Unit 7 — Errata, Corrections, and Limits of Certainty

Establishing defects, bounding their impact, and correcting without erasing history

A practical unit on classifying mathematical and computational defects, separating severity from confidence, constructing minimal reproducers, and writing auditable erratum drafts.

1 Learning outcomes

This unit has the stable identifier O017-U07. On completing it, you will be able to:

distinguish typographical errors, local exposition defects, missing hypotheses, theorems that are false as written, ambiguous locators, and computational defects;
separate impact severity from confidence level in a diagnosis;
freeze the version, copy, locator, and wording actually examined;
construct a minimal reproducer that checks every hypothesis before demonstrating failure of the conclusion;
distinguish what is observed, proved, provisionally inferred, and unknown;
formulate a sufficiently narrow correction and prove that the repair is indeed adequate;
preserve correction records as an append-only history without overwriting earlier claims or versions;
write one concise, specific, actionable issue or erratum draft without contacting anyone outside the project; and
stop or escalate an examination when the evidence, authority, or extent of the impact is not yet sufficient.

Prerequisites, continuity, and scope boundaries

Unit 2 supplies work and version identities; Unit 3 supplies methods for examining proof gaps; Unit 4 supplies records of claims, locators, and provenance; Unit 5 supplies auditable exposition; and Unit 6 supplies computational packages that can be rerun. Unit 7 uses all those outputs for a narrower task: establishing what is defective, how far its impact extends, how confident the diagnosis is, and what form of correction has been justified.

This unit does not teach issue-platform syntax, Git commands, program debugging, environment installation, or how to run notebooks. Nor does it authorize sending reports. Every issue formulation here remains a local draft until separate authorization and checks are in place.

2 An erratum is a bounded claim, not an expression of dislike

The sentence “this section seems wrong” is not yet a formal correction (erratum). An auditable erratum states at least five things:

the exact object examined: work, version, copy, and location;
the exact wording or output being questioned;
evidence of failure that can be checked with minimal effort;
the scope of consequences: which sentence, lemma, theorem, exercise, computational result, or downstream result is affected; and
the justified correction, including what remains unknown.

An erratum need not be long. It needs an intact chain of inference. Conversely, a long report may still be weak if it omits the version, presents only a numerical example that does not satisfy the hypotheses, or proposes new wording without proving it.

Original O017 bridge

The two-layer taxonomy separating statement status from defect type, the separate severity and confidence axes, the minimal-reproducer protocol, the append-only correction records, the connectedness case, the uncertainty ledger, and the stop/escalation gates in this unit are original O017 material. This unit does not copy prose, examples, exercises, or a sequence of presentation from an external donor.

3 Six defect types that must not be conflated

Classification helps select an action. It does not replace evidence. A finding may have one primary type and several secondary consequences, but every label must have a justification.

Type	Diagnostic test	Brief example	Reasonable initial action
Typographical error	Does an almost mechanical character change restore consistent wording without changing the idea?	the index $a_{n-1}$ is printed as $a_{n+1}$ although all surrounding lines use $a_{n-1}$	record both forms, show the context, and avoid guessing when more than one repair is plausible
Local exposition defect	Could the argument be correct, while one transition, definition, back-reference, or notation change is insufficiently explained?	“then clearly” skips a set inclusion that requires proof	supply the local step and check that no new hypothesis has been smuggled in
Missing hypothesis	Does the statement become true once a particular assumption is stated, while a counterexample exists without it?	a constant-sign conclusion lacks the assumption that the domain is connected	give a counterexample, state the candidate assumption, then prove the repaired version
Theorem false as written	Is there an object satisfying every written hypothesis but violating the conclusion?	a function on a disconnected domain takes positive and negative values without a zero	mark the published statement’s status as false; the root-cause diagnosis can be recorded separately
Ambiguous locator	Can readers using the same version still choose more than one plausible location?	“see the preceding theorem” when two theorems lie between the paragraph and the claim	provide a section/label/page locator that distinguishes the location and name the version used
Computational defect	Does the code, input, environment, procedure, or output fail to match the computational claim being made?	the manifest names program revision `r2`, but the log records execution of `r1`	freeze the artifacts and execution record; do not promote the output to evidence before resolving the mismatch

3.1 Falsity and its cause are separate fields

“Missing hypothesis” and “false theorem” appear to overlap because both can be true at once, but they answer different questions.

statement_status = false_as_written states that the published wording has a counterexample.
defect_kind = missing_hypothesis records a diagnosis of how that wording might be repaired.

Use status values explicitly: supported_as_written when adequate evidence supports the claim within its scope; false_as_written when exact evidence refutes it; unsupported_as_written when the basis provided is insufficient even though the truth of the claim has not been decided; unresolved when the claim audit is unfinished; and not_applicable when the defect record concerns only a non-claim object, such as an ambiguous locator or the form of a character. If a typo occurs within a formula, record the typographical finding and the mathematical claim’s status as two objects; do not force one value to answer two questions.

