Unit 5 — Writing Auditable Mathematical Exposition

Turning a valid argument into a clear, bounded, and checkable reading path

A practice unit on specifying readers and scope, arranging definitions–theorems–proofs–examples, maintaining notation, linking claims to locators, and revising exposition through a dependency audit.

1 Learning outcomes

This unit has the stable identifier O017-U05. After completing it, you will be able to:

specify the intended readers, prerequisites, purpose, and boundaries of an exposition before choosing its level of detail;
arrange definitions, theorems, proofs, examples, and counterexamples as components with distinct functions;
use notation whose types are specified, which is introduced before use, and which remains consistent throughout the document;
give important claims local locators and use external citations only for work that the cited source actually does;
add local signposts so that readers know the purpose, why a step appears, and where the argument will return;
revise a draft through an audit of claim dependencies without concealing assumptions, limits, or uncertainty;
distinguish proved statements from illustrations, counterexamples, heuristics, and historical commentary; and
turn written exposition into seminar notes that can be followed, questioned, and corrected honestly.

Prerequisites, continuity, and limits of focus

This unit assumes that you can already read a network of claims (Unit 1), that sources and their versions have been selected (Unit 2), that proof gaps have been reconstructed (Unit 3), and that locators and source decisions are available (Unit 4). Unit 5 does not repeat that work. Its input is an audited argument; its output is an exposition that lets other readers check that argument without guessing its structure or scope.

Readers of the main case are assumed to know positive integers, divisibility, contradiction, and the well-ordering principle for natural numbers. Neither analytic number theory nor primality-testing algorithms are required.

2 Exposition is an interface to an argument

A correct proof is not necessarily good exposition. A manuscript may contain every valid step yet remain difficult to audit because symbols appear before they are defined, a paragraph’s purpose is unstated, examples are presented as though they prove a theorem, or a citation is placed at the end of a paragraph without making clear which claim it supports.

Auditable exposition gives readers three paths at once:

a mathematical path: from assumptions to conclusion;
a rhetorical path: from the question to the reason each part is needed;
a checking path: from a claim to the definition, proof, example, or source that supports it.

These three paths are related, but they must not be conflated. The sentence “see the following example” helps the rhetorical path; it is not a mathematical justification. A citation provides a path for tracing and attribution; it does not automatically replace checking assumptions. A theorem number helps with checking; it does not make the theorem true.

O017 original bridge

Unit 3 asks, “How do we fill in a missing step?” Unit 4 asks, “Which version, location, and basis for a decision must be preserved?” Unit 5 asks, “How do we present an already audited result so that particular readers can follow, find, and criticize it?” This change of question prevents the editorial audit in this unit from becoming a repetition of proof reconstruction or source notebook design.

3 Begin with a reader and scope contract

Before writing the first paragraph, complete the following short contract.

Field	Question to answer	Main-case contract
Readers	What knowledge can genuinely be assumed?	divisibility, contradiction, well-ordering
Purpose	What should readers be able to do afterward?	explain why there are infinitely many primes and reject one mistaken interpretation
Core claim	What exact statement will be proved?	there are infinitely many prime numbers
Boundaries	Which adjacent claims are not promised?	the distribution of primes, uniqueness of factorization, and primality algorithms
Form	What use is the manuscript being prepared for?	independent reading and a 10–12-minute seminar
Admitted results	Which results may be used without proving them again?	only the basic facts explicitly stated below

This contract is not a ceremonial introduction. It determines the detail needed. For readers who know divisibility but not factorization, the sentence “take a prime divisor” requires a lemma. For advanced number-theory readers, the same lemma can be given a locator and summarized. Because the readers of the main case have not been given the prime factorization theorem, this unit proves the necessary lemma by elementary means.

3.1 Four scope tests

Every promise made by an exposition must pass four tests.

Subject test: are the objects and their domain clear?
Strength test: are the words “there exists,” “unique,” “for all,” and “only if” used precisely?
Prerequisite test: are all admitted results available to the intended readers?
Stopping test: does the author know which adjacent claims will not be pursued?

Failing the stopping test often produces a manuscript that is longer but no more auditable. Explaining the infinitude of primes does not oblige an author to explain how densely primes are distributed or how a machine finds large primes.

