The complete finite-quotient system}\n\\begin{proposition}[Profinite completion of the direct sum]\n\\label{thm:Dcompletion}\nThe canonical embedding of $D$ into $P$ identifies $P$ with $\\widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed cofinality statements in this book are proved for the specific groups under consideration.","widehat D$.\n\\end{proposition}\n\\begin{proof}\nLet $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed cofinality statements in this book are proved for the specific groups under consideration.","t $f:D\\to F$ be a homomorphism to a finite group. The image of a prime-order factor $C_p$ is trivial unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed cofinality statements in this book are proved for the specific groups under consideration.","unless $p$ divides $\\lvert F\\rvert$. Thus $f$ vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of $D$. Taking their inverse limit gives exactly $P$, with the canonical map supplied by the coordinates.\n\\end{proof}\nThis proof does not assume that $F$ is abelian. The image is abelian because the source is, and the order argument applies to each factor. The definition and general properties of profinite completion may be found in \\cite{Wilkes}. All needed cofinality statements in this book are proved for the specific groups under consideration.","\\section{A rational space behind the torsion}\nPut $V=P/D$ and let $q:P\\to V$ be the quotient map.","\\begin{theorem}[Unique division in the ambient quotient]\n\\label{thm:Vrational}\nMultiplication by every positive integer is bijective on $V$. Thus $V$ has a unique rational vector-space structure extending its additive group structure. The product $P$ has no nonzero divisible subgroup.\n\\end{theorem}\n\\begin{proof}\nFix $m\\ge1$ and $x\\in P$. Define $y_p=m^{-1}x_p$ when $p\\nmid m$, and $y_p=0$ when $p\\mid m$. Then $x-my$ has finite support. Thus $q(x)=m q(y)$, proving surjectivity on $V$.","}[Unique division in the ambient quotient]\n\\label{thm:Vrational}\nMultiplication by every positive integer is bijective on $V$. Thus $V$ has a unique rational vector-space structure extending its additive group structure. The product $P$ has no nonzero divisible subgroup.\n\\end{theorem}\n\\begin{proof}\nFix $m\\ge1$ and $x\\in P$. Define $y_p=m^{-1}x_p$ when $p\\nmid m$, and $y_p=0$ when $p\\mid m$. Then $x-my$ has finite support. Thus $q(x)=m q(y)$, proving surjectivity on $V$.","quotient]\n\\label{thm:Vrational}\nMultiplication by every positive integer is bijective on $V$. Thus $V$ has a unique rational vector-space structure extending its additive group structure. The product $P$ has no nonzero divisible subgroup.\n\\end{theorem}\n\\begin{proof}\nFix $m\\ge1$ and $x\\in P$. Define $y_p=m^{-1}x_p$ when $p\\nmid m$, and $y_p=0$ when $p\\mid m$. Then $x-my$ has finite support. Thus $q(x)=m q(y)$, proving surjectivity on $V$.","lication by every positive integer is bijective on $V$. Thus $V$ has a unique rational vector-space structure extending its additive group structure. The product $P$ has no nonzero divisible subgroup.\n\\end{theorem}\n\\begin{proof}\nFix $m\\ge1$ and $x\\in P$. Define $y_p=m^{-1}x_p$ when $p\\nmid m$, and $y_p=0$ when $p\\mid m$. Then $x-my$ has finite support. Thus $q(x)=m q(y)$, proving surjectivity on $V$.","For injectivity, suppose $mx\\in D$. Outside the union of the finite support of $mx$ and the prime divisors of $m$, multiplication by $m$ is invertible and $x_p=0$. Hence $x\\in D$ and $q(x)=0$. Unique division defines $(a/b)v$ as the unique solution $w$ of $bw=av$. The vector-space laws follow by clearing positive denominators and using injectivity of integer multiplication.","Finally, a homomorphism from a divisible group to $C_p$ is zero. For each source element $x$, write $x=py$ and observe that its image is $p$ times an element of $C_p$. Every coordinate of a divisible subgroup of $P$ therefore vanishes, so that subgroup is zero.\n\\end{proof}\nA group with no nonzero divisible subgroup is called \\emph{reduced}. Thus $P$, and every subgroup of $P$, is reduced. Nevertheless $P/D$ is divisible. Quotients can possess division that no subgroup of the original group supports. The nonsplitting theorem later turns this distinc","Finally, a homomorphism from a divisible group to $C_p$ is zero. For each source element $x$, write $x=py$ and observe that its image is $p$ times an element of $C_p$. Every coordinate of a divisible subgroup of $P$ therefore vanishes, so that subgroup is zero.\n\\end{proof}\nA group with no nonzero divisible subgroup is called \\emph{reduced}. Thus $P$, and every subgroup of $P$, is reduced. Nevertheless $P/D$ is divisible. Quotients can possess division that no subgroup of the original group supports. The nonsplitting theorem later turns this distinction into an exact obstruction.","y, a homomorphism from a divisible group to $C_p$ is zero. For each source element $x$, write $x=py$ and observe that its image is $p$ times an element of $C_p$. Every coordinate of a divisible subgroup of $P$ therefore vanishes, so that subgroup is zero.\n\\end{proof}\nA group with no nonzero divisible subgroup is called \\emph{reduced}. Thus $P$, and every subgroup of $P$, is reduced. Nevertheless $P/D$ is divisible. Quotients can possess division that no subgroup of the original group supports. The nonsplitting theorem later turns this distinction into an exact obstruction.","\\section{The entire subgroup lattice}\nFor $B\\subseteq\\N$, set\n\\[\nD_B=\\bigoplus_{i\\in B}C_{p_i}\\le D.\n\\]\nA prime spectrum records which prime orders occur. Define $\\sigma_p(M)=1$ when $M$ contains a nonzero element annihilated by $p$, and set it to zero otherwise. Since $p$ is prime, a nonzero such element has order exactly $p$.","\\begin{theorem}[Subgroup and isomorphism classification]\n\\label{thm:subD}\nThe map $B\\mapsto D_B$ is an order isomorphism from $\\mathcal P(\\N)$ onto $\\Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","ation]\n\\label{thm:subD}\nThe map $B\\mapsto D_B$ is an order isomorphism from $\\mathcal P(\\N)$ onto $\\Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","The map $B\\mapsto D_B$ is an order isomorphism from $\\mathcal P(\\N)$ onto $\\Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","rphism from $\\mathcal P(\\N)$ onto $\\Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","m $\\mathcal P(\\N)$ onto $\\Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","Sub(D)$. Every subgroup is fully invariant under all endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","ll endomorphisms of $D$, and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.",", and\n\\[\nD_B\\cong D_C\\quad\\Longleftrightarrow\\quad B=C.\n\\]\n\\end{theorem}\n\\begin{proof}\nLet $H\\le D$ and let $x\\in H$. Its support is a finite set $F$ of primes. For $p\\in F$, choose by the Chinese remainder theorem an integer $k$ that is one modulo $p$ and zero modulo every other prime in $F$. Then $kx$ is supported only at $p$, where it is nonzero. It generates the entire $C_p$ coordinate. Thus $H$ contains every coordinate factor appearing in one of its elements.","Let $B$ be the set of indices of these factors. Every element of $H$ lies in $D_B$, and every finite sum of the factors in $B$ lies in $H$. Hence $H=D_B$. Uniqueness follows by testing membership of $e_{p_i}$. Proposition~\\ref{thm:endD} proves full invariance. Finally $\\sigma_{p_i}(D_B)=1$ exactly when $i\\in B$, and these values are isomorphism invariants.\n\\end{proof}\nThe lattice operations have literal set-theoretic descriptions:\n\\[\nD_B+D_C=D_{B\\cup C},\\qquad D_B\\cap D_C=D_{B\\cap C},\n\\qquad D=D_B\\oplus D_{\\N\\setminus B}.\n\\]\nThe infinite support choice is therefore already visible in an elementary algebraic lattice. No exotic group operation is required.","e factors. Every element of $H$ lies in $D_B$, and every finite sum of the factors in $B$ lies in $H$. Hence $H=D_B$. Uniqueness follows by testing membership of $e_{p_i}$. Proposition~\\ref{thm:endD} proves full invariance. Finally $\\sigma_{p_i}(D_B)=1$ exactly when $i\\in B$, and these values are isomorphism invariants.\n\\end{proof}\nThe lattice operations have literal set-theoretic descriptions:\n\\[\nD_B+D_C=D_{B\\cup C},\\qquad D_B\\cap D_C=D_{B\\cap C},\n\\qquad D=D_B\\oplus D_{\\N\\setminus B}.\n\\]\nThe infinite support choice is therefore already visible in an elementary algebraic lattice. No exotic group operation is required.","Give $\\Sub(D)$ the topology inherited from $\\{0,1\\}^D$ by its membership functions. Membership of a vector with finite support depends on finitely many selector bits. Conversely, membership of $e_{p_i}$ recovers the $i$th bit. Consequently the order isomorphism above is also a homeomorphism from Cantor space $\\bits$ to $\\Sub(D)$.","\\begin{corollary}[Finite observations leave continuum many possibilities]\n\\label{thm:finiteviews}\nFix finitely many prime indices and their membership values. There are continuum many pairwise nonisomorphic $D_B$ consistent with those values. A full compatible sequence of finite membership patterns determines one $B$, but no fixed finite prefix does.\n\\end{corollary}\n\\begin{proof}\nThe unspecified bits form a countably infinite set of independent binary choices. Distinct completions give nonisomorphic groups by Theorem~\\ref{thm:subD}. A full compatible family fixes every coordinate and hence fixes the subset.\n\\end{proof}","\\section{Enumeration is enough for a computable copy}\nA subset of $\\N$ is computably enumerable if a program can list exactly its elements. The program need not decide that an element will never appear. All these notions relativize to an oracle $X$.","\\begin{theorem}[The prime-spectrum criterion]\n\\label{thm:spectrumce}\nFor every oracle $X$,\n\\[\nD_B\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad\nB\\text{ is computably enumerable relative to }X.\n\\]\n\\end{theorem}\n\\begin{proof}\nSuppose first that a copy is computable in $X$. For each $i$, search its domain for a nonzero element $x$ with $p_i x=0$. Each individual test is decidable relative to $X$. Such a witness exists exactly when $i\\in B$. Dovetailing the searches enumerates $B$.","The prime-spectrum criterion]\n\\label{thm:spectrumce}\nFor every oracle $X$,\n\\[\nD_B\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad\nB\\text{ is computably enumerable relative to }X.\n\\]\n\\end{theorem}\n\\begin{proof}\nSuppose first that a copy is computable in $X$. For each $i$, search its domain for a nonzero element $x$ with $p_i x=0$. Each individual test is decidable relative to $X$. Such a witness exists exactly when $i\\in B$. Dovetailing the searches enumerates $B$.",":spectrumce}\nFor every oracle $X$,\n\\[\nD_B\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad\nB\\text{ is computably enumerable relative to }X.\n\\]\n\\end{theorem}\n\\begin{proof}\nSuppose first that a copy is computable in $X$. For each $i$, search its domain for a nonzero element $x$ with $p_i x=0$. Each individual test is decidable relative to $X$. Such a witness exists exactly when $i\\in B$. Dovetailing the searches enumerates $B$.","{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad\nB\\text{ is computably enumerable relative to }X.\n\\]\n\\end{theorem}\n\\begin{proof}\nSuppose first that a copy is computable in $X$. For each $i$, search its domain for a nonzero element $x$ with $p_i x=0$. Each individual test is decidable relative to $X$. Such a witness exists exactly when $i\\in B$. Dovetailing the searches enumerates $B$.","Conversely, suppose $B$ is infinite and enumerable in $X$. List its distinct elements as $b_0,b_1,\\ldots$. To compute $b_j$, wait for the $(j+1)$st distinct enumerated value. This procedure terminates for every $j$. Use canonical finite tuples with the $j$th coordinate in $\\F_{p_{b_j}}$. Coordinatewise addition, normalization, and equality are computable in $X$. If $B$ is finite, $D_B$ is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration has finished.\n\\end{proof}\nThis construction is classical. Montalb\\'an gives the prime direct-sum example and its enumeration-degree interpretation explicitly \\cite[Example III.39]{Montalban}.","Conversely, suppose $B$ is infinite and enumerable in $X$. List its distinct elements as $b_0,b_1,\\ldots$. To compute $b_j$, wait for the $(j+1)$st distinct enumerated value. This procedure terminates for every $j$. Use canonical finite tuples with the $j$th coordinate in $\\F_{p_{b_j}}$. Coordinatewise addition, normalization, and equality are computable in $X$. If $B$ is finite, $D_B$ is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration has finished.\n\\end{proof}\nThis construction is classical. Montalb\\'an gives the prime direct-sum example and its enumeration-degree interpretation explicitly \\cite[Example III.39]{Montalban}. It is included here because its c","Conversely, suppose $B$ is infinite and enumerable in $X$. List its distinct elements as $b_0,b_1,\\ldots$. To compute $b_j$, wait for the $(j+1)$st distinct enumerated value. This procedure terminates for every $j$. Use canonical finite tuples with the $j$th coordinate in $\\F_{p_{b_j}}$. Coordinatewise addition, normalization, and equality are computable in $X$. If $B$ is finite, $D_B$ is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration has finished.\n\\end{proof}\nThis construction is classical. Montalb\\'an gives the prime direct-sum example and its enumeration-degree interpretation explicitly \\cite[Example III.39]{Montalban}. It is included here because its contrast with the later common-completion family","Conversely, suppose $B$ is infinite and enumerable in $X$. List its distinct elements as $b_0,b_1,\\ldots$. To compute $b_j$, wait for the $(j+1)$st distinct enumerated value. This procedure terminates for every $j$. Use canonical finite tuples with the $j$th coordinate in $\\F_{p_{b_j}}$. Coordinatewise addition, normalization, and equality are computable in $X$. If $B$ is finite, $D_B$ is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration has finished.\n\\end{proof}\nThis construction is classical. Montalb\\'an gives the prime direct-sum example and its enumeration-degree interpretation explicitly \\cite[Example III.39]{Montalban}. It is included here because its contrast with the later common-completion family is mathematically","Conversely, suppose $B$ is infinite and enumerable in $X$. List its distinct elements as $b_0,b_1,\\ldots$. To compute $b_j$, wait for the $(j+1)$st distinct enumerated value. This procedure terminates for every $j$. Use canonical finite tuples with the $j$th coordinate in $\\F_{p_{b_j}}$. Coordinatewise addition, normalization, and equality are computable in $X$. If $B$ is finite, $D_B$ is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration has finished.\n\\end{proof}\nThis construction is classical. Montalb\\'an gives the prime direct-sum example and its enumeration-degree interpretation explicitly \\cite[Example III.39]{Montalban}. It is included here because its contrast with the later common-completion family is mathematically informative.","\\section{A finite definition without computable equality}\nLet $K=\\{n:\\text{program }n\\text{ halts on input }n\\}$ be the diagonal halting set. It is enumerable. It is not decidable: a decision procedure would permit a program that halts on its own index precisely when the decision procedure says it does not. Its complement is not enumerable either, since simultaneous enumeration of a set and its complement decides membership.","Define\n\\[\nB_K=\\{2n:n\\in\\N\\}\\cup\\{2n+1:n\\notin K\\},\n\\qquad H=D_{B_K}.\n\\]\nThis is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.","\\begin{corollary}\n\\label{thm:noncompD}\nThe group $H$ has no computable copy. It nevertheless has an effectively enumerable infinite list of generators and relators.\n\\end{corollary}\n\\begin{proof}\nIf $B_K$ were enumerable, filtering its odd entries would enumerate the complement of $K$. Theorem~\\ref{thm:spectrumce} therefore excludes a computable copy.","For the relator description, introduce a commuting generator $x_i$ for each prime index, impose $p_i x_i=0$, and impose $x_{2n+1}=0$ whenever program $n$ is observed to halt on input $n$. The resulting quotient retains exactly the prime factors indexed by $B_K$. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not supplied by this enumeration.\n\\end{proof}\nThe conclusion is not absolute finite indescribability. The displayed definition is finite. What fails is the stronger contract of a countable copy with computable operations and decidable equality.","\\section{A purely finite-state boundary}\nConsider a restricted grammar in which a finite automaton reads the unary word $1^n$ and selects the factor $C_{p_n}$ exactly when it accepts.","\\begin{theorem}[Unary selector classification]\n\\label{thm:unary}\nA group $D_B$ is realizable by this unary-selector grammar if and only if $B$ is ultimately periodic.