REALIZATION CALCULUS




Realization
Boundaries
Finite Grammar, Infinite Groups,
and the Limits of Reconstruction




𝐷 ⊆ 𝐺𝐴 ⊆ 𝑃
One ambient completion.
Different algebraic realizations.
Precisely measured information.



Komorebi P.
Preface
An infinite object can be described by finite rules without being generated by finitely many
elements. It can be defined by a short mathematical sentence without admitting computable
operations. It can be separated point by point by finite observations without being
recoverable from the entire unmarked system of those observations. These distinctions
are the subject of this book.
  Realization Calculus studies the passage from specified data and admissible operations
to an actual mathematical object. Its boundary questions are therefore not external
qualifications. They ask which information a realization requires, which operations
preserve that information, and which observations discard it. Here these questions are
examined in a concrete universe of abelian groups. The constructions are elementary
enough to calculate with, yet expressive enough to carry every Turing degree.
   The starting object is the direct sum of one cyclic group of each prime order. Its finite
stages are cyclic. Its completion is topologically cyclic. The direct sum itself nevertheless
needs infinitely many seeds even when every group endomorphism is allowed. Its
subgroup lattice then supplies a transparent first model of finite description and effective
presentation.
  The central construction is more stringent. For every set 𝐴 ⊆ 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.


   The central separation. The compatible finite quotients determine the ambient
   completion. They do not determine which dense subgroup is the intended original
   realization. In this family the omitted choice has an exact divisibility signature and
   an exact computational cost.



How to read the book

Chapters 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

                                              i
Preface                                                                  REALIZATION CALCULUS


guidance, a theorem map, a formalization plan, and a glossary.
  Only elementary group theory is needed at the start. The later arguments introduce
profinite completion and the relevant computability notions before using them. Readers
interested chiefly in the common-completion theorem can begin with Chapters 4–6.
Readers interested in its effective meaning should then read Chapters 7–8, since the
distinction between a marked and an unmarked group is essential to the decoding proof.


Evidence and mathematical context

The main results are supported by the ordinary proofs printed here. The companion
programs use exact rational and modular arithmetic to check finite instances and to
reject selected false formulas. Those checks do not prove the infinite classification or
noncomputability statements. No Lean verification is claimed for this monograph.
   The prime-spectrum coding in Chapter 3 has an explicit classical antecedent in Montal-
bán’s text [1]. Degrees of isomorphism types, pure subgroups, cocycle extensions, and
profinite completion belong to established theories [2, 3, 4, 5]. The common-completion
family is proved directly rather than attributed to an unverified source. Historical priority
for that combined construction has not been established. The contribution presented here
is a self-contained theorem development organized around precise realization contracts,
not a claim that the classical ingredients originate in this work.


Conventions that prevent false conclusions

An unmarked finite-quotient system does not contain the original source group or a
distinguished coherent source element. An infinite parameter is not made finite by
assigning it a single letter. A computable copy is not the same as an enumerable list
of generators and relators. A nonzero divisible quotient does not imply a divisible
subgroup. These distinctions appear where they are used, rather than being left to a
general disclaimer.




                                             ii
Contents
Preface                                                                                        i

Notation and principal results                                                                vi

1 What a realization must supply                                                              1
  1.1 Four questions, not one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     1
  1.2 Finite descriptions and infinite information . . . . . . . . . . . . . . . . . .        2
  1.3 A reusable support obstruction . . . . . . . . . . . . . . . . . . . . . . . . .        3
  1.4 Observations and the original object . . . . . . . . . . . . . . . . . . . . . .        4

2 The squarefree ambient system                                                               5
  2.1 Finite stages and two connecting systems . . . . . . . . . . . . . . . . . . .          5
  2.2 Torsion and endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . .           5
  2.3 A single topological generator after completion . . . . . . . . . . . . . . . .         6
  2.4 The complete finite-quotient system . . . . . . . . . . . . . . . . . . . . . .         7
  2.5 A rational space behind the torsion . . . . . . . . . . . . . . . . . . . . . . .       8

3   Prime spectra and finite descriptions                                                      9
    3.1 The entire subgroup lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . .    9
    3.2 Enumeration is enough for a computable copy . . . . . . . . . . . . . . . .           10
    3.3 A finite definition without computable equality . . . . . . . . . . . . . . . .       10
    3.4 A purely finite-state boundary . . . . . . . . . . . . . . . . . . . . . . . . . .    11
    3.5 What the first family does not accomplish . . . . . . . . . . . . . . . . . . .       12

4   Pure preimages and common finite observations                                             13
    4.1 The preimage construction . . . . . . . . . . . . . . . . . . . . . . . . . . . .     13
    4.2 Root conditions do not depend on the subspace . . . . . . . . . . . . . . .           14
    4.3 Countability, rank, and nonsplitting . . . . . . . . . . . . . . . . . . . . . .      15
    4.4 What has been fixed and what still varies . . . . . . . . . . . . . . . . . . .       16

5 The paired-prime groups                                                                     17
  5.1 Choosing a rational line . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      17
  5.2 A section with finite arithmetic defects . . . . . . . . . . . . . . . . . . . . .      17
  5.3 The cocycle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     18
  5.4 Structure of the extension . . . . . . . . . . . . . . . . . . . . . . . . . . . .      19

6   Root certificates and finite quotients                                                    21
    6.1 An exact decision criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . .   21


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Contents                                                                      REALIZATION CALCULUS


    6.2    The quotient formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   22
    6.3    Why these are all finite quotients . . . . . . . . . . . . . . . . . . . . . . . .   23
    6.4    The information retained by a marked element . . . . . . . . . . . . . . . .         24

7   Classification, morphisms, and seed rank                                                    25
    7.1 The signature of an infinite-order element . . . . . . . . . . . . . . . . . . .        25
    7.2 Isomorphism means eventual agreement . . . . . . . . . . . . . . . . . . .              25
    7.3 All homomorphisms are diagonal . . . . . . . . . . . . . . . . . . . . . . . .          26
    7.4 The endomorphism ring and its units . . . . . . . . . . . . . . . . . . . . .           27
    7.5 Infinite seed rank before completion . . . . . . . . . . . . . . . . . . . . . .        28

8 The exact cost of an effective realization                                                    29
  8.1 Oracles, copies, and degrees . . . . . . . . . . . . . . . . . . . . . . . . . . .        29
  8.2 A presentation from the selector . . . . . . . . . . . . . . . . . . . . . . . .          29
  8.3 Decoding a marked copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          30
  8.4 Decoding an unmarked copy . . . . . . . . . . . . . . . . . . . . . . . . . .             30
  8.5 Every degree over one computable completion . . . . . . . . . . . . . . . .               31
  8.6 A sharper separation than prime-spectrum coding . . . . . . . . . . . . . .               32

9 What finite observations cannot reconstruct                                                   33
  9.1 The observation contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        33
  9.2 Element separation does not identify the source . . . . . . . . . . . . . . .             34
  9.3 Faithful on maps, but not a full reconstruction . . . . . . . . . . . . . . . .           35
  9.4 What a marked coherent point restores . . . . . . . . . . . . . . . . . . . . .           36
  9.5 Two different inverse-limit assertions . . . . . . . . . . . . . . . . . . . . . .        36

10 Escaping a fixed finite grammar                                                              37
   10.1 Fix the semantics before diagonalizing . . . . . . . . . . . . . . . . . . . . .        37
   10.2 Repeated diagonal coding . . . . . . . . . . . . . . . . . . . . . . . . . . . .        37
   10.3 The complexity consequence . . . . . . . . . . . . . . . . . . . . . . . . . . .        38
   10.4 Monotonicity under grammar extension . . . . . . . . . . . . . . . . . . . .            39
   10.5 Uniform families and individual instances . . . . . . . . . . . . . . . . . . .         39

11 A worked arithmetic atlas                                                                    40
   11.1 The empty selector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      40
   11.2 A successful composite-root calculation . . . . . . . . . . . . . . . . . . . .         40
   11.3 A torsion correction that cannot be discarded . . . . . . . . . . . . . . . . .         41
   11.4 Finite quotient representatives . . . . . . . . . . . . . . . . . . . . . . . . . .     41
   11.5 Changing one selector bit . . . . . . . . . . . . . . . . . . . . . . . . . . . .       42
   11.6 A finite tail exception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   42
   11.7 Reproducing the examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .        43

12 Boundaries as part of the calculus                                                           44
   12.1 The ambient object and its admissible realizations . . . . . . . . . . . . . .          44

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    12.2 Three certificates and one global obstruction . . . . . . . . . . . . . . . . .        44
    12.3 A realization profile rather than a single score . . . . . . . . . . . . . . . .       45
    12.4 What a machine-facing contract should retain . . . . . . . . . . . . . . . . .         45
    12.5 Further questions with exact starting points . . . . . . . . . . . . . . . . . .       46
    12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   46

A Exact computation and reproduction                                                            47
  A.1 A finite representation of each named element . . . . . . . . . . . . . . . .             47
  A.2 Root construction as an algorithm . . . . . . . . . . . . . . . . . . . . . . . .         48
  A.3 What the tests cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      48
  A.4 Reproduction commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           49
  A.5 Verification boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       49

B Theorem map and formalization interfaces                                                      50
  B.1 The proof dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          50
  B.2 Statement registry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      51
  B.3 A staged formalization plan . . . . . . . . . . . . . . . . . . . . . . . . . . .         52
  B.4 Negative statement tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        53

C Glossary                                                                                      54

References                                                                                      57




                                                 v
Notation and principal results
All 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

                        𝑝0 = 2,    𝑝1 = 3,   𝑝2 = 5,      𝑝3 = 7,   ....


 Symbol                 Meaning
 𝐶 𝑝 =É  F𝑝             The additive cyclic group of prime order 𝑝.
 𝐷=         𝐶𝑝          Finite-support prime-coordinate vectors.
       Î 𝑝
 𝑃 = 𝑝 𝐶𝑝               All prime-coordinate vectors, with product topology.
 𝑞:𝑃→𝑉                  The quotient map to 𝑉 = 𝑃/𝐷.
 𝐷𝐵                     The subgroup supported on prime indices in 𝐵 ⊆ N.
 𝐻𝑊 = 𝑞 −1 (𝑊)          The intermediate group associated to a rational subspace 𝑊 ≤ 𝑉.
 𝑆𝐴                     The prime set {𝑝 2𝑛+1𝐴 (𝑛) : 𝑛 ∈ N}.
 𝑢𝐴                     The coordinate indicator of 𝑆𝐴 in 𝑃.
 𝐺𝐴                     The group 𝑞 −1 (Q 𝑞(𝑢𝐴 )).
 𝑠 𝐴 (𝑟)                The specified rational lift of 𝑟 𝑞(𝑢𝐴 ) to 𝑃.
 𝑐 𝐴 (𝑟, 𝑡)             The finite cocycle 𝑠 𝐴 (𝑟) + 𝑠 𝐴 (𝑡) − 𝑠 𝐴 (𝑟 + 𝑡).
 𝑒𝑝                     The vector supported at 𝑝 with value one.
 𝑝 |𝐺 𝑔                 There exists 𝑦 ∈ 𝐺 with 𝑝 𝑦 = 𝑔.
 𝐴 =∗ 𝐵                 The symmetric difference 𝐴△𝐵 is finite.
 rad(𝑚)                 The product of the distinct prime divisors of 𝑚, with rad(1) = 1.
 𝐺
 b                      The profinite completion, not the Pontryagin dual.
 𝐴 ≤𝑇 𝑋                 Membership in 𝐴 is computable using oracle 𝑋.
 𝐾 𝔊 (𝐺)                Least admissible finite description length, or +∞.


The main statements have the following form:

                                   Tor(𝐺 𝐴 ) = 𝐷,                          𝐺 𝐴 /𝐷  Q,
                                  𝐺 𝐴 /𝑚𝐺 𝐴  𝐶rad(𝑚) ,                      𝐺
                                                                             c𝐴  𝑃,
                                  𝐺 𝐴  𝐺 𝐵 ⇐⇒ 𝐴 =∗ 𝐵,
          𝐺 𝐴 has an 𝑋-computable copy ⇐⇒ 𝐴 ≤𝑇 𝑋.

The 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.

                                              vi
CHAPTER 1



What a realization must supply
Before constructing a boundary, one must specify what is allowed to cross it. A finite seed family, a
finite description, and an effective presentation impose different requirements.


