Chapter map

1. 1. Case eligibility
2. 2. Time origin
3. 3. Period partition
4. 4. Exposure probabilities
5. 5. Occupancy vector
6. 6. M1 maximum
7. 7. Adjacent sums
8. 8. M2 maximum
9. 9. Fixed-total null
10. 10. Exact enumeration
11. 11. Simulation reference
12. 12. Joint statistic
13. 13. Multiple groups
14. 14. Calendar sensitivity
15. 15. Seasonality audit
16. 16. Independent follow-up

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1. Case eligibility

A small example makes the mechanism visible. Every included case follows one definition and observation frame.

A concrete situation

Twelve cases belong to one cohort.

What the mathematics does not license

Changing inclusion dates changes occupancy. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Freeze the protocol. 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. Time origin

This is where modelling choices begin to affect evidence. Use one calendar and time zone.

A concrete situation

Dates become quarter numbers 1 to 4.

What the mathematics does not license

Mixed zones move boundary cases. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Standardize timestamps. 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. Period partition

The central mathematical idea is concrete. Divide time into declared intervals.

A concrete situation

A year becomes four quarters.

What the mathematics does not license

Bin width determines visible bursts. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Choose before inspection. 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. Exposure probabilities

Here the geometry becomes operational. Assign null probability to each period.

A concrete situation

Equal quarters use .25 each only if exposure is equal.

What the mathematics does not license

Population and ascertainment may vary. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use weighted probabilities. 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. Occupancy vector

This step deserves a careful pause. Count cases in every period.

A concrete situation

Counts are 1,6,3,2.

What the mathematics does not license

Reporting backlog can manufacture a peak. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit delay distributions. 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. M1 maximum

The useful question is not only how to calculate, but what the calculation assumes. Take the largest single-period count.

A concrete situation

M1 equals 6.

What the mathematics does not license

The selected period is data-dependent. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use the maximum distribution. 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. Adjacent sums

A small example makes the mechanism visible. Add every permitted pair of successive periods.

A concrete situation

Sums are 7,9,5.

What the mathematics does not license

Circular wraparound is a separate assumption. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Declare calendar topology. 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. M2 maximum

This is where modelling choices begin to affect evidence. Take the largest adjacent-pair total.

A concrete situation

M2 equals 9.

What the mathematics does not license

M2 shares cases with M1. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Model joint dependence. 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. Fixed-total null

The central mathematical idea is concrete. Condition on the group total and randomize timing.

A concrete situation

Twelve cases follow a multinomial allocation.

What the mathematics does not license

Independent Poisson sampling answers another question. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State conditioning. 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. Exact enumeration

Here the geometry becomes operational. Enumerate occupancy vectors when totals are modest.

A concrete situation

All four-cell allocations summing to 12 are considered.

What the mathematics does not license

Unequal probabilities change weights. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use correct multinomial masses. 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. Simulation reference

This step deserves a careful pause. Draw null allocations and recompute both maxima.

A concrete situation

9999 vectors estimate the joint tail.

What the mathematics does not license

Monte Carlo error affects small p-values. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report seed and precision. 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 statistic

The useful question is not only how to calculate, but what the calculation assumes. Combine M1 and M2 with a valid bivariate rule.

A concrete situation

A calibrated score respects their correlation.

What the mathematics does not license

Minimum unadjusted p is invalid. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use published or simulated calibration. 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. Multiple groups

A small example makes the mechanism visible. Aggregate evidence across communities or cohorts.

A concrete situation

Each town contributes an observed-minus-expected maximum.

What the mathematics does not license

Groups with different totals have different nulls. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Standardize by their own moments. 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. Calendar sensitivity

This is where modelling choices begin to affect evidence. Shift only predeclared origins or use rolling methods.

A concrete situation

Monthly and quarterly analyses answer different scales.

What the mathematics does not license

Searching all origins inflates evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Adjust the search. 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. Seasonality audit

The central mathematical idea is concrete. Compare with expected seasonal timing.

A concrete situation

Winter has a higher baseline probability.

What the mathematics does not license

Equal cells create false clustering. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Model known seasonality. 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. Independent follow-up

Here the geometry becomes operational. Test later years without changing bins.

A concrete situation

A later cohort repeats the peak scale.

What the mathematics does not license

Replication remains descriptive. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Investigate exposures and mechanisms. 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

The Ederer–Myers–Mantel procedure studies temporal concentration through occupancy maxima. Cases are grouped into comparable time periods within each spatial unit or study group. M1 is the largest count in any one period. M2 is the largest count across any two successive periods. These maxima are designed to notice a sharp burst or a burst that straddles a calendar boundary.

Under a null of random timing, each group’s fixed total is allocated across periods according to declared probabilities, often equal only when exposure time and ascertainment are comparable. The observed maxima are compared with exact tables, formulas or simulation. M1 and M2 are dependent because the largest single period usually contributes to the largest adjacent pair, so separate unadjusted tests do not form a valid joint conclusion.

For twelve cases across four quarters with counts 1, 6, 3 and 2, M1 equals 6. The adjacent two-quarter sums are 7, 9 and 5, so M2 equals 9 when the calendar is treated as non-circular. Those values describe concentration. Their evidence depends on the multinomial null, the number of units being combined, the period probabilities and a joint calibration that respects dependence.

The procedure can reveal unusual temporal bunching across several communities or cohorts, but period choice, reporting delay, seasonality and changing population size can manufacture maxima. Analysts should predefine time bins, include unequal exposure probabilities where necessary, avoid rotating the calendar after seeing the data, and examine exact dates or independent periods before interpreting a burst as an epidemic process.