\n\\end{theorem}\n\\begin{proof}\nOn a unary input, the state after $n$ letters is obtained by iterating a single map of a finite state set. The orbit eventually repeats a state, and from that point it is periodic. Its acceptance sequence is therefore ultimately periodic.","Conversely, any finite prefix followed by a periodic binary pattern is recognized by a finite directed path entering a cycle. The prime-spectrum isomorphism criterion ensures that a different accepted set cannot accidentally realize an isomorphic $D_B$.\n\\end{proof}","\\begin{example}[A computable group outside the unary grammar]\nLet $B=\\{2^k:k\\ge0\\}$. It is decidable and enumerable, so $D_B$ has a computable copy. It is not ultimately periodic. Indeed, if $d>0$ were an eventual period, choose $2^k$ beyond the threshold with $2^k>d$. Periodicity would force $2^k+d$ to be a power of two, although it lies strictly between $2^k$ and $2^{k+1}$. Thus $D_B$ is outside the unary grammar. A binary-input automaton recognizes the strings $10^k$, so changing the input representation removes this particular obstruction.\n\\end{example}","\\{2^k:k\\ge0\\}$. It is decidable and enumerable, so $D_B$ has a computable copy. It is not ultimately periodic. Indeed, if $d>0$ were an eventual period, choose $2^k$ beyond the threshold with $2^k>d$. Periodicity would force $2^k+d$ to be a power of two, although it lies strictly between $2^k$ and $2^{k+1}$. Thus $D_B$ is outside the unary grammar. A binary-input automaton recognizes the strings $10^k$, so changing the input representation removes this particular obstruction.\n\\end{example}","\\section{What the first family does not accomplish}\nThe groups $D_B$ have distinct prime spectra, and those differences are already detectable by finite quotients. They do not yet show a failure of reconstruction from the full unmarked finite-quotient system. They isolate a simpler issue: which prime factors have been admitted in the first place.","The main family keeps every prime torsion factor present. It then places the selector in the divisibility of infinite-order elements instead. As a result, all finite quotients remain fixed while the effective cost of the source group varies. The rational space $P/D$ is the mechanism that permits this stronger separation.","\\section{The preimage construction}\nKeep $D\\subseteq P$ and $V=P/D$ from Chapter~\\ref{ch:ambient}. For a rational vector subspace $W\\le V$, define\n\\[\nH_W=q^{-1}(W).\n\\]\nThis is an actual subgroup of $P$ containing $D$. Its quotient by $D$ is canonically $W$. It is dense because it contains the dense subgroup $D$.","The construction separates the ambient coordinate space from the admissible realization. The quotient $V$ records infinite coordinate tails modulo finite changes. A rational subspace $W$ selects which such tails are allowed. The preimage restores all finite-support corrections.","\\begin{definition}[Purity]\nA subgroup $H$ of an abelian group $M$ is pure when\n\\[\nH\\cap mM=mH\\qquad(m\\ge1).\n\\]\nEquivalently, if an equation $my=h$ with $h\\in H$ has a solution in $M$, then it has a solution in $H$.\n\\end{definition}\nPurity is a standard algebraic notion \\cite{Fuchs}. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","\\begin{theorem}[Pure-preimage correspondence]\n\\label{thm:purecorrespondence}\nThe map\n\\[\nW\\longmapsto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","eorem}[Pure-preimage correspondence]\n\\label{thm:purecorrespondence}\nThe map\n\\[\nW\\longmapsto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","correspondence]\n\\label{thm:purecorrespondence}\nThe map\n\\[\nW\\longmapsto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","hm:purecorrespondence}\nThe map\n\\[\nW\\longmapsto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","The map\n\\[\nW\\longmapsto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","sto H_W=q^{-1}(W)\n\\]\nis an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","an inclusion-preserving bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","bijection between rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","n rational subspaces of $V$ and pure subgroups $H$ of $P$ containing $D$.\n\\end{theorem}\n\\begin{proof}\nFirst let $W$ be rational and suppose $h\\in H_W\\cap mP$. Because $W$ is closed under division by $m$, choose $x\\in H_W$ with $q(x)=q(h)/m$. Then $d=h-mx$ belongs to $D$. At a prime dividing $m$, the coordinate of $h$ is zero because $h\\in mP$, and that of $mx$ is also zero. Thus $d_p=0$ whenever $p\\mid m$.","Define $e_p=m^{-1}d_p$ at every other prime and $e_p=0$ at primes dividing $m$. The vector $e$ has finite support and satisfies $me=d$. Hence $h=m(x+e)$ with $x+e\\in H_W$. This proves purity.","Conversely, let $D\\subseteq H\\subseteq P$ be pure. Its image $q(H)$ is an additive subgroup of $V$. Given $h\\in H$ and $m\\ge1$, let $d\\in D$ agree with $h$ on the primes dividing $m$ and vanish elsewhere. Then $h-d\\in mP\\cap H$. Purity gives $h-d=mx$ for some $x\\in H$, so $q(x)=q(h)/m$. Thus $q(H)$ is closed under rational scalars. Since $D\\subseteq H$, the equality $H=q^{-1}(q(H))$ holds. This proves that the constructions are inverse. Both preserve inclusion.\n\\end{proof}\nThe bottom object is $H_{\\{0\\}}=D$, and the top object is $H_V=P$. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with $D$ serving as the bottom of the interval of intermediate groups.","Given $h\\in H$ and $m\\ge1$, let $d\\in D$ agree with $h$ on the primes dividing $m$ and vanish elsewhere. Then $h-d\\in mP\\cap H$. Purity gives $h-d=mx$ for some $x\\in H$, so $q(x)=q(h)/m$. Thus $q(H)$ is closed under rational scalars. Since $D\\subseteq H$, the equality $H=q^{-1}(q(H))$ holds. This proves that the constructions are inverse. Both preserve inclusion.\n\\end{proof}\nThe bottom object is $H_{\\{0\\}}=D$, and the top object is $H_V=P$. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with $D$ serving as the bottom of the interval of intermediate groups.","D$ agree with $h$ on the primes dividing $m$ and vanish elsewhere. Then $h-d\\in mP\\cap H$. Purity gives $h-d=mx$ for some $x\\in H$, so $q(x)=q(h)/m$. Thus $q(H)$ is closed under rational scalars. Since $D\\subseteq H$, the equality $H=q^{-1}(q(H))$ holds. This proves that the constructions are inverse. Both preserve inclusion.\n\\end{proof}\nThe bottom object is $H_{\\{0\\}}=D$, and the top object is $H_V=P$. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with $D$ serving as the bottom of the interval of intermediate groups.","so $q(x)=q(h)/m$. Thus $q(H)$ is closed under rational scalars. Since $D\\subseteq H$, the equality $H=q^{-1}(q(H))$ holds. This proves that the constructions are inverse. Both preserve inclusion.\n\\end{proof}\nThe bottom object is $H_{\\{0\\}}=D$, and the top object is $H_V=P$. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with $D$ serving as the bottom of the interval of intermediate groups.","\\section{Root conditions do not depend on the subspace}\n\\begin{theorem}[Ambient root criterion]\n\\label{thm:ambientroot}\nFor every rational subspace $W$, every $h\\in H_W$, and every $m\\ge1$,\n\\[\nh\\in mH_W\\quad\\Longleftrightarrow\\quad\nh_p=0\\text{ for all primes }p\\mid m.\n\\]\n\\end{theorem}\n\\begin{proof}\nMultiplication by $m$ on $P$ is zero at primes dividing $m$ and invertible on every other coordinate. Thus the right-hand condition is equivalent to $h\\in mP$. Apply purity from Theorem~\\ref{thm:purecorrespondence}.\n\\end{proof}\nThe condition is independent of $W$, but the set of admissible $h$ is not. This distinction is the reason identical quotient laws can coexist with different original groups.","pace}\n\\begin{theorem}[Ambient root criterion]\n\\label{thm:ambientroot}\nFor every rational subspace $W$, every $h\\in H_W$, and every $m\\ge1$,\n\\[\nh\\in mH_W\\quad\\Longleftrightarrow\\quad\nh_p=0\\text{ for all primes }p\\mid m.\n\\]\n\\end{theorem}\n\\begin{proof}\nMultiplication by $m$ on $P$ is zero at primes dividing $m$ and invertible on every other coordinate. Thus the right-hand condition is equivalent to $h\\in mP$. Apply purity from Theorem~\\ref{thm:purecorrespondence}.\n\\end{proof}\nThe condition is independent of $W$, but the set of admissible $h$ is not. This distinction is the reason identical quotient laws can coexist with different original groups.","\\in H_W$, and every $m\\ge1$,\n\\[\nh\\in mH_W\\quad\\Longleftrightarrow\\quad\nh_p=0\\text{ for all primes }p\\mid m.\n\\]\n\\end{theorem}\n\\begin{proof}\nMultiplication by $m$ on $P$ is zero at primes dividing $m$ and invertible on every other coordinate. Thus the right-hand condition is equivalent to $h\\in mP$. Apply purity from Theorem~\\ref{thm:purecorrespondence}.\n\\end{proof}\nThe condition is independent of $W$, but the set of admissible $h$ is not. This distinction is the reason identical quotient laws can coexist with different original groups.","h\\in mH_W\\quad\\Longleftrightarrow\\quad\nh_p=0\\text{ for all primes }p\\mid m.\n\\]\n\\end{theorem}\n\\begin{proof}\nMultiplication by $m$ on $P$ is zero at primes dividing $m$ and invertible on every other coordinate. Thus the right-hand condition is equivalent to $h\\in mP$. Apply purity from Theorem~\\ref{thm:purecorrespondence}.\n\\end{proof}\nThe condition is independent of $W$, but the set of admissible $h$ is not. This distinction is the reason identical quotient laws can coexist with different original groups.","\\begin{theorem}[One finite-quotient system for every preimage]\n\\label{thm:generalquotients}\nFor every rational subspace $W$ and every $m\\ge1$, the coordinate map induces an isomorphism\n\\[\nH_W/mH_W\\ \\cong\\ \\prod_{p\\mid m}C_p.\n\\]\nFor $m\\mid n$, the quotient transition map is the coordinate restriction from primes dividing $n$ to primes dividing $m$. These quotients are cofinal among all finite quotients, and\n\\[\n\\widehat{H_W}\\cong P.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe coordinate map onto the displayed finite product is surjective already on $D$. Its kernel is $mH_W$ by Theorem~\\ref{thm:ambientroot}. The transition assertion follows by evaluating the same coordinates.","em}[One finite-quotient system for every preimage]\n\\label{thm:generalquotients}\nFor every rational subspace $W$ and every $m\\ge1$, the coordinate map induces an isomorphism\n\\[\nH_W/mH_W\\ \\cong\\ \\prod_{p\\mid m}C_p.\n\\]\nFor $m\\mid n$, the quotient transition map is the coordinate restriction from primes dividing $n$ to primes dividing $m$. These quotients are cofinal among all finite quotients, and\n\\[\n\\widehat{H_W}\\cong P.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe coordinate map onto the displayed finite product is surjective already on $D$. Its kernel is $mH_W$ by Theorem~\\ref{thm:ambientroot}. The transition assertion follows by evaluating the same coordinates.","e coordinate map induces an isomorphism\n\\[\nH_W/mH_W\\ \\cong\\ \\prod_{p\\mid m}C_p.\n\\]\nFor $m\\mid n$, the quotient transition map is the coordinate restriction from primes dividing $n$ to primes dividing $m$. These quotients are cofinal among all finite quotients, and\n\\[\n\\widehat{H_W}\\cong P.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe coordinate map onto the displayed finite product is surjective already on $D$. Its kernel is $mH_W$ by Theorem~\\ref{thm:ambientroot}. The transition assertion follows by evaluating the same coordinates.","duces an isomorphism\n\\[\nH_W/mH_W\\ \\cong\\ \\prod_{p\\mid m}C_p.\n\\]\nFor $m\\mid n$, the quotient transition map is the coordinate restriction from primes dividing $n$ to primes dividing $m$. These quotients are cofinal among all finite quotients, and\n\\[\n\\widehat{H_W}\\cong P.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe coordinate map onto the displayed finite product is surjective already on $D$. Its kernel is $mH_W$ by Theorem~\\ref{thm:ambientroot}. The transition assertion follows by evaluating the same coordinates.","If $H_W\\to F$ is a homomorphism to a finite group, then the image is abelian and is killed by $\\lvert F\\rvert$. Its kernel therefore contains $\\lvert F\\rvert H_W$. Hence the displayed kernels are cofinal. Their inverse limit is the product of all prime coordinates, which is $P$. Under this identification the completion map is the original inclusion $H_W\\hookrightarrow P$.\n\\end{proof}\nThere is a stronger natural statement. It compares maps to every finite target rather than merely comparing the quotient groups.","\\begin{proposition}[Finite-target equivalence]\n\\label{thm:finitehom}\nFor every finite group $F$, restriction along $D\\hookrightarrow H_W$ gives a natural bijection\n\\[\n\\Hom(H_W,F)\\longrightarrow\\Hom(D,F).\n\\]\n\\end{proposition}\n\\begin{proof}\nPut $m=\\lvert F\\rvert$. Every $h\\in H_W$ agrees modulo $mH_W$ with an element of $D$, namely its finite coordinate vector on primes dividing $m$. Every homomorphism to $F$ kills $mH_W$. Thus two maps agreeing on $D$ agree everywhere.","roposition}[Finite-target equivalence]\n\\label{thm:finitehom}\nFor every finite group $F$, restriction along $D\\hookrightarrow H_W$ gives a natural bijection\n\\[\n\\Hom(H_W,F)\\longrightarrow\\Hom(D,F).\n\\]\n\\end{proposition}\n\\begin{proof}\nPut $m=\\lvert F\\rvert$. Every $h\\in H_W$ agrees modulo $mH_W$ with an element of $D$, namely its finite coordinate vector on primes dividing $m$. Every homomorphism to $F$ kills $mH_W$. Thus two maps agreeing on $D$ agree everywhere.","}[Finite-target equivalence]\n\\label{thm:finitehom}\nFor every finite group $F$, restriction along $D\\hookrightarrow H_W$ gives a natural bijection\n\\[\n\\Hom(H_W,F)\\longrightarrow\\Hom(D,F).\n\\]\n\\end{proposition}\n\\begin{proof}\nPut $m=\\lvert F\\rvert$. Every $h\\in H_W$ agrees modulo $mH_W$ with an element of $D$, namely its finite coordinate vector on primes dividing $m$. Every homomorphism to $F$ kills $mH_W$. Thus two maps agreeing on $D$ agree everywhere.","l{thm:finitehom}\nFor every finite group $F$, restriction along $D\\hookrightarrow H_W$ gives a natural bijection\n\\[\n\\Hom(H_W,F)\\longrightarrow\\Hom(D,F).\n\\]\n\\end{proposition}\n\\begin{proof}\nPut $m=\\lvert F\\rvert$. Every $h\\in H_W$ agrees modulo $mH_W$ with an element of $D$, namely its finite coordinate vector on primes dividing $m$. Every homomorphism to $F$ kills $mH_W$. Thus two maps agreeing on $D$ agree everywhere.","Conversely, a map $D\\to F$ kills every prime factor whose order does not divide $m$. It factors through the finite coordinate product on primes dividing $m$. Composing that finite-coordinate rule with the coordinate projection of $H_W$ extends it. The extension is unique by the first part. Postcomposition with a map of finite groups commutes with restriction, giving naturality.\n\\end{proof}\nThe argument never subtracts maps to a nonabelian group. The assertion is a bijection of sets of homomorphisms, and naturality is with respect to postcomposition.","\\section{Countability, rank, and nonsplitting}\n\\begin{proposition}[Structure of an ambient preimage]\n\\label{thm:HWstructure}\nFor every rational subspace $W$,\n\\[\n\\Tor(H_W)=D,\\qquad H_W/D\\cong W.\n\\]\nThe group $H_W$ is reduced and residually finite. It is countable exactly when $W$ has finite or countably infinite dimension over $\\Q$. If $W\\ne0$, the sequence\n\\[\n0\\longrightarrow D\\longrightarrow H_W\\longrightarrow W\\longrightarrow0\n\\]\ndoes not split.\n\\end{proposition}\n\\begin{proof}\nThe torsion equality follows from $D\\subseteq H_W\\subseteq P$ and Theorem~\\ref{thm:torP}. The quotient assertion is the first isomorphism theorem. Reducedness follows from reducedness of $P$. A nonzero element has a nonzero finite coordinate, which detects it and proves residual finiteness.","ture}\nFor every rational subspace $W$,\n\\[\n\\Tor(H_W)=D,\\qquad H_W/D\\cong W.\n\\]\nThe group $H_W$ is reduced and residually finite. It is countable exactly when $W$ has finite or countably infinite dimension over $\\Q$. If $W\\ne0$, the sequence\n\\[\n0\\longrightarrow D\\longrightarrow H_W\\longrightarrow W\\longrightarrow0\n\\]\ndoes not split.\n\\end{proposition}\n\\begin{proof}\nThe torsion equality follows from $D\\subseteq H_W\\subseteq P$ and Theorem~\\ref{thm:torP}. The quotient assertion is the first isomorphism theorem. Reducedness follows from reducedness of $P$. A nonzero element has a nonzero finite coordinate, which detects it and proves residual finiteness.","hen $W$ has finite or countably infinite dimension over $\\Q$. If $W\\ne0$, the sequence\n\\[\n0\\longrightarrow D\\longrightarrow H_W\\longrightarrow W\\longrightarrow0\n\\]\ndoes not split.\n\\end{proposition}\n\\begin{proof}\nThe torsion equality follows from $D\\subseteq H_W\\subseteq P$ and Theorem~\\ref{thm:torP}. The quotient assertion is the first isomorphism theorem. Reducedness follows from reducedness of $P$. A nonzero element has a nonzero finite coordinate, which detects it and proves residual finiteness.","A rational vector space on a finite or countable basis is countable, since each vector has a finite rational linear expression. Each fiber of $H_W\\to W$ is a coset of the countable group $D$. Thus $H_W$ is countable in this case. Conversely, a quotient of a countable group is countable and has at most a countable basis.","A splitting would embed the nonzero divisible group $W$ into $P$. Theorem~\\ref{thm:Vrational} excludes every such subgroup.