1.1 Four questions, not one

The expression “finitely realizable” is incomplete unless it names a contract. A group
may be infinite but cyclic. A group may require infinitely many algebraic generators but
have a short recursive construction. A description may select an exact group through
a noncomputable set. Finally, a topology may allow limits that are not present in the
underlying algebraic construction.
  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.
Definition 1.1 (Operator-seed realization). Let 𝑀 be an abelian group and let 𝒪 ⊆ End(𝑀)
be a declared collection of endomorphisms. For 𝑆 ⊆ 𝑀, its operator hull is the subgroup
generated by all 𝑇(𝑠), where 𝑠 ∈ 𝑆 and 𝑇 is a finite composition of members of 𝒪, including
the identity. A finite operator-seed realization requires a finite 𝑆 whose hull is 𝑀.

   The 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 srankEnd(𝑀) (𝑀) for its least generating cardinal. An infinite value can therefore
express a strong obstruction even when the operator collection itself is unrestricted.
Definition 1.2 (Finite semantic description). A finite-description 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.

   This 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.

                                                 1
1   What a realization must supply                                                REALIZATION CALCULUS


Definition 1.3 (Computable copy). A computable copy of a countable group is an isomor-
phic 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.

   A 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.

Definition 1.4 (Topological generation). For 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.

  The closure is taken in the stated topology. A change of topology is a change of contract,
not an innocuous notational adjustment.


1.2 Finite descriptions and infinite information

There are countably many finite strings over a countable alphabet. Consequently a grammar
with single-valued semantics realizes at most countably many group isomorphism types.
This observation requires neither an algorithm for checking validity nor an algorithm for
evaluating the semantics.

Proposition 1.5 (Countable-code bound). Let 𝔊 be a finite-description grammar as above.
Then its class of realized group isomorphism types has cardinal at most ℵ0 .

Proof. For each length 𝑛, the set of strings of length 𝑛 is countable. The union over 𝑛 ∈ N
is countable. The valid strings form a subset, and a single-valued image of a countable set
is countable.                                                                             □

   This 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.
  For quantitative description length, fix an injective encoding of descriptions by finite
binary strings. A symbol from a countable alphabet must carry its finite code rather
than count as a free unit of information. This encoding convention does not require the
semantic evaluation to be computable.

Definition 1.6 (Grammar-relative description complexity). For a fixed binary coding of 𝔊,
let |𝑑| be its bit length and set

                           𝐾𝔊 (𝐺) = inf{|𝑑| : 𝑑 is valid and Realize𝔊 (𝑑)  𝐺},

with value +∞ when there is no such description.

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1   What a realization must supply                                              REALIZATION CALCULUS


  The value depends on the grammar and the coding. A fixed finite description can be
long without being impossible. Conversely, a group with 𝐾 𝔊 (𝐺) = +∞ may have a short
description in an expanded language.

Boundary 1.7 (No uncharged infinite parameters). The 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.

  This boundary will become exact: the computational information in a selected 𝑢𝐴 is
precisely what is needed to compute the corresponding group.


1.3 A reusable support obstruction

A finite-seed failure can sometimes be certified without examining every possible operator
word. The packet structure supplies a witness.

Theorem 1.8 (Reachability obstruction). Let 𝑀 =          𝑖∈𝐼 𝑀 𝑖 be a direct sum of abelian groups
                                                          É

with coordinate projections 𝑃𝑖 . Suppose every primitive operator 𝑇 is an endomorphism and its
transfer block 𝑃 𝑗 𝑇𝑃𝑖 vanishes unless 𝑖 → 𝑗 is an allowed edge of a directed graph 𝐸. If 𝐹 is the
union of the supports of a seed family 𝑆, then
                                                       Ê
                                     Hull𝒪 (𝑆) ⊆                  𝑀𝑗 .
                                                   𝑗∈Reach𝐸 (𝐹)

If 𝑀 𝑗 ≠ 0 and 𝑗 is not reachable, 𝑃 𝑗 annihilates the entire hull but not 𝑀.

Proof. An 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.                                              □

  Reachability 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
graph is connected.
  If the packet projections are continuous into Hausdorff groups, the kernel of an
omitted projection is closed. The same witness then obstructs topological generation in
that topology. Later we will use precisely this argument after allowing every abstract
endomorphism of the group.




                                                   3
1   What a realization must supply                                        REALIZATION CALCULUS


1.4 Observations and the original object

An observation system also needs a contract. For a residually finite group, finite quotients
separate distinct elements. This is an elementwise assertion: if two elements differ, at least
one quotient distinguishes them. It does not assert that the unmarked family of quotients
identifies the source isomorphism type.
    The distinction is easiest to express in terms of an embedding

                                      𝐺 −→ lim 𝐺/𝑁 .
                                           ←−−
Residual 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.
   The later examples do not contradict the construction of an inverse limit. They show that
reconstructing the complete ambient realization and identifying a specified noncomplete
realization are different tasks. This is the point at which compatibility must be separated
from admissibility.




                                              4
CHAPTER 2



The squarefree ambient system
One sequence of finite cyclic groups has two natural limits. Their generation properties differ
because the connecting maps and the permitted elements differ.


2.1 Finite stages and two connecting systems

For 𝑁 ≥ 1, define                     Ö                         Ö
                               𝐴𝑁 =         𝐶𝑝𝑖 ,        𝑚𝑁 =         𝑝𝑖 .
                                      𝑖<𝑁                       𝑖<𝑁
The Chinese remainder map sends 𝑘 mod 𝑚 𝑁 to (𝑘 mod 𝑝 𝑖 )𝑖<𝑁 . It is an isomorphism, so
𝐴 𝑁 is cyclic and its all-one vector is a generator.

                          𝑁    Prime factors            𝑚𝑁   Finite group
                          1           2               2            𝐶2
                          2          2, 3             6            𝐶6
                          3        2, 3, 5           30            𝐶30
                          4       2, 3, 5, 7        210           𝐶210
                          5     2, 3, 5, 7, 11      2310          𝐶2310

  Zero extension gives injections 𝐴 𝑁 → 𝐴 𝑁+1 . Forgetting the last coordinate gives
surjections 𝐴 𝑁+1 → 𝐴 𝑁 . These are different diagrams. Their limits are
                                      Ê                                      Ö
                     𝐷 = lim 𝐴 𝑁 =          𝐶𝑝 ,         𝑃 = lim 𝐴 𝑁 =           𝐶𝑝 .
                         −−→                                 ←−−
                           𝑁           𝑝                      𝑁              𝑝

The 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.


2.2 Torsion and endomorphisms

Theorem 2.1 (The torsion subgroup of the product). The subgroup 𝐷 is dense and proper in
𝑃, and
                                    Tor(𝑃) = 𝐷.

                                                    5
2   The squarefree ambient system                                           REALIZATION CALCULUS


Proof. A 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 ∤ 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 ∈ 𝐷. The all-one vector has infinite support, so it belongs to
𝑃 \ 𝐷.                                                                                         □

Proposition 2.2 (Diagonal endomorphisms of the direct sum). Every endomorphism of 𝐷
acts by a scalar on each prime coordinate, and
                                                   Ö
                                       End(𝐷)          F𝑝
                                                    𝑝

as rings.

Proof. The 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 F𝑝 . 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.             □

Corollary 2.3 (An infinite seed obstruction). The endomorphism-seed rank of 𝐷 is ℵ0 . No
finite set topologically generates 𝐷, even after all endomorphisms are admitted.

Proof. The 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 1.8. 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.                                                                                      □

  Notice 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.


2.3 A single topological generator after completion

Theorem 2.4 (Topological cyclicity). Let 𝑢 = (1, 1, 1, . . .) ∈ 𝑃. Then

                                            ⟨𝑢⟩ = 𝑃.

In particular, trank(𝑃) = 1.



                                               6
2   The squarefree ambient system                                           REALIZATION CALCULUS




    Zero-extension maps



        A1                  A2         A3           ⋯                  D = ⨁Cp
                                                                              p

                                                                endomorphism-seed rank ℵ0

    Coordinate projections



        A1                  A2         A3           ⋯                   P = ∏Cp
                                                                              p

                                                                     topological rank 1




Figure 2.1: 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.


Proof. A 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.                                            □

  The 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.


2.4 The complete finite-quotient system

Proposition 2.5 (Profinite completion of the direct sum). The canonical embedding of 𝐷 into
𝑃 identifies 𝑃 with 𝐷.
                    b

Proof. Let 𝑓 : 𝐷 → 𝐹 be a homomorphism to a finite group. The image of a prime-order
factor 𝐶 𝑝 is trivial unless 𝑝 divides |𝐹|. 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. □

  This 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



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2   The squarefree ambient system                                             REALIZATION CALCULUS


profinite completion may be found in [5]. All needed cofinality statements in this book are
proved for the specific groups under consideration.


2.5 A rational space behind the torsion

Put 𝑉 = 𝑃/𝐷 and let 𝑞 : 𝑃 → 𝑉 be the quotient map.

Theorem 2.6 (Unique division in the ambient quotient). Multiplication 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.

Proof. Fix 𝑚 ≥ 1 and 𝑥 ∈ 𝑃. Define 𝑦 𝑝 = 𝑚 −1 𝑥 𝑝 when 𝑝 ∤ 𝑚, and 𝑦 𝑝 = 0 when 𝑝 | 𝑚. Then
𝑥 − 𝑚𝑦 has finite support. Thus 𝑞(𝑥) = 𝑚𝑞(𝑦), proving surjectivity on 𝑉.
  For injectivity, suppose 𝑚𝑥 ∈ 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 ∈ 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.              □

  A group with no nonzero divisible subgroup is called 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.




                                                 8
CHAPTER 3



Prime spectra and finite descriptions
The direct sum has a completely transparent subgroup universe. It separates the complexity of
selecting prime components from the complexity of performing the group operation.


3.1 The entire subgroup lattice

For 𝐵 ⊆ N, set                             Ê
                                    𝐷𝐵 =          𝐶 𝑝 𝑖 ≤ 𝐷.
                                           𝑖∈𝐵

A prime spectrum records which prime orders occur. Define 𝜎𝑝 (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.

Theorem 3.1 (Subgroup and isomorphism classification). The map 𝐵 ↦→ 𝐷𝐵 is an order
isomorphism from 𝒫(N) onto Sub(𝐷). Every subgroup is fully invariant under all endomorphisms
of 𝐷, and
                               𝐷𝐵  𝐷𝐶 ⇐⇒ 𝐵 = 𝐶.

Proof. Let 𝐻 ≤ 𝐷 and let 𝑥 ∈ 𝐻. Its support is a finite set 𝐹 of primes. For 𝑝 ∈ 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 𝐷𝐵 , and every
finite sum of the factors in 𝐵 lies in 𝐻. Hence 𝐻 = 𝐷𝐵 . Uniqueness follows by testing
membership of 𝑒 𝑝 𝑖 . Proposition 2.2 proves full invariance. Finally 𝜎𝑝 𝑖 (𝐷𝐵 ) = 1 exactly
when 𝑖 ∈ 𝐵, and these values are isomorphism invariants.                                  □

  The lattice operations have literal set-theoretic descriptions:

              𝐷𝐵 + 𝐷𝐶 = 𝐷𝐵∪𝐶 ,        𝐷𝐵 ∩ 𝐷𝐶 = 𝐷𝐵∩𝐶 ,         𝐷 = 𝐷𝐵 ⊕ 𝐷N\𝐵 .

The 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. Mem-
bership of a vector with finite support depends on finitely many selector bits. Conversely,

                                              9
3   Prime spectra and finite descriptions                                   REALIZATION CALCULUS


membership of 𝑒 𝑝 𝑖 recovers the 𝑖th bit. Consequently the order isomorphism above is also
a homeomorphism from Cantor space {0, 1}N to Sub(𝐷).

Corollary 3.2 (Finite observations leave continuum many possibilities). Fix finitely many
prime indices and their membership values. There are continuum many pairwise nonisomorphic 𝐷𝐵
consistent with those values. A full compatible sequence of finite membership patterns determines
one 𝐵, but no fixed finite prefix does.

Proof. The unspecified bits form a countably infinite set of independent binary choices.
Distinct completions give nonisomorphic groups by Theorem 3.1. A full compatible family
fixes every coordinate and hence fixes the subset.                                    □

3.2 Enumeration is enough for a computable copy

A 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.

