The Lumenary · Research pathways

Research ethics and attribution

Name: ______________________

Date: ______________________

You will learn to

Cite sources accurately, including results learned from talks, software, databases and AI systems
Decide authorship and acknowledgements fairly in a collaborative project, knowing that mathematics commonly lists authors alphabetically
Respond appropriately when an error is found in one's own published work or in someone else's
Label every claim in your writing as proved, cited, conjectured, heuristic or computational evidence
Explain, with a simple probabilistic model, why agreement among AI systems is no substitute for checking a proof

1.
A student finds a neat proof of a lemma in a recent preprint, rewrites it carefully in his own words and notation, and includes it in his thesis without mentioning the preprint. Which description is accurate?
- A.This is plagiarism: the idea of the proof is someone else's, whatever the wording
- B.It is fine, because none of the words are copied
- C.It is fine, because preprints are not published
- D.It is fine, because it is only a lemma, not a main result
2.
Which of the following should be cited in a research paper? (Choose all that apply.)
- A.A lemma you use from a 2023 arXiv preprint
- B.A table of values computed with a computer algebra system
- C.A result you heard in a seminar talk and have permission to use
- D.A counting argument you devised yourself for this paper
- E.The fact that $2 + 2 = 4$
3.
A mathematics paper by three authors lists them as Bose, Okafor, Zhang. What can a reader conclude from the order?
- A.Nothing about contributions: the names are in alphabetical order, the usual convention in mathematics
- B.Bose did most of the work
- C.Zhang is the senior author who supervised the work
- D.The authors certainly contributed equally
4.
You discover that the main lemma of your paper published last year has a fatal gap, and the main theorem may be false. Which actions are appropriate? (Choose all that apply.)
- A.Inform the journal's editor and publish a correction or, if the main result fails, a retraction
- B.Post a revised preprint with a clear note explaining what is wrong
- C.Tell the people you know are building on the result
- D.Fix it quietly in a later paper without mentioning the earlier error
- E.Wait and see whether anyone notices
5.
An AI system has produced a proof of a lemma you need, and three other AI systems say the proof is correct. What should you do before using the lemma in a paper? (Choose all that apply.)
- A.Check the proof yourself line by line, or verify it formally
- B.Disclose in the paper how AI systems were used
- C.Verify that every reference the AI system gave exists and says what is claimed
- D.List the AI system as a co-author
- E.State that the proof was verified by four independent AI systems
6.
In the lesson's model take prior $\pi_0 = \tfrac12$ , $q = \tfrac35$ and $\rho = 1$ (the checkers always share one verdict). What is the probability that the proof is correct after $10$ checkers all accept it?
- A. $\tfrac58$
- B.about $0.994$
- C. $\tfrac12$
- D. $1$
7.
The Lean theorem cauchy_sum_theorem_false refutes the claim that a pointwise limit of continuous functions $\R \to \R$ is continuous. Which hypothesis, added to the claim, gives a true theorem (Mathlib's TendstoUniformly.continuous)?
- A.The convergence is uniform: TendstoUniformly f g atTop
- B.Each $f_n$ is bounded
- C.Each $f_n$ is monotone
- D.The $f_n$ converge at every rational point
8.
Mersenne (1644) claimed that for primes $p \le 257$ , $2^p - 1$ is prime exactly for $p \in \{2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257\}$ . The true list is $\{2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127\}$ . How many exponents did Mersenne get wrong (wrongly included or wrongly omitted)?
9.
A "picture frame" polyhedron (a polyhedral torus) has $V - E + F = 0$ , contradicting the published claim that $V - E + F = 2$ for every polyhedron. Following Lakatos, which response repairs the theorem in the way the history of topology vindicated?
- A.Add the hypothesis the proof actually uses: the polyhedron's surface is homeomorphic to a sphere (and in general $V - E + F = 2 - 2g$ )
- B.Declare that a picture frame is not a real polyhedron
- C.Abandon Euler's formula as false
- D.Ignore the counterexample, since the formula holds in most cases
10.
Write posterior(prior, q, rho, k) for the lesson's model, and checkers_needed(prior, q, target) returning the least k >= 1 for which independent checkers (rho = 0) give a posterior of at least target. The tests also check that correlated checkers never pass the ceiling.
Computational — Practice

Answer key — Research ethics and attribution

1.
Plagiarism is presenting another person's words, ideas or results as one's own. Rewriting a proof in new words without credit still presents someone else's idea as one's own. The fix is easy: cite the preprint.
2.
Cite the preprint, the software (with version) and the result from the talk (as a personal communication, with permission). Your own new argument and elementary facts need no citation.
3.
The names are alphabetical, so the order says nothing about who did what. All three are authors because each made a substantial intellectual contribution and takes responsibility for the whole paper.
4.
Correct the record where it was made (editor; corrigendum or retraction), update the preprint with an explanatory note, and inform those who rely on the result. Quiet fixes and waiting both leave others building on a false statement.
5.
Check (or formally verify) the proof, disclose the use of AI, and verify every reference. An AI system cannot be an author, and agreement among AI systems is not a verification: their errors are correlated, so agreement carries limited evidence and never amounts to proof.
6.
With $\rho = 1$ the checkers return one common verdict, so all ten accept a flawed proof with probability $q = 3/5$ , exactly as one checker would. The posterior is $\frac{1/2}{1/2 + (1/2)(3/5)} = \frac{5}{8}$ , the same as after a single acceptance.
7.
Uniform convergence repairs the theorem: given $\varepsilon$ , pick $n$ with $\sup_x |f_n - g| < \varepsilon/3$ and use the continuity of $f_n$ . The counterexample $f_n(x) = c(x)^n$ is bounded and monotone and converges everywhere, so the other options cannot help — its convergence is not uniform near $x = 1$ .
8.
Wrongly included: $67$ (Cole factored $2^{67} - 1$ in 1903) and $257$ . Wrongly omitted: $61$ (Pervushin, 1883), $89$ and $107$ (Powers, 1911 and 1914). Five errors in all; the lesson's experiment and the Lean theorem mersenne_list_corrected recheck them.
9.
The proof uses the fact that the surface can be cut open and flattened like a sphere. Adding that hypothesis gives a true theorem, and the general formula $V - E + F = 2 - 2g$ for surfaces of genus $g$ explains the picture frame ( $g = 1$ ). Honest repair credits the counterexample by incorporating its lesson.
10.
```
def posterior(prior, q, rho, k):
    fooled = rho * q + (1 - rho) * q ** k
    return prior / (prior + (1 - prior) * fooled)


def checkers_needed(prior, q, target):
    k = 1
    while posterior(prior, q, 0, k) < target:
        k += 1
    return k
```
With $\pi_0 = \tfrac12$ and $q = \tfrac35$ , nine independent checkers give at least $0.99$ (since $(3/5)^9 \approx 0.0101 < 1/99$ ). Correlated checkers with $\rho = \tfrac12$ never pass $\tfrac{10}{13}$ , however many there are — the proposition in the lesson, proved in Lean as posterior_le_ceiling.