A counterexample suffices to establish the first field. It does not automatically establish the second. Perhaps the conclusion is too strong, the domain is wrong, the quantifier is mistaken, or an earlier definition does not fit. Therefore, do not replace “false as written” with “clearly just a typo” before a unique repair is genuinely supported by the context.

3.2 A typo can have mathematical consequences

Defect type does not measure impact. A missing minus sign may be an easily diagnosed typo, yet reverse the main theorem. Conversely, a false theorem in one optional exercise may not affect any other chapter. This is why classification and severity must be stored in separate fields.

3.3 An exposition defect is not permission to rewrite the source

If a missing step can be reconstructed from the written hypotheses, record the reconstruction as a new argument and link it to the old location. Do not alter the source to make it appear that the step was always there. If the reconstruction requires a new assumption, the finding has crossed the boundary of a local exposition defect and must be examined as a missing hypothesis or a false statement.

4 Severity and confidence are two separate axes

Severity answers “if this diagnosis is correct, how extensive are its consequences?” Confidence answers “how strong is the evidence that this diagnosis is correct?” They must not be averaged into a single number.

Code	Name	Impact criterion
`S0`	editorial	mathematical meaning, traceability, and computational results are unchanged
`S1`	local	one sentence, step, example, or locator is affected; the main result can still be maintained with the available evidence
`S2`	material	one lemma, exercise, table, or argument branch fails; its impact has been bounded and does not extend to the whole work
`S3`	major	the main theorem, many downstream results, or a principal public result may be invalid; the full extent may not yet be known
`S4`	critical	the defect affects the foundations of many results or high-risk uses and requires temporary suspension of use

Code	Name	Available basis
`C0`	unassessed	a report exists, but the version, location, or wording has not yet been frozen
`C1`	reasoned suspicion	the symptom is real, but a reproducer, context, or one audit step is still missing
`C2`	strong	the reproducer and context support the diagnosis; an independent check or formal proof is still worth seeking
`C3`	established	a counterexample or complete proof has been checked against the exact version and locator

The pair S3/C1 means “potentially large impact, incomplete diagnosis”; it is not grounds for declaring a theorem certainly false. The pair S0/C3 means “the typo is confirmed, but does not change the meaning.” A finding at S2/C3 may be more ready for correction than one at S4/C1, despite its lower severity number.

Do not multiply the two axes

A score such as severity × confidence = priority hides the reasoning. A potentially critical finding supported by preliminary evidence can receive the same score as a certain local finding, although they require different actions. Preserve the code pair, the basis for each axis, and the action rule explicitly.

4.1 An action matrix that does not alter the evidence

Situation	Local action	Wording to avoid
low impact, high confidence	prepare a concise correction and check cross-consistency	“the entire work is unreliable”
high impact, high confidence	stop relying on the claim, audit downstream results, and prepare escalation	“perhaps it is only a matter of style”
high impact, low confidence	freeze the evidence, flag the risk, and seek context or a suitable reviewer	“the theorem has been proved false”
unknown impact	do not fill in severity by guessing; map dependencies first	“it affects only one line”

Priority may also depend on active use, deadlines, and the ability to withhold a release. Those factors are recorded separately; they do not change the mathematical facts.

5 Freeze the object before diagnosing it

A report must distinguish work, version, copy, and location. The following minimum fields prevent corrections to the wrong object.

Field	Content	Completeness test
`work_id`	identity of the intellectual work	other versions can be linked without treating them as identical bytes
`version_id`	edition, release, date, commit, or state label	“latest” is not accepted as an identity
`copy_id`	the copy actually examined	format, completeness, and byte identity are recorded where available
`locator`	section, theorem, equation, page, paragraph, line, or anchor	readers with the same copy reach the same wording
`observed_text`	a controlled short quotation or exact transcription	the difference between source text and the report’s paraphrase is visible
`observed_at`	time of examination	not conflated with the publication date
`context_scope`	conventions, definitions, or assumptions applying at the location	the examiner knows whether the excerpt is self-contained

If only a PDF is available, printed page numbers and file page numbers may differ; record both where necessary. If HTML changes, an anchor alone may not suffice; pair it with a version or archive identity. If wording comes from an image, distinguish visible characters from conjectured characters. Do not turn “the letter appears to be $O$ ” into “the numeral is certainly $0$ ” without grounds.

6 A minimal reproducer for a mathematical defect

A minimal reproducer, or minimal reproduction case, is a small object that demonstrates a claim’s failure without requiring readers to trust the reporter’s narrative. It has five parts.

Frozen claim. Restate the quantifiers, domain, hypotheses, and conclusion without strengthening or weakening the source text.
Candidate object. Choose the simplest possible object that still lies within the claim’s domain.
Hypothesis checks. Verify them one by one; do not stop at “clearly satisfies them.”
Failure of the conclusion. Explicitly calculate or prove the negation of the conclusion.
Reproducer boundary. State the exact claim refuted and the nearby claims left untouched.

Small does not mean vague. A two-point example can be stronger than a large simulation when all its hypotheses can be checked exactly.