4 Claim architecture and locators

In this unit, every main component has a stable locator. A locator is an address; a status states its epistemic role. Both need to be explicit.

Claim ID	Status	Local locator	Role
`DEF-01`	definition	definition	establishes the meaning of “prime”
`LEM-01`	proved lemma	lemma	guarantees a prime divisor for an integer $>1$
`THM-01`	proved theorem	theorem	establishes the infinitude of primes
`EXM-01`	illustration	example	shows the mechanism using three primes
`CTR-01`	counterexample	counterexample	refutes the stronger claim that the product plus one is always prime

The list deliberately does not call the example a proof of THM-01. Readers can find the example and check its calculation, but the universal support for THM-01 remains in its proof.

4.1 Citations and locators are not interchangeable decorations

Use a local locator when the support is in the document being read. Use an external citation when an identity, statement, data, or idea comes from another work. If a source has several versions, identify the version actually checked and give a sufficiently precise location. General principles about metadata and versions of citable objects are adapted in a bounded way from The Turing Way (The Turing Way Community 2025, 2026). Those citations support these documentation principles; they are not used as a proof of the theorem about prime numbers in this unit.

Before keeping a citation, complete the following sentence:

I cite [version of the work], at [location], to support [scoped claim], on the basis of [text or evidence actually checked].

If any bracket cannot be filled in, narrow the claim or mark the uncertainty. Do not place one citation at the end of a paragraph containing four claims and leave readers to guess its scope. Do not cite a methodological source to give the impression that it proves a mathematical result it does not discuss. Unit 4 keeps the record; Unit 5 places the citation precisely at the sentence it supports.

5 Notation discipline before the proof

Compact notation helps only when its contract is visible. For the main case, we use the following dictionary.

Symbol	Type and meaning	First use
$a\mid b$	$a,b\in\mathbb Z$ and there is a $c\in\mathbb Z$ with $b=ac$	definition of divisibility below
$p$ or $q$	positive prime number	definition and lemma
$p_1,\ldots,p_k$	hypothetical list of all primes, without repetitions	proof of the theorem
$P$	$\prod_{i=1}^{k}p_i$	proof of the theorem
$N$	$P+1$	proof of the theorem

The same letter will not be used for both a number and a set. The index $i$ always runs from $1$ to $k$ when it appears in the product. The word “prime” always means a positive prime. These conventions avoid two side questions— negative primes and the empty product—that the core claim does not require.

Definition 1 For $a,b\in\mathbb Z$ , we write $a\mid b$ if there is a $c\in\mathbb Z$ such that $b=ac$ . An integer $p$ is called prime if $p>1$ and its only positive divisors are $1$ and $p$ .

The definition comes before the theorem so that the word “prime” does not depend on readers remembering a potentially different convention. However, we do not repeat the entire theory of divisibility. Only the following fact is needed: if $d\mid a$ and $d\mid b$ , then $d\mid(a-b)$ .

6 Worked case: writing the theorem on the infinitude of primes

This section shows the order that readers will encounter: a lemma licenses a step, the theorem uses that license, an example shows the mechanism, and a counterexample limits the interpretation. Signposts at the beginning of each part state why it is there.

6.1 The lemma that licenses “take a prime divisor”

The proof of the theorem will construct a number $N>1$ and then choose a prime divisor. To ensure that this choice has a basis, we prove the following result that licenses it.

Lemma 1 For every integer $n>1$ , there is a prime number $q$ with $q\mid n$ .

Proof 1. Fix $n>1$ and form

$D=\{d\in\mathbb N: d>1\text{ and }d\mid n\}.$

The set $D$ is nonempty because $n\in D$ . By the well-ordering principle, $D$ has a least element; call it $q$ .

Suppose that $q$ is not prime. Since $q>1$ , there are integers $a,b$ with $q=ab$ and $1<a<q$ and $1<b<q$ . From $a\mid q$ and $q\mid n$ , we obtain $a\mid n$ . Thus $a\in D$ , contradicting the minimality of $q$ . Hence $q$ is prime, and by its choice, $q\mid n$ .

The final sentence establishes the lemma’s two outputs separately: $q$ is prime and $q$ divides $n$ . Both will be used in the theorem.

6.2 The theorem and its proof

We can now state the main promise without mentioning an algorithm or the growth rate of the number of primes.