Theorem 3.3 (The prime-spectrum criterion). For every oracle 𝑋,

         𝐷𝐵 has an 𝑋-computable copy        ⇐⇒   𝐵 is computably enumerable relative to 𝑋.

Proof. Suppose 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 ∈ 𝐵. Dovetailing the searches enumerates 𝐵.
   Conversely, suppose 𝐵 is infinite and enumerable in 𝑋. List its distinct elements
as 𝑏 0 , 𝑏1 , . . .. 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 𝑗th coordinate in
F𝑝𝑏 𝑗 . Coordinatewise addition, normalization, and equality are computable in 𝑋. If 𝐵 is
finite, 𝐷𝐵 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.               □

   This construction is classical. Montalbán gives the prime direct-sum example and its
enumeration-degree interpretation explicitly [1, Example III.39]. It is included here because
its contrast with the later common-completion family is mathematically informative.
  A noncomputable enumerable selector can therefore produce a group with computable
multiplication and equality. The computable copy labels coordinates by their order of
discovery. It does not supply a decision procedure for membership in the original selector.


3.3 A finite definition without computable equality

Let 𝐾 = {𝑛 : program 𝑛 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

                                                 10
3   Prime spectra and finite descriptions                                  REALIZATION CALCULUS


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
                           𝐵𝐾 = {2𝑛 : 𝑛 ∈ N} ∪ {2𝑛 + 1 : 𝑛 ∉ 𝐾},   𝐻 = 𝐷𝐵𝐾 .
This is an explicit mathematical definition of a countably infinite, locally cyclic abelian
group.

Corollary 3.4. The group 𝐻 has no computable copy. It nevertheless has an effectively enumerable
infinite list of generators and relators.

Proof. If 𝐵𝐾 were enumerable, filtering its odd entries would enumerate the complement
of 𝐾. Theorem 3.3 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 𝐵𝐾 . All relators
are effectively enumerable. Deciding whether an odd generator has become trivial is not
supplied by this enumeration.                                                          □

  The 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.


3.4 A purely finite-state boundary

Consider a restricted grammar in which a finite automaton reads the unary word 1𝑛 and
selects the factor 𝐶 𝑝 𝑛 exactly when it accepts.

Theorem 3.5 (Unary selector classification). A group 𝐷𝐵 is realizable by this unary-selector
grammar if and only if 𝐵 is ultimately periodic.

Proof. On 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 𝐷𝐵 .           □

Example 3.6 (A computable group outside the unary grammar). Let 𝐵 = {2 𝑘 : 𝑘 ≥ 0}. It
is decidable and enumerable, so 𝐷𝐵 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 𝐷𝐵 is outside the unary grammar. A binary-input automaton recognizes the
strings 10 𝑘 , so changing the input representation removes this particular obstruction.

                                                   11
3   Prime spectra and finite descriptions                                REALIZATION CALCULUS


3.5 What the first family does not accomplish

The groups 𝐷𝐵 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.




                                             12
CHAPTER 4



Pure preimages and common finite
observations
The common-completion phenomenon is not confined to one selector construction. It follows from a
general correspondence between rational subspaces and pure intermediate groups.


4.1 The preimage construction

Keep 𝐷 ⊆ 𝑃 and 𝑉 = 𝑃/𝐷 from Chapter 2. For a rational vector subspace 𝑊 ≤ 𝑉, define

                                        𝐻𝑊 = 𝑞 −1 (𝑊).

This 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.
Definition 4.1 (Purity). A subgroup 𝐻 of an abelian group 𝑀 is pure when

                                𝐻 ∩ 𝑚𝑀 = 𝑚𝐻           (𝑚 ≥ 1).

Equivalently, if an equation 𝑚 𝑦 = ℎ with ℎ ∈ 𝐻 has a solution in 𝑀, then it has a solution
in 𝐻.

   Purity is a standard algebraic notion [3]. The proof below uses only the displayed
criterion. In particular, no structure theorem for mixed abelian groups is required.

Theorem 4.2 (Pure-preimage correspondence). The map

                                    𝑊 ↦−→ 𝐻𝑊 = 𝑞 −1 (𝑊)

is an inclusion-preserving bĳection between rational subspaces of 𝑉 and pure subgroups 𝐻 of 𝑃
containing 𝐷.

Proof. First let 𝑊 be rational and suppose ℎ ∈ 𝐻𝑊 ∩ 𝑚𝑃. Because 𝑊 is closed under
division by 𝑚, choose 𝑥 ∈ 𝐻𝑊 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 ∈ 𝑚𝑃, and that of 𝑚𝑥 is also zero.
Thus 𝑑 𝑝 = 0 whenever 𝑝 | 𝑚.

                                              13
4   Pure preimages and common finite observations                              REALIZATION CALCULUS


  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 ∈ 𝐻𝑊 . This proves
purity.
  Conversely, let 𝐷 ⊆ 𝐻 ⊆ 𝑃 be pure. Its image 𝑞(𝐻) is an additive subgroup of 𝑉. Given
ℎ ∈ 𝐻 and 𝑚 ≥ 1, let 𝑑 ∈ 𝐷 agree with ℎ on the primes dividing 𝑚 and vanish elsewhere.
Then ℎ − 𝑑 ∈ 𝑚𝑃 ∩ 𝐻. Purity gives ℎ − 𝑑 = 𝑚𝑥 for some 𝑥 ∈ 𝐻, so 𝑞(𝑥) = 𝑞(ℎ)/𝑚. Thus
𝑞(𝐻) is closed under rational scalars. Since 𝐷 ⊆ 𝐻, the equality 𝐻 = 𝑞 −1 (𝑞(𝐻)) holds. This
proves that the constructions are inverse. Both preserve inclusion.                       □

  The bottom object is 𝐻{0} = 𝐷, and the top object is 𝐻𝑉 = 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.


4.2 Root conditions do not depend on the subspace

Theorem 4.3 (Ambient root criterion). For every rational subspace 𝑊, every ℎ ∈ 𝐻𝑊 , and
every 𝑚 ≥ 1,
                  ℎ ∈ 𝑚𝐻𝑊 ⇐⇒ ℎ 𝑝 = 0 for all primes 𝑝 | 𝑚.

Proof. Multiplication 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 ∈ 𝑚𝑃. Apply purity from
Theorem 4.2.                                                                            □

  The 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.

Theorem 4.4 (One finite-quotient system for every preimage). For every rational subspace
𝑊 and every 𝑚 ≥ 1, the coordinate map induces an isomorphism
                                                           Ö
                                              𝐻𝑊 /𝑚𝐻𝑊           𝐶𝑝 .
                                                           𝑝|𝑚

For 𝑚 | 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

                                                    𝐻
                                                    d𝑊  𝑃.


Proof. The coordinate map onto the displayed finite product is surjective already on 𝐷. Its
kernel is 𝑚𝐻𝑊 by Theorem 4.3. The transition assertion follows by evaluating the same
coordinates.
  If 𝐻𝑊 → 𝐹 is a homomorphism to a finite group, then the image is abelian and is killed
by |𝐹|. Its kernel therefore contains |𝐹|𝐻𝑊 . 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 𝐻𝑊 ↩→ 𝑃.                                      □

                                                      14
4   Pure preimages and common finite observations                         REALIZATION CALCULUS


 There is a stronger natural statement. It compares maps to every finite target rather than
merely comparing the quotient groups.

Proposition 4.5 (Finite-target equivalence). For every finite group 𝐹, restriction along 𝐷 ↩→
𝐻𝑊 gives a natural bĳection

                                        Hom(𝐻𝑊 , 𝐹) −→ Hom(𝐷, 𝐹).

Proof. Put 𝑚 = |𝐹|. Every ℎ ∈ 𝐻𝑊 agrees modulo 𝑚𝐻𝑊 with an element of 𝐷, namely
its finite coordinate vector on primes dividing 𝑚. Every homomorphism to 𝐹 kills 𝑚𝐻𝑊 .
Thus two maps agreeing on 𝐷 agree everywhere.
   Conversely, a map 𝐷 → 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 𝐻𝑊 extends it. The extension is
unique by the first part. Postcomposition with a map of finite groups commutes with
restriction, giving naturality.                                                       □

   The argument never subtracts maps to a nonabelian group. The assertion is a bĳection
of sets of homomorphisms, and naturality is with respect to postcomposition.


4.3 Countability, rank, and nonsplitting

Proposition 4.6 (Structure of an ambient preimage). For every rational subspace 𝑊,

                                       Tor(𝐻𝑊 ) = 𝐷,        𝐻𝑊 /𝐷  𝑊 .

The group 𝐻𝑊 is reduced and residually finite. It is countable exactly when 𝑊 has finite or
countably infinite dimension over Q. If 𝑊 ≠ 0, the sequence

                                        0 −→ 𝐷 −→ 𝐻𝑊 −→ 𝑊 −→ 0

does not split.

Proof. The torsion equality follows from 𝐷 ⊆ 𝐻𝑊 ⊆ 𝑃 and Theorem 2.1. 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 𝐻𝑊 → 𝑊 is a coset of the countable
group 𝐷. Thus 𝐻𝑊 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 2.6 excludes
every such subgroup.                                                              □


                                                       15
4   Pure preimages and common finite observations                        REALIZATION CALCULUS


   The rank of the torsion-free quotient is exactly dimQ 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.


4.4 What has been fixed and what still varies

The 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 𝐻𝑊 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.




                                                    16
CHAPTER 5



The paired-prime groups
A binary selector chooses one prime from each pair. Its indicator determines a rational line in the
ambient quotient, and an explicit finite cocycle turns that line into a countable group.


5.1 Choosing a rational line

Let 𝐴 ⊆ N, write 𝑎 𝑛 = 1𝐴 (𝑛), and define

                                      𝑆𝐴 = {𝑝2𝑛+𝑎 𝑛 : 𝑛 ∈ N}.

Exactly one prime from each pair is selected. Both 𝑆𝐴 and its complement are infinite. Let
𝑢𝐴 ∈ 𝑃 be its indicator:                   (
                                             1, 𝑝 ∈ 𝑆𝐴 ,
                                 (𝑢𝐴 )𝑝 =
                                             0, 𝑝 ∉ 𝑆𝐴 .
Because 𝑢𝐴 has infinite support, 𝑞(𝑢𝐴 ) ≠ 0. We can therefore define a rational line and its
preimage by
                           𝑊𝐴 = Q 𝑞(𝑢𝐴 ),      𝐺 𝐴 = 𝑞 −1 (𝑊𝐴 ).
This definition produces an actual group for every set 𝐴. It does not claim an effective
method for obtaining the selector bits of an arbitrary 𝐴.


5.2 A section with finite arithmetic defects

For a reduced rational 𝑟 = 𝑎/𝑏, with 𝑏 > 0, define
                                      (
                                          𝑎𝑏 −1   (mod 𝑝),     𝑝 ∈ 𝑆𝐴 and 𝑝 ∤ 𝑏,
                     (𝑠 𝐴 (𝑎/𝑏))𝑝 =                                                           (5.1)
                                          0,                   𝑝 ∉ 𝑆𝐴 or 𝑝 | 𝑏.

The inverse is used only when the denominator is a unit in F𝑝 . 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 𝑏𝑠 𝐴 (𝑎/𝑏) − 𝑎𝑢𝐴 is supported on the prime divisors of 𝑏. It follows that

                                           𝑞(𝑠 𝐴 (𝑟)) = 𝑟 𝑞(𝑢𝐴 ).

Thus every element of 𝐺 𝐴 is 𝑠 𝐴 (𝑟) + 𝑑 for some 𝑟 ∈ Q and 𝑑 ∈ 𝐷.


                                                    17
5    The paired-prime groups                                                              REALIZATION CALCULUS



       an          0           1            0            0            1              1     0          1

       p2n
                   2           5           11           17           23              31    41        47



    p2n + 1
                   3           7           13           19           29              37    43        53

                   0           1            2            3            4              5     6          7
                                                        pair index n

Figure 5.1: 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.


Theorem 5.1 (Unique normal form). Every element of 𝐺 𝐴 has a unique expression

                                     𝑔 = 𝑠 𝐴 (𝑟) + 𝑑,          (𝑟, 𝑑) ∈ Q × 𝐷.

Consequently 𝐺 𝐴 is countably infinite, with a countable normal-form domain that does not depend
on the selector.

Proof. Existence follows from the preimage definition and the displayed quotient identity.
If two forms are equal, applying 𝑞 gives (𝑟 − 𝑟 ′)𝑞(𝑢𝐴 ) = 0. The vector 𝑞(𝑢𝐴 ) 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.                                       □

  The domain can be named without knowing 𝐴. The group operation on those names
will contain the selector information.