For a computational defect, the parallel structure consists of code version, input, dependencies, command or procedure, actual output, claimed output, and the observed difference. Environment construction and execution belong to B80 and Unit 6; Unit 7 judges whether the resulting evidence suffices to formulate a correction.

6.1 Test the reproducer before relying on it

Does the example object actually lie in the domain and satisfy every explicit hypothesis?
Have the conventions applying in the source section been included?
Is the failure exact, or does it arise only from rounding and display?
Does the example refute the source claim rather than a strengthening invented by the reporter?
Have parts that can be removed without eliminating the failure been removed?
Can readers check the reproducer without hidden access?

7 Worked case: constant sign without connectedness

The following case is entirely synthetic. Its local packet states that the proposition is self-contained; no section-level assumptions are inherited. The finding can therefore be audited without seeking an external source.

7.1 Version and locator examined

O017-U07-V01 — Proposition 7.3, synthetic version 1.0

V01 wording. Let $X$ be a topological space and let $f:X\to\mathbb R$ be continuous. If $f(x)\neq0$ for every $x\in X$ , then $f$ has constant sign: either $f(x)>0$ for every $x\in X$ or $f(x)<0$ for every $x\in X$ holds.

The local source record for this case is:

Field	Value
`work_id`	`O017-U07-W01`, Synthetic Topology Packet
`version_id`	`O017-U07-V01`, state label `1.0-sintetis`
`copy_id`	`O017-U07-CP01`, the complete excerpt embedded in this unit
`locator`	`#o017-u07-packet-v1`, Proposition 7.3
`claim_id`	`O017-U07-C01`
`context_scope`	self-contained statement; no inherited connectedness convention
`observed_text`	the entire wording in the V01 box above

“Constant sign” in V01 means a disjunction of two universal claims, not merely that each point has one of the two signs. The nowhere-zero hypothesis already ensures a sign at each point; the additional conclusion requires the same sign throughout the domain.

7.2 A minimal two-point reproducer

Take

$X=\{-1,1\}\subset\mathbb R$

with the subspace topology, and define $f:X\to\mathbb R$ by $f(x)=x$ . We check every hypothesis.

$X$ is a topological space with the subspace topology. In fact, both singletons are open in $X$ because $\{-1\}=X\cap(-3/2,-1/2)$ and $\{1\}=X\cap(1/2,3/2)$ .
$f$ is the restriction of the continuous identity function on $\mathbb R$ , so it is continuous on $X$ .
For every $x\in X$ , we have $f(x)\neq0$ because its only values are $-1$ and $1$ .
The first conclusion fails because $f(-1)=-1<0$ . The second conclusion fails because $f(1)=1>0$ .

Thus both alternative conclusions of V01 fail although every written hypothesis is satisfied. This establishes statement_status(O017-U07-C01) = false_as_written with confidence C3. The reproducer requires only two points; a one-point domain cannot contain both positive and negative values, so this example is also minimal in its number of points.

The finding refutes V01. It does not refute a statement about connected domains, does not assert that every function on a disconnected domain changes sign, and does not reveal the synthetic author’s intention.

7.3 Diagnosis: a missing connectedness hypothesis

The counterexample uses two components of the domain. A natural candidate repair is to add connectedness, not compactness, path-connectedness, or the assumption that $X$ is an interval.

O017-U07-V02 — Proposition 7.3, candidate corrected wording 1.1

V02 wording. Let $X$ be a connected topological space and let $f:X\to\mathbb R$ be continuous. If $f(x)\neq0$ for every $x\in X$ , then $f(x)>0$ for every $x\in X$ or $f(x)<0$ for every $x\in X$ .

7.4 Proof of the repaired version

Define two relatively open sets

$U=f^{-1}((-\infty,0)) \qquad\text{and}\qquad V=f^{-1}((0,\infty)).$

Both are open in $X$ because $f$ is continuous. They are disjoint. The hypothesis $f(x)\neq0$ for every $x$ gives $X=U\cup V$ . If $U$ and $V$ were both nonempty, they would separate $X$ into two disjoint nonempty open sets, contradicting the connectedness of $X$ . Hence one of $U$ or $V$ is empty.

If $U$ is empty, every value of $f$ is nonnegative; since no value is zero, every value is positive. If $V$ is empty, the same argument shows that every value is negative. Thus one of the alternatives in V02 holds. $\square$

The proof uses connectedness and continuity exactly once, through the preimages of two open rays. It does not use a metric, compactness, path-connectedness, differentiability, or the interval-specific form of the Intermediate Value Theorem.

7.5 Why the repair is minimal in the relevant sense

“Minimal” must specify a comparison class. Here we seek a condition on the domain that guarantees the conclusion for every nowhere-zero continuous real-valued function. In that class, connectedness is not only sufficient but necessary.

Suppose $X$ is disconnected. Then there are nonempty disjoint open sets $A$ and $B$ with $X=A\cup B$ . Since each is the complement of the other, they are also closed. Define

$g(x)= \begin{cases} -1,&x\in A,\\ 1,&x\in B. \end{cases}$

The function $g$ is continuous: the preimage of every open set in $\mathbb R$ is one of $\varnothing$ , $A$ , $B$ , or $X$ , all of which are open in $X$ . The function is nowhere zero but takes both signs. Therefore,

$X$ is connected if and only if every nowhere-zero continuous function $f:X\to\mathbb R$ has constant sign.

