Chapter map

1. 1. Sample size
2. 2. Neighbourhood
3. 3. Missing neighbours
4. 4. First extrusion
5. 5. Second extrusion
6. 6. Third extrusion
7. 7. Penalty total
8. 8. Normaliser
9. 9. Scaled loss
10. 10. Continuity
11. 11. Competitor
12. 12. Direction
13. 13. Pair with trust
14. 14. Tie audit
15. 15. Global audit
16. 16. Claim

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Sample size

This is where modelling choices begin to affect evidence. Set n=10.

A concrete situation

Ten observations are compared.

What the mathematics does not license

Use the same item set in both spaces. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record n. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Neighbourhood

The central mathematical idea is concrete. Set k=2.

A concrete situation

Original top-two neighbours are the target.

What the mathematics does not license

k defines locality. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record k. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Missing neighbours

Here the geometry becomes operational. Use embedded ranks 4, 5 and 6.

A concrete situation

Three true neighbours fell outside embedded top two.

What the mathematics does not license

Confirm each was originally within k. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

List ranks. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. First extrusion

This step deserves a careful pause. Compute 4-2.

A concrete situation

Penalty=2.

What the mathematics does not license

Use embedded rank for continuity. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Show step. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Second extrusion

The useful question is not only how to calculate, but what the calculation assumes. Compute 5-2.

A concrete situation

Penalty=3.

What the mathematics does not license

Keep direction clear. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Show step. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Third extrusion

A small example makes the mechanism visible. Compute 6-2.

A concrete situation

Penalty=4.

What the mathematics does not license

Farther loss costs more. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Show step. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Penalty total

This is where modelling choices begin to affect evidence. Add 2+3+4.

A concrete situation

Total extrusion penalty=9.

What the mathematics does not license

Audit all observations. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record sum. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Normaliser

The central mathematical idea is concrete. Compute 10x2x13.

A concrete situation

Denominator=260.

What the mathematics does not license

Same formula needs same n and k. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Scaled loss

Here the geometry becomes operational. Compute 18/260.

A concrete situation

Loss=.069231.

What the mathematics does not license

Do not divide by pair count. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record loss. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Continuity

This step deserves a careful pause. Compute 1-.069231.

A concrete situation

C(2)=.930769.

What the mathematics does not license

Higher is better. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record score. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Competitor

The useful question is not only how to calculate, but what the calculation assumes. Use penalty total 4.

A concrete situation

Competitor C(2)=.969231.

What the mathematics does not license

Hold preprocessing fixed. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Direction

A small example makes the mechanism visible. Name missing original neighbours.

A concrete situation

Continuity controls extrusions.

What the mathematics does not license

It does not directly reject false displayed neighbours. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Label direction. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Pair with trust

This is where modelling choices begin to affect evidence. Calculate trustworthiness too.

A concrete situation

Two directions expose different errors.

What the mathematics does not license

A single average can hide trade-offs. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report both. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Tie audit

The central mathematical idea is concrete. Create equal distances.

A concrete situation

Rank assignments can change.

What the mathematics does not license

Document deterministic or average-rank rules. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Stress-test. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Global audit

Here the geometry becomes operational. Compare distant-pair behaviour.

A concrete situation

Local continuity says little about global scale.

What the mathematics does not license

Add a separate global criterion. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Measure. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Claim

This step deserves a careful pause. State metric, scaling, n, k and score.

A concrete situation

Competitor retains more original neighbours at k=2.

What the mathematics does not license

Do not claim every structure is preserved. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Bound conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

Continuity asks whether points that were genuine neighbours in the original space remain neighbours after projection. For every original k-neighbour that falls outside the embedded k-neighbourhood, continuity penalises the point by how far its embedded rank lies beyond k. It therefore focuses on missing local relationships, also called extrusions in co-ranking language.

Use n=10 and k=2, so the normalising denominator is again 260. Suppose three true neighbours are missing from the displayed k-neighbourhoods and receive low-dimensional ranks 4, 5 and 6. Their penalties are 2, 3 and 4, totalling 9. Continuity is 1-[2(9)/260]=0.930769.

