Chapter map

1. 1. Correspondence
2. 2. Reference triangle
3. 3. Transformed triangle
4. 4. Centering
5. 5. Norm scaling
6. 6. Orthogonal fit
7. 7. Perfect disparity
8. 8. Perturbation
9. 9. Standardized X
10. 10. Aligned Y₂
11. 11. Residuals
12. 12. Squared disparity
13. 13. Reflection policy
14. 14. Weights
15. 15. Uncertainty
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. Correspondence

This is where modelling choices begin to affect evidence. Match the same three landmarks in X and Y.

A concrete situation

Rows refer to the same objects.

What the mathematics does not license

Wrong matching makes alignment meaningless. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record IDs. 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. Reference triangle

The central mathematical idea is concrete. Use X={(0,0),(1,0),(0,1)}.

A concrete situation

A right triangle defines shape.

What the mathematics does not license

Coordinate units are declared. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record X. 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. Transformed triangle

Here the geometry becomes operational. Use Y={(5,3),(5,5),(3,3)}.

A concrete situation

Y is translated, scaled and rotated.

What the mathematics does not license

Confirm row correspondence. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record Y. 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. Centering

This step deserves a careful pause. Subtract each centroid.

A concrete situation

Translations disappear.

What the mathematics does not license

Weights alter centroids. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record means. 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. Norm scaling

The useful question is not only how to calculate, but what the calculation assumes. Divide each centred matrix by its Frobenius norm.

A concrete situation

Uniform scale disappears.

What the mathematics does not license

Full and partial Procrustes differ. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Name policy. 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. Orthogonal fit

A small example makes the mechanism visible. Find the rotation or reflection minimising squared residuals.

A concrete situation

The standardized triangles coincide.

What the mathematics does not license

Declare whether reflection is allowed. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record transform. 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. Perfect disparity

This is where modelling choices begin to affect evidence. Sum squared aligned differences.

A concrete situation

Disparity=0.

What the mathematics does not license

Zero reflects the chosen invariances. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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8. Perturbation

The central mathematical idea is concrete. Move Y third point to (3,3.4).

A concrete situation

Shape now differs.

What the mathematics does not license

Keep other points fixed. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record Y₂. 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. Standardized X

Here the geometry becomes operational. Compute rows approximately (-.288675,-.288675),(.577350,-.288675),(-.288675,.577350).

A concrete situation

Reference has unit Frobenius norm.

What the mathematics does not license

Round only for display. The same reasoning helps students move from an exercise to a real dataset.

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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10. Aligned Y₂

This step deserves a careful pause. Optimally align to X.

A concrete situation

Rows are approximately (-.326328,-.335742),(.567937,-.241609),(-.241609,.577350).

What the mathematics does not license

Use full precision for score. That distinction matters because a correct formula can still answer the wrong operational question.

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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11. Residuals

The useful question is not only how to calculate, but what the calculation assumes. Subtract aligned rows point by point.

A concrete situation

Third-point perturbation affects the joint fit.

What the mathematics does not license

Inspect every landmark. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record residuals. 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. Squared disparity

A small example makes the mechanism visible. Sum all squared residuals.

A concrete situation

M²=.008152.

What the mathematics does not license

Lower is better under this normalization. The same reasoning helps students move from an exercise to a real dataset.

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. Reflection policy

This is where modelling choices begin to affect evidence. Forbid reflection and repeat.

A concrete situation

Fit can change for mirror images.

What the mathematics does not license

Policy follows the scientific question. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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14. Weights

The central mathematical idea is concrete. Give one landmark more weight.

A concrete situation

Alignment moves toward it.

What the mathematics does not license

Weights need external justification. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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15. Uncertainty

Here the geometry becomes operational. Bootstrap natural sampling units.

A concrete situation

Disparity becomes a distribution.

What the mathematics does not license

Do not resample coordinates as independent if grouped. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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16. Claim

This step deserves a careful pause. State invariances, matching and score.

A concrete situation

Perturbed triangle differs slightly after full alignment.

What the mathematics does not license

Do not claim same mechanism. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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A seven-day practice lab

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

Questions for a student, parent or mentor

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

A method-specific audit before deployment

Procrustes analysis compares configurations after removing transformations that are irrelevant to the question. A common full Procrustes workflow centres corresponding points, scales configurations to a declared norm, and finds the rotation or reflection that minimises the sum of squared coordinate differences. The remaining disparity describes shape disagreement under that policy.