Adding “ $X$ is connected” therefore repairs the universal theorem exactly. Adding “ $X$ is path-connected,” “ $X$ is an interval,” or “ $X$ is compact and connected” also suffices, but is stronger than necessary. For one particular function, connectedness of the image $f(X)$ is also sufficient; this does not replace the diagnosis of a domain condition guaranteeing the result for all continuous functions.

7.6 Severity, confidence, and extent

Field	Value	Basis
statement status	`false_as_written`	the two-point reproducer satisfies every hypothesis and violates both alternatives
primary type	`missing_hypothesis`	the connected version is proved true, and that condition is necessary for the universal guarantee on the domain
secondary consequence	theorem false as written	the status of the V01 text, not speculation about the author’s intention
severity	`S2 material` for this packet	one proposition fails; the packet supplies no other downstream results
confidence	`C3 established`	a complete counterexample and proof of the correction are available
unknown extent	uses outside the packet	the synthetic packet contains no external dependency list

Severity is not raised to S3 merely because the word “theorem” appears. The available packet establishes only that one proposition is affected. Conversely, if a later dependency audit finds many principal results using V01 on disconnected domains, severity can be changed through a new event without altering the original evidence.

8 A correction must preserve history

Replacing V01 with V02 in place and then deleting V01 removes the answers to important questions: which version once contained the false claim, what evidence exposed it, and which outputs might depend on it. An auditable correction uses new identities and directed relationships.

V01 --contains--> C01 --refutedBy--> CE01
 |                    |
 |                    +--diagnosedAs--> missing_hypothesis
 |
 +--succeededBy--> V02 --contains--> C02 --provedBy--> PR01

COR01 --corrects--> C01
COR01 --proposes--> C02

C01 remains refuted; it does not become C02. V01 remains as a historical version; it is not labelled “never in force.” V02 receives its own evidence and status.

8.1 Event ledger for the worked case

The narrow ledger uses these exact locator abbreviations:

V1 = #o017-u07-packet-v1
CE = #o017-u07-two-point-counterexample
V2 = #o017-u07-packet-v2
P2 = #o017-u07-connected-proof

Event	Append-only action	Object	Reason and evidence
`O017-U07-EVT01`	`observed`	`C01 at V01/V1`	the packet’s wording and context are frozen
`O017-U07-EVT02`	`refuted`	`C01`	`O017-U07-CE01`, the two-point reproducer at `CE`
`O017-U07-EVT03`	`classified`	finding `F01`	false status; primary diagnosis of missing connectedness hypothesis; `S2/C3`
`O017-U07-EVT04`	`proposed`	`C02 in V02`	candidate wording at `V2`
`O017-U07-EVT05`	`proved`	`C02`	proof at `P2`
`O017-U07-EVT06`	`bounded`	impact of `F01`	the packet contains no audit of downstream results; the external extent remains unknown

If V02 later also needs correction, add V03 and new relationships. Do not delete EVT04 or EVT05; readers need to know what was proposed and the basis for accepting it at the time.

8.2 Minimum content of a correction record

Field	Question
`correction_id`	which correction is being discussed?
`target_version` and `target_locator`	which historical wording is affected?
`target_claim`	which normalized claim is being assessed?
`observation`	what is actually visible in the copy?
`reproducer`	what small piece of evidence demonstrates the failure?
`statement_status` and `defect_kind`	what are the formal consequence and the diagnosed cause?
`severity` and `confidence`	how extensive are the consequences if correct, and how strong is the evidence?
`replacement_claim`	what new wording is proposed?
`verification`	what evidence supports the new wording?
`supersession_relation`	which active use does the new object replace without erasing history?
`uncertainties`	what context, intention, or impact remains unknown?
`event_log`	when, and on what basis, did the status change?

9 Writing a concise issue or erratum draft

A good draft puts checkable facts first. Long greetings, judgments about the author, a history of the entire process, and speculation about intentions are unnecessary. The title names the location and problem type. The body states the version, wording, reproducer, candidate correction, and boundaries.

Local draft — not sent to anyone

Title: Proposition 7.3 in version 1.0 is missing a connectedness hypothesis

Version and location: O017-U07-V01, Proposition 7.3, #o017-u07-packet-v1.

Problem: The statement claims that every nowhere-zero continuous function $f:X\to\mathbb R$ has constant sign on an arbitrary topological space.

Minimal reproducer: Take $X=\{-1,1\}$ with the subspace topology and $f(x)=x$ . This function is continuous and nowhere zero, but $f(-1)<0<f(1)$ .

Candidate correction: Add that $X$ is connected. With $U=f^{-1}((-\infty,0))$ and $V=f^{-1}((0,\infty))$ , the two sets are open, disjoint, and cover $X$ ; connectedness forces one of them to be empty.

Scope: The counterexample establishes that the proposition is false as written. The examined packet is insufficient to determine all downstream results that might be affected.