Theorem 1 The set of prime numbers is infinite.

Proof 2. Suppose, for a contradiction, that there are only finitely many prime numbers. Since $2$ is prime, all the primes can be listed as

$p_1,p_2,\ldots,p_k$

for some $k\geq1$ . Form

$P=\prod_{i=1}^{k}p_i \qquad\text{and}\qquad N=P+1.$

Since $N>1$ , LEM-01 gives a prime number $q$ with $q\mid N$ . The assumption that the list contains all primes requires $q=p_j$ for some $j\in\{1,\ldots,k\}$ . Therefore $q\mid P$ . Together with $q\mid N$ , the basic divisibility fact gives

$q\mid(N-P)=1,$

which is impossible for $q>1$ . This contradiction rejects the assumption that the list of all primes is finite. Thus there are infinitely many prime numbers.

Notice the function of each paragraph. The first opens the proof by contradiction and specifies the type of list. The second constructs a new object. The third brings the lemma’s output into the assumption of a complete list. The last names exactly the assumption that is rejected.

6.3 Example: seeing the mechanism without turning it into a universal proof

EXM-01 — An example using three primes

Take the list $2,3,5$ . Its product is $P=30$ , so $N=P+1=31$ . None of $2,3,5$ divides $31$ . That fact is all this example needs; we do not need a primality test for $31$ . The calculation shows why adding one avoids every divisor in the list used.

The example helps readers see the remainder-one congruence. But three primes are not “all primes,” so this calculation is not a proof of the infinitude of primes. The universal proof remains at #prf-o017-u05-infinitely-many-primes.

6.4 Counterexample: the product plus one need not be prime

CTR-01 — Rejecting a false strengthening

For the first six primes,

$2\cdot3\cdot5\cdot7\cdot11\cdot13+1 =30031 =59\cdot509.$

Thus the statement “the product of all the primes in the list plus one must be prime” is false.

The counterexample does not damage the theorem. The proof requires only that $N>1$ has some prime divisor $q$ . Since none of the $p_i$ divides $N$ , every such prime divisor lies outside the list. Replacing the needed claim (“there is a new prime divisor”) with a stronger one (“ $N$ is prime”) makes the exposition not merely overconfident but wrong.

6.5 What has and has not been established

At this point, readers are entitled to conclude:

for every finite list claimed to contain all primes, a number can be constructed that has a prime divisor outside the list; and
consequently, no finite list contains all primes.

Readers are not yet entitled to conclude:

every number of the form $p_1\cdots p_k+1$ is prime;
exactly one new prime divides that number;
this method produces all primes without repetition; or
primes occur with a particular frequency among the integers.

The list of boundaries at the end of the argument prevents readers from carrying the rhetoric of a “new prime” beyond the strength of the proof.

7 Local signposting

Section headings help global navigation, but readers also need signposts within an argument. Three kinds of sentence suffice for many expositions.

Purpose: “To choose a prime divisor of $N$ , we first prove the existence lemma.”
Reason for a transition: “The lemma is now available, so we return to the hypothetical list of all primes.”
Scope closure: “The contradiction rejects finiteness; it does not assert that $N$ itself is prime.”

Signposts must convey information. Phrases such as “clearly,” “it is easy to see,” or “as usual” are not signposts because they state neither a purpose, a basis, nor a point of return. Conversely, do not precede every line of algebra with a lengthy summary. Put signs at changes of role: from definition to lemma, from lemma to theorem, and from proof to testing the boundaries.

7.1 One paragraph, one main burden

A paragraph may contain several sentences, but it should have one main burden that can be named: establishing context, constructing an object, applying a result, or limiting a conclusion. If a paragraph simultaneously introduces symbols, states a theorem, gives a proof, makes a historical claim, and offers an example, citation locators become unclear and readers find it hard to tell which part is being tested.

8 Revision through a dependency audit

The Unit 5 audit begins after the mathematical content is available. Its purpose is not to discover a proof from scratch, but to test whether already valid relationships are visible in the manuscript. Consider the following draft.

A draft that must not be published

Let all the primes be $p_1,\ldots,p_k$ . Multiply them all together and add one. The resulting number is prime because it is not divisible by any $p_i$ . So there is another prime, and the result is proved.