5.3 The cocycle law

Define
                                     𝑐 𝐴 (𝑟, 𝑡) = 𝑠 𝐴 (𝑟) + 𝑠 𝐴 (𝑡) − 𝑠 𝐴 (𝑟 + 𝑡).
The sum is taken in 𝑃, but its value belongs to 𝐷.

Lemma 5.2 (A finite support bound). For all 𝑟, 𝑡 ∈ Q,

                                   supp 𝑐 𝐴 (𝑟, 𝑡) ⊆ {𝑝 : 𝑝 | den(𝑟) den(𝑡)}.

Proof. At a prime not dividing either denominator, all three rationals have well-defined
reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On

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5   The paired-prime groups                                                               REALIZATION CALCULUS


an unselected coordinate, all section values are zero. The reduced denominator of 𝑟 + 𝑡
has no prime divisor absent from the two original denominators.                      □

Theorem 5.3 (Explicit group law). In normal-form coordinates, addition is

                                  (𝑟, 𝑑) + (𝑡, 𝑒) = (𝑟 + 𝑡, 𝑑 + 𝑒 + 𝑐 𝐴 (𝑟, 𝑡)).                         (5.2)

The identity is (0, 0) and the inverse of (𝑟, 𝑑) is (−𝑟, −𝑑). The law is associative and commutative,
and the map (𝑟, 𝑑) ↦→ 𝑠 𝐴 (𝑟) + 𝑑 is an isomorphism onto the subgroup 𝐺 𝐴 ≤ 𝑃.

Proof. Substitution into addition in 𝑃 gives the formula. The section satisfies 𝑠 𝐴 (0) = 0 and
𝑠 𝐴 (−𝑟) = −𝑠 𝐴 (𝑟), so 𝑐 𝐴 (𝑟, 0) = 0 and 𝑐 𝐴 (𝑟, −𝑟) = 0. These identities give the stated identity
and inverse.
    Symmetry is immediate. The cocycle equation

                              𝑐 𝐴 (𝑟, 𝑡) + 𝑐 𝐴 (𝑟 + 𝑡, 𝑣) = 𝑐 𝐴 (𝑡, 𝑣) + 𝑐 𝐴 (𝑟, 𝑡 + 𝑣)

follows by expanding and cancelling the section terms. Expanding the two associations
in (5.2) now proves associativity. Uniqueness of normal forms makes the ambient map
bĳective, and the addition formula makes it a homomorphism.                         □

   This is the standard symmetric cocycle construction of an abelian extension [4, Section
3.2]. Here the particular section makes each individual correction finitely supported and
exactly computable from finitely many selector queries.

Proposition 5.4 (One cocycle value reads one coordinate). For every prime 𝑝,

                                      𝑐 𝐴 (1/𝑝, (𝑝 − 1)/𝑝) = −1𝑆𝐴 (𝑝)𝑒 𝑝 .

Proof. The support bound leaves only the coordinate 𝑝. At that coordinate both fractional
lifts are zero, and the lift of their sum, which is one, is 1𝑆𝐴 (𝑝).                   □

  Dropping the cocycle is therefore not a harmless simplification. The direct product
operation on Q × 𝐷 removes exactly these defects and gives a different, split extension.


5.4 Structure of the extension

By Chapter 4, the group is pure in 𝑃, reduced, residually finite, and dense. Its torsion is 𝐷,
and the rational coordinate gives the exact sequence
                                               𝜌𝐴
                         0 −→ 𝐷 −→ 𝐺 𝐴 −−→ Q −→ 0,                   𝜌𝐴 (𝑠 𝐴 (𝑟) + 𝑑) = 𝑟.               (5.3)

It does not split.




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5   The paired-prime groups                                                   REALIZATION CALCULUS


  There is an equivalent cocycle obstruction. Suppose a function 𝑏 : Q → 𝐷 modified the
section to an additive map 𝑟 ↦→ 𝑠 𝐴 (𝑟) + 𝑏(𝑟). Then

                              𝑐 𝐴 (𝑟, 𝑡) + 𝑏(𝑟) + 𝑏(𝑡) − 𝑏(𝑟 + 𝑡) = 0

for every 𝑟, 𝑡. The modified section would split (5.3). 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.

Proposition 5.5 (Finitely generated subgroups). Every finitely generated subgroup of 𝐺 𝐴 is
either finite cyclic or isomorphic to Z ⊕ 𝐶 𝑚 for a squarefree positive integer 𝑚, with 𝐶1 trivial.

Proof. Let 𝐿 ≤ 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 ≤ 𝐷 and
its finite support places it inside a finite cyclic group.
  Otherwise choose a lift 𝑥 ∈ 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 → 𝜌𝐴 (𝐿), which is finite cyclic of squarefree order. The
subgroup ⟨𝑥⟩ intersects that torsion kernel trivially, giving the direct sum.           □

  The extensions over finitely generated rational images split, but the extension over all of
Q does not. This is another exact distinction between finite algebraic stages and the global
realization.




                                                20
CHAPTER 6



Root certificates and finite quotients
An integer-root problem has a complete answer: either a finite coordinate obstructs it, or a finite
correction constructs a root. This calculation controls every finite quotient.



                                             u∅ = (1, 0)




                m=2:       (u∅)2 = 1                             m=3:       (u∅)3 = 0




                       No root                                       y = (1/3, 0)
              certificate: the prime 2                                 3y = u∅



Figure 6.1: For the marked element 𝑢𝐴 , 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.


6.1 An exact decision criterion

The general purity argument already establishes the root condition. For 𝐺 𝐴 , normal forms
make the successful branch constructive.

Theorem 6.1 (Root or obstruction). Let 𝑔 = 𝑠 𝐴 (𝑟) + 𝑑 ∈ 𝐺 𝐴 and let 𝑚 ≥ 1. Then

                           𝑔 ∈ 𝑚𝐺 𝐴      ⇐⇒      𝑔𝑝 = 0    for every 𝑝 | 𝑚.

If the condition fails, a prime 𝑝 | 𝑚 with 𝑔𝑝 ≠ 0 is a nonexistence certificate. If it holds, a root is
obtained by the finite procedure in the proof.

Proof. Necessity follows because multiplication by 𝑚 is zero in every coordinate 𝐶 𝑝 with
𝑝 | 𝑚.


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6   Root certificates and finite quotients                                                     REALIZATION CALCULUS


    Assume the condition holds and put 𝑡 = 𝑟/𝑚. The vector

                                                        ℎ = 𝑔 − 𝑚𝑠 𝐴 (𝑡)

has 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 | 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 ∈ 𝐷 and 𝑚𝑒 = ℎ. Consequently

                                             𝑦 = 𝑠 𝐴 (𝑡) + 𝑒   satisfies      𝑚 𝑦 = 𝑔.

Every nonzero correction coordinate is found in the stated finite support set.                                   □

  A 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.

Proposition 6.2 (All roots and their number). If 𝑚 𝑦 = 𝑔 is solvable, its solution set is the coset
                                                                              Ê
                                             𝑦0 + 𝐺 𝐴 [𝑚],       𝐺 𝐴 [𝑚] =              𝐶𝑝 .
                                                                                  𝑝|𝑚

It has exactly rad(𝑚) elements.

Proof. Two 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 | 𝑚. All those finite coordinate factors lie in 𝐷 ⊆ 𝐺 𝐴 . The kernel is therefore the
displayed finite direct sum, whose cardinal is rad(𝑚).                                     □

  Repeated 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.


6.2 The quotient formula

Theorem 6.3 (All integer quotients). For 𝑚 ≥ 1, the coordinate map
                                                                       Ö
                                                     𝜋𝑚 : 𝐺 𝐴 −→             𝐶𝑝
                                                                       𝑝|𝑚

is surjective with kernel 𝑚𝐺 𝐴 . Thus
                                                               Ö
                                               𝐺 𝐴 /𝑚𝐺 𝐴            𝐶 𝑝  𝐶rad(𝑚) .
                                                               𝑝|𝑚

For 𝑚 | 𝑛, the transition map is the corresponding coordinate projection.

                                                                22
6   Root certificates and finite quotients                                         REALIZATION CALCULUS


Proof. A 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.       □

                        𝑚      rad(𝑚)        𝐺 𝐴 /𝑚𝐺 𝐴   Root-fiber size when nonempty
                         1             1            0                               1
                         4             2           𝐶2                               2
                         8             2           𝐶2                               2
                        12             6           𝐶6                               6
                        18             6           𝐶6                               6
                        30            30          𝐶30                              30
                        60            30          𝐶30                              30

    The kernel itself depends only on the distinct prime divisors:

                                                  𝑚𝐺 𝐴 = rad(𝑚)𝐺 𝐴 .

This is stronger than equality of the quotient orders. It is equality of the subgroups inside
the fixed source group.


6.3 Why these are all finite quotients

Let 𝑓 : 𝐺 𝐴 → 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 |𝐹|. Therefore 𝑓 kills |𝐹|𝐺 𝐴 and
factors through the quotient in Theorem 6.3.
  This proves that every finite quotient is cyclic of squarefree order. Conversely every
cyclic squarefree group occurs, by selecting its finite prime set. The quotient system, not
merely this list, is canonically the same for all selectors.

Corollary 6.4 (Common profinite completion). For every 𝐴 ⊆ N, the inclusion 𝐺 𝐴 ⊆ 𝑃
extends to a canonical isomorphism
                                   𝐺c𝐴  𝑃.

The induced topology on 𝐺 𝐴 is its full profinite topology.

Proof. 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.                                                                         □

   The 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

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6   Root certificates and finite quotients                                    REALIZATION CALCULUS


inverse-system meaning of a computable profinite presentation [4, Definition 3.3]. 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.


6.4 The information retained by a marked element

For the distinguished element 𝑢𝐴 = 𝑠 𝐴 (1), the coordinate criterion simplifies to

                                              𝑝 |𝐺 𝐴 𝑢𝐴      ⇐⇒    𝑝 ∉ 𝑆𝐴 .                  (6.1)

Consequently, for each 𝑛,

                                               𝑝 2𝑛 |𝐺𝐴 𝑢𝐴    ⇐⇒     𝑛 ∈ 𝐴,                  (6.2)
                                             𝑝 2𝑛+1 |𝐺𝐴 𝑢𝐴    ⇐⇒     𝑛 ∉ 𝐴.                  (6.3)

Exactly one root equation is solvable. This binary pattern is the selector written in intrinsic
group language.
  The finite quotient objects alone have not changed. What has been added is the coherent
image of a particular source element. It is therefore important not to paraphrase the
common-completion theorem as equality of every possible finite observation. Marked
source data is a stronger observation contract.




                                                             24
CHAPTER 7



Classification, morphisms, and seed
rank
The finite quotients are fixed, but the original groups are not. Divisibility classifies their isomorphism
types, and the coordinate structure determines every homomorphism.


7.1 The signature of an infinite-order element

The rational coordinate of a normal form detects torsion: 𝑠 𝐴 (𝑟) + 𝑑 has infinite order exactly
when 𝑟 ≠ 0.

Lemma 7.1 (Tail signature). If 𝑔 = 𝑠 𝐴 (𝑟) + 𝑑 has infinite order, then for all but finitely many
primes,
                               𝑝 |𝐺𝐴 𝑔 ⇐⇒ 𝑝 ∉ 𝑆𝐴 .
An explicit exception set consists of the primes dividing the nonzero numerator or denominator of 𝑟,
together with supp(𝑑).

Proof. Outside that set, the reduction of 𝑟 is defined and nonzero. The coordinate of 𝑔 is
therefore nonzero exactly at the selected primes. Apply Theorem 6.1 with multiplier 𝑝. □

   The 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.


7.2 Isomorphism means eventual agreement

Theorem 7.2 (Unmarked isomorphism classification). For all 𝐴, 𝐵 ⊆ N,

                                     𝐺𝐴  𝐺𝐵       ⇐⇒       𝐴 =∗ 𝐵.

When 𝐴 =∗ 𝐵, the two groups are equal as subgroups of the fixed product 𝑃.