The draft does not claim to know the author’s intention, does not request replacement of the entire work, and does not assert impact that has not been audited. If a report is addressed to a real project, the public version identity, locators accessible to the recipient, contribution policy, and authorization to send must be checked separately.

9.1 A six-sentence template

For a simple case, the following six sentences often suffice.

“I examined [version/copy] at [locator].”
“The text states [normalized claim].”
“Take [minimal object]; it satisfies [list of hypotheses].”
“However, [calculation or argument] violates [conclusion].”
“The wording [candidate correction] suffices because [short proof or proof locator].”
“I have not determined [actual uncertainty about extent, intention, or another matter].”

If the correction cannot be proved concisely, the fifth sentence should say “a correction has not yet been established” and link to the evidence dossier; do not compress a gap into false certainty.

9.2 Reports must remain accessible

Do not make color, visual position, or screenshots the sole evidence. Briefly transcribe the disputed wording, name every important table and formula, and explain differences in words. A locator such as “the red box on the right” must include a version and a label meaningful without the image. When the shape of a character is itself the problem—for example, the letter O resembling the numeral 0—record what is visible and provide a textual description; do not resolve ambiguity through color or typography alone. The draft’s order must remain logical when read linearly: target, claim, reproducer, candidate correction, then limits of certainty.

10 The uncertainty ledger

Uncertainty is not an apologetic paragraph. It is a working object with a question, available evidence, consequences if wrong, a next test, and a stop rule.

ID	Question	Known	Unknown	Impact	Next test or action
`UQ01`	is V01 false as a self-contained statement?	the packet states there are no section-level assumptions; an exact counterexample is available	no material uncertainty remains	status of C01	close at `C3`
`UQ02`	is connectedness a sufficient repair?	a complete proof is available	none concerning sufficiency	status of C02	close at `C3`
`UQ03`	is this the creator’s intended wording?	a natural and exact repair for the universal theorem	the creator’s intention is not recorded	editorial choice	do not claim to know the intention; seek the owner’s decision if actual publication is undertaken
`UQ04`	which results depend on V01?	the local packet contains no other dependencies	the entire extent outside the packet	severity	retain `S2` for the packet; audit dependencies before making broader claims
`UQ05`	has V02 been published?	V02 is only a candidate correction in this unit	there is no external release	version status	do not state that V01 has been replaced outside the packet

Here UQ01 through UQ05 abbreviate the corresponding stable identifiers O017-U07-UQ01 through O017-U07-UQ05 only within this narrow table.

Use statuses such as open, under-test, closed-supported, and closed-undetermined. “Closed-undetermined” does not mean “true”; that status preserves the limits of the examination.

10.1 Language matched to the strength of evidence

Evidence status	Appropriate language	Overstated language
text visible in a frozen copy	“copy CP01 contains …”	“all editions state …”
an exact reproducer has been checked	“the claim is false as written”	“the author does not understand …”
one correction has been proved sufficient	“adding connectedness produces a true statement”	“this is the only possible correction”
context is incomplete	“whether a section-level convention applies is not yet known”	“there are certainly no other assumptions”
some downstream results have been audited	“impact has been found in A and B”	“only A and B are affected”
computational output does not match	“the execution record is inconsistent”	“the algorithm is certainly wrong”

11 Stop and escalation gates

Care does not mean examining without limit. Use observable gates.

11.1 Stop before declaring an error

Stop before establishing the old claim’s status, and retain the finding at C0 or C1, if any of the following applies:

the version or location cannot be distinguished;
the wording is available only as a second-hand paraphrase;
the candidate example fails one hypothesis;
a section-level convention that might supply the hypothesis has not been checked; or
the numerical failure lies within unknown rounding bounds.

Stopping means not promoting the status, not discarding the record. Preserve the evidence, question, and next step.

The candidate correction’s status has its own gate. If an exact counterexample has established the old claim as false_as_written/C3, retain that status even if its replacement remains unproved. Mark the replacement unresolved, do not activate it as a correction, and stop before claiming that the problem has been repaired. Uncertainty about a repair does not weaken the evidence against the old claim.

11.2 Escalation without exaggeration

Escalate to a suitable reviewer or decision owner if:

actual or potential severity is S3 or S4, at any confidence level; C1/C2 requires further verification, whereas C3 requires a dependency audit and a decision about withholding or correcting active outputs;
the proof of the correction depends on expertise beyond the examiner’s;
two nonequivalent repairs both fit the context;
license identity or component rights affect the form of correction that may be published;
the dependency audit finds active outputs that may use a false claim; or
a computational mismatch cannot be disentangled among code, data, environment, and interpretation of results.

Escalation must bring a dossier, not merely an alarm: version, locator, reproducer, severity/confidence pair, open questions, and the requested decision. Contacting outside parties is a separate action and is not performed by this unit.

11.3 Closure conditions

A finding may be closed as corrected locally when:

the historical target remains identifiable;
the old claim’s status and evidence of failure are recorded;
the replacement claim has sufficient evidence;
the extent of the impact examined is stated;
remaining uncertainties are not concealed; and
the active artifact points to the new version without deleting the old one.