The central error is mathematical: the resulting number need not be prime, as CTR-01 shows. But merely replacing one word does not make the exposition auditable. Lay out its dependencies.

Draft claim	Required inputs	Status	Editorial revision
the list contains all primes	assumption for contradiction and $k\geq1$	implicit	state the framework and the type of list
$N=p_1\cdots p_k+1>1$	definition of $N$ and positivity of the factors	partial	name $P,N$ and state $N>1$
$N$ is prime	does not follow from a remainder of one	false	remove it; use `LEM-01` to choose a prime divisor $q$
no $p_i$ divides $N$	$p_i\mid P$ and $N-P=1$	reason missing	write the divisibility argument
there is a prime outside the list	$q$ is prime, $q\mid N$ , and $q\neq p_i$	becomes valid after revision	link to the lemma and contradiction
the theorem is complete	the finiteness assumption has actually been rejected	too vague	name the rejected assumption again

Revise in the following order.

Mark every sentence that carries a claim, not just formulas.
For each claim, write down the inputs it needs and the locators where those inputs are available.
Remove claims stronger than the need or the proof.
Move definitions before first use and make symbol types consistent.
Add signposts only at changes of role.
Read the conclusion against the scope contract, then remove promises that the argument does not fulfill.

The revision’s outcome is not the audit table. Its outcome is the fluent exposition in the worked case above; the table is only an editorial tool. The Unit 4 source notebook remains the permanent home for identities and decision history. Unit 5 does not duplicate that notebook within the prose.

9 Guided practice: a geometric sum without rhetorical gaps

Suppose a set of class notes says only:

$1+x+\cdots+x^n=(x^{n+1}-1)/(x-1)$ . Just multiply, and we are done.

The draft does not give the domains of $n$ and $x$ , uses division before excluding $x=1$ , and does not show what should be multiplied. We will repair it for readers who know school algebra.

9.1 Model revision

For $n\in\mathbb N_0:=\{0,1,2,\ldots\}$ and $x\in\mathbb R$ , define

$S_n(x)=\sum_{j=0}^{n}x^j.$

If $x\neq1$ , then

$S_n(x)=\frac{x^{n+1}-1}{x-1}. \tag{GS}$

Indeed, multiplying by $x-1$ produces successive cancellation:

$\begin{aligned} (x-1)S_n(x) &=\sum_{j=0}^{n}x^{j+1}-\sum_{j=0}^{n}x^j\\ &=x^{n+1}-1. \end{aligned}$

Since $x\neq1$ , division by $x-1$ is valid and gives (GS). For $x=1$ , the definition directly gives $S_n(1)=n+1$ ; the fractional form (GS) must not be used because its denominator is zero.

As an illustrative check, $S_3(2)=1+2+4+8=15$ , which agrees with $(2^4-1)/(2-1)$ . The calculation checks one instance, rather than proving the formula for every $n$ and $x$ . The value $x=1$ is an excluded domain case and shows that the claim of the fractional form for every real number is invalid, even though the sum $S_n(1)$ itself remains defined.

9.2 What changed

The readers and prerequisites determine the level of detail.
$n$ , $x$ , and $S_n$ have specified types before they are used.
The claim separates the case $x\neq1$ from $x=1$ .
“Just multiply” is replaced by an identity that can actually be checked.
The example is labeled illustrative, and the boundary case is not hidden.

This repair is not a new lesson on how to discover the geometric identity. It shows how to present an already known identity completely and honestly.

10 From manuscript to auditable seminar

Seminar notes are not a manuscript cut down at random. The order of the argument must remain visible when some details are moved to a companion sheet. For the infinitude-of-primes case, the following 10–12-minute plan suffices.

Time	Role	What is shown or said
0–2 minutes	question and contract	core claim, prerequisites, and three things that will not be discussed
2–4 minutes	definition and license	definition of a prime and statement of `LEM-01`; the lemma’s proof is available at its locator
4–8 minutes	main argument	hypothetical list, $P$ , $N$ , divisor $q$ , and contradiction $q\mid1$
8–10 minutes	misinterpretation	the example $31$ , followed by the counterexample $30031=59\cdot509$
10–12 minutes	boundaries and questions	exact conclusion, claim locators, and open questions

To make the presentation easy to follow:

say what a symbol means at first use, rather than merely displaying it;
do not rely on color as the only distinction between assumptions and conclusions;
keep the definition and main claim visible while discussing the proof;
tell readers when a detail is available at a written locator; and
if a question exposes an error, record the claim and its location, correct only as far as can be established, and separate the correction from any extension.