Proof. Suppose 𝑓 : 𝐺 𝐴 → 𝐺 𝐵 is an isomorphism. The image of 𝑢𝐴 has infinite order. An
isomorphism preserves the existence of a 𝑝th root for every prime 𝑝. Combining the exact
signature (6.1) with Lemma 7.1 for 𝑓 (𝑢𝐴 ) gives 𝑆𝐴 =∗ 𝑆𝐵 . Since each set chooses exactly
one prime from each pair, the corresponding sets of bits differ finitely.


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7   Classification, morphisms, and seed rank                                      REALIZATION CALCULUS


  Conversely, suppose 𝐴 =∗ 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. □

    The normal forms need not be identical. Their coordinate change is

                                         (𝑟, 𝑑) ↦−→ 𝑟, 𝑑 + 𝑠 𝐴 (𝑟) − 𝑠 𝐵 (𝑟) .
                                                                            
                                                                                                 (7.1)

It fixes the represented ambient element. The correction is finite exactly because the
selectors differ only finitely.

Corollary 7.3 (Marked classification). An isomorphism (𝐺 𝐴 , 𝑢𝐴 )  (𝐺 𝐵 , 𝑢𝐵 ) exists if and only
if 𝐴 = 𝐵.

Proof. A marked isomorphism preserves the exact two-prime root test (6.2)–(6.3) at every
index. Therefore every bit agrees. The converse is the identity map.                   □

Example 7.4 (The same group with a different marked point). Let 𝐴 = ∅ and 𝐵 = {0}.
Only the first prime pair changes. Then 𝐺 𝐴 = 𝐺 𝐵 inside 𝑃, but 𝑢𝐴 − 𝑢𝐵 = 𝑒2 − 𝑒3 . The
distinguished element is not preserved. In particular, the statement 𝐺 𝐴  𝐺 𝐵 ⇐⇒ 𝐴 = 𝐵
would be false for unmarked groups.


7.3 All homomorphisms are diagonal

The product 𝑃 is also a commutative ring under coordinatewise multiplication. Multi-
plication by any 𝜆 ∈ 𝑃 preserves 𝐷, so it induces an additive endomorphism 𝑀 𝜆 of 𝑉.
Because division in 𝑉 is unique, this induced endomorphism is Q-linear.

Theorem 7.5 (Complete homomorphism description). Every homomorphism 𝑓 : 𝐺 𝐴 → 𝐺 𝐵
has the form
                           𝑓 (𝑔)𝑝 = 𝜆 𝑝 𝑔𝑝 (𝑝 prime)
for a unique 𝜆 ∈ 𝑃. Conversely, coordinate multiplication by 𝜆 defines such a homomorphism
exactly when
                                     𝑞(𝜆𝑢𝐴 ) ∈ Q 𝑞(𝑢𝐵 ).                              (7.2)
Equivalently, there is a rational 𝑟 for which 𝜆𝑢𝐴 − 𝑠 𝐵 (𝑟) has finite support.

Proof. For a homomorphism 𝑓 , the composite 𝐺 𝐴 → 𝐺 𝐵 → 𝐶 𝑝 kills 𝑝𝐺 𝐴 . The quotient
𝐺 𝐴 /𝑝𝐺 𝐴 is canonically 𝐶 𝑝 . Every endomorphism of this cyclic additive group is multipli-
cation by a unique scalar 𝜆 𝑝 . Hence the displayed coordinate rule holds. Uniqueness also
follows by evaluating 𝑓 (𝑒 𝑝 ).
   Conversely, multiplication by any 𝜆 is a group endomorphism of 𝑃 and preserves 𝐷.
Since its induced map on 𝑉 is rational-linear, it carries Q 𝑞(𝑢𝐴 ) into Q 𝑞(𝑢𝐵 ) exactly when
it carries the generating vector into that line. This is (7.2). Taking preimages proves that


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7   Classification, morphisms, and seed rank                                  REALIZATION CALCULUS


the restriction maps 𝐺 𝐴 into 𝐺 𝐵 . The equivalent finite-support condition follows from the
definition of the quotient.                                                                □

  No 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.

Theorem 7.6 (Torsion-image obstruction). If 𝐴 ≠∗ 𝐵, every homomorphism 𝐺 𝐴 → 𝐺 𝐵 has
image contained in 𝐷. Consequently neither group embeds in the other, and neither is a quotient of
the other.

Proof. A homomorphism preserves torsion and induces a map Q → Q on the quotients,
necessarily multiplication by a rational 𝑟. If 𝑟 = 0, its image lies in 𝐷.
   Assume 𝑟 ≠ 0. Then 𝑓 (𝑢𝐴 ) has the tail signature of 𝑆𝐵 . For every 𝑝 ∉ 𝑆𝐴 , the element
𝑢𝐴 has a 𝑝th root, and its image has a 𝑝th root as well. Lemma 7.1 implies that 𝑆𝐵 \ 𝑆𝐴
is finite. Each differing prime pair contributes one element to this difference, so 𝐴 =∗ 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  Q ≠ 0.                                    □

  A useful precise consequence of Theorem 7.5 is that, when 𝐴 ≠∗ 𝐵, the possible diagonal
sequences are exactly those for which 𝜆 𝑝 = 0 at all but finitely many primes of 𝑆𝐴 . Their
values on the complementary primes are unrestricted. These maps can be nonzero, and
their torsion images need not be finite.


7.4 The endomorphism ring and its units

Write 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.

Corollary 7.7 (Endomorphism ring). Under coordinate multiplication,

                End(𝐺 𝐴 )  𝐸𝐴 = {𝜆 ∈ 𝑃 : ∃𝑟 ∈ Q, 𝜆 𝑝 = 𝑟 𝑝 for almost every 𝑝 ∈ 𝑆𝐴 }.

The rational 𝑟 is unique. It is the scalar induced on 𝐺 𝐴 /𝐷, and the map 𝐸𝐴 → Q sending 𝜆 to 𝑟 is
a surjective ring homomorphism.

Proof. Apply Theorem 7.5 with 𝐵 = 𝐴. At unselected coordinates, both 𝜆𝑢𝐴 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.

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7   Classification, morphisms, and seed rank                                   REALIZATION CALCULUS


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.                                                        □

   The arbitrary coordinates outside 𝑆𝐴 show why the endomorphism ring may contain
infinitely specified operators. The seed-rank obstruction below remains true even when
all these operators are available.

Corollary 7.8 (Automorphisms). An endomorphism with diagonal 𝜆 ∈ 𝐸𝐴 is an automorphism
if and only if every coordinate 𝜆 𝑝 is nonzero. In that case its rational quotient scalar is nonzero,
and its inverse has diagonal 𝜆−1
                               𝑝 .


Proof. An 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 𝑆𝐴 is infinite. Outside finitely many selected primes, the inverse diagonal
is reduction of its reciprocal. Thus the inverse diagonal belongs to 𝐸𝐴 and defines an
inverse endomorphism.                                                                       □

7.5 Infinite seed rank before completion

Theorem 7.9 (Seed-rank separation). For every 𝐴,

                                               srankEnd(𝐺𝐴 ) (𝐺 𝐴 ) = ℵ0 .

No finite family topologically generates 𝐺 𝐴 , even after arbitrary endomorphisms are admitted. Yet
its completion has ordinary topological generating rank one.

Proof. Each element 𝑠 𝐴 (𝑟) + 𝑑 has only finitely many nonzero coordinates outside 𝑆𝐴 . A
finite family therefore omits infinitely many unselected prime coordinates. By Theorem 7.5,
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 ∈ 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 2.4.     □

   The 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.




                                                           28
CHAPTER 8



The exact cost of an effective realization
The selector does not change the completion, but it determines exactly which oracles can compute the
original group. The proof must distinguish a named element from an unmarked isomorphism type.


8.1 Oracles, copies, and degrees

An oracle for a set 𝑋 ⊆ N answers membership questions about 𝑋. The notation 𝐴 ≤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.
   The degree spectrum of a countable structure is the set of degrees of its copies. When
that spectrum has a least degree, it is called the degree of the isomorphism type. This
terminology is standard in computable structure theory [1, 2]. Here we prove an exact
oracle criterion that specifies which oracles compute at least one copy. We do not identify
this statement with an unproved claim that every such oracle degree is realized as the
exact degree of a particular copy.
   For a group on a countable coded domain, knowledge of its atomic diagram is equivalent,
at the level needed here, to knowledge of its group operation and equality. A computation
of integer multiples uses finitely many additions. A search for a root can therefore be
performed in any oracle-computable copy.


8.2 A presentation from the selector

Reduced 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.

Proposition 8.1 (Uniform construction relative to the selector). If 𝐴 ≤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 deg𝑇 (𝐴).

Proof. To 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 5.2 guarantees that every other correction coordinate is zero.
Formula (5.2) then gives the sum. Negation and equality are computable without querying
𝐴.

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   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 5.4.
For 𝑝 = 𝑝 2𝑛 , that value is zero exactly when 𝑛 ∈ 𝐴. Thus the operation computes every
selector bit.                                                                               □

  The 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.


8.3 Decoding a marked copy

Proposition 8.2 (Uniform marked decoding). Given an 𝑋-computable copy of 𝐺 𝐴 together
with the name of 𝑢𝐴 , one can uniformly compute 𝐴 in 𝑋.

Proof. For input 𝑛, search the domain simultaneously for solutions of

                                                 𝑝 2𝑛 𝑦 = 𝑢𝐴 ,        𝑝2𝑛+1 𝑧 = 𝑢𝐴 .

Each candidate test is decidable in 𝑋. Exactly one equation has a solution by (6.2)–(6.3).
Dovetailing the searches therefore terminates. A solution to the first equation means
𝑛 ∈ 𝐴, and a solution to the second means 𝑛 ∉ 𝐴.                                       □

   The 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.


8.4 Decoding an unmarked copy

Theorem 8.3 (Exact oracle criterion). For every 𝐴, 𝑋 ⊆ N,

                                𝐺 𝐴 has an 𝑋-computable copy                ⇐⇒         𝐴 ≤𝑇 𝑋.

In particular, the isomorphism type of 𝐺 𝐴 has least presentation degree deg𝑇 (𝐴).

Proof. The reverse implication is Proposition 8.1. 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 7.1 gives a finite exception set. Choose 𝑁 so that every prime in the pairs
indexed by 𝑛 ≥ 𝑁 avoids that set. For these 𝑛, exactly one of

                                                  𝑝2𝑛 𝑦 = ℎ,          𝑝2𝑛+1 𝑧 = ℎ

has a solution, and the first is solvable exactly when 𝑛 ∈ 𝐴.

                                                                 30
8   The exact cost of an effective realization                                       REALIZATION CALCULUS




                                 finite cocycle                       same type

             Selector                             Normal-form copy                 Marked copy
                A                                      ℚ×D                            (H, h)



                              Two complementary root searches recover every bit



                     Unmarked copies: a fixed element and finite exception data are needed.




Figure 8.1: The two effective directions. Selector access computes the finite cocycle and hence a
normal-form copy. A marked copy computes the selector by two complementary root searches.
The middle-to-right arrow records an abstract isomorphism, not an algorithm for finding one. The
unmarked converse uses a fixed infinite-order element and finite exception data.


  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 deg𝑇 (𝐴) by Proposition 8.1, while
every copy computes 𝐴 by the forward implication. Hence this degree is the least one. □

  There 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 8.2 gives the genuinely uniform version.
  The distinction also matches the algebra. If 𝐴 =∗ 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.


8.5 Every degree over one computable completion

Corollary 8.4 (Degree variation with fixed observations). Every 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 6.

Proof. Choose any set 𝐴 of the desired degree and apply Theorem 8.3. All the structural
and completion properties were proved in Chapters 4–6.                               □

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  For 𝐴 = ∅, the copy is computable. For 𝐴 = 𝐾, the group has no computable copy
and its least degree is the halting degree 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.


8.6 A sharper separation than prime-spectrum coding

For the initial groups 𝐷𝐵 , 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.
   It resides in the relationship between torsion coordinates and infinite-order elements.
All finite quotient groups remain fixed, but the root pattern of an admissible marked
element distinguishes the selectors. This is why the construction measures a realization
cost rather than merely relabeling the finite quotient data.




                                                 32
CHAPTER 9



What finite observations cannot
reconstruct
The obstruction is not merely computational. The observation system itself has forgotten which
dense realization was intended. No decoder using only that system can restore the missing choice.


9.1 The observation contract

Let ℱ (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          Ö
                                             𝐶𝑝
                                             𝑝|𝑚

with the same coordinate projections form a cofinal model for every ℱ (𝐺 𝐴 ). Proposition 4.5
also identifies the homomorphism sets into every finite target by a natural bĳection induced
by restriction to 𝐷.