Local closure does not mean that someone else has accepted or published the correction.

12 Audit and correction protocol

Use the following sequence for mathematical or computational findings.

Register the finding. Assign an ID and record who or what triggered the examination; do not immediately issue a verdict.
Freeze the target. Record work, version, copy, locator, wording, time, and context scope.
Normalize the claim. State quantifiers, types, hypotheses, and conclusion.
Classify provisionally. Separate statement status from defect type.
Build a minimal reproducer. Check every hypothesis and show the exact failure.
Assess both axes. Assign severity and confidence with separate justifications; mark the extent not yet audited.
Seek the narrowest correction. Test reducing assumptions or weakening the conclusion; do not guess the intention.
Prove the correction. Preserve the new proof as a separate object.
Audit dependencies. Determine which results actually use the old claim and on what domains.
Append events. Link the old version, evidence of failure, correction, and candidate version; do not overwrite.
Write a concise draft. Include facts, reproducer, correction, and boundaries; keep external contact outside this process.
Apply the gates. Close, hold, or escalate according to the available evidence and authority.

The sequence may return to earlier steps. Discovering a new section-level convention, for example, changes claim normalization and may invalidate the reproducer. The event log must preserve that change rather than present the final decision as though it had been the first.

13 Guided practice: one finding, two axes, three claims

A synthetic manuscript at version 2.1 states in Lemma 4:

For every real number $x$ , $\sqrt{x^2}=x$ .

Take $x=-2$ . The hypothesis “ $x$ is real” is satisfied, whereas $\sqrt{(-2)^2}=\sqrt4=2\neq-2$ . Thus the lemma is false as written. The repair $\sqrt{x^2}=|x|$ is true for every $x\in\mathbb R$ by the definition of the principal square root. Another repair, “if $x\geq0$ , then $\sqrt{x^2}=x$ ,” is also true but narrows the domain.

Three claims must be separated:

C-old: the old universal statement, refuted by $x=-2$ ;
C-abs: the identity with absolute value, accepted with a definitional proof;
C-nonneg: the domain restriction, accepted but narrower.

We can assign confidence C3. Severity cannot yet be inferred from the lemma alone: if no other result uses it, it may be S1; if many downstream results replace $\sqrt{x^2}$ with $x$ for negative values, it may be S3. Until the dependency graph has been examined, record an unknown severity rather than choose a number that feels reasonable.

14 Checklist before finalizing a draft

Are the examined version, copy, locator, and wording exact?
Has the claim been normalized without silent changes?
Has every reproducer hypothesis actually been checked?
Is the failure of the conclusion exact and relevant?
Is statement status distinguished from the defect-type diagnosis?
Do severity and confidence have separate grounds?
Is the candidate correction proved sufficient and not stronger than necessary without justification?
Are old versions and claims still preserved in the history?
Are unaudited impact and unknown intentions still described as unknown?
Does the draft request one clear action without personal judgments?
Are stop, escalation, and contact-authorization gates respected?

15 Exercises

O017-U07-X01 — Two-layer classification. For each case below, give the statement status, the primary defect kind, and one secondary consequence if any: (a) the index $n+1$ is printed as $n-1$ , but the substitution on the next line is consistent only with $n+1$ ; (b) the proof uses unstated compactness, and the theorem fails on a noncompact domain; (c) “see Lemma 2” when two appendices each have a Lemma 2; (d) the log names code revision r3, while the manifest and output name r2;
1. an implication is true, but one paragraph uses its converse; and
2. a definition is correct but its symbol changes type without explanation. Link your answer to O017-U07-H01.
O017-U07-X02 — Severity is not confidence. Give code pairs and reasons for: (a) a certain title typo that does not affect retrieval; (b) a suspicion that the main theorem is false because one simulation is unstable; (c) an exact counterexample to a standalone exercise used nowhere else; and (d) a code-version mismatch in a principal result whose extent has not been audited. Do not produce a combined score. Link your answer to O017-U07-H02.
O017-U07-X03 — Version, locator, and reproducer. A report says only “new edition, monotonicity theorem wrong for $x=-1$ .” List the minimum questions that must be answered before the status can be raised above C0. Then write a reproducer skeleton that separates hypotheses, object, failure of the conclusion, and boundaries. Link your answer to O017-U07-H03.
O017-U07-X04 — The connectedness repair. Prove that if $X$ is connected and $f:X\to\mathbb R\setminus\{0\}$ is continuous, then the image $f(X)$ lies entirely within one component of $\mathbb R\setminus\{0\}$ . Then, for an arbitrary disconnected space, construct a continuous function into $\{-1,1\}$ taking both values. Explain why “path-connected” is an unnecessarily strong repair for this claim. Link your answer to O017-U07-H04.
O017-U07-X05 — Correction history. A version V1 contains claim CLM1; counterexample CE1 refutes it; V2 contains replacement claim CLM2; then V3 narrows the notation without changing the meaning of CLM2. Design events and directed relationships that preserve all three versions. Explain why changing the content of CLM1 into the content of CLM2 damages the audit. Link your answer to O017-U07-H05.
O017-U07-X06 — Drafts and gates. Write a six-sentence draft for the $\sqrt{x^2}=x$ case in the guided practice. Add two uncertainty entries: the extent of downstream results and the choice between an absolute-value correction and a domain restriction. Specify a stop condition and one decision that must be escalated. The draft remains local. Link your answer to O017-U07-H06.