The answer “I cannot yet establish that step from the available material” is better than naming a theorem that has not been checked. Unit 8 will go further into reviewer reports and responses to criticism; the focus here is the architecture of exposition and the traceability of a seminar.

11 Checklist before submitting a manuscript

Use the following list once the mathematical content is stable.

Are the readers, prerequisites, purpose, and boundaries stated?
Is every symbol introduced before its first use, and does it keep the same type?
Do definitions, theorems, proofs, examples, and counterexamples have distinct statuses and locators?
Does every universal claim actually receive a universal proof, rather than merely an example?
Does every citation have an honest scope, version, and location?
Do transitions state a purpose or reason, rather than merely “clearly”?
Does the dependency audit confirm that no unsupported nodes remain?
Does the conclusion restate the theorem’s exact strength and name its limits?
Does the seminar version preserve the path of the argument and provide locators for details moved elsewhere?

One “no” answer does not always mean the theorem is false, but it means the exposition is not yet ready to be called auditable.

12 Exercises

O017-U05-E01 — Reader contract. An author wants to explain that $\sqrt2$ is irrational to students who know even and odd numbers but have not encountered a prime-divisibility lemma. Write a six-field contract: readers, purpose, core claim, boundaries, form, and admitted results. Name one detail that must be added and one adjacent topic that must be excluded.
O017-U05-E02 — Dissecting notation. Audit the sentence “Take $a_n\in A$ , and since $a$ converges, its limit is $A$ .” Find at least four type or notation problems. Rewrite it as a meaningful statement about a sequence $(a_n)$ in a set $A\subseteq \mathbb R$ and a limit $L$ ; do not add the conclusion $L\in A$ without appropriate assumptions.
O017-U05-E03 — Claim–locator map. For the infinitude-of-primes case, make a directed graph with nodes DEF-01, LEM-01, THM-01, the lemma’s proof, the theorem’s proof, EXM-01, and CTR-01. Give every arrow a type (defines, proves, uses, illustrates, or refutes-strengthening). Explain why there is no proves arrow from EXM-01 to THM-01.
O017-U05-E04 — Honest citation. Write two sentences with pointers about this unit: one citing The Turing Way for the metadata/version principle, and one stating that the proof of the infinitude of primes is self-contained at a local locator. Then write one dishonest cited sentence and explain exactly what work it wrongly assigns to the donor source.
O017-U05-E05 — Revision audit. An author changes CTR-01 to the sentence “the product method fails because $30031$ is composite.” Revise that sentence to distinguish the rejected claim from the mechanism that remains valid. Include the counterexample’s locator, the correct claim, and one sentence stating a boundary.
O017-U05-E06 — Seminar plan. Turn the geometric-sum model into notes for a five-minute seminar. State what stays on screen, what is spoken, what is available through a locator, when the case $x=1$ appears, and how you will answer the question “Doesn’t the example $x=2$ already prove the formula?”

