Chapter map

1. 1. Matrices
2. 2. Trace K
3. 3. Trace L
4. 4. Square root K
5. 5. Sandwich
6. 6. Inner square root
7. 7. Numerator
8. 8. Denominator
9. 9. Similarity
10. 10. Identity check
11. 11. Orthogonal support
12. 12. Orthogonal score
13. 13. Rotation check
14. 14. Regularization
15. 15. Dual distance
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. Matrices

The central mathematical idea is concrete. Use K=diag(4,1) and L=diag(1,4).

A concrete situation

Both matrices are positive semidefinite.

What the mathematics does not license

State whether matrices are Gram or covariance objects. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record K,L. 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. Trace K

Here the geometry becomes operational. Add K diagonal entries.

A concrete situation

trK=5.

What the mathematics does not license

Trace normalization carries overall scale. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record trace. 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. Trace L

This step deserves a careful pause. Add L diagonal entries.

A concrete situation

trL=5.

What the mathematics does not license

Use the same preprocessing. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record trace. 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. Square root K

The useful question is not only how to calculate, but what the calculation assumes. Take the positive semidefinite square root.

A concrete situation

K^(1/2)=diag(2,1).

What the mathematics does not license

Do not take entrywise roots for non-diagonal matrices. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record matrix. 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. Sandwich

A small example makes the mechanism visible. Form K^(1/2)LK^(1/2).

A concrete situation

The result is diag(4,4).

What the mathematics does not license

Matrix multiplication order matters. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record matrix. 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. Inner square root

This is where modelling choices begin to affect evidence. Take the PSD square root of the sandwich.

A concrete situation

The result is diag(2,2).

What the mathematics does not license

Clip tiny negative roundoff only with a declared tolerance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record matrix. 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. Numerator

The central mathematical idea is concrete. Take the trace.

A concrete situation

The Bures overlap numerator is four.

What the mathematics does not license

Use full eigenvalue precision. Keeping this boundary clear makes the result easier to audit and transfer.

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. Denominator

Here the geometry becomes operational. Compute sqrt(trK trL).

A concrete situation

sqrt(5×5)=5.

What the mathematics does not license

Zero-trace cases need separate handling. 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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9. Similarity

This step deserves a careful pause. Divide four by five.

A concrete situation

Normalized Bures similarity=.8.

What the mathematics does not license

Report the exact normalization. Evidence becomes stronger when the limitation is tested instead of hidden.

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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10. Identity check

The useful question is not only how to calculate, but what the calculation assumes. Set L=K.

A concrete situation

Numerator and denominator both equal five.

What the mathematics does not license

This validates the implementation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record similarity one. 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. Orthogonal support

A small example makes the mechanism visible. Use diag(1,0) and diag(0,1).

A concrete situation

The sandwich is zero.

What the mathematics does not license

Rank deficiency is legitimate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record matrices. 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. Orthogonal score

This is where modelling choices begin to affect evidence. Take the normalized overlap.

A concrete situation

Similarity=0.

What the mathematics does not license

Zero describes support geometry, not causal independence. Evidence becomes stronger when the limitation is tested instead of hidden.

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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13. Rotation check

The central mathematical idea is concrete. Rotate one covariance basis.

A concrete situation

Similarity responds to relative spectral geometry.

What the mathematics does not license

State whether rotation should count. Keeping this boundary clear makes the result easier to audit and transfer.

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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14. Regularization

Here the geometry becomes operational. Add εI only when needed and repeat.

A concrete situation

Scores can move, especially near zero eigenvalues.

What the mathematics does not license

Report ε and unregularized diagnostics. A trustworthy workflow records the choice before seeing the final test result.

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. Dual distance

This step deserves a careful pause. Translate the overlap into the corresponding Bures or shape-distance expression.

A concrete situation

Dual views agree under the theorem assumptions.

What the mathematics does not license

Do not mix squared and square-root conventions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Name convention. 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

The useful question is not only how to calculate, but what the calculation assumes. State matrix construction, centring, rank and normalization.

A concrete situation

The diagonal example has normalized similarity .8.