Theorem 9.1 (No decoder from the unmarked finite-quotient system). There is no
isomorphism-invariant assignment 𝑅 on these observation systems such that

                            𝑅(ℱ (𝐺 𝐴 ))  𝐺 𝐴        for every 𝐴 ⊆ N.

This remains impossible without any computability restriction on 𝑅.

Proof. Take 𝐴 = ∅ and 𝐵 = N. Their observation systems are isomorphic by Theorem 6.3.
Their selectors differ infinitely, so their groups are not isomorphic by Theorem 7.2. An
isomorphism-invariant assignment has one output isomorphism type on that input
observation type. It cannot recover both nonisomorphic source types.                   □

  The 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.

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9   What finite observations cannot reconstruct                                REALIZATION CALCULUS



                                     Different original dense realizations


             G∅                              Gℕ                 GK                   GA




                                                  P = ∏Cp
                                                          p

                          The same unmarked finite quotients and transition maps


Figure 9.1: Different selectors, including computable and noncomputable ones, lead to the same
complete finite-quotient system and the same product completion. The arrows are completion
maps. They are not isomorphisms from the countable source groups to the uncountable product.


9.2 Element separation does not identify the source

Every 𝐺 𝐴 is residually finite. Its coordinate maps distinguish any two distinct elements.
They give an injective map
                                      𝐺 𝐴 ↩→ lim 𝐴 𝑁 = 𝑃.
                                             ←−−
                                                      𝑁

There is no conflict with Theorem 9.1. 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.

Proposition 9.2 (A proper map invisible after completion). The inclusion 𝐷 ↩→ 𝐺 𝐴 is not
an isomorphism, but it induces an isomorphism of profinite completions.

Proof. The quotient 𝐺 𝐴 /𝐷  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.                                                                    □

   This supplies a map-level failure: completion need not reflect isomorphisms. It is
stronger than merely exhibiting two abstract groups with isomorphic completions.


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9.3 Faithful on maps, but not a full reconstruction

There is a useful categorical distinction. A functor is faithful when different maps remain
different. This does not require different objects to remain nonisomorphic, and it does not
require every map between observed objects to lift.

Proposition 9.3 (Faithfulness and failure of fullness). Profinite 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.

Proof. Let 𝑓 , 𝑔 : 𝐻 → 𝐾 be distinct homomorphisms between residually finite groups.
Choose ℎ with 𝑓 (ℎ) ≠ 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 ≠∗ 𝐵. Under the canonical identifications of the comple-
tions with 𝑃, consider the identity map 𝑃 → 𝑃. If it were induced by a homomorphism
𝐺 𝐴 → 𝐺 𝐵 , that homomorphism would be the identity on every coordinate and hence
would include 𝐺 𝐴 into 𝐺 𝐵 . Theorem 7.6 excludes such an embedding. After adjoining
𝐷 to the source category, the proper inclusion of Proposition 9.2 proves failure to reflect
isomorphisms.                                                                              □

  The functorial extension of a homomorphism to profinite completions is standard [5,
Proposition 3.1.2]. The distinction between the two source categories is essential. Within
the paired-prime family alone, a supplied map can be recovered more strongly.

Proposition 9.4 (Reflection within the paired-prime family). For a homomorphism 𝑓 : 𝐺 𝐴 →
𝐺 𝐵 , the induced completion map b
                                 𝑓 is an isomorphism if and only if 𝑓 is an isomorphism.

                                                                       𝑓 is an isomorphism.
Proof. One implication follows from functoriality. Conversely, suppose b
By Theorem 7.5, its coordinate action is multiplication by a diagonal sequence 𝜆. Each
𝜆 𝑝 must be nonzero, so 𝑓 is injective. If 𝐴 ≠∗ 𝐵, Theorem 7.6 would force its image to
be torsion, contradicting injectivity on an infinite-order element. Hence 𝐴 =∗ 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 7.8 makes it an automorphism. □

  Thus 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.




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9   What finite observations cannot reconstruct                          REALIZATION CALCULUS


9.4 What a marked coherent point restores

The coherent image of 𝑢𝐴 in the finite quotients records, at prime 𝑝, whether its coordinate
is zero or one. Within the declared family, these zero patterns recover 𝑆𝐴 and therefore 𝐴.
   This observation should be read in two directions. A marked coherent point can restore
information discarded by the unmarked system. But specifying that infinite coherent point
can itself require a noncomputable selector. The finite quotient tables remain computable
while the chosen point does not.
  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.


9.5 Two different inverse-limit assertions

For the subgroup family 𝐷𝐵 , a complete coherent sequence of finite membership decisions
determines 𝐵 exactly. For the family 𝐺 𝐴 , the unmarked finite quotient system is indepen-
dent of 𝐴. These are different observations, even though both involve finite structures and
inverse limits.
   The first system remembers which coordinate factors were selected. The second
remembers the finite quotient objects but forgets how a distinguished infinite-order source
element sits in them. Treating these as the same notion of “all finite information” would
erase the very distinction that the construction proves.


    Reconstruction boundary. Compatibility identifies the complete ambient object.
    Reconstruction of a specified noncomplete source additionally requires its admissible
    image. A finite or effective realization must account for the information in that
    image.




                                                  36
CHAPTER 10



Escaping a fixed finite grammar
A relative diagonal argument can be carried out entirely inside one common-completion class.
Because finite selector changes leave the group unchanged, the diagonal must disagree infinitely
often.


10.1 Fix the semantics before diagonalizing

Let 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

                                          𝑑0 , 𝑑1 , 𝑑2 , . . .

and 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 ⊆ N such that 𝑅 𝑗  𝐺 𝐵 𝑗 . For other
𝑗, set 𝐵 𝑗 = ∅. 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.


10.2 Repeated diagonal coding

Fix a computable pairing bĳection ⟨𝑗, 𝑘⟩ : N2 → N. For a freely chosen set 𝐶 ⊆ N, define a
new selector 𝐴𝐶 by

                                  1𝐴𝐶 (2⟨𝑗, 𝑘⟩) = 1 − 1𝐵 𝑗 (2⟨𝑗, 𝑘⟩),                     (10.1)
                              1𝐴𝐶 (2⟨𝑗, 𝑘⟩ + 1) = 1𝐶 (𝑗).                                 (10.2)

Every 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.

Theorem 10.1 (Escape within one profinite-completion class). For every 𝐶 ⊆ N, the group
𝐺 𝐴𝐶 is not isomorphic to any output of 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.

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10    Escaping a fixed finite grammar                                           REALIZATION CALCULUS



                    Even block: defeat an output             Odd block: preserve a free bit



     j=0                    1 − B0(2⟨0, k⟩)                      C(0), C(0), C(0), …



     j=1                    1 − B1(2⟨1, k⟩)                      C(1), C(1), C(1), …



     j=2                    1 − B2(2⟨2, k⟩)                      C(2), C(2), C(2), …



     j=3                    1 − B3(2⟨3, k⟩)                      C(3), C(3), C(3), …



Figure 10.1: 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.


Proof. Fix a description index 𝑗. If 𝑅 𝑗 is outside the family, it cannot be isomorphic to
𝐺 𝐴𝐶 . Otherwise rule (10.1) makes 𝐴𝐶 disagree with 𝐵 𝑗 at every index 2⟨𝑗, 𝑘⟩. These are
infinitely many distinct indices, so 𝐴𝐶 ≠∗ 𝐵 𝑗 . Theorem 7.2 excludes the isomorphism.
  If 𝐶 ≠ 𝐶 ′, choose 𝑗 at which they differ. Rule (10.2) makes 𝐴𝐶 and 𝐴𝐶 ′ differ at infinitely
many odd indices. Theorems 7.2 and 7.6 then give nonisomorphism and nonembeddability.
The remaining properties hold for every paired-prime group.                                  □


10.3 The complexity consequence

The theorem implies
                                              𝐾 𝔊 (𝐺 𝐴𝐶 ) = +∞
for 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 𝐴𝐶
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.
  The same distinction applies to a language extended by a new constant. A constant
naming one previously absent group enlarges the language. If it is only a semantic constant,

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it need not give algorithms for that group’s operations. A new realization rule must be
assessed under its actual semantic and effective contract.


10.4 Monotonicity under grammar extension

Proposition 10.2 (Admissible expansion). Suppose every valid description in 𝔊 translates to a
valid description in ℌ with the same semantic group. Then every 𝔊-realizable group is ℌ-realizable.
If every translated description has length at most 𝑓 (|𝑑|), then

                                       𝐾 ℌ (𝐺) ≤ 𝑓 (𝐾 𝔊 (𝐺))

whenever 𝐾 𝔊 (𝐺) is finite.

Proof. Translate a shortest valid 𝔊 description of 𝐺. Its semantic output remains 𝐺, and its
length has the stated bound.                                                              □

  An obstruction to a small grammar can therefore disappear after an admissible expansion.
The power-of-two selector from Chapter 3 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.


10.5 Uniform families and individual instances

The expression 𝐴 ↦→ 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.
  These are exact distinctions between a construction schema, its parameters, and an
individual realized object. They are the appropriate replacement for an absolute label
such as “non-Komorebi group.”




                                                39
CHAPTER 11



A worked arithmetic atlas
The examples below retain the infinite definition while calculating only the finite coordinates that a
particular operation needs. They make the cocycle and the root certificates visible.


11.1 The empty selector

Take 𝐴 = ∅. The selected primes are

                              𝑆𝐴 = {2, 5, 11, 17, 23, 31, 41, 47, . . .},

namely the even-indexed primes. The first coordinates of three rational lifts are

                                    𝑝           2    3   5    7     11    13
                               𝑢𝐴 = 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

Each 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 · 3 = 1 in F5 .
  Doubling the half-lift agrees with 𝑢𝐴 at every prime except 2. Therefore

                                           2(1/2, 0) = (1, 𝑒2 ).

The element 𝑢𝐴 has no half in the group: there is no 𝑦 with 2𝑦 = 𝑢𝐴 , 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 𝑢𝐴 is zero. In this case

                                           3(1/3, 0) = (1, 0).

The three roots are
                              (1/3, 0),        (1/3, 𝑒3 ),        (1/3, 2𝑒3 ).
Their differences form 𝐺 𝐴 [3] = 𝐶3 , as predicted by Proposition 6.2.


11.2 A successful composite-root calculation

Let
                                        𝑔 = 𝑠 𝐴 (1) + 𝑒2 = (1, 𝑒2 ).

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11   A worked arithmetic atlas                                             REALIZATION CALCULUS


Its 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 𝑢𝐴 because the denominator is invertible, except possibly at 3,
which is unselected. At 2, both sides are zero. Hence

                                      6(1/6, 0) = (1, 𝑒2 ) = 𝑔.

Every other root is obtained by adding an arbitrary element of 𝐶2 ⊕ 𝐶3 . There are exactly
six roots. This calculation exhibits a positive certificate without searching an unbounded
domain.
  Now contrast 𝑢𝐴 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.


11.3 A torsion correction that cannot be discarded

Take the purely torsion target 𝑒3 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:
                                    2(0, 2𝑒3 ) = (0, 𝑒3 ).
In F3 , 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.


11.4 Finite quotient representatives

For any target 𝑔 and positive multiplier 𝑚, define the finite vector
                                                    Õ
                                         𝑑𝑚 (𝑔) =         𝑔𝑝 𝑒 𝑝 .
                                                    𝑝|𝑚

Then 𝑔 − 𝑑𝑚 (𝑔) has zero coordinates at every prime dividing 𝑚. The root theorem provides
a witness that it belongs to 𝑚𝐺 𝐴 . Thus 𝑑𝑚 (𝑔) is an exact finite representative of the coset
of 𝑔.
     For 𝑚 = 12, only coordinates 2 and 3 matter. The six representatives are

                                 0, 𝑒2 , 𝑒3 , 2𝑒3 , 𝑒2 + 𝑒3 , 𝑒2 + 2𝑒3 .

The vector 𝑒2 + 𝑒3 generates this quotient, which is 𝐶6 , not 𝐶12 . Repeated prime powers in
the multiplier do not create additional coordinates.



                                                   41
11              A worked arithmetic atlas                                       REALIZATION CALCULUS



                     30

                     25
 |GA/mGA| = rad(m)




                     20

                     15

                     10

                      5

                      0
                          1            5    10             15              20    25            30
                                                        multiplier m

Figure 11.1: The exact order of 𝐺 𝐴 /𝑚𝐺 𝐴 for 1 ≤ 𝑚 ≤ 30 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.