\n\\end{proof}\nThe rank of the torsion-free quotient is exactly $\\dim_{\\Q}W$. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it admits an especially sharp classification and computability theorem.","\\section{What has been fixed and what still varies}\nThe entire product $P$, all finite quotient objects, their transition maps, and the maps to arbitrary finite targets are fixed throughout the construction. The subspace $W$ remains free. This is an ambient realization theorem, not a method for discovering $W$ from those observations.","The preimage formula also identifies the correct role of completion. Every $H_W$ contains the same dense torsion subgroup. Its completion admits every compatible coordinate tuple. Its original image admits only those tuples whose tail class lies in $W$. The missing information is an admissibility condition on the quotient, not a defect in the finite coordinates themselves.","\\section{Choosing a rational line}\nLet $A\\subseteq\\N$, write $a_n=\\one_A(n)$, and define\n\\[\nS_A=\\{p_{2n+a_n}:n\\in\\N\\}.\n\\]\nExactly one prime from each pair is selected. Both $S_A$ and its complement are infinite. Let $u_A\\in P$ be its indicator:\n\\[\n(u_A)_p=\\begin{cases}1,&p\\in S_A,","\\section{Choosing a rational line}\nLet $A\\subseteq\\N$, write $a_n=\\one_A(n)$, and define\n\\[\nS_A=\\{p_{2n+a_n}:n\\in\\N\\}.\n\\]\nExactly one prime from each pair is selected. Both $S_A$ and its complement are infinite. Let $u_A\\in P$ be its indicator:\n\\[\n(u_A)_p=\\begin{cases}1,&p\\in S_A,\\\\0,&p\\notin S_A.\\end{cases}\n\\]\nBecause $u_A$ has infinite support, $q(u_A)\\ne0$. We can therefore define a rational line and its preimage by\n\\[\nW_A=\\Q q(u_A),\\qquad G_A=q^{-1}(W_A).\n\\]\nThis definition produces an actual group for every set $A$. It does not claim an effective method for obtaining the selector bits of an arbitrary $A$.","0,&p\\notin S_A.\\end{cases}\n\\]\nBecause $u_A$ has infinite support, $q(u_A)\\ne0$. We can therefore define a rational line and its preimage by\n\\[\nW_A=\\Q q(u_A),\\qquad G_A=q^{-1}(W_A).\n\\]\nThis definition produces an actual group for every set $A$. It does not claim an effective method for obtaining the selector bits of an arbitrary $A$.","\\begin{figure}[tbp]\n\\centering\\includegraphics[width=\\textwidth]{\\figfile{02_prime_pairs.pdf}}\n\\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.}\n\\label{fig:primepairs}\n\\end{figure}","\\section{A section with finite arithmetic defects}\nFor a reduced rational $r=a/b$, with $b>0$, define\n\\begin{equation}\n\\label{eq:section}\n(s_A(a/b))_p=\n\\begin{cases}\na b^{-1}\\pmod p,&p\\in S_A\\text{ and }p\\nmid b,","\\section{A section with finite arithmetic defects}\nFor a reduced rational $r=a/b$, with $b>0$, define\n\\begin{equation}\n\\label{eq:section}\n(s_A(a/b))_p=\n\\begin{cases}\na b^{-1}\\pmod p,&p\\in S_A\\text{ and }p\\nmid b,\\\\\n0,&p\\notin S_A\\text{ or }p\\mid b.\n\\end{cases}\n\\end{equation}\nThe 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 $C_p$.","0,&p\\notin S_A\\text{ or }p\\mid b.\n\\end{cases}\n\\end{equation}\nThe 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 $C_p$.","The difference $b s_A(a/b)-a u_A$ is supported on the prime divisors of $b$. It follows that\n\\[\nq(s_A(r))=r q(u_A).\n\\]\nThus every element of $G_A$ is $s_A(r)+d$ for some $r\\in\\Q$ and $d\\in D$.","\\begin{theorem}[Unique normal form]\n\\label{thm:normalform}\nEvery element of $G_A$ has a unique expression\n\\[\ng=s_A(r)+d,\\qquad (r,d)\\in\\Q\\times D.\n\\]\nConsequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","em}[Unique normal form]\n\\label{thm:normalform}\nEvery element of $G_A$ has a unique expression\n\\[\ng=s_A(r)+d,\\qquad (r,d)\\in\\Q\\times D.\n\\]\nConsequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","normal form]\n\\label{thm:normalform}\nEvery element of $G_A$ has a unique expression\n\\[\ng=s_A(r)+d,\\qquad (r,d)\\in\\Q\\times D.\n\\]\nConsequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","alform}\nEvery element of $G_A$ has a unique expression\n\\[\ng=s_A(r)+d,\\qquad (r,d)\\in\\Q\\times D.\n\\]\nConsequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","ion\n\\[\ng=s_A(r)+d,\\qquad (r,d)\\in\\Q\\times D.\n\\]\nConsequently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","uently $G_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","_A$ is countably infinite, with a countable normal-form domain that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","in that does not depend on the selector.\n\\end{theorem}\n\\begin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","gin{proof}\nExistence 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 $\\Q$.\n\\end{proof}\nThe domain can be named without knowing $A$. The group operation on those names will contain the selector information.","\\section{The cocycle law}\nDefine\n\\[\nc_A(r,t)=s_A(r)+s_A(t)-s_A(r+t).\n\\]\nThe sum is taken in $P$, but its value belongs to $D$.","\\begin{lemma}[A finite support bound]\n\\label{thm:cocyclesupport}\nFor all $r,t\\in\\Q$,\n\\[\n\\supp c_A(r,t)\\subseteq\n\\{p:p\\mid\\den(r)\\den(t)\\}.\n\\]\n\\end{lemma}\n\\begin{proof}\nAt 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.\n\\end{proof}","port bound]\n\\label{thm:cocyclesupport}\nFor all $r,t\\in\\Q$,\n\\[\n\\supp c_A(r,t)\\subseteq\n\\{p:p\\mid\\den(r)\\den(t)\\}.\n\\]\n\\end{lemma}\n\\begin{proof}\nAt 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.\n\\end{proof}","\\begin{theorem}[Explicit group law]\n\\label{thm:cocyclelaw}\nIn normal-form coordinates, addition is\n\\begin{equation}\n\\label{eq:grouplaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","law]\n\\label{thm:cocyclelaw}\nIn normal-form coordinates, addition is\n\\begin{equation}\n\\label{eq:grouplaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","ordinates, addition is\n\\begin{equation}\n\\label{eq:grouplaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","tion is\n\\begin{equation}\n\\label{eq:grouplaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","n}\n\\label{eq:grouplaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","plaw}\n(r,d)+(t,e)=(r+t,d+e+c_A(r,t)).\n\\end{equation}\nThe 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 $G_A\\le P$.\n\\end{theorem}\n\\begin{proof}\nSubstitution 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.","Symmetry is immediate. The cocycle equation\n\\[\nc_A(r,t)+c_A(r+t,v)=c_A(t,v)+c_A(r,t+v)\n\\]\nfollows 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.\n\\end{proof}\nThis 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.","\\begin{proposition}[One cocycle value reads one coordinate]\n\\label{thm:cocyclebit}\nFor every prime $p$,\n\\[\nc_A(1/p,(p-1)/p)=-\\one_{S_A}(p)e_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe 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 $\\one_{S_A}(p)$.\n\\end{proof}\nDropping 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.","ordinate]\n\\label{thm:cocyclebit}\nFor every prime $p$,\n\\[\nc_A(1/p,(p-1)/p)=-\\one_{S_A}(p)e_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe 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 $\\one_{S_A}(p)$.\n\\end{proof}\nDropping 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.","\\section{Structure of the extension}\nBy 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\n\\begin{equation}\n\\label{eq:extension}\n0\\longrightarrow D\\longrightarrow G_A\\xrightarrow{\\rho_A}\\Q\\longrightarrow0,\n\\qquad \\rho_A(s_A(r)+d)=r.\n\\end{equation}\nIt does not split.","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\n\\[\nc_A(r,t)+b(r)+b(t)-b(r+t)=0\n\\]\nfor 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 $D$.","\\begin{proposition}[Finitely generated subgroups]\n\\label{thm:fgsubgroups}\nEvery 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.\n\\end{proposition}\n\\begin{proof}\nLet $L\\le G_A$ 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.","subgroups]\n\\label{thm:fgsubgroups}\nEvery 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.\n\\end{proposition}\n\\begin{proof}\nLet $L\\le G_A$ 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.","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.\n\\end{proof}\nThe extensions over finitely generated rational images split, but the extension over all of $\\Q$ does not. This is another exact distinction between f","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.\n\\end{proof}\nThe 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.","tracting 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.\n\\end{proof}\nThe 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.","\\begin{figure}[H]\n\\centering\\includegraphics[width=0.94\\textwidth]{\\figfile{03_root_certificates.pdf}}\n\\caption{For the marked element $u_A$, one prime in each pair gives an obstruction and the other gives a root. A nonzero coordinate at a prime dividing the multiplier certifies impossibility. The displayed pattern is an exact worked selector, not a statistical sample.}\n\\label{fig:rootcertificates}\n\\end{figure}","\\section{An exact decision criterion}\nThe general purity argument already establishes the root condition. For $G_A$, normal forms make the successful branch constructive.","\\begin{theorem}[Root or obstruction]\n\\label{thm:rootcertificate}\nLet $g=s_A(r)+d\\in G_A$ and let $m\\ge1$. Then\n\\[\ng\\in mG_A\\quad\\Longleftrightarrow\\quad\n g_p=0\\quad\\text{for every }p\\mid m.\n\\]\nIf the condition fails, a prime $p\\mid m$ with $g_p\\ne0$ is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof.\n\\end{theorem}\n\\begin{proof}\nNecessity follows because multiplication by $m$ is zero in every coordinate $C_p$ with $p\\mid m$.","obstruction]\n\\label{thm:rootcertificate}\nLet $g=s_A(r)+d\\in G_A$ and let $m\\ge1$. Then\n\\[\ng\\in mG_A\\quad\\Longleftrightarrow\\quad\n g_p=0\\quad\\text{for every }p\\mid m.\n\\]\nIf the condition fails, a prime $p\\mid m$ with $g_p\\ne0$ is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof.\n\\end{theorem}\n\\begin{proof}\nNecessity follows because multiplication by $m$ is zero in every coordinate $C_p$ with $p\\mid m$.","ificate}\nLet $g=s_A(r)+d\\in G_A$ and let $m\\ge1$. Then\n\\[\ng\\in mG_A\\quad\\Longleftrightarrow\\quad\n g_p=0\\quad\\text{for every }p\\mid m.\n\\]\nIf the condition fails, a prime $p\\mid m$ with $g_p\\ne0$ is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof.\n\\end{theorem}\n\\begin{proof}\nNecessity follows because multiplication by $m$ is zero in every coordinate $C_p$ with $p\\mid m$.","t $g=s_A(r)+d\\in G_A$ and let $m\\ge1$. Then\n\\[\ng\\in mG_A\\quad\\Longleftrightarrow\\quad\n g_p=0\\quad\\text{for every }p\\mid m.\n\\]\nIf the condition fails, a prime $p\\mid m$ with $g_p\\ne0$ is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof.\n\\end{theorem}\n\\begin{proof}\nNecessity follows because multiplication by $m$ is zero in every coordinate $C_p$ with $p\\mid m$.","Assume the condition holds and put $t=r/m$. The vector\n\\[\nh=g-ms_A(t)\n\\]\nhas zero image in $V$ and therefore lies in $D$. Its support is contained in the union of the support of $d$ and the prime divisors of $\\den(r)\\den(t)$. Indeed, at any other selected coordinate, reduction gives $r-mt=0$, and unselected section values are zero.","For $p\\mid m$, the coordinate $h_p$ is zero. At every other prime put $e_p=m^{-1}h_p$, and at primes dividing $m$ put $e_p=0$. Then $e\\in D$ and $me=h$. Consequently\n\\[\ny=s_A(t)+e\\quad\\text{satisfies}\\quad my=g.\n\\]\nEvery nonzero correction coordinate is found in the stated finite support set.\n\\end{proof}\nA failed root is witnessed by one coordinate. A successful root requires a rational division and a finite-support adjustment. This is a simple instance of a realization problem with both positive and negative certificates.","\\begin{proposition}[All roots and their number]\n\\label{thm:rootfiber}\nIf $my=g$ is solvable, its solution set is the coset\n\\[\ny_0+G_A[m],\\qquad\nG_A[m]=\\bigoplus_{p\\mid m}C_p.\n\\]\nIt has exactly $\\rad(m)$ elements.\n\\end{proposition}\n\\begin{proof}\nTwo solutions differ by an element annihilated by $m$, and adding such an element to a solution gives another. The ambient product has a nonzero $m$-torsion coordinate only when $p\\mid m$. All those finite coordinate factors lie in $D\\subseteq G_A$. The kernel is therefore the displayed finite direct sum, whose cardinal is $\\rad(m)$.\n\\end{proof}\nRepeated prime powers do not enlarge the root kernel. For example, a solvable fourth-root equation has two solutions, not four. The squarefree coordinate structure is responsible for th","\\begin{proposition}[All roots and their number]\n\\label{thm:rootfiber}\nIf $my=g$ is solvable, its solution set is the coset\n\\[\ny_0+G_A[m],\\qquad\nG_A[m]=\\bigoplus_{p\\mid m}C_p.\n\\]\nIt has exactly $\\rad(m)$ elements.\n\\end{proposition}\n\\begin{proof}\nTwo solutions differ by an element annihilated by $m$, and adding such an element to a solution gives another. The ambient product has a nonzero $m$-torsion coordinate only when $p\\mid m$. All those finite coordinate factors lie in $D\\subseteq G_A$. The kernel is therefore the displayed finite direct sum, whose cardinal is $\\rad(m)$.\n\\end{proof}\nRepeated prime powers do not enlarge the root kernel. For example, a solvable fourth-root equation has two solutions, not four. The squarefree coordinate structure is responsible for this behavior.","roots and their number]\n\\label{thm:rootfiber}\nIf $my=g$ is solvable, its solution set is the coset\n\\[\ny_0+G_A[m],\\qquad\nG_A[m]=\\bigoplus_{p\\mid m}C_p.\n\\]\nIt has exactly $\\rad(m)$ elements.\n\\end{proposition}\n\\begin{proof}\nTwo solutions differ by an element annihilated by $m$, and adding such an element to a solution gives another. The ambient product has a nonzero $m$-torsion coordinate only when $p\\mid m$. All those finite coordinate factors lie in $D\\subseteq G_A$. The kernel is therefore the displayed finite direct sum, whose cardinal is $\\rad(m)$.\n\\end{proof}\nRepeated prime powers do not enlarge the root kernel. For example, a solvable fourth-root equation has two solutions, not four. The squarefree coordinate structure is responsible for this behavior.","$ is solvable, its solution set is the coset\n\\[\ny_0+G_A[m],\\qquad\nG_A[m]=\\bigoplus_{p\\mid m}C_p.\n\\]\nIt has exactly $\\rad(m)$ elements.\n\\end{proposition}\n\\begin{proof}\nTwo solutions differ by an element annihilated by $m$, and adding such an element to a solution gives another. The ambient product has a nonzero $m$-torsion coordinate only when $p\\mid m$. All those finite coordinate factors lie in $D\\subseteq G_A$. The kernel is therefore the displayed finite direct sum, whose cardinal is $\\rad(m)$.\n\\end{proof}\nRepeated prime powers do not enlarge the root kernel. For example, a solvable fourth-root equation has two solutions, not four. The squarefree coordinate structure is responsible for this behavior.","_p.\n\\]\nIt has exactly $\\rad(m)$ elements.\n\\end{proposition}\n\\begin{proof}\nTwo solutions differ by an element annihilated by $m$, and adding such an element to a solution gives another. The ambient product has a nonzero $m$-torsion coordinate only when $p\\mid m$. All those finite coordinate factors lie in $D\\subseteq G_A$. The kernel is therefore the displayed finite direct sum, whose cardinal is $\\rad(m)$.\n\\end{proof}\nRepeated prime powers do not enlarge the root kernel. For example, a solvable fourth-root equation has two solutions, not four. The squarefree coordinate structure is responsible for this behavior.","\\section{The quotient formula}\n\\begin{theorem}[All integer quotients]\n\\label{thm:quotientformula}\nFor $m\\ge1$, the coordinate map\n\\[\n\\pi_m:G_A\\longrightarrow\\prod_{p\\mid m}C_p\n\\]\nis surjective with kernel $mG_A$. Thus\n\\[\nG_A/mG_A\\cong\\prod_{p\\mid m}C_p\\cong C_{\\rad(m)}.\n\\]\nFor $m\\mid n$, the transition map is the corresponding coordinate projection.\n\\end{theorem}\n\\begin{proof}\nA vector supported on the prime divisors of $m$ realizes any tuple in the target, proving surjectivity. The root criterion identifies the kernel. The Chinese remainder theorem identifies the finite product with the squarefree cyclic group. The transition assertion follows by taking the same coordinates before and after quotienting.\n\\end{proof}","mula}\n\\begin{theorem}[All integer quotients]\n\\label{thm:quotientformula}\nFor $m\\ge1$, the coordinate map\n\\[\n\\pi_m:G_A\\longrightarrow\\prod_{p\\mid m}C_p\n\\]\nis surjective with kernel $mG_A$. Thus\n\\[\nG_A/mG_A\\cong\\prod_{p\\mid m}C_p\\cong C_{\\rad(m)}.\n\\]\nFor $m\\mid n$, the transition map is the corresponding coordinate projection.\n\\end{theorem}\n\\begin{proof}\nA vector supported on the prime divisors of $m$ realizes any tuple in the target, proving surjectivity. The root criterion identifies the kernel. The Chinese remainder theorem identifies the finite product with the squarefree cyclic group. The transition assertion follows by taking the same coordinates before and after quotienting.