Chapter map

1. 1. Pair table
2. 2. P-positive total
3. 3. Q-positive total
4. 4. P to Q numerator
5. 5. P to Q score
6. 6. Q to P numerator
7. 7. Q to P score
8. 8. Asymmetry
9. 9. Competitor counts
10. 10. Competitor forward
11. 11. Competitor reverse
12. 12. Joint decision
13. 13. Zero denominator
14. 14. Chance
15. 15. Cluster sizes
16. 16. Claim

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1. Pair table

This step deserves a careful pause. Use a=4,b=8,c=5,d=11.

A concrete situation

Twenty-eight pairs are classified.

What the mathematics does not license

Define every cell. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record counts. 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. P-positive total

The useful question is not only how to calculate, but what the calculation assumes. Compute a+b.

A concrete situation

P joins 12 pairs.

What the mathematics does not license

This is the first denominator. Evidence becomes stronger when the limitation is tested instead of hidden.

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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3. Q-positive total

A small example makes the mechanism visible. Compute a+c.

A concrete situation

Q joins 9 pairs.

What the mathematics does not license

This is the reverse denominator. Keeping this boundary clear makes the result easier to audit and transfer.

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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4. P to Q numerator

This is where modelling choices begin to affect evidence. Use a=4.

A concrete situation

Four P-positive pairs survive in Q.

What the mathematics does not license

Keep the conditioning set visible. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record numerator. 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. P to Q score

The central mathematical idea is concrete. Compute 4/12.

A concrete situation

Directional Wallace=.333333.

What the mathematics does not license

It resembles pair recall under this orientation. Evidence becomes stronger when the limitation is tested instead of hidden.

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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6. Q to P numerator

Here the geometry becomes operational. Use a=4 again.

A concrete situation

Four Q-positive pairs were also in P.

What the mathematics does not license

Same numerator, new denominator. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record numerator. 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. Q to P score

This step deserves a careful pause. Compute 4/9.

A concrete situation

Reverse Wallace=.444444.

What the mathematics does not license

It resembles pair precision under this orientation. A trustworthy workflow records the choice before seeing the final test result.

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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8. Asymmetry

The useful question is not only how to calculate, but what the calculation assumes. Compare .333333 and .444444.

A concrete situation

Directions answer different questions.

What the mathematics does not license

Do not average automatically. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Interpret. 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. Competitor counts

A small example makes the mechanism visible. Use a=8,b=2,c=4,d=14.

A concrete situation

Counts total 28.

What the mathematics does not license

Check feasibility. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record counts. 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. Competitor forward

This is where modelling choices begin to affect evidence. Compute 8/10.

A concrete situation

Forward score=.80.

What the mathematics does not license

Same direction is required. A trustworthy workflow records the choice before seeing the final test result.

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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11. Competitor reverse

The central mathematical idea is concrete. Compute 8/12.

A concrete situation

Reverse score=.666667.

What the mathematics does not license

Keep denominator visible. Evidence becomes stronger when the limitation is tested instead of hidden.

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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12. Joint decision

Here the geometry becomes operational. Both scores improve.

A concrete situation

Competitor is preferred in both directions.

What the mathematics does not license

Trade-offs can occur in other cases. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State result. 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. Zero denominator

This step deserves a careful pause. Let P have no positive pairs.

A concrete situation

P-to-Q coefficient is undefined.

What the mathematics does not license

Do not set it to one silently. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Flag failure. 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. Chance

The useful question is not only how to calculate, but what the calculation assumes. Random partitions can yield positive scores.

A concrete situation

Raw Wallace is not chance adjusted.

What the mathematics does not license

Add adjusted or resampled evidence. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit. 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. Cluster sizes

A small example makes the mechanism visible. Create one dominant cluster.

A concrete situation

Positive-pair totals can surge.

What the mathematics does not license

Report margins. Keeping this boundary clear makes the result easier to audit and transfer.

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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16. Claim

This is where modelling choices begin to affect evidence. Name orientation,a,b,c,d and both ratios.

A concrete situation

Competitor preserves more positive pairs both ways.

What the mathematics does not license

Do not claim identical partitions. A trustworthy workflow records the choice before seeing the final test result.

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

Wallace coefficients deliberately keep the two directions of partition agreement separate. In pair-count notation, a is the number of pairs joined in both partitions, b the pairs joined only in P, and c the pairs joined only in Q. The directional scores are W(P to Q)=a/(a+b) and W(Q to P)=a/(a+c). They answer different conditional questions.

For the eight-observation pair table a=4, b=8, c=5 and d=11, W(P to Q)=4/12=0.333333. Only one third of the pairs joined in P remain joined in Q. In the reverse direction, W(Q to P)=4/9=0.444444. About four ninths of the pairs joined in Q are also joined in P.

A competitor with a=8,b=2,c=4,d=14 gives W(P to Q)=8/10=0.80 and W(Q to P)=8/12=0.666667. It improves both directions. If one coefficient rose while the other fell, the conclusion would depend on whether preserving P-groups or avoiding extra Q-merges mattered more.

Software names Wallace I and Wallace II are not always intuitive, so label each score by its denominator and direction. A zero positive-pair denominator makes that direction undefined. The coefficients are not chance-adjusted and can be affected by cluster-size imbalance. Report both directions, not their favourite-looking one.

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Frequently asked questions

What are Wallace coefficients?

Two directional conditional probabilities of preserving positive pair relationships between partitions.