11.5 Changing one selector bit

Let 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:

                                            𝑠 𝐴 (1/2) − 𝑠 𝐵 (1/2) = 𝑒3 .

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

                                             (1/2, 0)𝐴 = (1/2, 𝑒3 )𝐵 .

Doubling the right-hand side in the 𝐵 group gives

                                                      (1, 2𝑒3 )𝐵 ,

which is the transported form of (1, 𝑒2 )𝐴 . The correction in (7.1) is required to compare
the two group laws. Merely reusing the same pair (𝑟, 𝑑) under a new selector changes the
represented element.


11.6 A finite tail exception

Consider an infinite-order element

                                                 𝑥 = 𝑠 𝐴 (5/6) + 2𝑒7 .

An 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 𝑢𝐴 in every later prime pair.

                                                          42
11   A worked arithmetic atlas                                         REALIZATION CALCULUS


  This example illustrates the nonuniform step in Theorem 8.3. In known normal-form
coordinates the exception set is computable. In an arbitrary unmarked copy, a reduction
may use a finite exception bound without uniformly discovering it from the copy.


11.7 Reproducing the examples

From the package root, the following uses the supplied standard-library implementation:

 from fractions import Fraction
 from src.dense_lift_groups import DenseLiftGroup, Element

 G = DenseLiftGroup(lambda n: False)
 u = Element.make(1)
 half = Element.make(Fraction(1, 2))

 assert G.scale(2, half) == Element.make(1, {2: 1})
 assert G.root(u, 2).obstructing_prime == 2
 third = G.root(u, 3).root
 assert third is not None and G.scale(3, third) == u

 x = Element.make(1, {2: 1})
 sixth = G.root(x, 6).root
 assert sixth is not None and G.scale(6, sixth) == x


Only computable selector examples are executed. A callable supplied as an oracle is
assumed to return the correct selector bit. The implementation does not pretend to evaluate
the halting set or an arbitrary noncomputable parameter.




                                            43
CHAPTER 12



Boundaries as part of the calculus
The constructions identify specific obligations for realization: compatibility, admissibility, generation,
and effective evaluation. A failure at one level need not be a failure at another.


12.1 The ambient object and its admissible realizations

The chain
                                             𝐷 ⊆ 𝐺𝐴 ⊆ 𝑃
contains three different mathematical roles. The direct sum provides the finite-support
corrections. The product supplies every compatible prime-coordinate tuple. The interme-
diate 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 construction does not assert that every Komorebi Space model is a prime-coordinate
product. It isolates a theorem-level phenomenon that any broad realization language must
be able to express: the space of compatible observations can be larger than the intended
source image.


12.2 Three certificates and one global obstruction

The finite-seed argument supplies a projection certificate. An omitted prime coordinate
vanishes on every admissible endomorphism orbit but not on the target group. This is a
structural failure of generation under the fixed operator action.
  The root algorithm supplies a coordinate certificate of impossibility or a finite correction
witnessing success. It is a fully local algebraic realization problem. The successful and
unsuccessful branches are both explicit.
  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.
  The grammar theorem is different. It excludes every finite description in a fixed language
by an infinite diagonal comparison. It need not supply a terminating rejection procedure

                                                   44
12   Boundaries as part of the calculus                                            REALIZATION CALCULUS


for individual candidate descriptions. Conflating this global semantic obstruction with
the finite root certificates would overstate its algorithmic content.


12.3 A realization profile rather than a single score

One scalar cannot capture all the boundaries exhibited here. The following profile retains
the distinctions:
        seed rank, grammar complexity, least presentation degree, observation type .
                                                                                                 

For every 𝐺 𝐴 , the seed rank is ℵ0 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.
   The profile is not proposed as a complete classification invariant for arbitrary mathemat-
ical objects. It is a disciplined way to avoid substituting one question for another. A theory
may have exact coordinates without a finite generator, a finite formula without computable
equality, or an effectively presented inverse-limit completion without a computable copy
of the intended dense realization.


12.4 What a machine-facing contract should retain

A reusable construction record should specify its input type, its permitted parameters,
its output object, and the sense in which the output is determined. It should also state
whether it returns a concrete representation, an abstract isomorphism type, or an existence
proof. When completion is used, the intended image and the completed ambient object
should be separate fields.
     For the present family, a compact exact record is:

Field                           Content
Ambient system                  𝐷 ⊆ 𝑃 and the quotient 𝑞 : 𝑃 → 𝑃/𝐷.
Parameter                       The selector 𝐴, with its declared oracle or finite semantic description.
Admissible image                𝑞 −1 (Q 𝑞(𝑢𝐴 )).
Concrete names                  Reduced rationals paired with normalized finite-support vectors.
Operation                       The finite cocycle law (5.2).
Positive certificate            A root and its exact multiplication check.
Negative certificate            A prime coordinate obstructing the required root.
Global invariants               Selector modulo finite changes and least presentation degree.
Observation bound-              Unmarked finite quotients forget the admissible source image.
ary

Such a record does not solve arbitrary group-theoretic problems. It makes the semantics of
this construction inspectable and prevents a machine from promoting a finite observation
theorem into a source reconstruction theorem.

                                                      45
12   Boundaries as part of the calculus                                  REALIZATION CALCULUS


12.5 Further questions with exact starting points

The general correspondence of Chapter 4 extends the ambient construction to higher-
dimensional rational subspaces 𝑊 ≤ 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.
   A third question concerns enriched observations. A coherent marked point restores the
selector in the declared rank-one family. A useful general theorem would characterize
which additional marks or admissibility predicates suffice for reconstruction in larger
classes. Such a result must state whether its reconstruction is semantic, effective, or
functorial.
   Finally, the proof development is suitable for formalization. The arithmetic kernel,
quotient lemmas, tail-signature argument, and computability reduction have different
proof obligations. Appendix B separates these tasks. A successful Lean build of the
algebraic layer would not, by itself, verify the oracle theorem unless its effective semantics
were also formalized.


12.6 Conclusion

The 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.
  Realization is therefore not exhausted by compatibility. It includes the selection of
admissible objects, the transport of their meaning, and an account of the information
needed to evaluate them. The groups in this book turn that distinction into explicit
constructions, exact classifications, and reproducible finite certificates.




                                             46
APPENDIX A



Exact computation and reproduction
The computational companion evaluates the finite arithmetic of the construction. It does not stand
in for the infinite proofs. This appendix states its contracts, its checks, and its limits.


A.1 A finite representation of each named element

An element is stored as a reduced rational and a normalized finite-support tuple of
prime residues. The tuple is ordered by prime label, has no repeated labels, and omits
zero residues. Use the constructor Element.make to obtain this canonical form. Direct
construction enforces the same normalization. Rational inputs must be integers or
Fraction values. Residues, prime labels, and multipliers must be integers. Floating-point
values are rejected rather than rounded or silently truncated.
   The selector is supplied as a callable taking a nonnegative integer 𝑛 to the Boolean value
of 1𝐴 (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.


Interface                            Contract
Element.make(r, d)                   Normalize a rational and finite torsion data.
section_value(r, p)                  Evaluate one coordinate of the specified section.
cocycle(r, t)                        Return the exact finite correction for rational addition.
add(x, y)                            Apply the cocycle group law.
scale(m, x)                          Integer multiplication by repeated doubling.
root(x, m)                           For 𝑚 ≥ 1, return a root or an obstructing prime.
finite_change_transport              Change normal forms when a complete finite difference
                                     set is supplied.

   The finite-change transport routine has a global precondition: the selectors differ only
at the supplied indices. No finite examination of arbitrary oracle callables can verify that
precondition. The executable tests use selectors constructed to satisfy it exactly.




                                                47
A   Exact computation and reproduction                                        REALIZATION CALCULUS


A.2 Root construction as an algorithm

The proof of Theorem 6.1 gives the following procedure.

1. Factor the positive integer 𝑚 and inspect the target coordinates at its prime divisors. If
   any is nonzero, return that prime as the obstruction.
2. Divide the rational coordinate by 𝑚 to obtain a candidate section lift.
3. 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.
4. 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.

The 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 | 𝑚 and a nonzero target coordinate.
   Nonpositive or noninteger root multipliers are rejected. The routine does not enumerate
all elements of the group. Its finite support bound comes from the theorem. This
distinguishes a certified construction from an unbounded search that might fail to
terminate when no root exists.


A.3 What the tests cover

The core suite checks cocycle symmetry, support, and coordinate formulas over a specified
finite rational set. It checks the cocycle identity on all triples from that set for eight periodic
selector masks. It also tests group identities on reproducibly generated exact elements, root
witnesses and obstructions, quotient representatives, marked bit decoding, tail signatures
outside explicit exceptions, and finite-change transport.
   The supplemental suite checks the monograph’s additional calculations. It enumerates
entire finite root fibers and verifies their cardinalities. It checks explicit diagonal endo-
morphisms against coordinate multiplication and addition, and reproduces the worked
sixth-root, torsion-root, and coordinate-change examples.
  The verification run passed 114,738 core assertions, 53,284 supplemental assertions, and
21,047 release-regression assertions, for a total of 189,069 exact finite assertions. Seven
named mathematical negative controls reject the false simplifications below. The release
suite additionally checks malformed-input rejection, normal-form normalization, and
diagonal automorphism inversion. Run reports record the executed source hashes and
parameters.




                                                48
A   Exact computation and reproduction                                   REALIZATION CALCULUS



False simplification rejected            Reason for rejection
Omit the cocycle                    Twice the half-lift can differ from 𝑢𝐴 at prime 2.
Every rational division lifts       A nonzero divisor-prime coordinate obstructs a root.
Reuse coordinates after a selector  The same pair may represent a different ambient
change                              element.
Discard torsion correction in a rootA pure torsion target may need a nonzero finite
                                    correction.
Use 𝐶 𝑚 for every integer quotient Prime powers collapse to 𝐶rad(𝑚) .
Allow a zero root multiplier        The root contract requires a positive integer.
Treat rational diagonal representa- Exceptional denominator coordinates can destroy
tives as a ring embedding           multiplicativity of a chosen section.

   These checks exercise intended failure mechanisms. They are not a random test of
historical originality or an attempt to compute a halting oracle. The universal results about
all selectors, isomorphism classes, and Turing degrees are supported by the printed proofs.


A.4 Reproduction commands

Python 3.10 or later is sufficient for the exact arithmetic. From the package root, run

python src/dense_lift_groups.py
python src/exact_checks.py --output verification/local_core.json
python src/supplemental_checks.py --output verification/local_extra.json
python src/release_checks.py --output verification/local_release.json


The tests use only the standard library. Matplotlib is needed only to regenerate the vector
figures:

python src/make_figures.py


Rebuild the monograph with python build.py. The README lists the required TeX
packages. Once dependencies are installed, the build uses no network access. The package
retains the editable manuscript, figures, tests, theorem map, and original research archives.


A.5 Verification boundary

No Lean compilation was performed. Appendix B is a formalization plan, not a certificate.
The infinite classification, noncomputability, and grammar-escape results rest on the
printed proofs. Finite arithmetic tests and typesetting checks support the implementation
and publication, not those universal conclusions. The verification records distinguish
these forms of evidence.



                                               49
APPENDIX B



Theorem map and formalization
interfaces
The logical dependencies are explicit enough to support a modular formalization. The interfaces
below record what each stage must actually prove and what it may not silently assume.


B.1 The proof dependencies




             Ambient quotient                                   Pure preimages




          Normal forms / cocycle                                 Root criterion




       Tail signature / classification                         All finite quotients




       Exact oracle cost                 Grammar escape                       No decoder




Figure B.1: Principal dependencies of the proof development. The finite arithmetic checks are not
nodes from which the infinite theorems are inferred. Effective results require their own oracle and
presentation semantics after the algebraic construction has been established.

  The early direct-sum family can be formalized independently of the common-completion
construction. Its key ingredients are coordinate projections, prime-order torsion, and the
Chinese remainder theorem. The main family then needs a quotient model for 𝑃/𝐷, a
proof of unique divisibility, and a rational-line preimage.

                                                50
B    Theorem map and formalization interfaces                                      REALIZATION CALCULUS


  A concrete normal-form model and an ambient preimage model should both be retained.
The former supports executable arithmetic. The latter makes the completion and purity
arguments transparent. Their equivalence is Theorem 5.1 together with the cocycle law.