\n\\end{proof}","\\begin{center}\n\\begin{tabular}{@{}rrrr@{}}\n\\toprule\n$m$&$\\rad(m)$&$G_A/mG_A$&Root-fiber size when nonempty","midrule\n1&1&$0$&1","4&2&$C_2$&2","8&2&$C_2$&2","12&6&$C_6$&6","18&6&$C_6$&6","30&30&$C_{30}$&30","60&30&$C_{30}$&30","The kernel itself depends only on the distinct prime divisors:\n\\[\nmG_A=\\rad(m)G_A.\n\\]\nThis is stronger than equality of the quotient orders. It is equality of the subgroups inside the fixed source group.","\\section{Why these are all finite quotients}\nLet $f:G_A\\to F$ be any homomorphism to a finite group. Its image is an abelian subgroup of $F$, and every element of the image is annihilated by $\\lvert F\\rvert$. Therefore $f$ kills $\\lvert F\\rvert G_A$ and factors through the quotient in Theorem~\\ref{thm:quotientformula}.","\\begin{corollary}[Common profinite completion]\n\\label{thm:GAcompletion}\nFor every $A\\subseteq\\N$, the inclusion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite","\\begin{corollary}[Common profinite completion]\n\\label{thm:GAcompletion}\nFor every $A\\subseteq\\N$, the inclusion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3","\\begin{corollary}[Common profinite completion]\n\\label{thm:GAcompletion}\nFor every $A\\subseteq\\N$, the inclusion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product","\\begin{corollary}[Common profinite completion]\n\\label{thm:GAcompletion}\nFor every $A\\subseteq\\N$, the inclusion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective i","mmon profinite completion]\n\\label{thm:GAcompletion}\nFor every $A\\subseteq\\N$, the inclusion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","sion $G_A\\subseteq P$ extends to a canonical isomorphism\n\\[\n\\widehat{G_A}\\cong P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","P.\n\\]\nThe induced topology on $G_A$ is its full profinite topology.\n\\end{corollary}\n\\begin{proof}\nThe coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","The coordinate kernels are cofinal among finite-index normal subgroups by the preceding factorization argument. Their inverse limit is $P$. Neighborhoods defined by finitely many coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","coordinate zeros coincide with the kernels $mG_A$, so the induced and profinite topologies agree.\n\\end{proof}\nThe product $P$ has a computable profinite presentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","resentation: list the primes, form the finite addition tables of $A_N$, and give the coordinate projections. This is the standard inverse-system meaning of a computable profinite presentation \\cite[Definition 3.3]{Lupini}. It is independent of $A$. The uncountable product is presented by an effective inverse system, not by a computable listing of all its elements. This does not turn an arbitrary $G_A$ into a computable discrete group.","\\section{The information retained by a marked element}\nFor the distinguished element $u_A=s_A(1)$, the coordinate criterion simplifies to\n\\begin{equation}\n\\label{eq:markroot}\np\\mid_{G_A}u_A\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\end{equation}\nConsequently, for each $n$,\n\\begin{align}\n p_{2n}\\mid_{G_A}u_A&\\quad\\Longleftrightarrow\\quad n\\in A,\\label{eq:evenroot}","\\section{The information retained by a marked element}\nFor the distinguished element $u_A=s_A(1)$, the coordinate criterion simplifies to\n\\begin{equation}\n\\label{eq:markroot}\np\\mid_{G_A}u_A\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\end{equation}\nConsequently, for each $n$,\n\\begin{align}\n p_{2n}\\mid_{G_A}u_A&\\quad\\Longleftrightarrow\\quad n\\in A,\\label{eq:evenroot}\\\\\n p_{2n+1}\\mid_{G_A}u_A&\\quad\\Longleftrightarrow\\quad n\\notin A.\\label{eq:oddroot}\n\\end{align}\nExactly one root equation is solvable. This binary pattern is the selector written in intrinsic group language.","\\section{The signature of an infinite-order element}\nThe rational coordinate of a normal form detects torsion: $s_A(r)+d$ has infinite order exactly when $r\\ne0$.","\\begin{lemma}[Tail signature]\n\\label{thm:tailsignature}\nIf $g=s_A(r)+d$ has infinite order, then for all but finitely many primes,\n\\[\np\\mid_{G_A}g\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\]\nAn explicit exception set consists of the primes dividing the nonzero numerator or denominator of $r$, together with $\\supp(d)$.\n\\end{lemma}\n\\begin{proof}\nOutside that set, the reduction of $r$ is defined and nonzero. The coordinate of $g$ is therefore nonzero exactly at the selected primes. Apply Theorem~\\ref{thm:rootcertificate} with multiplier $p$.\n\\end{proof}\nThe full signature of the distinguished element is exact. The signature of an arbitrary infinite-order element is exact only in the tail. This finite discrepancy is what determines the correct unmarked classifica","\\begin{lemma}[Tail signature]\n\\label{thm:tailsignature}\nIf $g=s_A(r)+d$ has infinite order, then for all but finitely many primes,\n\\[\np\\mid_{G_A}g\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\]\nAn explicit exception set consists of the primes dividing the nonzero numerator or denominator of $r$, together with $\\supp(d)$.\n\\end{lemma}\n\\begin{proof}\nOutside that set, the reduction of $r$ is defined and nonzero. The coordinate of $g$ is therefore nonzero exactly at the selected primes. Apply Theorem~\\ref{thm:rootcertificate} with multiplier $p$.\n\\end{proof}\nThe full signature of the distinguished element is exact. The signature of an arbitrary infinite-order element is exact only in the tail. This finite discrepancy is what determines the correct unmarked classification.","\\label{thm:tailsignature}\nIf $g=s_A(r)+d$ has infinite order, then for all but finitely many primes,\n\\[\np\\mid_{G_A}g\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\]\nAn explicit exception set consists of the primes dividing the nonzero numerator or denominator of $r$, together with $\\supp(d)$.\n\\end{lemma}\n\\begin{proof}\nOutside that set, the reduction of $r$ is defined and nonzero. The coordinate of $g$ is therefore nonzero exactly at the selected primes. Apply Theorem~\\ref{thm:rootcertificate} with multiplier $p$.\n\\end{proof}\nThe full signature of the distinguished element is exact. The signature of an arbitrary infinite-order element is exact only in the tail. This finite discrepancy is what determines the correct unmarked classification.","_A}g\\quad\\Longleftrightarrow\\quad p\\notin S_A.\n\\]\nAn explicit exception set consists of the primes dividing the nonzero numerator or denominator of $r$, together with $\\supp(d)$.\n\\end{lemma}\n\\begin{proof}\nOutside that set, the reduction of $r$ is defined and nonzero. The coordinate of $g$ is therefore nonzero exactly at the selected primes. Apply Theorem~\\ref{thm:rootcertificate} with multiplier $p$.\n\\end{proof}\nThe full signature of the distinguished element is exact. The signature of an arbitrary infinite-order element is exact only in the tail. This finite discrepancy is what determines the correct unmarked classification.","\\section{Isomorphism means eventual agreement}\n\\begin{theorem}[Unmarked isomorphism classification]\n\\label{thm:isoclass}\nFor all $A,B\\subseteq\\N$,\n\\[\nG_A\\cong G_B\\quad\\Longleftrightarrow\\quad A\\eqfin B.\n\\]\nWhen $A\\eqfin B$, the two groups are equal as subgroups of the fixed product $P$.\n\\end{theorem}\n\\begin{proof}\nSuppose $f:G_A\\to G_B$ is an isomorphism. The image of $u_A$ has infinite order. An isomorphism preserves the existence of a $p$th root for every prime $p$. Combining the exact signature \\eqref{eq:markroot} with Lemma~\\ref{thm:tailsignature} for $f(u_A)$ gives $S_A\\eqfin S_B$. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.","ent}\n\\begin{theorem}[Unmarked isomorphism classification]\n\\label{thm:isoclass}\nFor all $A,B\\subseteq\\N$,\n\\[\nG_A\\cong G_B\\quad\\Longleftrightarrow\\quad A\\eqfin B.\n\\]\nWhen $A\\eqfin B$, the two groups are equal as subgroups of the fixed product $P$.\n\\end{theorem}\n\\begin{proof}\nSuppose $f:G_A\\to G_B$ is an isomorphism. The image of $u_A$ has infinite order. An isomorphism preserves the existence of a $p$th root for every prime $p$. Combining the exact signature \\eqref{eq:markroot} with Lemma~\\ref{thm:tailsignature} for $f(u_A)$ gives $S_A\\eqfin S_B$. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.","ed isomorphism classification]\n\\label{thm:isoclass}\nFor all $A,B\\subseteq\\N$,\n\\[\nG_A\\cong G_B\\quad\\Longleftrightarrow\\quad A\\eqfin B.\n\\]\nWhen $A\\eqfin B$, the two groups are equal as subgroups of the fixed product $P$.\n\\end{theorem}\n\\begin{proof}\nSuppose $f:G_A\\to G_B$ is an isomorphism. The image of $u_A$ has infinite order. An isomorphism preserves the existence of a $p$th root for every prime $p$. Combining the exact signature \\eqref{eq:markroot} with Lemma~\\ref{thm:tailsignature} for $f(u_A)$ gives $S_A\\eqfin S_B$. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.","quad\\Longleftrightarrow\\quad A\\eqfin B.\n\\]\nWhen $A\\eqfin B$, the two groups are equal as subgroups of the fixed product $P$.\n\\end{theorem}\n\\begin{proof}\nSuppose $f:G_A\\to G_B$ is an isomorphism. The image of $u_A$ has infinite order. An isomorphism preserves the existence of a $p$th root for every prime $p$. Combining the exact signature \\eqref{eq:markroot} with Lemma~\\ref{thm:tailsignature} for $f(u_A)$ gives $S_A\\eqfin S_B$. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.","ghtarrow\\quad A\\eqfin B.\n\\]\nWhen $A\\eqfin B$, the two groups are equal as subgroups of the fixed product $P$.\n\\end{theorem}\n\\begin{proof}\nSuppose $f:G_A\\to G_B$ is an isomorphism. The image of $u_A$ has infinite order. An isomorphism preserves the existence of a $p$th root for every prime $p$. Combining the exact signature \\eqref{eq:markroot} with Lemma~\\ref{thm:tailsignature} for $f(u_A)$ gives $S_A\\eqfin S_B$. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.","Conversely, suppose $A\\eqfin B$. For every rational $r$, the vectors $s_A(r)$ and $s_B(r)$ differ only on the finitely many prime pairs where the selectors differ. Thus their difference belongs to $D$. The descriptions $G_A=s_A(\\Q)+D$ and $G_B=s_B(\\Q)+D$ define the same subset of $P$.\n\\end{proof}\nThe normal forms need not be identical. Their coordinate change is\n\\begin{equation}\n\\label{eq:transport}\n(r,d)\\longmapsto\n\\bigl(r,d+s_A(r)-s_B(r)\\bigr).\n\\end{equation}\nIt fixes the represented ambient element. The correction is finite exactly because the selectors differ only finitely.","\\begin{corollary}[Marked classification]\n\\label{thm:markedclass}\nAn isomorphism $(G_A,u_A)\\cong(G_B,u_B)$ exists if and only if $A=B$.\n\\end{corollary}\n\\begin{proof}\nA marked isomorphism preserves the exact two-prime root test \\eqref{eq:evenroot}--\\eqref{eq:oddroot} at every index. Therefore every bit agrees. The converse is the identity map.\n\\end{proof}","\\begin{example}[The same group with a different marked point]\nLet $A=\\varnothing$ and $B=\\{0\\}$. Only the first prime pair changes. Then $G_A=G_B$ inside $P$, but $u_A-u_B=e_2-e_3$. The distinguished element is not preserved. In particular, the statement $G_A\\cong G_B\\iff A=B$ would be false for unmarked groups.\n\\end{example}","\\section{All homomorphisms are diagonal}\nThe product $P$ is also a commutative ring under coordinatewise multiplication. Multiplication by any $\\lambda\\in P$ preserves $D$, so it induces an additive endomorphism $\\overline M_\\lambda$ of $V$. Because division in $V$ is unique, this induced endomorphism is $\\Q$-linear.","\\begin{theorem}[Complete homomorphism description]\n\\label{thm:allhom}\nEvery homomorphism $f:G_A\\to G_B$ has the form\n\\[\nf(g)_p=\\lambda_p g_p\\qquad(p\\text{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","{theorem}[Complete homomorphism description]\n\\label{thm:allhom}\nEvery homomorphism $f:G_A\\to G_B$ has the form\n\\[\nf(g)_p=\\lambda_p g_p\\qquad(p\\text{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","lete homomorphism description]\n\\label{thm:allhom}\nEvery homomorphism $f:G_A\\to G_B$ has the form\n\\[\nf(g)_p=\\lambda_p g_p\\qquad(p\\text{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","the form\n\\[\nf(g)_p=\\lambda_p g_p\\qquad(p\\text{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","\\lambda_p g_p\\qquad(p\\text{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","{ prime})\n\\]\nfor a unique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","ique $\\lambda\\in P$. Conversely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","rsely, coordinate multiplication by $\\lambda$ defines such a homomorphism exactly when\n\\begin{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","{equation}\n\\label{eq:homcondition}\nq(\\lambda u_A)\\in\\Q q(u_B).\n\\end{equation}\nEquivalently, there is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","ere is a rational $r$ for which $\\lambda u_A-s_B(r)$ has finite support.\n\\end{theorem}\n\\begin{proof}\nFor a homomorphism $f$, the composite $G_A\\to G_B\\to C_p$ kills $pG_A$. The quotient $G_A/pG_A$ is canonically $C_p$. Every endomorphism of this cyclic additive group is multiplication by a unique scalar $\\lambda_p$. Hence the displayed coordinate rule holds. Uniqueness also follows by evaluating $f(e_p)$.","Conversely, multiplication by any $\\lambda$ is a group endomorphism of $P$ and preserves $D$. Since its induced map on $V$ is rational-linear, it carries $\\Q q(u_A)$ into $\\Q q(u_B)$ exactly when it carries the generating vector into that line. This is \\eqref{eq:homcondition}. Taking preimages proves that the restriction maps $G_A$ into $G_B$. The equivalent finite-support condition follows from the definition of the quotient.\n\\end{proof}\nNo continuity assumption was imposed. Every resulting homomorphism is nevertheless continuous for the induced profinite topologies, because each output coordinate depends only on the corresponding input coordinate. Its extension to the completion is the same coordinatewise scalar map on $P$.","$ into $G_B$. The equivalent finite-support condition follows from the definition of the quotient.\n\\end{proof}\nNo continuity assumption was imposed. Every resulting homomorphism is nevertheless continuous for the induced profinite topologies, because each output coordinate depends only on the corresponding input coordinate. Its extension to the completion is the same coordinatewise scalar map on $P$.","\\begin{theorem}[Torsion-image obstruction]\n\\label{thm:homobstruction}\nIf $A\\neqfin B$, every homomorphism $G_A\\to G_B$ has image contained in $D$. Consequently neither group embeds in the other, and neither is a quotient of the other.\n\\end{theorem}\n\\begin{proof}\nA homomorphism preserves torsion and induces a map $\\Q\\to\\Q$ on the quotients, necessarily multiplication by a rational $r$. If $r=0$, its image lies in $D$.","orsion-image obstruction]\n\\label{thm:homobstruction}\nIf $A\\neqfin B$, every homomorphism $G_A\\to G_B$ has image contained in $D$. Consequently neither group embeds in the other, and neither is a quotient of the other.\n\\end{theorem}\n\\begin{proof}\nA homomorphism preserves torsion and induces a map $\\Q\\to\\Q$ on the quotients, necessarily multiplication by a rational $r$. If $r=0$, its image lies in $D$.","Assume $r\\ne0$. Then $f(u_A)$ has the tail signature of $S_B$. For every $p\\notin S_A$, the element $u_A$ has a $p$th root, and its image has a $p$th root as well. Lemma~\\ref{thm:tailsignature} implies that $S_B\\setminus S_A$ is finite. Each differing prime pair contributes one element to this difference, so $A\\eqfin B$, contrary to the assumption.","An embedding cannot send an infinite-order element to torsion. A surjection onto $G_B$ cannot have torsion image, since $G_B/D\\cong\\Q\\ne0$.\n\\end{proof}\nA useful precise consequence of Theorem~\\ref{thm:allhom} is that, when $A\\neqfin B$, the possible diagonal sequences are exactly those for which $\\lambda_p=0$ at all but finitely many primes of $S_A$. Their values on the complementary primes are unrestricted. These maps can be nonzero, and their torsion images need not be finite.","\\section{The endomorphism ring and its units}\nWrite $\\overline r_p$ for reduction of a rational $r$ at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.","\\begin{corollary}[Endomorphism ring]\n\\label{thm:endGA}\nUnder coordinate multiplication,\n\\[\n\\End(G_A)\\cong\nE_A=\\{\\lambda\\in P:\\exists r\\in\\Q,\\ \n\\lambda_p=\\overline r_p\\text{ for almost every }p\\in S_A\\}.\n\\]\nThe rational $r$ is unique. It is the scalar induced on $G_A/D$, and the map $E_A\\to\\Q$ sending $\\lambda$ to $r$ is a surjective ring homomorphism.\n\\end{corollary}\n\\begin{proof}\nApply Theorem~\\ref{thm:allhom} with $B=A$. At unselected coordinates, both $\\lambda u_A$ and $s_A(r)$ vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.","omorphism ring]\n\\label{thm:endGA}\nUnder coordinate multiplication,\n\\[\n\\End(G_A)\\cong\nE_A=\\{\\lambda\\in P:\\exists r\\in\\Q,\\ \n\\lambda_p=\\overline r_p\\text{ for almost every }p\\in S_A\\}.