A competing embedding has total extrusion penalty 4. Its continuity is 1-[2(4)/260]=0.969231, so it retains more original local structure at k=2. Yet it might still introduce false neighbours. A map can score well on continuity and less well on trustworthiness, which is why both directions should be read together.

Continuity is not a guarantee that global geometry, cluster gaps, angles or densities are faithful. It depends on the original distance definition and on k. Ties, duplicates, missing values, mixed units and categorical encodings can change ranks before any projection occurs. Publish the pair of curves, preprocessing choices and uncertainty across seeds.

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Frequently asked questions

What is continuity in dimensionality reduction?

A normalized rank-based score that penalises original k-neighbours that fall outside the embedded k-neighbourhood.

Which direction is better?

Higher is better, with one indicating no missing original neighbours at the chosen k.

What is an extrusion?

A genuine original neighbour is pushed beyond the chosen neighbourhood in the embedding.

Does continuity reject false neighbours?

Not directly. Trustworthiness handles that complementary direction.

Does it measure global geometry?

No. Add a global or task-specific criterion when that matters.

What must be reported?

n, k, both distance definitions, scaling, tie rule, preprocessing, seeds, both directional curves and uncertainty.

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Primary source and fact-check note

Jarkko Venna and Samuel Kaski, “Neighborhood Preservation in Nonlinear Projection Methods: An Experimental Study,” ICANN 2001, DOI 10.1007/3-540-44668-0_68; co-ranking formulation checked against Lee and Verleysen (2009), DOI 10.1016/j.neucom.2008.12.017. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Trustworthiness and false-neighbour errors
• How lossy projection works
• Vector spaces in AI
• Locally Linear Embedding
• How Mathematics Works: Statistics

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: rank evidence, scale and responsible use

Retention audit

For each original top-k neighbour, record its embedded rank and mark every extrusion.

Two-direction audit

Place continuity beside trustworthiness so missing true neighbours and false displayed neighbours remain separate.

Global audit

Add a criterion for global distances or downstream performance when local retention is not the whole objective.

Stability audit

Repeat across stochastic seeds, samples and preprocessing choices; report the spread and failure cases.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild one complete example from identifiers. Record original neighbour sets, embedded ranks, extrusion penalties, normaliser and C(k). Explain each denominator before using software.

One-change experiment

Move exactly one neighbour rank across the k boundary. Recalculate the affected cell, penalty or overlap and state why the score changes.

Tie experiment

Create two equal distances, apply a documented tie rule and show whether the conclusion is stable.

Scale experiment

Repeat at three values of k. Describe what each neighbourhood size means for the actual application.

Metric experiment

Compare Euclidean distance with one justified alternative after appropriate preprocessing. Attribute changes to ranks, not to vague algorithm behaviour.

Seed experiment

Run a stochastic embedding repeatedly. Report median, range and a visibly poor seed instead of publishing only the best map.

Parent explanation

Explain why a beautiful two-dimensional picture can contain false neighbours or hide true ones without implying that all projections are useless.

Student transfer

Apply the same ranking logic to a school map, music recommendations or image retrieval. Identify what counts as a true neighbour and what evidence could refute it.

Limit statement

Write one supported sentence, one overclaim and one repaired claim that names n, k, distance, preprocessing and uncertainty.

Next measurement

Choose a complementary local, global or task-specific check and explain the distinct question it answers.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Sample size

Notebook step 1 isolates sample size. Set n=10. The auditable result is: Ten observations are compared. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Use the same item set in both spaces. Record n. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 2: Neighbourhood

Notebook step 2 isolates neighbourhood. Set k=2. The auditable result is: Original top-two neighbours are the target. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: k defines locality. Record k. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 3: Missing neighbours

Notebook step 3 isolates missing neighbours. Use embedded ranks 4, 5 and 6. The auditable result is: Three true neighbours fell outside embedded top two. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Confirm each was originally within k. List ranks. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 4: First extrusion

Notebook step 4 isolates first extrusion. Compute 4-2. The auditable result is: Penalty=2. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Use embedded rank for continuity. Show step. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 5: Second extrusion

Notebook step 5 isolates second extrusion. Compute 5-2. The auditable result is: Penalty=3. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Keep direction clear. Show step. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 6: Third extrusion