16 Hints and answer guidance

O017-U07-H01. (a) Primarily a typographical error; the mathematical status depends on which wording is considered active, so do not change it without recording the contextual evidence; use not_applicable when the finding concerns only the printed form, then assess the formula’s claim as a separate object. (b) The theorem is false as written, with a diagnosis of a missing compactness hypothesis.
1. An ambiguous locator with claim status not_applicable. (d) A computational/provenance defect; the claim that revision r3 was run remains unsupported as written, and the evidence does not yet suffice to declare the algorithm wrong. (e) The local use is wrong even though the source implication may remain true. (f) A local exposition defect with not_applicable if no claim has yet been assessed, unless the type change makes a particular claim false. Keep status and cause as separate fields.
O017-U07-H02. Reasonable assignments are (a) S0/C3; (b) potential S3/C1, because the simulation is not yet an exact counterexample; (c) S1 or S2 with C3, depending on the exercise’s role, but with bounded extent; and (d) unknown severity or potential S3, with strong confidence only in the artifact mismatch, not in the falsity of the mathematical result. The justification for each axis matters more than any single code.
O017-U07-H03. Ask for the work’s title, edition number or version identity, copy, format, page/section/label, full wording, definition of “monotone,” domain, section-level assumptions, and time of observation. The reproducer must state the quantified claim, check that $-1$ lies in the domain and satisfies the hypotheses, calculate the exact failure, and then state whether it refutes the source claim or a strengthening invented by the reporter.
O017-U07-H04. The continuous image of a connected space is connected, whereas the two components of $\mathbb R\setminus\{0\}$ are $(-\infty,0)$ and $(0,\infty)$ . Alternatively, use preimages as in the unit’s proof. If $X=A\cup B$ is a separation, the function with value $-1$ on $A$ and $1$ on $B$ is continuous because $A$ and $B$ are both open and closed. Connectedness alone suffices, so path-connectedness adds unused structure.
O017-U07-H05. Preserve at least V1 contains CLM1, CLM1 refutedBy CE1, V2 succeeds V1, V2 contains CLM2, CLM2 provedBy P2, V3 succeeds V2, and V3 restates CLM2, or a new claim declared equivalent with proof. Observation, refutation, proposal, proof, and notation-change events remain ordered. Overwriting CLM1 removes the object that CE1 actually refuted and makes historical downstream results appear to use a claim that did not yet exist.
O017-U07-H06. The draft must name the version and location, the universal wording, $x=-2$ , the hypothesis check, the value $\sqrt{(-2)^2}=2$ , and two valid corrections. Stop before selecting one correction as the source’s intention if the intended context is unavailable. Escalate the editorial choice to the work’s owner; audit the dependency graph before assigning severity. Do not send the exercise draft.

17 Unit completion task

Prepare a complete erratum dossier for the following synthetic packet. The packet is the entire available context; do not invent authors, DOIs, pages, commits, execution results, or additional versions.

O017-U07-PKT-META — Packet identity

work_id: O017-U07-PKT-W01, Notes on Extrema on Intervals.
version_id: O017-U07-PKT-V1, label 1.2-sintetis.
copy_id: O017-U07-PKT-CP1, all five packet excerpts in this unit.
observed_at: 2026-08-21.
There are no section-level assumptions beyond those written in each excerpt.

O017-U07-PKT-S01 — Proposition 6.2

If $f:[0,1]\to\mathbb R$ is continuous, then there is exactly one $c\in(0,1)$ at which $f$ attains its maximum, and $f'(c)=0$ .

O017-U07-PKT-P01 — Proof excerpt

“The Extreme Value Theorem gives a maximum point $c$ . Since $c$ is a maximum point, its derivative is zero. Uniqueness follows because that maximum point has been chosen.”

O017-U07-PKT-T01 — Figure caption

“Graph of a continuous function on the interval $[O,1]$ .” The first character inside the interval appears as the capital letter O; the packet supplies neither a source file nor an editor’s note.

O017-U07-PKT-L01 — Back-reference

The next paragraph says “this uniqueness will be used again; see the theorem above.” Two other results lie between the paragraph and Proposition 6.2. There is no other label or locator.

O017-U07-PKT-C01 — Attached computational record

The manifest names program=gridmax.py@r2, step=0.01, and domain [0,1]. The log header names program=gridmax.py@r1; the sole output line is argmax_grid=0.99. The prose after the log states “the computation proves that the exact maximum point is $c=1$ .” The code, environment identity, and other outputs are unavailable in the packet.