\n\\]\nThe rational $r$ is unique. It is the scalar induced on $G_A/D$, and the map $E_A\\to\\Q$ sending $\\lambda$ to $r$ is a surjective ring homomorphism.\n\\end{corollary}\n\\begin{proof}\nApply Theorem~\\ref{thm:allhom} with $B=A$. At unselected coordinates, both $\\lambda u_A$ and $s_A(r)$ vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.","coordinate multiplication,\n\\[\n\\End(G_A)\\cong\nE_A=\\{\\lambda\\in P:\\exists r\\in\\Q,\\ \n\\lambda_p=\\overline r_p\\text{ for almost every }p\\in S_A\\}.\n\\]\nThe rational $r$ is unique. It is the scalar induced on $G_A/D$, and the map $E_A\\to\\Q$ sending $\\lambda$ to $r$ is a surjective ring homomorphism.\n\\end{corollary}\n\\begin{proof}\nApply Theorem~\\ref{thm:allhom} with $B=A$. At unselected coordinates, both $\\lambda u_A$ and $s_A(r)$ vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.","lication,\n\\[\n\\End(G_A)\\cong\nE_A=\\{\\lambda\\in P:\\exists r\\in\\Q,\\ \n\\lambda_p=\\overline r_p\\text{ for almost every }p\\in S_A\\}.\n\\]\nThe rational $r$ is unique. It is the scalar induced on $G_A/D$, and the map $E_A\\to\\Q$ sending $\\lambda$ to $r$ is a surjective ring homomorphism.\n\\end{corollary}\n\\begin{proof}\nApply Theorem~\\ref{thm:allhom} with $B=A$. At unselected coordinates, both $\\lambda u_A$ and $s_A(r)$ vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.","If two different rationals agreed modulo almost every selected prime, their nonzero difference would have a numerator divisible by infinitely many primes, which is impossible. Thus $r$ is unique. Addition and composition induce addition and multiplication of the quotient scalars, giving a ring homomorphism. The diagonal sequence $s_A(r)$ realizes every rational $r$, proving surjectivity.\n\\end{proof}\nThe arbitrary coordinates outside $S_A$ show why the endomorphism ring may contain infinitely specified operators. The seed-rank obstruction below remains true even when all these operators are available.","lo almost every selected prime, their nonzero difference would have a numerator divisible by infinitely many primes, which is impossible. Thus $r$ is unique. Addition and composition induce addition and multiplication of the quotient scalars, giving a ring homomorphism. The diagonal sequence $s_A(r)$ realizes every rational $r$, proving surjectivity.\n\\end{proof}\nThe arbitrary coordinates outside $S_A$ show why the endomorphism ring may contain infinitely specified operators. The seed-rank obstruction below remains true even when all these operators are available.","\\begin{corollary}[Automorphisms]\n\\label{thm:autGA}\nAn endomorphism with diagonal $\\lambda\\in E_A$ is an automorphism if and only if every coordinate $\\lambda_p$ is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal $\\lambda_p^{-1}$.\n\\end{corollary}\n\\begin{proof}\nAn automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, the eventual rational scalar cannot be zero because $S_A$ is infinite. Outside finitely many selected primes, the inverse diagonal is reduction of its reciprocal. Thus the inverse diagonal belongs to $E_A$ and defines an inverse endomorphism.\n\\end{proof}","_A$ is an automorphism if and only if every coordinate $\\lambda_p$ is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal $\\lambda_p^{-1}$.\n\\end{corollary}\n\\begin{proof}\nAn automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, the eventual rational scalar cannot be zero because $S_A$ is infinite. Outside finitely many selected primes, the inverse diagonal is reduction of its reciprocal. Thus the inverse diagonal belongs to $E_A$ and defines an inverse endomorphism.\n\\end{proof}","has diagonal $\\lambda_p^{-1}$.\n\\end{corollary}\n\\begin{proof}\nAn automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, the eventual rational scalar cannot be zero because $S_A$ is infinite. Outside finitely many selected primes, the inverse diagonal is reduction of its reciprocal. Thus the inverse diagonal belongs to $E_A$ and defines an inverse endomorphism.\n\\end{proof}","\\section{Infinite seed rank before completion}\n\\begin{theorem}[Seed-rank separation]\n\\label{thm:rankGA}\nFor every $A$,\n\\[\n\\srank_{\\End(G_A)}(G_A)=\\aleph_0.\n\\]\nNo finite family topologically generates $G_A$, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one.\n\\end{theorem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not re","\\section{Infinite seed rank before completion}\n\\begin{theorem}[Seed-rank separation]\n\\label{thm:rankGA}\nFor every $A$,\n\\[\n\\srank_{\\End(G_A)}(G_A)=\\aleph_0.\n\\]\nNo finite family topologically generates $G_A$, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one.\n\\end{theorem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not repair the failure.","nk before completion}\n\\begin{theorem}[Seed-rank separation]\n\\label{thm:rankGA}\nFor every $A$,\n\\[\n\\srank_{\\End(G_A)}(G_A)=\\aleph_0.\n\\]\nNo finite family topologically generates $G_A$, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one.\n\\end{theorem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not repair the failure.","its completion has ordinary topological generating rank one.\n\\end{theorem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not repair the failure.","ary topological generating rank one.\n\\end{theorem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not repair the failure.","orem}\n\\begin{proof}\nEach element $s_A(r)+d$ has only finitely many nonzero coordinates outside $S_A$. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~\\ref{thm:allhom}, every endomorphism preserves a zero coordinate as zero. Finite sums of orbit elements still vanish at any omitted prime $p$. The projection onto $C_p$ is a non-generation certificate, since $e_p\\in G_A$ has nonzero projection. Its kernel is closed, so taking closure does not repair the failure.","Countably many seeds suffice because $G_A$ is countable. For ordinary topological generation, the coordinate vectors already generate the dense subgroup $D$. Finally, the completion is $P$, whose all-one vector is a topological generator by Theorem~\\ref{thm:procyclic}.\n\\end{proof}\nThe same infinite seed rank occurs for every selector. It does not measure the varying effective complexity. Algebraic generation, finite description, and computable presentation remain separate invariants.","\\section{Oracles, copies, and degrees}\nAn oracle for a set $X\\subseteq\\N$ answers membership questions about $X$. The notation $A\\le_T X$ means that a terminating oracle procedure decides membership in $A$. Two sets have the same Turing degree when each is computable from the other.","\\section{A presentation from the selector}\nReduced rationals have canonical integer codes. Finite-support vectors in $D$ have canonical codes given by increasing prime labels and nonzero residues. Their Cartesian product is a computably named countable domain with decidable equality and a computable bijection to $\\N$.","\\begin{proposition}[Uniform construction relative to the selector]\n\\label{thm:computeupper}\nIf $A\\le_T X$, then the normal-form construction gives an $X$-computable copy of $G_A$. In its canonical normal-form presentation, the group operation has Turing degree exactly $\\deg_T(A)$.\n\\end{proposition}\n\\begin{proof}\nTo add $(r,d)$ and $(t,e)$, compute the prime divisors of $\\den(r)\\den(t)$. At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~\\ref{thm:cocyclesupport} guarantees that every other correction coordinate is zero. Formula~\\eqref{eq:grouplaw} then gives the sum. Negation and equality are computable without querying $A$.","\\begin{proof}\nTo add $(r,d)$ and $(t,e)$, compute the prime divisors of $\\den(r)\\den(t)$. At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~\\ref{thm:cocyclesupport} guarantees that every other correction coordinate is zero. Formula~\\eqref{eq:grouplaw} then gives the sum. Negation and equality are computable without querying $A$.","For the lower bound on this particular presentation, compute the sum of the coded elements $(1/p,0)$ and $((p-1)/p,0)$. Its finite torsion part is the value in Proposition~\\ref{thm:cocyclebit}. For $p=p_{2n}$, that value is zero exactly when $n\\in A$. Thus the operation computes every selector bit.\n\\end{proof}\nThe number of oracle questions used by one addition is finite and bounded by the number of distinct prime divisors of the denominator product. This is a locality statement about the operation, not a claim of a polynomial-time implementation. Factoring integers and finding their prime indices contribute ordinary computational costs.","\\section{Decoding a marked copy}\n\\begin{proposition}[Uniform marked decoding]\n\\label{thm:markeddecode}\nGiven an $X$-computable copy of $G_A$ together with the name of $u_A$, one can uniformly compute $A$ in $X$.\n\\end{proposition}\n\\begin{proof}\nFor input $n$, search the domain simultaneously for solutions of\n\\[\np_{2n}y=u_A,\\qquad p_{2n+1}z=u_A.\n\\]\nEach candidate test is decidable in $X$. Exactly one equation has a solution by \\eqref{eq:evenroot}--\\eqref{eq:oddroot}. Dovetailing the searches therefore terminates. A solution to the first equation means $n\\in A$, and a solution to the second means $n\\notin A$.\n\\end{proof}\nThe procedure does not decide a negative existential statement directly. It uses two complementary positive searches. The paired selector is what converts enumerable root information into","\\section{Decoding a marked copy}\n\\begin{proposition}[Uniform marked decoding]\n\\label{thm:markeddecode}\nGiven an $X$-computable copy of $G_A$ together with the name of $u_A$, one can uniformly compute $A$ in $X$.\n\\end{proposition}\n\\begin{proof}\nFor input $n$, search the domain simultaneously for solutions of\n\\[\np_{2n}y=u_A,\\qquad p_{2n+1}z=u_A.\n\\]\nEach candidate test is decidable in $X$. Exactly one equation has a solution by \\eqref{eq:evenroot}--\\eqref{eq:oddroot}. Dovetailing the searches therefore terminates. A solution to the first equation means $n\\in A$, and a solution to the second means $n\\notin A$.\n\\end{proof}\nThe procedure does not decide a negative existential statement directly. It uses two complementary positive searches. The paired selector is what converts enumerable root information into a total decision procedur","\\section{Decoding a marked copy}\n\\begin{proposition}[Uniform marked decoding]\n\\label{thm:markeddecode}\nGiven an $X$-computable copy of $G_A$ together with the name of $u_A$, one can uniformly compute $A$ in $X$.\n\\end{proposition}\n\\begin{proof}\nFor input $n$, search the domain simultaneously for solutions of\n\\[\np_{2n}y=u_A,\\qquad p_{2n+1}z=u_A.\n\\]\nEach candidate test is decidable in $X$. Exactly one equation has a solution by \\eqref{eq:evenroot}--\\eqref{eq:oddroot}. Dovetailing the searches therefore terminates. A solution to the first equation means $n\\in A$, and a solution to the second means $n\\notin A$.\n\\end{proof}\nThe procedure does not decide a negative existential statement directly. It uses two complementary positive searches. The paired selector is what converts enumerable root information into a total decision procedure.","e name of $u_A$, one can uniformly compute $A$ in $X$.\n\\end{proposition}\n\\begin{proof}\nFor input $n$, search the domain simultaneously for solutions of\n\\[\np_{2n}y=u_A,\\qquad p_{2n+1}z=u_A.\n\\]\nEach candidate test is decidable in $X$. Exactly one equation has a solution by \\eqref{eq:evenroot}--\\eqref{eq:oddroot}. Dovetailing the searches therefore terminates. A solution to the first equation means $n\\in A$, and a solution to the second means $n\\notin A$.\n\\end{proof}\nThe procedure does not decide a negative existential statement directly. It uses two complementary positive searches. The paired selector is what converts enumerable root information into a total decision procedure.","$ in $X$.\n\\end{proposition}\n\\begin{proof}\nFor input $n$, search the domain simultaneously for solutions of\n\\[\np_{2n}y=u_A,\\qquad p_{2n+1}z=u_A.\n\\]\nEach candidate test is decidable in $X$. Exactly one equation has a solution by \\eqref{eq:evenroot}--\\eqref{eq:oddroot}. Dovetailing the searches therefore terminates. A solution to the first equation means $n\\in A$, and a solution to the second means $n\\notin A$.\n\\end{proof}\nThe procedure does not decide a negative existential statement directly. It uses two complementary positive searches. The paired selector is what converts enumerable root information into a total decision procedure.","\\section{Decoding an unmarked copy}\n\\begin{theorem}[Exact oracle criterion]\n\\label{thm:degree}\nFor every $A,X\\subseteq\\N$,\n\\[\nG_A\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","rked copy}\n\\begin{theorem}[Exact oracle criterion]\n\\label{thm:degree}\nFor every $A,X\\subseteq\\N$,\n\\[\nG_A\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","\\begin{theorem}[Exact oracle criterion]\n\\label{thm:degree}\nFor every $A,X\\subseteq\\N$,\n\\[\nG_A\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","on]\n\\label{thm:degree}\nFor every $A,X\\subseteq\\N$,\n\\[\nG_A\\text{ has an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","as an }X\\text{-computable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","mputable copy}\n\\quad\\Longleftrightarrow\\quad A\\le_T X.\n\\]\nIn particular, the isomorphism type of $G_A$ has least presentation degree $\\deg_T(A)$.\n\\end{theorem}\n\\begin{proof}\nThe reverse implication is Proposition~\\ref{thm:computeupper}. For the forward implication, let $H$ be an $X$-computable copy. Choose the name $h$ of an infinite-order element. Such an element exists because $H$ has quotient $\\Q$ by its torsion subgroup.","Under an isomorphism to $G_A$, the element $h$ has a normal form with nonzero rational part. Lemma~\\ref{thm:tailsignature} gives a finite exception set. Choose $N$ so that every prime in the pairs indexed by $n\\ge N$ avoids that set. For these $n$, exactly one of\n\\[\np_{2n}y=h,\\qquad p_{2n+1}z=h\n\\]\nhas a solution, and the first is solvable exactly when $n\\in A$.","Dovetailing the two searches decides every bit beyond $N$. Hard-code the name of $h$, the number $N$, and the finitely many bits below $N$ into the oracle procedure. These are finite constants, so the result is a Turing reduction of $A$ to $X$.","The canonical normal-form copy has exact degree $\\deg_T(A)$ by Proposition~\\ref{thm:computeupper}, while every copy computes $A$ by the forward implication. Hence this degree is the least one.\n\\end{proof}\nThere is no claim of a uniform algorithm finding an infinite-order element or its exception bound in every unmarked copy. The theorem is an existence statement about Turing reductions, and a finite list of constants is legitimate in such a reduction. For marked copies, Proposition~\\ref{thm:markeddecode} gives the genuinely uniform version.","The distinction also matches the algebra. If $A\\eqfin B$, then the unmarked groups are identical inside $P$ and $A$ and $B$ have the same Turing degree. Recovering a literal selector uniformly from the unmarked group would demand more than the isomorphism type remembers.","\\section{Every degree over one computable completion}\n\\begin{corollary}[Degree variation with fixed observations]\n\\label{thm:alldegrees}\nEvery Turing degree occurs as the least presentation degree of a countable, reduced, residually finite abelian group whose profinite completion is $P$, equipped with the fixed computable inverse-system presentation of Chapter~\\ref{ch:roots}.\n\\end{corollary}\n\\begin{proof}\nChoose any set $A$ of the desired degree and apply Theorem~\\ref{thm:degree}. All the structural and completion properties were proved in Chapters~\\ref{ch:purity}--\\ref{ch:roots}.\n\\end{proof}\nFor $A=\\varnothing$, the copy is computable. For $A=K$, the group has no computable copy and its least degree is the halting degree $\\mathbf 0'$. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their transition maps.","r one computable completion}\n\\begin{corollary}[Degree variation with fixed observations]\n\\label{thm:alldegrees}\nEvery Turing degree occurs as the least presentation degree of a countable, reduced, residually finite abelian group whose profinite completion is $P$, equipped with the fixed computable inverse-system presentation of Chapter~\\ref{ch:roots}.\n\\end{corollary}\n\\begin{proof}\nChoose any set $A$ of the desired degree and apply Theorem~\\ref{thm:degree}. All the structural and completion properties were proved in Chapters~\\ref{ch:purity}--\\ref{ch:roots}.\n\\end{proof}\nFor $A=\\varnothing$, the copy is computable. For $A=K$, the group has no computable copy and its least degree is the halting degree $\\mathbf 0'$. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their transition maps.","ntable, reduced, residually finite abelian group whose profinite completion is $P$, equipped with the fixed computable inverse-system presentation of Chapter~\\ref{ch:roots}.