13 Hints and answer guidance

O017-U05-H01. An adequate contract names proof by contradiction as the form, the claim “there is no ratio of two integers with a nonzero denominator whose square is $2$ ,” and the need to explain why an even square requires its base to be even. Full uniqueness of factorization or decimal approximations to $\sqrt2$ can be excluded.
O017-U05-H02. $a_n$ is a term, not a sequence; “ $a$ converges” has no specified type; the limit must be a number, not a set; the quantification of $n$ and the meaning of membership are unclear; and $L\in A$ requires, for example, that $A$ be closed. A safe form is: “Let $A\subseteq\mathbb R$ , and let $(a_n)$ be a real sequence with $a_n\in A$ for every $n$ , and suppose $a_n\to L$ for some $L\in\mathbb R$ .” Stop there if $A$ is not known to be closed.
O017-U05-H03. The definition gives meaning to the symbols in the lemma and theorem; the lemma’s proof proves LEM-01; the theorem’s proof uses LEM-01 and proves THM-01; the example illustrates theorem THM-01; the counterexample refutes-strengthening but does not refute THM-01. One instance does not discharge a universal quantifier.
O017-U05-H04. The donor citation can support the recommendation to record metadata and versions. For the mathematics, state that the complete argument is available at #prf-o017-u05-infinitely-many-primes, without claiming that the donor proves it. A dishonest sentence would be “According to The Turing Way, there are infinitely many primes”; the methodological source has not been given a mathematical locator and was not checked for that work.
O017-U05-H05. An accurate form is: “CTR-01 refutes the strengthening that $P+1$ is always prime; it does not refute the construction, because every prime divisor of $P+1$ still lies outside the list of factors.” Add that the theorem guarantees neither a unique new divisor nor an enumeration procedure without repetition.
O017-U05-H06. Keep the definition of $S_n(x)$ and the two-case claim visible. Explain the cancellation of terms aloud while pointing to the two lines of algebra; show $x=1$ before the numerical example so that the boundary does not look like a footnote. The honest answer is that the example $x=2,n=3$ checks only one pair of inputs, whereas the algebraic identity handles all $n\in\mathbb N_0$ and $x\neq1$ .

14 Unit completion task

Write a mathematical exposition of 1,000–1,400 words and an 8-minute seminar plan based on the following synthetic packet. The packet is local class material, not an external publication; use its locators, and do not invent authors, DOIs, years, or pages.

Packet O017-U05-PKT — Odd-degree polynomials

PKT-DEF-01. An odd-degree real polynomial has the form $p(x)=a_dx^d+\cdots+a_0$ with $a_d\neq0$ and $d$ odd.
PKT-LEM-01. Every real polynomial is continuous on $\mathbb R$ .
PKT-LEM-02. If $d$ is odd, there is an $R>0$ such that $p(-R)$ and $p(R)$ have opposite signs. Which one is positive depends on $a_d$ .
PKT-LEM-03. If a function is continuous on $[-R,R]$ and its values at the two endpoints have opposite signs, there is a $c\in(-R,R)$ where the function’s value is $0$ .
PKT-EXM-01. For $p(x)=x^3+x+1$ , we have $p(-1)=-1$ and $p(0)=1$ .
PKT-CTR-01. The even-degree polynomial $x^2+1$ has no real root.
PKT-CTR-02. The odd-degree polynomial $x^3-x$ has three real roots; odd degree guarantees existence, not uniqueness.

The final product must contain:

a reader contract for beginning calculus students who know continuity and the Intermediate Value Theorem but have not studied formal asymptotic analysis;
definitions and the main theorem with stable locators of your own;
a notation dictionary distinguishing the degree $d$ , leading coefficient $a_d$ , bound $R$ , and root $c$ ;
a complete proof that explicitly links PKT-LEM-01, PKT-LEM-02, and PKT-LEM-03, including checks of the interval and signs;
one example that brackets a root and two counterexamples that limit two different strengthenings;
a claim–locator list distinguishing packet material, your statements, proofs, examples, and boundaries;
a paragraph explaining why packet locators are not bibliographic citations and why unavailable metadata must still be marked tidak diketahui (“unknown”);
a dependency audit of the first draft, followed by a clean revised manuscript without an audit table interrupting the flow;
signposts at the transitions definition → lemma → theorem → example → boundaries;
an eight-minute seminar plan that keeps the main claim on screen, provides locators for details, and names two things that have not been proved; and
a short revision note naming one sentence removed, one sentence narrowed, and one dependency made explicit.

14.1 Rubric

Each criterion is scored 0, 1, or 2.

Criterion	0	1	2
Readers and scope	readers or promises unstated	some prerequisites/boundaries visible	readers, purpose, prerequisites, form, and boundaries consistently determine the detail
Claims and notation	objects, quantifiers, or symbols change meaning	some types/locators implicit	definitions, theorems, symbols, quantifiers, and locators are precise before use
Proof path	signs, interval, or use of a lemma incorrect	correct idea with one implicit arrow	all three lemmas are used with the correct contracts and yield a real root without a gap
Examples and boundaries	example used as proof or counterexample aimed at the wrong claim	example status stated but one strengthening unclear	illustration and two counterexamples precisely delimit existence, degree parity, and uniqueness
Locators and citations	metadata fabricated or citation substituted for proof	locators present but scope of support unclear	every claim can be found, local material is distinguished from external sources, and uncertainty is honest
Revision and seminar	raw draft transferred to speech or revision note absent	some audit, revision note, or oral flow present	audit produces a clean manuscript; revision note identifies removal, narrowing, and an explicit dependency; eight-minute plan uses signposts, names two things not proved, and gives access to details