B.2 Statement registry


    Statement            Conclusion                                     Key dependency
    1.8                  An inaccessible packet obstructs generation.   Typed block-support
                                                                        condition.
    2.1                  Tor(𝑃) = 𝐷, and 𝐷 is dense.                    Distinct prime coordinates.
    2.2                  All endomorphisms of 𝐷 are diagonal.           Prime annihilators.
    2.3                  𝐷 has infinite endomorphism-seed rank.         Finite support.
    2.4                  𝑃 is topologically cyclic.                     Chinese remainder theorem.
    2.5                  𝐷
                         b = 𝑃.                                         Cofinal finite coordinate
                                                                        kernels.
    2.6                  𝑃/𝐷 is a rational vector space.                Unique integer division
                                                                        modulo 𝐷.
    3.1                  Sub(𝐷)  𝒫(N).                                 Coordinate isolation by CRT.
    3.3                  Computable copies of 𝐷𝐵 correspond to          Prime-order witness searches.
                         enumerable selectors.
    3.5                  Unary selector groups correspond to            Finite-state orbit.
                         ultimately periodic sets.
    4.2                  Pure intermediate groups are rational          Finite adjustment at divisor
                         preimages.                                     primes.
    4.4                  All preimages 𝐻𝑊 have completion 𝑃.            Purity and finite-target
                                                                        exponent.
    4.5                  Restriction preserves all maps to finite       Finite coordinate factorization.
                         targets.
    5.1                  𝐺 𝐴 has unique (𝑟, 𝑑) names.                   Nonzero rational line.
    5.2                  Each cocycle value has controlled finite       Reduction outside
                         support.                                       denominator primes.
    5.3                  The normal forms form the actual group 𝐺 𝐴 .   Cocycle identity.
    5.4                  A cocycle value reads one selector             The chosen section at one
                         coordinate.                                    prime.
    6.1                  A root or a prime obstruction is               Finite residual correction.
                         constructible.
    6.2                  A nonempty root fiber has rad(𝑚) elements.     The prime-coordinate kernel.
    6.3                  𝐺 𝐴 /𝑚𝐺 𝐴 = 𝐶rad(𝑚) .                          Root kernel and finite
                                                                        surjectivity.
    7.1                  Every nontorsion element has the same tail     Finite numerator,
                         signature.                                     denominator, and support
                                                                        exceptions.
    7.2                  𝐺 𝐴  𝐺 𝐵 iff 𝐴 =∗ 𝐵.                          Intrinsic divisibility and finite
                                                                        transport.
    7.5                  Every homomorphism is a diagonal               The quotients 𝐺 𝐴 /𝑝𝐺 𝐴 .
                         multiplier.

                                                     51
B    Theorem map and formalization interfaces                                     REALIZATION CALCULUS



    Statement            Conclusion                                     Key dependency
    7.6                  Different eventual classes admit only          One selected prime per pair.
                         torsion-image maps.
    7.7                  Exact endomorphism-ring description.           Diagonal action on the
                                                                        rational line.
    7.9                  Every 𝐺 𝐴 has infinite seed rank.              Infinitely many omitted
                                                                        unselected coordinates.
    8.3                  An oracle computes a copy iff it computes 𝐴.   Cocycle evaluation and
                                                                        complementary root searches.
    9.1                  No recovery from the unmarked quotient         Two nonisomorphic
                         system.                                        common-completion groups.
    9.3                  Completion is faithful but not full on the     Finite separation and the
                         paired family.                                 torsion-image obstruction.
    9.4                  Completion reflects isomorphisms between       Diagonal units and the exact
                         paired-prime groups.                           source category.
    10.1                 Continuum many groups escape a fixed           Repeated diagonal
                         countable grammar.                             disagreement.



B.3 A staged formalization plan

B.3.1 The algebraic ambient layer

Represent prime-indexed factors using finite additive groups. Construct the direct sum,
the direct product, the canonical inclusion, and the quotient. Prove that a product vector
is torsion exactly when it has finite support. The proof that multiplication by a positive
integer is bĳective on the quotient must retain the finite exceptional set of prime divisors.
  Only after unique divisibility is proved should the rational module structure be installed.
A declaration that merely assumes a rational vector space equivalent to the quotient would
bypass the first structural theorem.

B.3.2 The section and cocycle layer

Define reduced rational section coordinates with an explicit branch for denominator
primes. Establish the finite support bound before treating the cocycle as an element of the
direct sum. Prove the normalized cocycle identity and the equivalence between normal
forms and the rational-line preimage.
   A finite-vector implementation alone does not prove that its values coincide with the
intended infinite-coordinate section. The coordinate identity is the essential bridge.

B.3.3 Root, quotient, and completion layer

Formalize both directions of the integer-root criterion. The constructive direction should
expose the finite correction, so the certificate theorem can be reused. The quotient theorem
requires surjectivity of the coordinate map, not only a kernel calculation.

                                                     52
B   Theorem map and formalization interfaces                             REALIZATION CALCULUS


  For profinite completion, prove cofinality among all finite-index normal subgroups
using the finite-target exponent. A proof about a selected inverse system is not yet a proof
that it is the full profinite completion.

B.3.4 Classification and morphism layer

Define the tail signature with a finite exceptional prime set. The isomorphism theorem
must return eventual equality, not literal equality. The converse should include the actual
finite-change normal-form map.
   The homomorphism theorem should derive diagonal action from 𝐺 𝐴 /𝑝𝐺 𝐴  𝐶 𝑝
rather than assume diagonal operators at the start. This distinction is what makes the
full-endomorphism seed obstruction stronger than a restricted-grammar example. The
completion interfaces must also distinguish the subcategory consisting only of the groups
𝐺 𝐴 from the larger category that includes 𝐷. Reflection of isomorphisms holds in the
former and fails in the latter.

B.3.5 Effective layer

The oracle theorem requires a formal notion of a computable group copy, enumeration
of candidate roots, and dovetailing of the two searches. The unmarked implication must
explicitly quantify over the chosen name of a nontorsion element and a finite exception
bound. It must not assert a uniform procedure for obtaining those constants.
  Algebraic formalization does not automatically verify this effective semantics. Con-
versely, a program that computes normal-form addition for several masks does not establish
the universal oracle criterion.


B.4 Negative statement tests

A future formal development should reject the following strengthened statements under
the current hypotheses: finite endomorphism-seed generation of 𝐷, universal root existence
in 𝐺 𝐴 , replacement of 𝐶rad(𝑚) by 𝐶 𝑚 , literal unmarked selector recovery, splitting of the
rational extension, and recovery of the original group from the unmarked finite-quotient
system alone.
  These tests should fail for the intended mathematical reason. An unrelated missing
import is not a successful negative control. None of these proposed formal controls is
represented as already executed in Lean.




                                               53
APPENDIX C



Glossary
Admissible realization
The particular object or image allowed by a construction contract, as distinct from every
compatible tuple in an ambient completion.

Atomic diagram
The effective record of the basic relations and operations of a countable named structure.
It is the standard object used to measure the degree of a copy.

Cocycle
Here, a symmetric map 𝑐 : Q × Q → 𝐷 satisfying the identity that makes the twisted
addition on Q × 𝐷 associative.

Cofinal quotient system
A family of finite quotients through which every finite quotient factors. Its inverse limit
computes the full profinite completion.

Computable copy
A countable realization with effective names, decidable equality, and computable opera-
tions. An oracle version permits the stated oracle.

Computable profinite presentation
An effective inverse system of finite groups and surjective transition maps whose limit
is the given profinite group. This is not a computable enumeration of all elements of an
uncountable product.

Computably enumerable set
A set whose members can be listed by a procedure. Nonmembership need not be decidable.

Completion
An ambient object obtained by admitting compatible limits in a specified topology. In this
book the relevant completion is profinite.

Dense subgroup
A subgroup meeting every nonempty basic open set. Density does not imply equality
with the ambient group.




                                            54
Glossary                                                                REALIZATION CALCULUS


Divisible abelian group
An abelian group in which every equation 𝑚 𝑦 = 𝑥 with 𝑚 ≥ 1 is solvable. A divisible
quotient need not split off as a subgroup.

Endomorphism-seed rank
The minimum number of elements generating a group as a module over its full endomor-
phism ring.

Eventual equality
Equality modulo finitely many coordinates, written 𝐴 =∗ 𝐵. This is the unmarked selector
invariant of 𝐺 𝐴 .

Finite description
A valid finite code in a specified language with declared semantics. It need not provide
computable operations.

Finite support
A vector has only finitely many nonzero coordinates. Such vectors form 𝐷 inside 𝑃.

Fully invariant subgroup
A subgroup preserved by every endomorphism of its ambient group. Every subgroup of
the squarefree direct sum has this property.

Grammar-relative nonrealizability
Failure to have an admissible description in one fixed grammar. It is not absolute
mathematical nonexistence or indescribability.

Marked group
A group together with named elements that isomorphisms must preserve. The mark 𝑢𝐴
retains the literal selector.

Normal form
The unique pair (𝑟, 𝑑) representing 𝑠 𝐴 (𝑟) + 𝑑. Equality of these names is simpler than the
selector-dependent addition law.

Oracle
An external set-membership resource available to a computation. Its information is
charged to the effective realization contract.

Prime spectrum
The prime orders of nonzero torsion elements. It classifies the groups 𝐷𝐵 but is constant
across all 𝐺 𝐴 .




                                            55
Glossary                                                                 REALIZATION CALCULUS


Profinite completion
The inverse limit of all finite quotient groups, denoted 𝐺.
                                                         b It is not the Pontryagin dual.

Procyclic group
A topological group with a dense cyclic subgroup. The product 𝑃 is procyclic.

Pure subgroup
A subgroup 𝐻 ≤ 𝑀 for which 𝐻 ∩ 𝑚𝑀 = 𝑚𝐻 for all positive integers 𝑚.

Rational line
A one-dimensional subspace over Q. The preimage of Q 𝑞(𝑢𝐴 ) is 𝐺 𝐴 .

Reduced group
A group with no nonzero divisible subgroup. This property does not exclude divisible
quotients.

Residual finiteness
Every nonzero element is detected by some finite quotient. It does not imply source
reconstruction from the unmarked quotient system.

Root certificate
Either an exact element 𝑦 with 𝑚 𝑦 = 𝑔, or a prime divisor of 𝑚 at which the target has a
nonzero coordinate.

Selector
A subset 𝐴 ⊆ N whose bits choose one prime from each consecutive pair. It determines
the admissible rational line.

Squarefree radical
The integer rad(𝑚) formed by retaining each prime divisor of 𝑚 once.

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.

Torsion
Elements of finite order. In the product and in every 𝐺 𝐴 , the torsion subgroup is exactly 𝐷.

Turing degree
A measure of relative computability under mutual oracle computation. The least presenta-
tion degree of 𝐺 𝐴 is the degree of 𝐴.

Unmarked observations
Finite quotient objects and transition maps without a named coherent source element or
the original admissible image.




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References
[1] Antonio Montalbán. Computable Structure Theory: Within the Arithmetic. Perspectives in
    Logic, Cambridge University Press, 2021. For the cited numbering, see the author’s
    draft of June 25, 2019, Example III.39 and the following paragraph, p. 50. https:
    //math.berkeley.edu/~antonio/CSTpart1.pdf.

[2] Wesley Calvert, Valentina Harizanov, and Alexandra Shlapentokh. Turing degrees of
    isomorphism types of algebraic objects. Journal of the London Mathematical Society 75(2)
    (2007), 273–286. doi:10.1112/jlms/jdl012. Preprint: arXiv:math/0507128.

[3] László Fuchs. Abelian Groups. Springer Monographs in Mathematics, Springer, 2015.
    doi:10.1007/978-3-319-19422-6. General background for the terminology of pure
    subgroups and mixed abelian groups. The arguments used here are proved in the text.

[4] Martino Lupini, Alexander Melnikov, and André Nies. Computable topological abelian
    groups. arXiv:2105.12897v2, 2021. Definition 3.3 supplies the computable inverse-system
    convention. Section 3.2 describes symmetric cocycles and abelian extensions.

[5] Gareth Wilkes. Part III Profinite Groups. Lecture notes, Lent Term 2020. Sections 1.2 and
    3.1 discuss profinite completion and its functoriality. https://www.dpmms.cam.ac.
    uk/~grw46/LectureNotes.pdf.




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