Use the function $f(x)=x$ , the constant function $g(x)=0$ , and $h(x)=-|x-1/2|$ as candidate reproducers, but check for yourself which claims each function refutes. The final product must include:

a frozen-target table with the version, copy, and locator for every excerpt, and normalized wording for Proposition 6.2;
a classification of every finding with statement_status, primary type, secondary consequence, severity, confidence, and the basis for each;
minimal reproducers checking continuity, maximum location, interiority, uniqueness, and existence of a derivative where relevant: $f$ tests a boundary maximum, $g$ tests uniqueness, and $h$ tests differentiability at an interior maximum;
separation of the correction into two valid results: the Extreme Value Theorem guarantees only a maximum on $[0,1]$ , whereas the zero-derivative condition requires differentiability at an interior local maximum;
complete proofs or justifications for each replacement result, without concluding uniqueness;
separate diagnoses of the letter O, the reference “the theorem above,” and the computational-revision mismatch; do not guess that the algorithm is wrong;
an append-only correction record with the old claim’s ID, two replacement claims, evidence, a candidate version, succession relationships, and events; V1 must not be overwritten;
an impact audit of the paragraph relying on uniqueness, with status unknown for uses outside the packet;
a ledger of at least five uncertainties, including the intended character O, the choice of replacement wording, the code actually run, and the extent of downstream results;
an issue draft of no more than 180 words naming the version, location, two reproducers, separate corrections, and the evidence boundary; the draft is not sent;
stop and escalation rules explaining what can be closed on the packet’s evidence and what requires the work’s owner or a computational examiner; and
a four-sentence executive summary using no stronger language than the dossier.

17.1 Rubric

Each criterion is scored 0, 1, or 2.

Criterion	0	1	2
Targets and locators	version/location is invented or missing	some excerpts can be found	every finding points to the version, copy, packet locator, and exact wording
Mathematical reproducers	examples fail the hypotheses or do not demonstrate failure	one consequence is correctly refuted	$f(x)=x$ , $g(x)=0$ , and $h(x)=-\|x-1/2\|$ separate failures of interiority, uniqueness, and differentiability, with every hypothesis checked
Corrections and proofs	a patched wording remains false	two results are named but one condition or proof is unclear	existence of a maximum and Fermat’s condition are separated, bounded, and justified without claiming uniqueness
Two-axis classification	type, status, severity, and confidence are conflated	some fields are separated	every finding has separate fields and grounds, including unknown extent
History and uncertainty	V1 is overwritten or uncertainty is filled with guesses	the old version remains but events/questions are insufficient	claims, versions, evidence, succession, events, and at least five uncertainties are preserved
Draft and gates	the draft accuses, overstates, or is treated as already sent	main facts are present but actions/boundaries are unclear	the draft is at most 180 words, specific, and local; close, hold, and escalation rules are executable

The passing score is at least 10 out of 12, with scores of 2 in Mathematical reproducers, Corrections and proofs, and History and uncertainty. Inventing metadata, calling grid computation proof of an exact maximum, inferring a unique maximum from the selection of one point, applying the derivative condition at an interval endpoint, or deleting V1 requires revision even if the total score is sufficient.

18 Boundaries with B80 and other units

This unit does not teach programming, package APIs, root-finding or optimization algorithms, numerical error analysis, software testing, environment management, notebooks, checksums, or system commands. All that implementation work belongs to B80. When a computational defect is found, Unit 7 freezes the claim, compares the contract with the output, assesses the evidence, and formulates a correction; Unit 6 supplies the package and execution record to be examined.

Unit 2 handles searching and source authority, Unit 3 reconstructs missing proof steps, Unit 4 designs provenance, and Unit 5 develops exposition. Unit 7 does not repeat that work: it uses the results to make erratum decisions. If a missing step can be supplied without changing the claim, the reconstruction comes from Unit 3; Unit 7 determines only whether the source needs a correction note. If a paragraph needs rewriting for readability but is not wrong, its expository revision belongs to Unit 5.

Unit 8 handles reviewer reports and responses to criticism as interpersonal exchanges. Unit 7 stops at the dossier and local draft; it does not send messages, negotiate responses, or judge authors’ conduct. Unit 9 handles genuinely authorized, bounded community contributions.

19 Sources, provenance, changes, and rights

All original Indonesian prose, the operational taxonomy, severity/confidence scales, protocols, templates, tables, synthetic packets, worked cases, reproducers, proofs, erratum drafts, 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 while preserving its mathematical content, stable identifiers, and attribution.

The characterization of connectedness through nowhere-zero continuous functions, the fact that the continuous image of a connected space is connected, the Extreme Value Theorem, and the zero-derivative condition at an interior local maximum are classical mathematical results. O017 does not claim to have discovered them. The wording, pedagogical ordering, two-point example, proofs presented, and their audit function were written specifically for this unit and were not copied from a particular source.

Packets O017-U07-W01 and O017-U07-PKT-W01 are synthetic local materials, not DOIs, publisher identities, or evidence of external works bearing those titles. Unit 7 adapts no external donor and copies no donor prose. CC BY-SA 4.0 covers only the original O017 material, not external sources readers may use for independent exercises.