\n\\end{corollary}\n\\begin{proof}\nChoose any set $A$ of the desired degree and apply Theorem~\\ref{thm:degree}. All the structural and completion properties were proved in Chapters~\\ref{ch:purity}--\\ref{ch:roots}.\n\\end{proof}\nFor $A=\\varnothing$, the copy is computable. For $A=K$, the group has no computable copy and its least degree is the halting degree $\\mathbf 0'$. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their transition maps.","oup whose profinite completion is $P$, equipped with the fixed computable inverse-system presentation of Chapter~\\ref{ch:roots}.\n\\end{corollary}\n\\begin{proof}\nChoose any set $A$ of the desired degree and apply Theorem~\\ref{thm:degree}. All the structural and completion properties were proved in Chapters~\\ref{ch:purity}--\\ref{ch:roots}.\n\\end{proof}\nFor $A=\\varnothing$, the copy is computable. For $A=K$, the group has no computable copy and its least degree is the halting degree $\\mathbf 0'$. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their transition maps.","ystem presentation of Chapter~\\ref{ch:roots}.\n\\end{corollary}\n\\begin{proof}\nChoose any set $A$ of the desired degree and apply Theorem~\\ref{thm:degree}. All the structural and completion properties were proved in Chapters~\\ref{ch:purity}--\\ref{ch:roots}.\n\\end{proof}\nFor $A=\\varnothing$, the copy is computable. For $A=K$, the group has no computable copy and its least degree is the halting degree $\\mathbf 0'$. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their transition maps.","The group $G_K$ still has a finite mathematical definition. Its noncomputability is not a proof that it cannot be named in a finite semantic language. The theorem concerns effective copies with computable operations and equality.","\\section{A sharper separation than prime-spectrum coding}\nFor the initial groups $D_B$, an enumeration of the selected prime set suffices. For $G_A$, the paired complementary tests decide the selector. Meanwhile, every prime torsion factor is present in every $G_A$, so the information is no longer stored in which finite-order elements exist.","\\section{The observation contract}\nLet $\\Obs(G)$ denote the unmarked inverse system of finite quotient groups and their transition maps, considered up to isomorphism as a pro-object. The source group $G$ is not supplied again as part of the input. Nor do we retain evaluations of named source elements. Supplying the source itself would make a reconstruction problem vacuous, and supplying coherent marked elements is a stronger contract.","For the family of this book, there is no ambiguity about the comparison of pro-objects. The same concrete finite products\n\\[\n\\prod_{p\\mid m}C_p\n\\]\nwith the same coordinate projections form a cofinal model for every $\\Obs(G_A)$. Proposition~\\ref{thm:finitehom} also identifies the homomorphism sets into every finite target by a natural bijection induced by restriction to $D$.","\\begin{theorem}[No decoder from the unmarked finite-quotient system]\n\\label{thm:nodecoder}\nThere is no isomorphism-invariant assignment $R$ on these observation systems such that\n\\[\nR(\\Obs(G_A))\\cong G_A\n\\qquad\\text{for every }A\\subseteq\\N.\n\\]\nThis remains impossible without any computability restriction on $R$.\n\\end{theorem}\n\\begin{proof}\nTake $A=\\varnothing$ and $B=\\N$. Their observation systems are isomorphic by Theorem~\\ref{thm:quotientformula}. Their selectors differ infinitely, so their groups are not isomorphic by Theorem~\\ref{thm:isoclass}. An isomorphism-invariant assignment has one output isomorphism type on that input observation type. It cannot recover both nonisomorphic source types.\n\\end{proof}\nThe proof uses two objects, not a counting argument. The full family shows that the ambiguity is much larger: one observ","\\begin{theorem}[No decoder from the unmarked finite-quotient system]\n\\label{thm:nodecoder}\nThere is no isomorphism-invariant assignment $R$ on these observation systems such that\n\\[\nR(\\Obs(G_A))\\cong G_A\n\\qquad\\text{for every }A\\subseteq\\N.\n\\]\nThis remains impossible without any computability restriction on $R$.\n\\end{theorem}\n\\begin{proof}\nTake $A=\\varnothing$ and $B=\\N$. Their observation systems are isomorphic by Theorem~\\ref{thm:quotientformula}. Their selectors differ infinitely, so their groups are not isomorphic by Theorem~\\ref{thm:isoclass}. An isomorphism-invariant assignment has one output isomorphism type on that input observation type. It cannot recover both nonisomorphic source types.\n\\end{proof}\nThe proof uses two objects, not a counting argument. The full family shows that the ambiguity is much larger: one observation system supports continuum many isomorphism types, including examples with every possible least pr","\\begin{theorem}[No decoder from the unmarked finite-quotient system]\n\\label{thm:nodecoder}\nThere is no isomorphism-invariant assignment $R$ on these observation systems such that\n\\[\nR(\\Obs(G_A))\\cong G_A\n\\qquad\\text{for every }A\\subseteq\\N.\n\\]\nThis remains impossible without any computability restriction on $R$.\n\\end{theorem}\n\\begin{proof}\nTake $A=\\varnothing$ and $B=\\N$. Their observation systems are isomorphic by Theorem~\\ref{thm:quotientformula}. Their selectors differ infinitely, so their groups are not isomorphic by Theorem~\\ref{thm:isoclass}. An isomorphism-invariant assignment has one output isomorphism type on that input observation type. It cannot recover both nonisomorphic source types.\n\\end{proof}\nThe proof uses two objects, not a counting argument. The full family shows that the ambiguity is much larger: one observation system supports continuum many isomorphism types, including examples with every possible least presentation degree.","\\section{Element separation does not identify the source}\nEvery $G_A$ is residually finite. Its coordinate maps distinguish any two distinct elements. They give an injective map\n\\[\nG_A\\hookrightarrow\\varprojlim_N A_N=P.\n\\]\nThere is no conflict with Theorem~\\ref{thm:nodecoder}. The injective map identifies each element once the domain is known. The unmarked inverse system does not identify the image of that domain among all compatible tuples.","The complete product is uncountable. Each $G_A$ is countable and dense, hence proper and noncomplete. In particular, a theorem asserting that a complete compatible realization equals its inverse limit has not failed. These examples deliberately retain a different admissible subspace before completion.","\\begin{proposition}[A proper map invisible after completion]\n\\label{thm:propercompletion}\nThe inclusion $D\\hookrightarrow G_A$ is not an isomorphism, but it induces an isomorphism of profinite completions.\n\\end{proposition}\n\\begin{proof}\nThe quotient $G_A/D\\cong\\Q$ is nonzero, so the inclusion is proper. Both completions are canonically $P$, and the coordinate inclusions agree, so the induced completion map is the identity on $P$.\n\\end{proof}\nThis supplies a map-level failure: completion need not reflect isomorphisms. It is stronger than merely exhibiting two abstract groups with isomorphic completions.","p invisible after completion]\n\\label{thm:propercompletion}\nThe inclusion $D\\hookrightarrow G_A$ is not an isomorphism, but it induces an isomorphism of profinite completions.\n\\end{proposition}\n\\begin{proof}\nThe quotient $G_A/D\\cong\\Q$ is nonzero, so the inclusion is proper. Both completions are canonically $P$, and the coordinate inclusions agree, so the induced completion map is the identity on $P$.\n\\end{proof}\nThis supplies a map-level failure: completion need not reflect isomorphisms. It is stronger than merely exhibiting two abstract groups with isomorphic completions.","\\begin{proposition}[Faithfulness and failure of fullness]\n\\label{thm:faithful}\nProfinite completion is faithful on the category of residually finite groups and group homomorphisms. Its restriction to the full subcategory with objects $G_A$ is not full. On the larger full subcategory with objects $D$ and all $G_A$, it does not reflect isomorphisms.\n\\end{proposition}\n\\begin{proof}\nLet $f,g:H\\to K$ be distinct homomorphisms between residually finite groups. Choose $h$ with $f(h)\\ne g(h)$. Some finite quotient of $K$ distinguishes those elements. Therefore the induced completion maps differ on the image of $h$, proving faithfulness.","m:faithful}\nProfinite completion is faithful on the category of residually finite groups and group homomorphisms. Its restriction to the full subcategory with objects $G_A$ is not full. On the larger full subcategory with objects $D$ and all $G_A$, it does not reflect isomorphisms.\n\\end{proposition}\n\\begin{proof}\nLet $f,g:H\\to K$ be distinct homomorphisms between residually finite groups. Choose $h$ with $f(h)\\ne g(h)$. Some finite quotient of $K$ distinguishes those elements. Therefore the induced completion maps differ on the image of $h$, proving faithfulness.","l}\nProfinite completion is faithful on the category of residually finite groups and group homomorphisms. Its restriction to the full subcategory with objects $G_A$ is not full. On the larger full subcategory with objects $D$ and all $G_A$, it does not reflect isomorphisms.\n\\end{proposition}\n\\begin{proof}\nLet $f,g:H\\to K$ be distinct homomorphisms between residually finite groups. Choose $h$ with $f(h)\\ne g(h)$. Some finite quotient of $K$ distinguishes those elements. Therefore the induced completion maps differ on the image of $h$, proving faithfulness.","he category of residually finite groups and group homomorphisms. Its restriction to the full subcategory with objects $G_A$ is not full. On the larger full subcategory with objects $D$ and all $G_A$, it does not reflect isomorphisms.\n\\end{proposition}\n\\begin{proof}\nLet $f,g:H\\to K$ be distinct homomorphisms between residually finite groups. Choose $h$ with $f(h)\\ne g(h)$. Some finite quotient of $K$ distinguishes those elements. Therefore the induced completion maps differ on the image of $h$, proving faithfulness.","bcategory with objects $G_A$ is not full. On the larger full subcategory with objects $D$ and all $G_A$, it does not reflect isomorphisms.\n\\end{proposition}\n\\begin{proof}\nLet $f,g:H\\to K$ be distinct homomorphisms between residually finite groups. Choose $h$ with $f(h)\\ne g(h)$. Some finite quotient of $K$ distinguishes those elements. Therefore the induced completion maps differ on the image of $h$, proving faithfulness.","For failure of fullness, choose $A\\neqfin B$. Under the canonical identifications of the completions with $P$, consider the identity map $P\\to P$. If it were induced by a homomorphism $G_A\\to G_B$, that homomorphism would be the identity on every coordinate and hence would include $G_A$ into $G_B$. Theorem~\\ref{thm:homobstruction} excludes such an embedding. After adjoining $D$ to the source category, the proper inclusion of Proposition~\\ref{thm:propercompletion} proves failure to reflect isomorphisms.\n\\end{proof}\nThe functorial extension of a homomorphism to profinite completions is standard \\cite[Proposition 3.1.2]{Wilkes}. The distinction between the two source categories is essential. Within the paired-prime family alone, a suppl","For failure of fullness, choose $A\\neqfin B$. Under the canonical identifications of the completions with $P$, consider the identity map $P\\to P$. If it were induced by a homomorphism $G_A\\to G_B$, that homomorphism would be the identity on every coordinate and hence would include $G_A$ into $G_B$. Theorem~\\ref{thm:homobstruction} excludes such an embedding. After adjoining $D$ to the source category, the proper inclusion of Proposition~\\ref{thm:propercompletion} proves failure to reflect isomorphisms.\n\\end{proof}\nThe functorial extension of a homomorphism to profinite completions is standard \\cite[Proposition 3.1.2]{Wilkes}. The distinction between the two source categories is essential. Within the paired-prime family alone, a supplied map can be recovered more strongly.","\\begin{proposition}[Reflection within the paired-prime family]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{t","\\begin{proposition}[Reflection within the paired-prime family]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{","\\begin{proposition}[Reflection within the paired-prime family]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distin","\\begin{proposition}[Reflection within the paired-prime family]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two memb","red-prime family]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an iso…","]\n\\label{thm:rankonereflection}\nFor a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This d…","For a homomorphism $f:G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This does not reconstruct an unknown s…",":G_A\\to G_B$, the induced completion map $\\widehat f$ is an isomorphism if and only if $f$ is an isomorphism.\n\\end{proposition}\n\\begin{proof}\nOne implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This does not reconstruct an unknown source from its comple…","implication follows from functoriality. Conversely, suppose $\\widehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This does not reconstruct an unknown source from its completion alone.","ehat f$ is an isomorphism. By Theorem~\\ref{thm:allhom}, its coordinate action is multiplication by a diagonal sequence $\\lambda$. Each $\\lambda_p$ must be nonzero, so $f$ is injective. If $A\\neqfin B$, Theorem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This does not reconstruct an unknown source from its completion alone.","rem~\\ref{thm:homobstruction} would force its image to be torsion, contradicting injectivity on an infinite-order element. Hence $A\\eqfin B$, and the source and target are equal as subgroups of $P$. The map is now an endomorphism of that group with every diagonal entry nonzero. Corollary~\\ref{thm:autGA} makes it an automorphism.\n\\end{proof}\nThus completion forgets distinctions between the objects $G_A$, and some maps between their completions have no lift. But when a homomorphism between two members of this family is already supplied, an isomorphism after completion certifies that the supplied map was an isomorphism. This does not reconstruct an unknown source from its completion alone.","\\section{What a marked coherent point restores}\nThe coherent image of $u_A$ in the finite quotients records, at prime $p$, whether its coordinate is zero or one. Within the declared family, these zero patterns recover $S_A$ and therefore $A$.","One point is sufficient to identify our family because the construction contract says that the rational line through its tail class is the admissible quotient. Without that contract, a point in $P$ would not identify an arbitrary dense subgroup containing it. The realization rule and its parameter must be retained together.","\\section{Two different inverse-limit assertions}\nFor the subgroup family $D_B$, a complete coherent sequence of finite membership decisions determines $B$ exactly. For the family $G_A$, the unmarked finite quotient system is independent of $A$. These are different observations, even though both involve finite structures and inverse limits.","\\section{Fix the semantics before diagonalizing}\nLet $\\mathfrak G$ be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as\n\\[\nd_0,d_1,d_2,\\ldots\n\\]\nand write $R_j$ for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.","For each $j$ whose output lies in the family, choose $B_j\\subseteq\\N$ such that $R_j\\cong G_{B_j}$. For other $j$, set $B_j=\\varnothing$. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","The isomorphism criterion says that a selector must avoid the whole eventual-equivalence class of $B_j$, not merely disagree at one coordinate. A naive Cantor diagonal with one disagreement for each $j$ would not establish that requirement.","\\section{Repeated diagonal coding}\nFix a computable pairing bijection $\\langle j,k\\rangle:\\N^2\\to\\N$. For a freely chosen set $C\\subseteq\\N$, define a new selector $A_C$ by\n\\begin{align}\n\\one_{A_C}(2\\langle j,k\\rangle)\n &=1-\\one_{B_j}(2\\langle j,k\\rangle),\\label{eq:diageven}","\\section{Repeated diagonal coding}\nFix a computable pairing bijection $\\langle j,k\\rangle:\\N^2\\to\\N$. For a freely chosen set $C\\subseteq\\N$, define a new selector $A_C$ by\n\\begin{align}\n\\one_{A_C}(2\\langle j,k\\rangle)\n &=1-\\one_{B_j}(2\\langle j,k\\rangle),\\label{eq:diageven}\\\\\n\\one_{A_C}(2\\langle j,k\\rangle+1)\n &=\\one_C(j).\\label{eq:diagodd}\n\\end{align}\nEvery natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.","\\begin{theorem}[Escape within one profinite-completion class]\n\\label{thm:escape}\nFor every $C\\subseteq\\N$, the group $G_{A_C}$ is not isomorphic to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","theorem}[Escape within one profinite-completion class]\n\\label{thm:escape}\nFor every $C\\subseteq\\N$, the group $G_{A_C}$ is not isomorphic to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","[Escape within one profinite-completion class]\n\\label{thm:escape}\nFor every $C\\subseteq\\N$, the group $G_{A_C}$ is not isomorphic to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","m:escape}\nFor every $C\\subseteq\\N$, the group $G_{A_C}$ is not isomorphic to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","C}$ is not isomorphic to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","c to any output of $\\mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","mathfrak G$. Distinct $C$ give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion $P$.