Use reference triangle X={(0,0),(1,0),(0,1)} and Y={(5,3),(5,5),(3,3)}. Y is a translated, uniformly scaled and rotated version of X. After centring, norm scaling and optimal orthogonal alignment, both standardized configurations coincide and the full Procrustes squared disparity is zero.

Now perturb the third Y point to (3,3.4). Standardized X is approximately [(-.288675,-.288675),(.577350,-.288675),(-.288675,.577350)]. After optimal alignment, Y is approximately [(-.326328,-.335742),(.567937,-.241609),(-.241609,.577350)]. Summed squared residuals are .008152.

A small disparity does not prove that configurations were generated by the same process. Correspondences, reflection policy, scaling, weights, dimensions and landmark coverage determine the question. If point matching is uncertain, alignment can be meaningless. Report the transformation model, residuals by point, uncertainty and whether the same observations informed selection and evaluation.

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Frequently asked questions

What is Procrustes analysis?

A family of methods that aligns corresponding point configurations under declared transformations and measures the residual mismatch.

What transformations can be removed?

Depending on the variant: translation, rotation, reflection and sometimes uniform scale.

Which direction is better for disparity?

Lower is better under the same normalization, weights, correspondence and transformation policy.

Why does reflection policy matter?

A mirror image may be equivalent for one task and impossible for another.

Does low disparity prove common cause?

No. It shows geometric similarity after allowed transformations.

What must be reported?

Correspondences, dimensions, centring, norm, scale and reflection policy, weights, transform, point residuals, disparity and uncertainty.

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Primary source and fact-check note

John C. Gower, “Generalized Procrustes Analysis,” Psychometrika 40(1), 33–51 (1975), DOI 10.1007/BF02291478. 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

• Kabsch best-fit rotations
• Iterative Closest Point
• Classical multidimensional scaling
• Co-ranking matrices
• 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: configuration evidence and responsible use

Correspondence audit

Freeze landmark identities, exclusions, dimensions and weights before alignment.

Transformation audit

State translation, scale, rotation and reflection freedoms and explain why each is scientifically acceptable.

Residual audit

Show aligned coordinates and residual vectors for every point so one landmark cannot hide inside a global disparity.

Uncertainty audit

Repeat across samples, starts, matching alternatives and held-out landmarks where possible.

Mechanism workshop: calculate, challenge and transfer

Manual reconstruction

Rebuild the example with identifiers and record correspondences, centroids, norms, orthogonal transform, aligned coordinates, point residuals and disparity. Explain every transformation before software.

One-entry experiment

Change one distance, coordinate, loading or row while holding the comparison rule fixed. Recalculate every affected matrix cell and score.

Scaling experiment

Apply uniform and unequal scaling. State which changes are removed by the method and which alter the evidence.

Sign or reflection experiment

Reverse an axis or mirror a configuration. Apply the declared invariance policy and explain the scientific consequence.

Correspondence experiment

Swap two object, landmark or variable labels. Show why arithmetic can remain tidy while the comparison becomes meaningless.

Stability experiment

Repeat samples, starts, rotations, factor matches or preprocessing choices and report a distribution.

Parent explanation

Explain why two maps can look different yet match after alignment, or look similar while their distance or loading structure differs.

Student transfer

Apply the mechanism to school maps, sensor tables, questionnaire factors or image embeddings using real identifiers.

Limit statement

Write one supported claim, one overclaim and one repaired claim naming preprocessing, invariances and uncertainty.

Next measurement

Choose a complementary local, global, predictive or substantive check and explain its separate question.