Why are there two values?

The denominators condition on different partitions, so splits and merges are not interchangeable.

Which direction is precision?

It depends on which partition is treated as reference; label the denominator and direction explicitly.

Are they chance adjusted?

The raw coefficients are not.

When is a coefficient undefined?

When its conditioning partition contains no co-clustered pairs.

What must be reported?

Orientation, a, b, c, d, both denominators, both values, cluster-size margins and any adjustment or resampling.

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Primary source and fact-check note

David L. Wallace, “A Method for Comparing Two Hierarchical Clusterings: Comment,” Journal of the American Statistical Association 78(383), 569–576 (1983), DOI 10.1080/01621459.1983.10478009. 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

• Fowlkes–Mallows geometric mean
• B-Cubed precision and recall
• Adjusted Rand Index and chance correction
• How Mathematics Works: Statistics
• Bukit Timah Mathematics learning routes

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: partition-comparison evidence and responsible use

Data audit

Freeze the shared item set, identifiers, exclusions, missing-label rules, weights, sample size and any reference partition.

Partition audit

Record both label vectors, cluster counts, sizes, unassigned items, overlaps and whether labels are hard or soft.

Table audit

Show the full contingency table and derive a, b, c, d, row maxima, column maxima and positive-pair totals.

Direction audit

Name reference and candidate, split and merge directions, denominators, normalization and lower-or-higher selection rule.

Arithmetic audit

Recalculate combinations, pair totals, component ratios, means, raw distances and normalized distances independently.

Uncertainty audit

Repeat samples, label perturbations, cluster counts, algorithms, starts and plausible matching or missing-item conventions.

Meaning audit

Compare partition agreement with geometry, stability and domain usefulness without assuming that either partition is true.

Final claim

State the exact pair or overlap evidence and stop before causal, semantic, career or guaranteed-performance claims.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Pair table

Begin with this mechanism: Use a=4,b=8,c=5,d=11. Recreate the concrete case: Twenty-eight pairs are classified. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Define every cell. Complete the transfer step: Record counts. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: P-positive total

Begin with this mechanism: Compute a+b. Recreate the concrete case: P joins 12 pairs. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: This is the first denominator. Complete the transfer step: Show step. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Q-positive total

Begin with this mechanism: Compute a+c. Recreate the concrete case: Q joins 9 pairs. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: This is the reverse denominator. Complete the transfer step: Show step. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: P to Q numerator

Begin with this mechanism: Use a=4. Recreate the concrete case: Four P-positive pairs survive in Q. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Keep the conditioning set visible. Complete the transfer step: Record numerator. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 5: P to Q score

Begin with this mechanism: Compute 4/12. Recreate the concrete case: Directional Wallace=.333333. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It resembles pair recall under this orientation. Complete the transfer step: Record value. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Q to P numerator

Begin with this mechanism: Use a=4 again. Recreate the concrete case: Four Q-positive pairs were also in P. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Same numerator, new denominator. Complete the transfer step: Record numerator. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Q to P score

Begin with this mechanism: Compute 4/9. Recreate the concrete case: Reverse Wallace=.444444. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It resembles pair precision under this orientation. Complete the transfer step: Record value. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Asymmetry

Begin with this mechanism: Compare .333333 and .444444. Recreate the concrete case: Directions answer different questions. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not average automatically. Complete the transfer step: Interpret. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 9: Competitor counts

Begin with this mechanism: Use a=8,b=2,c=4,d=14. Recreate the concrete case: Counts total 28. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Check feasibility. Complete the transfer step: Record counts. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Competitor forward

Begin with this mechanism: Compute 8/10. Recreate the concrete case: Forward score=.80. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Same direction is required. Complete the transfer step: Record value. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: Competitor reverse

Begin with this mechanism: Compute 8/12. Recreate the concrete case: Reverse score=.666667. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Keep denominator visible. Complete the transfer step: Record value. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Joint decision

Begin with this mechanism: Both scores improve. Recreate the concrete case: Competitor is preferred in both directions. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Trade-offs can occur in other cases. Complete the transfer step: State result. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 13: Zero denominator

Begin with this mechanism: Let P have no positive pairs. Recreate the concrete case: P-to-Q coefficient is undefined. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not set it to one silently. Complete the transfer step: Flag failure. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Chance

Begin with this mechanism: Random partitions can yield positive scores. Recreate the concrete case: Raw Wallace is not chance adjusted. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Add adjusted or resampled evidence. Complete the transfer step: Audit. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Cluster sizes

Begin with this mechanism: Create one dominant cluster. Recreate the concrete case: Positive-pair totals can surge. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Report margins. Complete the transfer step: Stress-test. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: Name orientation,a,b,c,d and both ratios. Recreate the concrete case: Competitor preserves more positive pairs both ways. Write a, b, c, d, both positive-pair denominators, both directional coefficients, orientation and candidate comparison with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not claim identical partitions. Complete the transfer step: Bound conclusion. Change exactly one item, label, cluster size, contingency-table cell, pair count, direction, denominator, normalization, matching rule, missing-item policy or random seed. Explain which mathematical object changes, which score moves and which interpretation must weaken. Keep pair agreement, splits, merges, geometry, stability, uncertainty and practical usefulness on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving all singletons, one giant cluster, zero positive-pair denominator, missing labels, unequal cluster sizes, duplicated identifiers or a nonmaximum table-cell change. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.