\n\\end{theorem}\n\\begin{proof}\nFix a description index $j$. If $R_j$ is outside the family, it cannot be isomorphic to $G_{A_C}$. Otherwise rule \\eqref{eq:diageven} makes $A_C$ disagree with $B_j$ at every index $2\\langle j,k\\rangle$. These are infinitely many distinct indices, so $A_C\\neqfin B_j$. Theorem~\\ref{thm:isoclass} excludes the isomorphism.","If $C\\ne C'$, choose $j$ at which they differ. Rule \\eqref{eq:diagodd} makes $A_C$ and $A_{C'}$ differ at infinitely many odd indices. Theorems~\\ref{thm:isoclass} and \\ref{thm:homobstruction} then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.\n\\end{proof}","\\begin{figure}[tbp]\n\\centering\\includegraphics[width=\\textwidth]{\\figfile{06_repeated_diagonal.pdf}}\n\\caption{Two infinite coordinate blocks are allocated to every $j$. On the even block the selector opposes the representative $B_j$ forever. On the odd block it repeats the free bit $C(j)$ forever. Both repetitions are necessary to survive the quotient by finite changes.}\n\\label{fig:diagonal}\n\\end{figure}","\\section{The complexity consequence}\nThe theorem implies\n\\[\nK_{\\mathfrak G}(G_{A_C})=+\\infty\n\\]\nfor every $C$. It produces continuum many distinct isomorphism types outside the grammar while preserving all the finite quotient data.","This is a statement relative to a specified language and semantics. The definition of $A_C$ may use the entire semantic table of that language in the surrounding mathematical theory. It does not yield an effective evaluator for the table. Nor does it produce a paradox by defining in one paragraph an object claimed to be beyond every possible finite definition. The object escapes the fixed internal grammar, not all larger metatheories.","\\section{Monotonicity under grammar extension}\n\\begin{proposition}[Admissible expansion]\n\\label{thm:grammarextension}\nSuppose every valid description in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selecto","\\section{Monotonicity under grammar extension}\n\\begin{proposition}[Admissible expansion]\n\\label{thm:grammarextension}\nSuppose every valid description in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result","\\section{Monotonicity under grammar extension}\n\\begin{proposition}[Admissible expansion]\n\\label{thm:grammarextension}\nSuppose every valid description in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the","\\section{Monotonicity under grammar extension}\n\\begin{proposition}[Admissible expansion]\n\\label{thm:grammarextension}\nSuppose every valid description in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the larger realization program.","oposition}[Admissible expansion]\n\\label{thm:grammarextension}\nSuppose every valid description in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the larger realization program.","on in $\\mathfrak G$ translates to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the larger realization program.","s to a valid description in $\\mathfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the larger realization program.","hfrak H$ with the same semantic group. Then every $\\mathfrak G$-realizable group is $\\mathfrak H$-realizable. If every translated description has length at most $f(\\lvert d\\rvert)$, then\n\\[\nK_{\\mathfrak H}(G)\\le f(K_{\\mathfrak G}(G))\n\\]\nwhenever $K_{\\mathfrak G}(G)$ is finite.\n\\end{proposition}\n\\begin{proof}\nTranslate a shortest valid $\\mathfrak G$ description of $G$. Its semantic output remains $G$, and its length has the stated bound.\n\\end{proof}\nAn obstruction to a small grammar can therefore disappear after an admissible expansion. The power-of-two selector from Chapter~\\ref{ch:spectrum} illustrates this concretely: a unary finite-state selector fails, while a binary one succeeds. Such a result characterizes a boundary rather than discrediting the larger realization program.","\\section{Uniform families and individual instances}\nThe expression $A\\mapsto G_A$ is one uniform mathematical construction. It is not a finite description of each arbitrary instance unless the selector is also described. A single family can contain uncountably many values even though a parameter-free finite syntax has only countably many strings.","For computable selectors, a program for $A$ provides a finite effective input to the normal-form construction. For noncomputable but finitely definable selectors, one obtains a finite semantic description without an effective copy. For selectors outside a chosen finite-description language, the family formula still proves existence relative to that selector, but does not internalize it into that language.","\\section{The empty selector}\nTake $A=\\varnothing$. The selected primes are\n\\[\nS_A=\\{2,5,11,17,23,31,41,47,\\ldots\\},\n\\]\nnamely the even-indexed primes. The first coordinates of three rational lifts are\n\\begin{center}\n\\begin{tabular}{@{}crrrrrr@{}}\n\\toprule\n$p$&2&3&5&7&11&13","midrule\n$u_A=s_A(1)$&1&0&1&0&1&0","$s_A(1/2)$&0&0&3&0&6&0","$s_A(1/3)$&1&0&2&0&4&0","bottomrule\n\\end{tabular}\n\\end{center}\nEach residue is exact. At $p=2$, the lift of $1/2$ is set to zero because the denominator is not invertible. At $p=5$, its value is $3$ because $2\\cdot3=1$ in $\\F_5$.","Doubling the half-lift agrees with $u_A$ at every prime except $2$. Therefore\n\\[\n2(1/2,0)=(1,e_2).\n\\]\nThe element $u_A$ has no half in the group: there is no $y$ with $2y=u_A$, because its $2$-coordinate is nonzero. The apparent candidate $(1/2,0)$ misses by precisely the certificate coordinate.","At the unselected prime $3$, the coordinate of $u_A$ is zero. In this case\n\\[\n3(1/3,0)=(1,0).\n\\]\nThe three roots are\n\\[\n(1/3,0),\\qquad(1/3,e_3),\\qquad(1/3,2e_3).\n\\]\nTheir differences form $G_A[3]=C_3$, as predicted by Proposition~\\ref{thm:rootfiber}.","\\section{A successful composite-root calculation}\nLet\n\\[\ng=s_A(1)+e_2=(1,e_2).\n\\]\nIts coordinate at $2$ is $1+1=0$, and its coordinate at $3$ is zero. Thus the multiplier $6$ has no obstructing coordinate.","The rational candidate is $t=1/6$. At a selected prime other than $2$, multiplication by $6$ recovers the coordinate of $u_A$ because the denominator is invertible, except possibly at $3$, which is unselected. At $2$, both sides are zero. Hence\n\\[\n6(1/6,0)=(1,e_2)=g.\n\\]\nEvery other root is obtained by adding an arbitrary element of $C_2\\oplus C_3$. There are exactly six roots. This calculation exhibits a positive certificate without searching an unbounded domain.","Now contrast $u_A$ itself. The same multiplier $6$ fails at the prime $2$. No change at the $3$-coordinate and no choice of rational representative can repair that obstruction while keeping the target fixed.","\\section{A torsion correction that cannot be discarded}\nTake the purely torsion target $e_3$ and multiplier $2$. The root criterion succeeds because the target has zero $2$-coordinate. Its rational part is zero, but the answer is not the zero element:\n\\[\n2(0,2e_3)=(0,e_3).\n\\]\nIn $\\F_3$, multiplication by $2$ sends $2$ to $1$. This is the simplest example showing why the finite correction in the root algorithm is necessary even when the rational coordinate causes no difficulty.","\\section{Finite quotient representatives}\nFor any target $g$ and positive multiplier $m$, define the finite vector\n\\[\nd_m(g)=\\sum_{p\\mid m}g_p e_p.\n\\]\nThen $g-d_m(g)$ has zero coordinates at every prime dividing $m$. The root theorem provides a witness that it belongs to $mG_A$. Thus $d_m(g)$ is an exact finite representative of the coset of $g$.","For $m=12$, only coordinates $2$ and $3$ matter. The six representatives are\n\\[\n0,\\ e_2,\\ e_3,\\ 2e_3,\\ e_2+e_3,\\ e_2+2e_3.\n\\]\nThe vector $e_2+e_3$ generates this quotient, which is $C_6$, not $C_{12}$. Repeated prime powers in the multiplier do not create additional coordinates.","\\begin{figure}[tbp]\n\\centering\\includegraphics[width=\\textwidth]{\\figfile{07_squarefree_quotients.pdf}}\n\\caption{The exact order of $G_A/mG_A$ for $1\\le m\\le30$ is $\\rad(m)$. The values are independent of the selector. This plot evaluates the theorem's arithmetic formula, rather than estimating the size of an unknown quotient.}\n\\label{fig:radical}\n\\end{figure}","\\section{Changing one selector bit}\nLet $B=\\{0\\}$, so the first pair changes from selecting $2$ to selecting $3$. The groups $G_A$ and $G_B$ are equal as subgroups of $P$. Their half-lift coordinates differ, however:\n\\[\ns_A(1/2)-s_B(1/2)=e_3.\n\\]\nAt $p=2$, both half-lifts are zero. At $p=3$, the $B$ lift is $2$, so its negative is $1$. Therefore the same ambient element has normal forms\n\\[\n(1/2,0)_A=(1/2,e_3)_B.\n\\]\nDoubling the right-hand side in the $B$ group gives\n\\[\n(1,2e_3)_B,\n\\]\nwhich is the transported form of $(1,e_2)_A$. The correction in \\eqref{eq:transport} is required to compare the two group laws. Merely reusing the same pair $(r,d)$ under a new selector changes the represented element.","he first pair changes from selecting $2$ to selecting $3$. The groups $G_A$ and $G_B$ are equal as subgroups of $P$. Their half-lift coordinates differ, however:\n\\[\ns_A(1/2)-s_B(1/2)=e_3.\n\\]\nAt $p=2$, both half-lifts are zero. At $p=3$, the $B$ lift is $2$, so its negative is $1$. Therefore the same ambient element has normal forms\n\\[\n(1/2,0)_A=(1/2,e_3)_B.\n\\]\nDoubling the right-hand side in the $B$ group gives\n\\[\n(1,2e_3)_B,\n\\]\nwhich is the transported form of $(1,e_2)_A$. The correction in \\eqref{eq:transport} is required to compare the two group laws. Merely reusing the same pair $(r,d)$ under a new selector changes the represented element.","nges from selecting $2$ to selecting $3$. The groups $G_A$ and $G_B$ are equal as subgroups of $P$. Their half-lift coordinates differ, however:\n\\[\ns_A(1/2)-s_B(1/2)=e_3.\n\\]\nAt $p=2$, both half-lifts are zero. At $p=3$, the $B$ lift is $2$, so its negative is $1$. Therefore the same ambient element has normal forms\n\\[\n(1/2,0)_A=(1/2,e_3)_B.\n\\]\nDoubling the right-hand side in the $B$ group gives\n\\[\n(1,2e_3)_B,\n\\]\nwhich is the transported form of $(1,e_2)_A$. The correction in \\eqref{eq:transport} is required to compare the two group laws. Merely reusing the same pair $(r,d)$ under a new selector changes the represented element.","ps $G_A$ and $G_B$ are equal as subgroups of $P$. Their half-lift coordinates differ, however:\n\\[\ns_A(1/2)-s_B(1/2)=e_3.\n\\]\nAt $p=2$, both half-lifts are zero. At $p=3$, the $B$ lift is $2$, so its negative is $1$. Therefore the same ambient element has normal forms\n\\[\n(1/2,0)_A=(1/2,e_3)_B.\n\\]\nDoubling the right-hand side in the $B$ group gives\n\\[\n(1,2e_3)_B,\n\\]\nwhich is the transported form of $(1,e_2)_A$. The correction in \\eqref{eq:transport} is required to compare the two group laws. Merely reusing the same pair $(r,d)$ under a new selector changes the represented element.","At $p=2$, both half-lifts are zero. At $p=3$, the $B$ lift is $2$, so its negative is $1$. Therefore the same ambient element has normal forms\n\\[\n(1/2,0)_A=(1/2,e_3)_B.\n\\]\nDoubling the right-hand side in the $B$ group gives\n\\[\n(1,2e_3)_B,\n\\]\nwhich is the transported form of $(1,e_2)_A$. The correction in \\eqref{eq:transport} is required to compare the two group laws. Merely reusing the same pair $(r,d)$ under a new selector changes the represented element.","\\section{A finite tail exception}\nConsider an infinite-order element\n\\[\nx=s_A(5/6)+2e_7.\n\\]\nAn exception set for its signature is $\\{2,3,5,7\\}$. Outside these primes, the denominator is invertible, the numerator is nonzero, and the torsion correction vanishes. Thus the root pattern of $x$ agrees with that of $u_A$ in every later prime pair.","\\section{The ambient object and its admissible realizations}\nThe chain\n\\[\nD\\subseteq G_A\\subseteq P\n\\]\ncontains three different mathematical roles. The direct sum provides the finite-support corrections. The product supplies every compatible prime-coordinate tuple. The intermediate group is the intended admissible realization, determined by a rational line in the quotient.","This arrangement gives a concrete model for a central question of Komorebi Space. Once the compatible ambient object exists, what additional data identifies the realization under study? Here the answer is exact: a rational subspace of $P/D$, or in the paired family its selector class. Existence of the ambient product does not eliminate that choice.","The nonsplitting result supplies an extension obstruction. A rational quotient exists, but no homomorphic section realizes it inside the reduced product. In cocycle coordinates, the obstruction is the impossibility of removing the defect by a $D$-valued correction.","\\section{A realization profile rather than a single score}\nOne scalar cannot capture all the boundaries exhibited here. The following profile retains the distinctions:\n\\[\n\\bigl(\\text{seed rank},\\ \\text{grammar complexity},\\\n\\text{least presentation degree},\\ \\text{observation type}\\bigr).\n\\]\nFor every $G_A$, the seed rank is $\\aleph_0$ and the observation type is the same. The least presentation degree varies with $A$. Grammar complexity depends on which selectors the chosen semantic language can name.","ion profile rather than a single score}\nOne scalar cannot capture all the boundaries exhibited here. The following profile retains the distinctions:\n\\[\n\\bigl(\\text{seed rank},\\ \\text{grammar complexity},\\\n\\text{least presentation degree},\\ \\text{observation type}\\bigr).\n\\]\nFor every $G_A$, the seed rank is $\\aleph_0$ and the observation type is the same. The least presentation degree varies with $A$. Grammar complexity depends on which selectors the chosen semantic language can name.","midrule\nAmbient system&$D\\subseteq P$ and the quotient $q:P\\to P/D$.","Parameter&The selector $A$, with its declared oracle or finite semantic description.","Admissible image&$q^{-1}(\\Q q(u_A))$.","\\section{Further questions with exact starting points}\nThe general correspondence of Chapter~\\ref{ch:purity} extends the ambient construction to higher-dimensional rational subspaces $W\\le P/D$. Its finite quotient theorem is already complete. What remains is to classify those subspaces under the diagonal multipliers that induce group homomorphisms. The paired rank-one family has an exact answer, but the book does not claim that its binary signature classifies higher rank.","A second question concerns effective selection. For $G_A$, the normal-form operation and the paired root tests give a matching upper and lower bound. For a more general countable $W$, one must specify how its rational dependencies are presented and how lifts are computed. A list of candidate spanning vectors is not enough if their dependence relations are unavailable.","\\section{Conclusion}\nThe principal boundary is now concrete. There is a fixed ambient completion, presented by a computable inverse system, and a family of countable, residually finite groups densely realized inside it. Their finite-quotient systems are canonically identified, and restriction to $D$ naturally identifies their homomorphism sets into every finite target. Yet the groups have distinct algebraic types and require arbitrarily different computational information.","The selector is supplied as a callable taking a nonnegative integer $n$ to the Boolean value of $\\one_A(n)$. Integer bits zero and one are also accepted. Other return values are rejected. The group object converts a prime label to its index, determines which pair it belongs to, and queries the corresponding selector bit. This is an oracle interface. A finite Python function implements only those selectors for which it can return the correct value. No program in the package claims to compute a noncomputable selector.","\\texttt{root(x, m)}&For $m\\ge1$, return a root or an obstructing prime.","\\section{Root construction as an algorithm}\nThe proof of Theorem~\\ref{thm:rootcertificate} gives the following procedure.\n\\begin{enumerate}\n\\item Factor the positive integer $m$ and inspect the target coordinates at its prime divisors. If any is nonzero, return that prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integ","\\section{Root construction as an algorithm}\nThe proof of Theorem~\\ref{thm:rootcertificate} gives the following procedure.\n\\begin{enumerate}\n\\item Factor the positive integer $m$ and inspect the target coordinates at its prime divisors. If any is nonzero, return that prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","n algorithm}\nThe proof of Theorem~\\ref{thm:rootcertificate} gives the following procedure.