Evidence notebook: reconstruct every mathematical step

Evidence notebook 1: Correspondence

Notebook step 1 isolates correspondence. Match the same three landmarks in X and Y. The reproducible result is: Rows refer to the same objects. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Wrong matching makes alignment meaningless. Record IDs. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 2: Reference triangle

Notebook step 2 isolates reference triangle. Use X={(0,0),(1,0),(0,1)}. The reproducible result is: A right triangle defines shape. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Coordinate units are declared. Record X. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 3: Transformed triangle

Notebook step 3 isolates transformed triangle. Use Y={(5,3),(5,5),(3,3)}. The reproducible result is: Y is translated, scaled and rotated. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Confirm row correspondence. Record Y. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 4: Centering

Notebook step 4 isolates centering. Subtract each centroid. The reproducible result is: Translations disappear. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Weights alter centroids. Record means. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 5: Norm scaling

Notebook step 5 isolates norm scaling. Divide each centred matrix by its Frobenius norm. The reproducible result is: Uniform scale disappears. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Full and partial Procrustes differ. Name policy. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 6: Orthogonal fit

Notebook step 6 isolates orthogonal fit. Find the rotation or reflection minimising squared residuals. The reproducible result is: The standardized triangles coincide. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Declare whether reflection is allowed. Record transform. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 7: Perfect disparity

Notebook step 7 isolates perfect disparity. Sum squared aligned differences. The reproducible result is: Disparity=0. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Zero reflects the chosen invariances. Record score. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 8: Perturbation

Notebook step 8 isolates perturbation. Move Y third point to (3,3.4). The reproducible result is: Shape now differs. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Keep other points fixed. Record Y₂. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 9: Standardized X

Notebook step 9 isolates standardized x. Compute rows approximately (-.288675,-.288675),(.577350,-.288675),(-.288675,.577350). The reproducible result is: Reference has unit Frobenius norm. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Round only for display. Record matrix. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 10: Aligned Y₂

Notebook step 10 isolates aligned y₂. Optimally align to X. The reproducible result is: Rows are approximately (-.326328,-.335742),(.567937,-.241609),(-.241609,.577350). The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Use full precision for score. Record matrix. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 11: Residuals

Notebook step 11 isolates residuals. Subtract aligned rows point by point. The reproducible result is: Third-point perturbation affects the joint fit. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Inspect every landmark. Record residuals. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 12: Squared disparity

Notebook step 12 isolates squared disparity. Sum all squared residuals. The reproducible result is: M²=.008152. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Lower is better under this normalization. Record score. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 13: Reflection policy

Notebook step 13 isolates reflection policy. Forbid reflection and repeat. The reproducible result is: Fit can change for mirror images. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Policy follows the scientific question. Compare. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 14: Weights

Notebook step 14 isolates weights. Give one landmark more weight. The reproducible result is: Alignment moves toward it. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Weights need external justification. Stress-test. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 15: Uncertainty

Notebook step 15 isolates uncertainty. Bootstrap natural sampling units. The reproducible result is: Disparity becomes a distribution. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Do not resample coordinates as independent if grouped. Summarise. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Evidence notebook 16: Claim

Notebook step 16 isolates claim. State invariances, matching and score. The reproducible result is: Perturbed triangle differs slightly after full alignment. The evidence concerns shape disagreement remaining after declared transformations. Keep object, landmark, variable or row identities beside every value. The safeguard is: Do not claim same mechanism. Bound conclusion. Change one input while holding centring, scaling, matching and formula conventions fixed. Recalculate the affected distance, Gram entry, eigenvalue, transform, residual, dot product, trace or coefficient. Explain whether the movement came from data, preprocessing, correspondence or model flexibility. If another reader cannot reconstruct the value, it is not ready for a claim. Finish with the narrow conclusion and the extra evidence still required.

Failure cases and responsible recovery

Wrong correspondences

A perfect-looking alignment can be meaningless when point identities are mismatched. Audit labels before optimisation. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Reflection ambiguity

Allowing a mirror can reduce disparity even when handedness matters physically. Set the policy from the application. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Scale removal

Full Procrustes treats uniform size as irrelevant. Use partial Procrustes when size itself is evidence. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Dominant landmark

One high-weight or distant point drives the fit. Show point residuals and weight sensitivity. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Different dimensions

Padding, projecting or truncating coordinates changes the comparison. Declare the common space. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Post-hoc matching

Trying many landmark subsets and keeping the best alignment inflates similarity. Separate exploration and evaluation. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Dependent points

Repeated landmarks within subjects require group-aware resampling. Do not bootstrap coordinates as independent. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.

Responsible conclusion

Describe geometric agreement after allowed transforms, not identity of mechanisms or outcomes. Recalculate a small counterexample, repeat under another defensible setting and state the extra evidence needed to decide.