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Frequently asked questions

What is M1?

M1 is the largest case count in any one declared time period.

What is M2?

M2 is the largest case count in any two successive declared periods.

Why use both?

M1 detects a sharp peak, while M2 can detect a burst split across a period boundary.

What is the null model?

A fixed group total is allocated across periods according to stated timing probabilities.

Why not test M1 and M2 separately?

They are dependent, so separate unadjusted conclusions have the wrong joint error rate.

Does a large maximum prove an epidemic?

No. Seasonality, reporting delays, exposure changes and retrospective bin choices can create peaks.

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Primary source and fact-check note

Fred Ederer, Max H. Myers and Nathan Mantel, “A Statistical Problem in Space and Time: Do Leukemia Cases Come in Clusters?” Biometrics 20(3), 626–638 (1964), DOI 10.2307/2528500. 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

• Knox space–time interaction tests
• Jacquez rank-overlap methods
• How Mathematics Works

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: topic-specific validation and responsible use

Eligibility audit

Freeze case definition and observation frame.

Calendar audit

Document bins, boundaries and time zones.

Exposure audit

Use unequal period probabilities when required.

Maximum audit

Reproduce M1 and M2 from the occupancy vector.

Dependence audit

Use a valid joint reference for both maxima.

Search audit

Control multiple groups, widths and origins.

Transfer audit

Reserve future periods before looking.

Final claim

A temporal maximum is concentration, not an epidemic cause.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Case eligibility

Begin with this mechanism: Every included case follows one definition and observation frame. Recreate the concrete case: Twelve cases belong to one cohort. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Changing inclusion dates changes occupancy. Complete the transfer step: Freeze the protocol. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Time origin

Begin with this mechanism: Use one calendar and time zone. Recreate the concrete case: Dates become quarter numbers 1 to 4. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Mixed zones move boundary cases. Complete the transfer step: Standardize timestamps. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 3: Period partition

Begin with this mechanism: Divide time into declared intervals. Recreate the concrete case: A year becomes four quarters. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Bin width determines visible bursts. Complete the transfer step: Choose before inspection. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Exposure probabilities

Begin with this mechanism: Assign null probability to each period. Recreate the concrete case: Equal quarters use .25 each only if exposure is equal. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Population and ascertainment may vary. Complete the transfer step: Use weighted probabilities. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Occupancy vector

Begin with this mechanism: Count cases in every period. Recreate the concrete case: Counts are 1,6,3,2. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Reporting backlog can manufacture a peak. Complete the transfer step: Audit delay distributions. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: M1 maximum

Begin with this mechanism: Take the largest single-period count. Recreate the concrete case: M1 equals 6. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The selected period is data-dependent. Complete the transfer step: Use the maximum distribution. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Adjacent sums

Begin with this mechanism: Add every permitted pair of successive periods. Recreate the concrete case: Sums are 7,9,5. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Circular wraparound is a separate assumption. Complete the transfer step: Declare calendar topology. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: M2 maximum

Begin with this mechanism: Take the largest adjacent-pair total. Recreate the concrete case: M2 equals 9. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: M2 shares cases with M1. Complete the transfer step: Model joint dependence. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Fixed-total null

Begin with this mechanism: Condition on the group total and randomize timing. Recreate the concrete case: Twelve cases follow a multinomial allocation. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Independent Poisson sampling answers another question. Complete the transfer step: State conditioning. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: Exact enumeration

Begin with this mechanism: Enumerate occupancy vectors when totals are modest. Recreate the concrete case: All four-cell allocations summing to 12 are considered. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Unequal probabilities change weights. Complete the transfer step: Use correct multinomial masses. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Simulation reference

Begin with this mechanism: Draw null allocations and recompute both maxima. Recreate the concrete case: 9999 vectors estimate the joint tail. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Monte Carlo error affects small p-values. Complete the transfer step: Report seed and precision. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Joint statistic

Begin with this mechanism: Combine M1 and M2 with a valid bivariate rule. Recreate the concrete case: A calibrated score respects their correlation. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Minimum unadjusted p is invalid. Complete the transfer step: Use published or simulated calibration. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: Multiple groups

Begin with this mechanism: Aggregate evidence across communities or cohorts. Recreate the concrete case: Each town contributes an observed-minus-expected maximum. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Groups with different totals have different nulls. Complete the transfer step: Standardize by their own moments. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Calendar sensitivity

Begin with this mechanism: Shift only predeclared origins or use rolling methods. Recreate the concrete case: Monthly and quarterly analyses answer different scales. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Searching all origins inflates evidence. Complete the transfer step: Adjust the search. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Seasonality audit

Begin with this mechanism: Compare with expected seasonal timing. Recreate the concrete case: Winter has a higher baseline probability. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Equal cells create false clustering. Complete the transfer step: Model known seasonality. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: Independent follow-up

Begin with this mechanism: Test later years without changing bins. Recreate the concrete case: A later cohort repeats the peak scale. Write case dates, group totals, period boundaries, exposure probabilities, occupancy vectors, one-period counts, adjacent sums, M1, M2, exact multinomial masses, simulated maxima and joint tail ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Replication remains descriptive. Complete the transfer step: Investigate exposures and mechanisms. Change exactly one event, coordinate, timestamp, period boundary, exposure probability, area centroid, adjacency rule, risk denominator, expected count, case target k, neighbour rank k, tie rule, permutation restriction, simulation count, bootstrap seed, global calibration, observation period, validation region or alternative hypothesis. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, representation, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.