\n\\begin{enumerate}\n\\item Factor the positive integer $m$ and inspect the target coordinates at its prime divisors. If any is nonzero, return that prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","inspect the target coordinates at its prime divisors. If any is nonzero, return that prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","rime divisors. If any is nonzero, return that prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","t prime as the obstruction.\n\\item Divide the rational coordinate by $m$ to obtain a candidate section lift.\n\\item Compute the finite residual between the target and $m$ times that lift. Its support is confined to the input torsion support and the denominator primes stated in the proof.\n\\item At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","At primes not dividing $m$, divide the residual coordinate by $m$ in the finite field. At primes dividing $m$, the residual is zero. Add this finite correction to the candidate lift.\n\\end{enumerate}\nThe procedure returns a complete normal-form element. Its certificate is checked by the independently implemented integer multiplication routine. The negative certificate is checked by verifying both $p\\mid m$ and a nonzero target coordinate.","midrule\nOmit the cocycle&Twice the half-lift can differ from $u_A$ at prime $2$.","Use $C_m$ for every integer quotient&Prime powers collapse to $C_{\\rad(m)}$.","\\ref{thm:torP}&$\\Tor(P)=D$, and $D$ is dense.&Distinct prime coordinates.","\\ref{thm:endD}&All endomorphisms of $D$ are diagonal.&Prime annihilators.","\\ref{thm:rankD}&$D$ has infinite endomorphism-seed rank.&Finite support.","\\ref{thm:procyclic}&$P$ is topologically cyclic.&Chinese remainder theorem.","\\ref{thm:Dcompletion}&$\\widehat D=P$.&Cofinal finite coordinate kernels.","\\ref{thm:Vrational}&$P/D$ is a rational vector space.&Unique integer division modulo $D$.","\\ref{thm:subD}&$\\Sub(D)\\cong\\mathcal P(\\N)$.&Coordinate isolation by CRT.","\\ref{thm:spectrumce}&Computable copies of $D_B$ correspond to enumerable selectors.&Prime-order witness searches.","\\ref{thm:generalquotients}&All preimages $H_W$ have completion $P$.&Purity and finite-target exponent.","\\ref{thm:normalform}&$G_A$ has unique $(r,d)$ names.&Nonzero rational line.","\\ref{thm:cocyclelaw}&The normal forms form the actual group $G_A$.&Cocycle identity.","\\ref{thm:rootfiber}&A nonempty root fiber has $\\rad(m)$ elements.&The prime-coordinate kernel.","\\ref{thm:quotientformula}&$G_A/mG_A=C_{\\rad(m)}$.&Root kernel and finite surjectivity.","\\ref{thm:isoclass}&$G_A\\cong G_B$ iff $A\\eqfin B$.&Intrinsic divisibility and finite transport.","\\ref{thm:allhom}&Every homomorphism is a diagonal multiplier.&The quotients $G_A/pG_A$.","\\ref{thm:rankGA}&Every $G_A$ has infinite seed rank.&Infinitely many omitted unselected coordinates.","\\ref{thm:degree}&An oracle computes a copy iff it computes $A$.&Cocycle evaluation and complementary root searches.","The traced implementation erases to the executable solver. If $B$ is its finite correction bound, it uses at most $m+1+2B$ selector queries and at most $m+1+B$ loop candidates. These are explicit oracle and iteration counts. They are not bounds on the bit cost of rational arithmetic, primality testing, division, or modular inversion. The Lean proofs certify the Lean implementation and its mathematical interpretation. They do not claim kernel verification of the separate Python runtime.","\\item[Cocycle] Here, a symmetric map $c:\\Q\\times\\Q\\to D$ satisfying the identity that makes the twisted addition on $\\Q\\times D$ associative.\n\\Needspace{5\\baselineskip}\\item[Cofinal quotient system] A family of finite quotients through which every finite quotient factors. Its inverse limit computes the full profinite completion.\n\\Needspace{5\\baselineskip}\\item[Computable copy] A countable realization with effective names, decidable equality, and computable operations. An oracle version permits the stated oracle.\n\\Needspace{5\\baselineskip}\\item[Compu","\\item[Cocycle] Here, a symmetric map $c:\\Q\\times\\Q\\to D$ satisfying the identity that makes the twisted addition on $\\Q\\times D$ associative.\n\\Needspace{5\\baselineskip}\\item[Cofinal quotient system] A family of finite quotients through which every finite quotient factors. Its inverse limit computes the full profinite completion.\n\\Needspace{5\\baselineskip}\\item[Computable copy] A countable realization with effective names, decidable equality, and computable operations. An oracle version permits the stated oracle.\n\\Needspace{5\\baselineskip}\\item[Computable profinite presentation] An effective inverse system of finite grou","\\item[Divisible abelian group] An abelian group in which every equation $my=x$ with $m\\ge1$ is solvable. A divisible quotient need not split off as a subgroup.\n\\Needspace{5\\baselineskip}\\item[Endomorphism-seed rank] The minimum number of elements generating a group as a module over its full endomorphism ring.\n\\Needspace{5\\baselineskip}\\item[Eventual equality] Equality modulo finitely many coordinates, written $A\\eqfin B$. This is the unmarked selector invariant of $G_A$.\n\\Needspace{5\\baselineskip}\\item[Finite description] A valid finite code in a specified language with","\\item[Divisible abelian group] An abelian group in which every equation $my=x$ with $m\\ge1$ is solvable. A divisible quotient need not split off as a subgroup.\n\\Needspace{5\\baselineskip}\\item[Endomorphism-seed rank] The minimum number of elements generating a group as a module over its full endomorphism ring.\n\\Needspace{5\\baselineskip}\\item[Eventual equality] Equality modulo finitely many coordinates, written $A\\eqfin B$. This is the unmarked selector invariant of $G_A$.\n\\Needspace{5\\baselineskip}\\item[Finite description] A valid finite code in a specified language with declared sema","\\item[Eventual equality] Equality modulo finitely many coordinates, written $A\\eqfin B$. This is the unmarked selector invariant of $G_A$.\n\\Needspace{5\\baselineskip}\\item[Finite description] A valid finite code in a specified language with declared semantics. It need not provide computable operations.\n\\Needspace{5\\baselineskip}\\item[Finite support] A vector has only finitely many nonzero coordinates. Such vectors form $D$ inside $P$.\n\\Needspace{5\\baselineskip}\\item[Fully invariant subgroup] A subgroup preserved by every endomorphism of its ambient group. Every subgroup of the squ","\\item[Eventual equality] Equality modulo finitely many coordinates, written $A\\eqfin B$. This is the unmarked selector invariant of $G_A$.\n\\Needspace{5\\baselineskip}\\item[Finite description] A valid finite code in a specified language with declared semantics. It need not provide computable operations.\n\\Needspace{5\\baselineskip}\\item[Finite support] A vector has only finitely many nonzero coordinates. Such vectors form $D$ inside $P$.\n\\Needspace{5\\baselineskip}\\item[Fully invariant subgroup] A subgroup preserved by every endomorphism of its ambient group. Every subgroup of the squarefree direct sum has this property.\n\\Needspace{5","\\item[Finite support] A vector has only finitely many nonzero coordinates. Such vectors form $D$ inside $P$.\n\\Needspace{5\\baselineskip}\\item[Fully invariant subgroup] A subgroup preserved by every endomorphism of its ambient group. Every subgroup of the squarefree direct sum has this property.\n\\Needspace{5\\baselineskip}\\item[Grammar-relative nonrealizability] Failure to have an admissible description in one fixed grammar. It is not absolute mathematical nonexistence or indescribability.\n\\Needspace{5\\baselineskip}\\item[Marked group] A group together with named elements that isomorphisms mu","\\item[Finite support] A vector has only finitely many nonzero coordinates. Such vectors form $D$ inside $P$.\n\\Needspace{5\\baselineskip}\\item[Fully invariant subgroup] A subgroup preserved by every endomorphism of its ambient group. Every subgroup of the squarefree direct sum has this property.\n\\Needspace{5\\baselineskip}\\item[Grammar-relative nonrealizability] Failure to have an admissible description in one fixed grammar. It is not absolute mathematical nonexistence or indescribability.\n\\Needspace{5\\baselineskip}\\item[Marked group] A group together with named elements that isomorphisms must preserve","\\item[Marked group] A group together with named elements that isomorphisms must preserve. The mark $u_A$ retains the literal selector.\n\\Needspace{5\\baselineskip}\\item[Normal form] The unique pair $(r,d)$ representing $s_A(r)+d$. Equality of these names is simpler than the selector-dependent addition law.\n\\Needspace{5\\baselineskip}\\item[Oracle] An external set-membership resource available to a computation. Its information is charged to the effective realization contract.\n\\Needspace{5\\baselineskip}\\item[Prime spectrum] The prime orders of nonzero torsion elements. It classifies the groups $D_B$ bu","\\item[Normal form] The unique pair $(r,d)$ representing $s_A(r)+d$. Equality of these names is simpler than the selector-dependent addition law.\n\\Needspace{5\\baselineskip}\\item[Oracle] An external set-membership resource available to a computation. Its information is charged to the effective realization contract.\n\\Needspace{5\\baselineskip}\\item[Prime spectrum] The prime orders of nonzero torsion elements. It classifies the groups $D_B$ but is constant across all $G_A$.\n\\Needspace{5\\baselineskip}\\item[Profinite completion] The inverse l","\\item[Normal form] The unique pair $(r,d)$ representing $s_A(r)+d$. Equality of these names is simpler than the selector-dependent addition law.\n\\Needspace{5\\baselineskip}\\item[Oracle] An external set-membership resource available to a computation. Its information is charged to the effective realization contract.\n\\Needspace{5\\baselineskip}\\item[Prime spectrum] The prime orders of nonzero torsion elements. It classifies the groups $D_B$ but is constant across all $G_A$.\n\\Needspace{5\\baselineskip}\\item[Profinite completion] The inverse limit of all finite quoti","\\item[Prime spectrum] The prime orders of nonzero torsion elements. It classifies the groups $D_B$ but is constant across all $G_A$.\n\\Needspace{5\\baselineskip}\\item[Profinite completion] The inverse limit of all finite quotient groups, denoted $\\widehat G$. It is not the Pontryagin dual.\n\\Needspace{5\\baselineskip}\\item[Procyclic group] A topological group with a dense cyclic subgroup. The product $P$ is procyclic.\n\\Needspace{5\\baselineskip}\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dime","\\item[Prime spectrum] The prime orders of nonzero torsion elements. It classifies the groups $D_B$ but is constant across all $G_A$.\n\\Needspace{5\\baselineskip}\\item[Profinite completion] The inverse limit of all finite quotient groups, denoted $\\widehat G$. It is not the Pontryagin dual.\n\\Needspace{5\\baselineskip}\\item[Procyclic group] A topological group with a dense cyclic subgroup. The product $P$ is procyclic.\n\\Needspace{5\\baselineskip}\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The p","\\item[Profinite completion] The inverse limit of all finite quotient groups, denoted $\\widehat G$. It is not the Pontryagin dual.\n\\Needspace{5\\baselineskip}\\item[Procyclic group] A topological group with a dense cyclic subgroup. The product $P$ is procyclic.\n\\Needspace{5\\baselineskip}\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. T","\\item[Procyclic group] A topological group with a dense cyclic subgroup. The product $P$ is procyclic.\n\\Needspace{5\\baselineskip}\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some f","\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotien","\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.\n\\Needspace{5\\b","\\item[Pure subgroup] A subgroup $H\\le M$ for which $H\\cap mM=mH$ for all positive integers $m$.\n\\Needspace{5\\baselineskip}\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.\n\\Needspace{5\\baselineskip}\\item[Root certifi","\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.\n\\Needspace{5\\baselineskip}\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the","\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.\n\\Needspace{5\\baselineskip}\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the target has a nonzero coordin","\\item[Rational line] A one-dimensional subspace over $\\Q$. The preimage of $\\Q q(u_A)$ is $G_A$.\n\\Needspace{5\\baselineskip}\\item[Reduced group] A group with no nonzero divisible subgroup. This property does not exclude divisible quotients.\n\\Needspace{5\\baselineskip}\\item[Residual finiteness] Every nonzero element is detected by some finite quotient. It does not imply source reconstruction from the unmarked quotient system.\n\\Needspace{5\\baselineskip}\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the target has a nonzero coordinate.\n\\Nee","\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the target has a nonzero coordinate.\n\\Needspace{5\\baselineskip}\\item[Selector] A subset $A\\subseteq\\N$ whose bits choose one prime from each consecutive pair. It determines the admissible rational line.\n\\Needspace{5\\baselineskip}\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Eve","\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the target has a nonzero coordinate.\n\\Needspace{5\\baselineskip}\\item[Selector] A subset $A\\subseteq\\N$ whose bits choose one prime from each consecutive pair. It determines the admissible rational line.\n\\Needspace{5\\baselineskip}\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-","\\item[Root certificate] Either an exact element $y$ with $my=g$, or a prime divisor of $m$ at which the target has a nonzero coordinate.\n\\Needspace{5\\baselineskip}\\item[Selector] A subset $A\\subseteq\\N$ whose bits choose one prime from each consecutive pair. It determines the admissible rational line.\n\\Needspace{5\\baselineskip}\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of $G_A$ has","\\item[Selector] A subset $A\\subseteq\\N$ whose bits choose one prime from each consecutive pair. It determines the admissible rational line.\n\\Needspace{5\\baselineskip}\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of $G_A$ has the selector's tail signature.\n\\Needspace{5\\baselineskip}\\item[Torsion] Elements of finite order. In the product","\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of $G_A$ has the selector's tail signature.\n\\Needspace{5\\baselineskip}\\item[Torsion] Elements of finite order. In the product and in every $G_A$, the torsion subgroup is exactly $D$.\n\\Needspace{5\\baselineskip}\\item[Turing degree] A measure of relative computability under mutual oracle computation.","\\item[Squarefree radical] The integer $\\rad(m)$ formed by retaining each prime divisor of $m$ once.\n\\Needspace{5\\baselineskip}\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of $G_A$ has the selector's tail signature.\n\\Needspace{5\\baselineskip}\\item[Torsion] Elements of finite order. In the product and in every $G_A$, the torsion subgroup is exactly $D$.\n\\Needspace{5\\baselineskip}\\item[Turing degree] A measure of relative computability under mutual oracle computation. The least presentation degree of $G_A$ is the","\\item[Tail signature] The pattern of prime-root solvability outside a finite exception set. Every infinite-order element of $G_A$ has the selector's tail signature.\n\\Needspace{5\\baselineskip}\\item[Torsion] Elements of finite order. In the product and in every $G_A$, the torsion subgroup is exactly $D$.\n\\Needspace{5\\baselineskip}\\item[Turing degree] A measure of relative computability under mutual oracle computation. The least presentation degree of $G_A$ is the degree of $A$.\n\\Needspace{5\\baselineskip}\\item[Unmarked observations] Finite quotient objects and transition maps without a named coherent source element or the o","\\item[Torsion] Elements of finite order. In the product and in every $G_A$, the torsion subgroup is exactly $D$.\n\\Needspace{5\\baselineskip}\\item[Turing degree] A measure of relative computability under mutual oracle computation. The least presentation degree of $G_A$ is the degree of $A$.\n\\Needspace{5\\baselineskip}\\item[Unmarked observations] Finite quotient objects and transition maps without a named coherent source element or the original admissible image.\n\\end{description}","\\item[Turing degree] A measure of relative computability under mutual oracle computation. The least presentation degree of $G_A$ is the degree of $A$.\n\\Needspace{5\\baselineskip}\\item[Unmarked observations] Finite quotient objects and transition maps without a named coherent source element or the original admissible image.\n\\end{description}","and the Limits of Reconstruction};\n\\node[anchor=west,text=white,font=\\fontsize{22}{28}\\selectfont] at ([xshift=0.82in,yshift=2.65in]current page.south west) {$D\\ \\subseteq\\ G_A\\ \\subseteq\\ P$};\n\\node[anchor=north west,text=CoverLight,text width=5.4in,align=left,font=\\fontsize{11}{16}\\selectfont] at ([xshift=0.82in,yshift=2.20in]current page.south west) {One ambient completion."]}