The passing score is at least 10 out of 12, with a score of 2 for Proof path and Locators and citations. Concluding that the root is unique, applying the Intermediate Value Theorem without continuity or a sign change, inventing packet metadata, or using a numerical example as a universal proof requires revision even if the total score is sufficient. The three-part revision note, signposts, two unproved boundaries, and eight-minute limit are also mandatory gates, not optional items that other scores can offset.

15 Boundaries with B80 and other units

This unit does not teach Python, SageMath, SymPy, or NumPy/SciPy syntax, factorization or root-finding algorithms, plotting, writing tests, notebooks, environment management, implementing exact or floating-point arithmetic, or error-certification techniques. All those implementation skills belong to B80. Here, readers only classify the epistemic status of already available outputs according to their contracts: the exact integer product $30031=59\cdot509$ is a finite certificate that can be checked as a proof, and exact sign evaluations can be valid parts of an Intermediate Value Theorem argument. By contrast, numerical output without a certificate, bracketing without error guarantees, or checks whose procedures and arithmetic have not been audited remain empirical evidence; they must not be promoted to universal proof.

Nor does this unit repeat Unit 1’s claim-network mapping, Unit 2’s source search and authority assessment, Unit 3’s reconstruction of missing steps, or Unit 4’s provenance notebook design. It uses their outputs to write a readable product. A rerunnable computational package is the subject of Unit 6; the erratum format, Unit 7; and the review process and detailed responses, Unit 8. Discussion of seminars here is limited to delivery architecture, not the peer-review process.

16 Sources, provenance, changes, and rights

The original Indonesian prose, reader contract, exposition architecture, notation dictionary, presentation of the lemma and theorem on the infinitude of primes, proofs, example $31$ , counterexample $30031$ , geometric-sum model, synthetic polynomial packet, exercises, answer guidance, completion task, and rubric in this unit are original O017 material by the O017 contributors, 2026, licensed under CC BY-SA 4.0. This English edition translates that original material and retains the same license, identifiers, mathematical content, and component credits.

The theorem on the infinitude of primes, the existence lemma for prime divisors, the geometric-sum identity, and the theorem on the existence of a real root for an odd-degree polynomial are classical mathematical results. O017 does not claim to have discovered them. The wording, pedagogical arrangement, proofs presented, examples, dependency audit, and use of locators were written specifically for this unit and were not copied from a particular mathematical source.

The general principles that scholarly objects need adequate metadata, that the version cited must be distinguishable, and that citations should be placed according to their role are adapted in a bounded way from The Turing Way (The Turing Way Community 2025, 2026), The Turing Way Community, at the fixed commit c98a0e6...131d1ce, specifically:

book/website/communication/citable/citable-metadata.md; and
book/website/communication/citable/citable-versioning.md.

That donor material is licensed under CC BY 4.0. Its concepts were summarized, bounded, rearranged, re-expressed in Indonesian, and contextualized for mathematical exposition; this English edition translates that original O017 treatment. No donor sentences, images, examples, or exercises were copied. Research Software Engineering with Python is not adapted in Unit 5; that frozen source remains relevant to the later computational unit, but is not credited for material that is not used here.

The Turing Way Community, other donor authors, publishers, and their affiliations do not endorse, approve, or sponsor O017. Each frozen source remains subject to its own license. The CC BY-SA 4.0 license for original O017 material does not replace or relicense donor components.

References

The Turing Way Community. 2025. The Turing Way Handbook for Reproducible, Ethical and Collaborative Research. Version 1.2.3. https://doi.org/10.5281/zenodo.3233853.

The Turing Way Community. 2026. The Turing Way Handbook for Reproducible, Ethical and Collaborative Research: Frozen Source Witness. https://github.com/the-turing-way/the-turing-way/tree/c98a0e6ca47450456cca7c5eedda2d5ee131d1ce.