What the mathematics does not license

Do not claim semantic or functional identity. Keeping this boundary clear makes the result easier to audit and transfer.

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

Normalized Bures similarity compares positive semidefinite representation matrices through matrix square roots. For Gram or covariance matrices K and L, the numerator is tr[(K^(1/2)LK^(1/2))^(1/2)] and normalization by sqrt(trK trL) places the similarity on an interpretable scale under the stated conditions. The construction connects Bures geometry with shape distances.

Take K=diag(4,1) and L=diag(1,4). Because the matrices commute, K^(1/2)LK^(1/2)=diag(4,4), whose positive square root is diag(2,2). The numerator is four, both traces are five, and normalized Bures similarity is 4/sqrt(5×5)=.8.

If L=K, the numerator equals trK and the normalized value is one. At the opposite extreme, K=diag(1,0) and L=diag(0,1) have orthogonal support; the sandwiched matrix is zero and the similarity is zero. These examples make shared spectral support, not elementwise matching, the central mechanism.

Matrix square roots require positive semidefinite inputs and careful eigenspectrum handling. Centring, covariance versus Gram construction, trace normalization, rank deficiency, negative roundoff eigenvalues and regularization all matter. The recent duality results clarify relations to shape comparison, but a high Bures similarity still describes geometry under a chosen representation, not semantic identity or causal equivalence.

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Frequently asked questions

What is normalized Bures similarity?

A trace-normalized overlap between positive semidefinite matrices based on a sandwiched matrix square root.

Why use a matrix square root?

It captures shared spectral geometry while respecting the positive semidefinite cone.

What range should it have?

Under the stated PSD construction and normalization it lies between zero and one, subject to numerical tolerance.

Is an entrywise square root valid?

No. Use the positive semidefinite matrix square root from an eigendecomposition or stable equivalent.

Does one mean identical raw activations?

No. It means maximal similarity for the chosen matrix representation and normalization.

What must be reported?

Matrix construction, centring, scaling, rank, trace, square-root method, tolerance, regularization, score convention, uncertainty and complementary checks.

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Primary source and fact-check note

William Harvey, Blake Larsen and Alex H. Williams, “Duality of Bures and Shape Distances with Implications for Comparing Neural Representations,” Proceedings of Machine Learning Research 243 (2024). 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

• Procrustes analysis
• Centered Kernel Alignment
• RV coefficient
• Gromov–Wasserstein distance
• Representational Similarity Analysis

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: representation evidence and responsible comparison

PSD audit

Check symmetry and eigenspectrum of every Gram or covariance matrix with a declared numerical tolerance.

Root audit

Verify the positive semidefinite square root by reconstructing the original matrix.

Normalization audit

Publish numerator, both traces, denominator and whether similarity or squared distance is reported.

Sensitivity audit

Repeat centring, rank, ε-regularization and sample splits and preserve unregularized diagnostics.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild the smallest example for PSD matrix square roots and shape geometry; label observations, matrices, norms and every convention.

One-entry experiment

Change one similarity, activation, singular value or eigenvalue while freezing every other decision. Recalculate all affected terms.

Invariance experiment

Apply a feature permutation, orthogonal rotation, isotropic scaling or invertible subspace transform and predict the result before computing it.

Rank experiment

Remove or add a low-energy direction and explain whether the score changes because of signal, regularization or threshold policy.

Orientation experiment

Swap the two representations. Identify which measures remain unchanged and why PWCCA can move.

Validation experiment

Choose a method on training observations and test the frozen choice on held-out examples without reusing targets.

Student transfer

Compare handwriting features, image layers, language embeddings or sensor representations while preserving row identities.

Parent explanation

Describe how two maps of the same examples can use different coordinates yet preserve similar relationships.

Limit statement

Write one supported claim, one tempting overclaim and one repaired claim that names sample, method and uncertainty.