Notebook step 6 isolates third extrusion. Compute 6-2. The auditable result is: Penalty=4. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Farther loss costs more. Show step. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 7: Penalty total

Notebook step 7 isolates penalty total. Add 2+3+4. The auditable result is: Total extrusion penalty=9. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Audit all observations. Record sum. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 8: Normaliser

Notebook step 8 isolates normaliser. Compute 10x2x13. The auditable result is: Denominator=260. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Same formula needs same n and k. Record value. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 9: Scaled loss

Notebook step 9 isolates scaled loss. Compute 18/260. The auditable result is: Loss=.069231. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Do not divide by pair count. Record loss. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 10: Continuity

Notebook step 10 isolates continuity. Compute 1-.069231. The auditable result is: C(2)=.930769. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Higher is better. Record score. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 11: Competitor

Notebook step 11 isolates competitor. Use penalty total 4. The auditable result is: Competitor C(2)=.969231. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Hold preprocessing fixed. Compare. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 12: Direction

Notebook step 12 isolates direction. Name missing original neighbours. The auditable result is: Continuity controls extrusions. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: It does not directly reject false displayed neighbours. Label direction. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 13: Pair with trust

Notebook step 13 isolates pair with trust. Calculate trustworthiness too. The auditable result is: Two directions expose different errors. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: A single average can hide trade-offs. Report both. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 14: Tie audit

Notebook step 14 isolates tie audit. Create equal distances. The auditable result is: Rank assignments can change. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Document deterministic or average-rank rules. Stress-test. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 15: Global audit

Notebook step 15 isolates global audit. Compare distant-pair behaviour. The auditable result is: Local continuity says little about global scale. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Add a separate global criterion. Measure. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Evidence notebook 16: Claim

Notebook step 16 isolates claim. State metric, scaling, n, k and score. The auditable result is: Competitor retains more original neighbours at k=2. Write the identifiers, rank lists and arithmetic on the page before reading a software score. In this article the evidence concerns true original neighbours that disappear beyond rank k after projection. The main safeguard is: Do not claim every structure is preserved. Bound conclusion. Then change one input while keeping n, k, preprocessing, distance metric and tie convention fixed. Recalculate the affected rank, count, penalty, overlap or matrix cell and explain the direction of movement. If a classmate cannot reproduce the value from the notebook, the value is not yet ready for a chart or recommendation. Finally, record which conclusion is supported at this neighbourhood scale and which wider conclusion still needs a global, task-specific or out-of-sample check.

Failure cases and responsible recovery

A true group fragments

Members that were close originally scatter across the display. Trace every missing original neighbour to its embedded rank and calculate extrusion penalties before calling the original group incoherent. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

High continuity, false neighbours

Most true neighbours survive, yet extra distant points enter the displayed neighbourhood. Pair continuity with trustworthiness so retention and contamination stay separate instead of being hidden in one attractive average. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Boundary ties

Equal embedded distances place a genuine neighbour just inside or outside k. Declare the tie rule and show the score under plausible alternatives before treating a small change as a scientific discovery. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Different meaningful scales

A small k protects immediate matches while a larger k protects community structure. Plot continuity across a prespecified range and explain which scale corresponds to the educational, engineering or scientific decision. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Preprocessing leak

Scaling parameters learned from the full dataset influence original ranks and the projection. Fit transformations within training data, carry them forward unchanged and reserve untouched observations for the final claim. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Global geometry is compressed

Continuity remains strong while distant groups move closer. Add an appropriate global criterion and tell readers that retained local neighbours do not validate angles, axes, densities or inter-cluster gaps. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Subgroup loss

The overall score looks good but a smaller subgroup loses many true neighbours. Report point-level or subgroup summaries with adequate uncertainty and investigate measurement coverage rather than hiding the failure in the mean. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.

Responsible conclusion

Say exactly which original relationships remain visible at specified n, k, metric and preprocessing. Avoid promising that the embedding preserves every structure or guarantees downstream accuracy. Work through one small counterexample by hand, then repeat it with a second k. Record what changes, what remains stable and which additional evidence would resolve the uncertainty. A student should be able to explain the mechanism without using the colour or shape of the final picture as proof.