Next measurement

Choose a complementary predictive, causal, local or domain check and explain the separate evidence it supplies.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Matrices

Step 1 isolates matrices. Use K=diag(4,1) and L=diag(1,4). The reproducible result is: Both matrices are positive semidefinite. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: State whether matrices are Gram or covariance objects. Record K,L. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 2: Trace K

Step 2 isolates trace k. Add K diagonal entries. The reproducible result is: trK=5. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Trace normalization carries overall scale. Record trace. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 3: Trace L

Step 3 isolates trace l. Add L diagonal entries. The reproducible result is: trL=5. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Use the same preprocessing. Record trace. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 4: Square root K

Step 4 isolates square root k. Take the positive semidefinite square root. The reproducible result is: K^(1/2)=diag(2,1). This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Do not take entrywise roots for non-diagonal matrices. Record matrix. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 5: Sandwich

Step 5 isolates sandwich. Form K^(1/2)LK^(1/2). The reproducible result is: The result is diag(4,4). This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Matrix multiplication order matters. Record matrix. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 6: Inner square root

Step 6 isolates inner square root. Take the PSD square root of the sandwich. The reproducible result is: The result is diag(2,2). This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Clip tiny negative roundoff only with a declared tolerance. Record matrix. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 7: Numerator

Step 7 isolates numerator. Take the trace. The reproducible result is: The Bures overlap numerator is four. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Use full eigenvalue precision. Record value. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 8: Denominator

Step 8 isolates denominator. Compute sqrt(trK trL). The reproducible result is: sqrt(5×5)=5. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Zero-trace cases need separate handling. Record value. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 9: Similarity

Step 9 isolates similarity. Divide four by five. The reproducible result is: Normalized Bures similarity=.8. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Report the exact normalization. Record score. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 10: Identity check

Step 10 isolates identity check. Set L=K. The reproducible result is: Numerator and denominator both equal five. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: This validates the implementation. Record similarity one. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 11: Orthogonal support

Step 11 isolates orthogonal support. Use diag(1,0) and diag(0,1). The reproducible result is: The sandwich is zero. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Rank deficiency is legitimate. Record matrices. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 12: Orthogonal score

Step 12 isolates orthogonal score. Take the normalized overlap. The reproducible result is: Similarity=0. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Zero describes support geometry, not causal independence. Record score. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 13: Rotation check

Step 13 isolates rotation check. Rotate one covariance basis. The reproducible result is: Similarity responds to relative spectral geometry. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: State whether rotation should count. Interpret. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 14: Regularization

Step 14 isolates regularization. Add εI only when needed and repeat. The reproducible result is: Scores can move, especially near zero eigenvalues. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Report ε and unregularized diagnostics. Stress-test. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 15: Dual distance

Step 15 isolates dual distance. Translate the overlap into the corresponding Bures or shape-distance expression. The reproducible result is: Dual views agree under the theorem assumptions. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Do not mix squared and square-root conventions. Name convention. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Evidence notebook 16: Claim

Step 16 isolates claim. State matrix construction, centring, rank and normalization. The reproducible result is: The diagonal example has normalized similarity .8. This is evidence about trace-normalized PSD spectral overlap. Keep observation identifiers beside every matrix and write the exact formula before entering numbers. The safeguard is: Do not claim semantic or functional identity. Bound conclusion. Next change one input while holding centring, rank, normalization and sample fixed. Recalculate the cells, singular values, canonical correlations, weights or matrix roots that must move. Explain whether the change came from data geometry, numerical stabilization or an analytical choice. Finish with the narrow claim supported by this step and the additional evidence still needed.

Failure cases and responsible recovery

Entrywise root

Taking square roots cell by cell is not a matrix square root. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Indefinite input

Negative eigenvalues outside tolerance violate the PSD construction. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Trace-zero case

Normalizing by zero is undefined and needs an explicit policy. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Silent clipping

Removing negative eigenvalues can materially change the geometry. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Convention mixing

Bures similarity, squared distance and square-root distance are distinct outputs. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Regularization hiding

Adding εI can inflate shared support and must be disclosed. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Semantic identity

Spectral overlap does not establish equal meaning. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.

Sample certainty

One representation batch does not establish population stability. Rebuild a counterexample, repeat under another defensible setting and state what additional evidence would repair the claim.